theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation...
Transcript of theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation...
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MotivationRepresentation theory and CFT
Quantum groups
AM Semikhatov LCFT and representation theory
![Page 2: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/2.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Logarithmic Conformal Field Theory
AM Semikhatov
Lebedev Physics Institute
Dubna Workshop on LCFTetc, June 2007
AM Semikhatov LCFT and representation theory
![Page 3: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/3.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Logarithmic Conformal Field Theory:How far can we go with representation theory?
AM Semikhatov
Lebedev Physics Institute
Dubna Workshop on LCFTetc, June 2007
AM Semikhatov LCFT and representation theory
![Page 4: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/4.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Plan of the Talk
1 Motivation
2 Representation theory and CFT
3 Quantum groups
AM Semikhatov LCFT and representation theory
![Page 5: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/5.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Plan of the Talk
1 Motivation
2 Representation theory and CFT
3 Quantum groups
AM Semikhatov LCFT and representation theory
![Page 6: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/6.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Plan of the Talk
1 Motivation
2 Representation theory and CFT
3 Quantum groups
AM Semikhatov LCFT and representation theory
![Page 7: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/7.jpg)
MotivationRepresentation theory and CFT
Quantum groups
AM Semikhatov LCFT and representation theory
![Page 8: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/8.jpg)
MotivationRepresentation theory and CFT
Quantum groups
1 Motivation
2 Representation theory and CFT
3 Quantum groups
AM Semikhatov LCFT and representation theory
![Page 9: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/9.jpg)
MotivationRepresentation theory and CFT
Quantum groups
A 2002 quotation
Logarithmic CFTs may be interesting from several standpoints.A Nichols, 2002:
LCFTs have now been studied for over ten years . . . in. . . : WZNWmodels [19–32], gravitational dressing [33,34], polymers andpercolation [35–38], 2d turbulence [39–43], certain limits ofQCD [44–46], the Seiberg–Witten solution of N = 2 supersymmetricYang–Mills [47,48], and the Abelian sand-pile model [49,50].. . . applications has been to disordered systems and the quantum Halleffect [18,51–60]. . . . to string theory [61–70] and in theAdS/CFT correspondence [71–78]. The holographic relation betweenlogarithmic operators and vacuum instability was considered in [79,80].An approach to LCFT using nilpotent dimensions was given in [81,82].. . . the appearance of a logarithmic partner of the stress tensor in c = 0LCFTs [37,83,84], . . . [85,86]. . . . of particular interest has been theanalysis of LCFTs in the presence of a boundary [87–91].
AM Semikhatov LCFT and representation theory
![Page 10: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/10.jpg)
MotivationRepresentation theory and CFT
Quantum groups
A 2002 quotation
Logarithmic CFTs may be interesting from several standpoints.A Nichols, 2002:
LCFTs have now been studied for over ten years . . . in. . . : WZNWmodels [19–32], gravitational dressing [33,34], polymers andpercolation [35–38], 2d turbulence [39–43], certain limits ofQCD [44–46], the Seiberg–Witten solution of N = 2 supersymmetricYang–Mills [47,48], and the Abelian sand-pile model [49,50].. . . applications has been to disordered systems and the quantum Halleffect [18,51–60]. . . . to string theory [61–70] and in theAdS/CFT correspondence [71–78]. The holographic relation betweenlogarithmic operators and vacuum instability was considered in [79,80].An approach to LCFT using nilpotent dimensions was given in [81,82].. . . the appearance of a logarithmic partner of the stress tensor in c = 0LCFTs [37,83,84], . . . [85,86]. . . . of particular interest has been theanalysis of LCFTs in the presence of a boundary [87–91].
AM Semikhatov LCFT and representation theory
![Page 11: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/11.jpg)
MotivationRepresentation theory and CFT
Quantum groups
A 2002 quotation
Logarithmic CFTs may be interesting from several standpoints.A Nichols, 2002:
LCFTs have now been studied for over ten years . . . in. . . : WZNWmodels [19–32], gravitational dressing [33,34], polymers andpercolation [35–38], 2d turbulence [39–43], certain limits ofQCD [44–46], the Seiberg–Witten solution of N = 2 supersymmetricYang–Mills [47,48], and the Abelian sand-pile model [49,50].. . . applications has been to disordered systems and the quantum Halleffect [18,51–60]. . . . to string theory [61–70] and in theAdS/CFT correspondence [71–78]. The holographic relation betweenlogarithmic operators and vacuum instability was considered in [79,80].An approach to LCFT using nilpotent dimensions was given in [81,82].. . . the appearance of a logarithmic partner of the stress tensor in c = 0LCFTs [37,83,84], . . . [85,86]. . . . of particular interest has been theanalysis of LCFTs in the presence of a boundary [87–91].
AM Semikhatov LCFT and representation theory
![Page 12: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/12.jpg)
MotivationRepresentation theory and CFT
Quantum groups
A 2002 quotation
Logarithmic CFTs may be interesting from several standpoints.A Nichols, 2002:
LCFTs have now been studied for over ten years . . . in. . . : WZNWmodels [19–32], gravitational dressing [33,34], polymers andpercolation [35–38], 2d turbulence [39–43], certain limits ofQCD [44–46], the Seiberg–Witten solution of N = 2 supersymmetricYang–Mills [47,48], and the Abelian sand-pile model [49,50].. . . applications has been to disordered systems and the quantum Halleffect [18,51–60]. . . . to string theory [61–70] and in theAdS/CFT correspondence [71–78]. The holographic relation betweenlogarithmic operators and vacuum instability was considered in [79,80].An approach to LCFT using nilpotent dimensions was given in [81,82].. . . the appearance of a logarithmic partner of the stress tensor in c = 0LCFTs [37,83,84], . . . [85,86]. . . . of particular interest has been theanalysis of LCFTs in the presence of a boundary [87–91].
AM Semikhatov LCFT and representation theory
![Page 13: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/13.jpg)
MotivationRepresentation theory and CFT
Quantum groups
A 2002 quotation
Logarithmic CFTs may be interesting from several standpoints.A Nichols, 2002:
LCFTs have now been studied for over ten years . . . in. . . : WZNWmodels [19–32], gravitational dressing [33,34], polymers andpercolation [35–38], 2d turbulence [39–43], certain limits ofQCD [44–46], the Seiberg–Witten solution of N = 2 supersymmetricYang–Mills [47,48], and the Abelian sand-pile model [49,50].. . . applications has been to disordered systems and the quantum Halleffect [18,51–60]. . . . to string theory [61–70] and in theAdS/CFT correspondence [71–78]. The holographic relation betweenlogarithmic operators and vacuum instability was considered in [79,80].An approach to LCFT using nilpotent dimensions was given in [81,82].. . . the appearance of a logarithmic partner of the stress tensor in c = 0LCFTs [37,83,84], . . . [85,86]. . . . of particular interest has been theanalysis of LCFTs in the presence of a boundary [87–91].
AM Semikhatov LCFT and representation theory
![Page 14: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/14.jpg)
MotivationRepresentation theory and CFT
Quantum groups
A 2002 quotation
Logarithmic CFTs may be interesting from several standpoints.A Nichols, 2002:
LCFTs have now been studied for over ten years . . . in. . . : WZNWmodels [19–32], gravitational dressing [33,34], polymers andpercolation [35–38], 2d turbulence [39–43], certain limits ofQCD [44–46], the Seiberg–Witten solution of N = 2 supersymmetricYang–Mills [47,48], and the Abelian sand-pile model [49,50].. . . applications has been to disordered systems and the quantum Halleffect [18,51–60]. . . . to string theory [61–70] and in theAdS/CFT correspondence [71–78]. The holographic relation betweenlogarithmic operators and vacuum instability was considered in [79,80].An approach to LCFT using nilpotent dimensions was given in [81,82].. . . the appearance of a logarithmic partner of the stress tensor in c = 0LCFTs [37,83,84], . . . [85,86]. . . . of particular interest has been theanalysis of LCFTs in the presence of a boundary [87–91].
AM Semikhatov LCFT and representation theory
![Page 15: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/15.jpg)
MotivationRepresentation theory and CFT
Quantum groups
A 2002 quotation
Logarithmic CFTs may be interesting from several standpoints.A Nichols, 2002:
LCFTs have now been studied for over ten years . . . in. . . : WZNWmodels [19–32], gravitational dressing [33,34], polymers andpercolation [35–38], 2d turbulence [39–43], certain limits ofQCD [44–46], the Seiberg–Witten solution of N = 2 supersymmetricYang–Mills [47,48], and the Abelian sand-pile model [49,50].. . . applications has been to disordered systems and the quantum Halleffect [18,51–60]. . . . to string theory [61–70] and in theAdS/CFT correspondence [71–78]. The holographic relation betweenlogarithmic operators and vacuum instability was considered in [79,80].An approach to LCFT using nilpotent dimensions was given in [81,82].. . . the appearance of a logarithmic partner of the stress tensor in c = 0LCFTs [37,83,84], . . . [85,86]. . . . of particular interest has been theanalysis of LCFTs in the presence of a boundary [87–91].
AM Semikhatov LCFT and representation theory
![Page 16: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/16.jpg)
MotivationRepresentation theory and CFT
Quantum groups
A 2002 quotation
Logarithmic CFTs may be interesting from several standpoints.A Nichols, 2002:
LCFTs have now been studied for over ten years . . . in. . . : WZNWmodels [19–32], gravitational dressing [33,34], polymers andpercolation [35–38], 2d turbulence [39–43], certain limits ofQCD [44–46], the Seiberg–Witten solution of N = 2 supersymmetricYang–Mills [47,48], and the Abelian sand-pile model [49,50].. . . applications has been to disordered systems and the quantum Halleffect [18,51–60]. . . . to string theory [61–70] and in theAdS/CFT correspondence [71–78]. The holographic relation betweenlogarithmic operators and vacuum instability was considered in [79,80].An approach to LCFT using nilpotent dimensions was given in [81,82].. . . the appearance of a logarithmic partner of the stress tensor in c = 0LCFTs [37,83,84], . . . [85,86]. . . . of particular interest has been theanalysis of LCFTs in the presence of a boundary [87–91].
AM Semikhatov LCFT and representation theory
![Page 17: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/17.jpg)
MotivationRepresentation theory and CFT
Quantum groups
A 2002 quotation
Logarithmic CFTs may be interesting from several standpoints.A Nichols, 2002:
LCFTs have now been studied for over ten years . . . in. . . : WZNWmodels [19–32], gravitational dressing [33,34], polymers andpercolation [35–38], 2d turbulence [39–43], certain limits ofQCD [44–46], the Seiberg–Witten solution of N = 2 supersymmetricYang–Mills [47,48], and the Abelian sand-pile model [49,50].. . . applications has been to disordered systems and the quantum Halleffect [18,51–60]. . . . to string theory [61–70] and in theAdS/CFT correspondence [71–78]. The holographic relation betweenlogarithmic operators and vacuum instability was considered in [79,80].An approach to LCFT using nilpotent dimensions was given in [81,82].. . . the appearance of a logarithmic partner of the stress tensor in c = 0LCFTs [37,83,84], . . . [85,86]. . . . of particular interest has been theanalysis of LCFTs in the presence of a boundary [87–91].
AM Semikhatov LCFT and representation theory
![Page 18: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/18.jpg)
MotivationRepresentation theory and CFT
Quantum groups
A 2002 quotation
Logarithmic CFTs may be interesting from several standpoints.A Nichols, 2002:
LCFTs have now been studied for over ten years . . . in. . . : WZNWmodels [19–32], gravitational dressing [33,34], polymers andpercolation [35–38], 2d turbulence [39–43], certain limits ofQCD [44–46], the Seiberg–Witten solution of N = 2 supersymmetricYang–Mills [47,48], and the Abelian sand-pile model [49,50].. . . applications has been to disordered systems and the quantum Halleffect [18,51–60]. . . . to string theory [61–70] and in theAdS/CFT correspondence [71–78]. The holographic relation betweenlogarithmic operators and vacuum instability was considered in [79,80].An approach to LCFT using nilpotent dimensions was given in [81,82].. . . the appearance of a logarithmic partner of the stress tensor in c = 0LCFTs [37,83,84], . . . [85,86]. . . . of particular interest has been theanalysis of LCFTs in the presence of a boundary [87–91].
AM Semikhatov LCFT and representation theory
![Page 19: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/19.jpg)
MotivationRepresentation theory and CFT
Quantum groups
A 2002 quotation
Logarithmic CFTs may be interesting from several standpoints.A Nichols, 2002:
LCFTs have now been studied for over ten years . . . in. . . : WZNWmodels [19–32], gravitational dressing [33,34], polymers andpercolation [35–38], 2d turbulence [39–43], certain limits ofQCD [44–46], the Seiberg–Witten solution of N = 2 supersymmetricYang–Mills [47,48], and the Abelian sand-pile model [49,50].. . . applications has been to disordered systems and the quantum Halleffect [18,51–60]. . . . to string theory [61–70] and in theAdS/CFT correspondence [71–78]. The holographic relation betweenlogarithmic operators and vacuum instability was considered in [79,80].An approach to LCFT using nilpotent dimensions was given in [81,82].. . . the appearance of a logarithmic partner of the stress tensor in c = 0LCFTs [37,83,84], . . . [85,86]. . . . of particular interest has been theanalysis of LCFTs in the presence of a boundary [87–91].
AM Semikhatov LCFT and representation theory
![Page 20: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/20.jpg)
MotivationRepresentation theory and CFT
Quantum groups
A 2002 quotation
Logarithmic CFTs may be interesting from several standpoints.A Nichols, 2002:
LCFTs have now been studied for over ten years . . . in. . . : WZNWmodels [19–32], gravitational dressing [33,34], polymers andpercolation [35–38], 2d turbulence [39–43], certain limits ofQCD [44–46], the Seiberg–Witten solution of N = 2 supersymmetricYang–Mills [47,48], and the Abelian sand-pile model [49,50].. . . applications has been to disordered systems and the quantum Halleffect [18,51–60]. . . . to string theory [61–70] and in theAdS/CFT correspondence [71–78]. The holographic relation betweenlogarithmic operators and vacuum instability was considered in [79,80].An approach to LCFT using nilpotent dimensions was given in [81,82].. . . the appearance of a logarithmic partner of the stress tensor in c = 0LCFTs [37,83,84], . . . [85,86]. . . . of particular interest has been theanalysis of LCFTs in the presence of a boundary [87–91].
AM Semikhatov LCFT and representation theory
![Page 21: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/21.jpg)
MotivationRepresentation theory and CFT
Quantum groups
A 2002 quotation
Logarithmic CFTs may be interesting from several standpoints.A Nichols, 2002:
LCFTs have now been studied for over ten years . . . in. . . : WZNWmodels [19–32], gravitational dressing [33,34], polymers andpercolation [35–38], 2d turbulence [39–43], certain limits ofQCD [44–46], the Seiberg–Witten solution of N = 2 supersymmetricYang–Mills [47,48], and the Abelian sand-pile model [49,50].. . . applications has been to disordered systems and the quantum Halleffect [18,51–60]. . . . to string theory [61–70] and in theAdS/CFT correspondence [71–78]. The holographic relation betweenlogarithmic operators and vacuum instability was considered in [79,80].An approach to LCFT using nilpotent dimensions was given in [81,82].. . . the appearance of a logarithmic partner of the stress tensor in c = 0LCFTs [37,83,84], . . . [85,86]. . . . of particular interest has been theanalysis of LCFTs in the presence of a boundary [87–91].
AM Semikhatov LCFT and representation theory
![Page 22: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/22.jpg)
MotivationRepresentation theory and CFT
Quantum groups
A 2002 quotation
Logarithmic CFTs may be interesting from several standpoints.A Nichols, 2002:
LCFTs have now been studied for over ten years . . . in. . . : WZNWmodels [19–32], gravitational dressing [33,34], polymers andpercolation [35–38], 2d turbulence [39–43], certain limits ofQCD [44–46], the Seiberg–Witten solution of N = 2 supersymmetricYang–Mills [47,48], and the Abelian sand-pile model [49,50].. . . applications has been to disordered systems and the quantum Halleffect [18,51–60]. . . . to string theory [61–70] and in theAdS/CFT correspondence [71–78]. The holographic relation betweenlogarithmic operators and vacuum instability was considered in [79,80].An approach to LCFT using nilpotent dimensions was given in [81,82].. . . the appearance of a logarithmic partner of the stress tensor in c = 0LCFTs [37,83,84], . . . [85,86]. . . . of particular interest has been theanalysis of LCFTs in the presence of a boundary [87–91].
AM Semikhatov LCFT and representation theory
![Page 23: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/23.jpg)
MotivationRepresentation theory and CFT
Quantum groups
A 2002 quotation
Logarithmic CFTs may be interesting from several standpoints.A Nichols, 2002:
LCFTs have now been studied for over ten years . . . in. . . : WZNWmodels [19–32], gravitational dressing [33,34], polymers andpercolation [35–38], 2d turbulence [39–43], certain limits ofQCD [44–46], the Seiberg–Witten solution of N = 2 supersymmetricYang–Mills [47,48], and the Abelian sand-pile model [49,50].. . . applications has been to disordered systems and the quantum Halleffect [18,51–60]. . . . to string theory [61–70] and in theAdS/CFT correspondence [71–78]. The holographic relation betweenlogarithmic operators and vacuum instability was considered in [79,80].An approach to LCFT using nilpotent dimensions was given in [81,82].. . . the appearance of a logarithmic partner of the stress tensor in c = 0LCFTs [37,83,84], . . . [85,86]. . . . of particular interest has been theanalysis of LCFTs in the presence of a boundary [87–91].
AM Semikhatov LCFT and representation theory
![Page 24: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/24.jpg)
MotivationRepresentation theory and CFT
Quantum groups
And more recently
PA Pearce, J Rasmussen, and JB Zuber (“Temperly–Lieb” approach)Pierce and Rasmussen (dense polymers)M Jeng, G Piroux, P Ruelle (sand-pile model)F Lesage, P Mathieu, J Rasmussen, H Saleur (WZW models)vertex-operator algebras with nonsemisimple representation categories:YZ Huang, J Lepowsky, and L Zhang;J Fuchs; M Miyamoto; A Milas;Flohr, N CarquevilleV Schomerus and H Saleur (supergeometry =⇒ logs)M Gaberdiel and I Runkel (boundary logarithmic theories)Flohr and Gaberdiel (torus amplitudes)H Eberle and Flohr (fusion)not to mention S Hwang, J Fuchs, Semikhatov, I Tipunin and B Feigin,A Gainutdinov, Semikhatov, Tipuninhep-th/0306274, hep-th/0504093, math.QA/0512621, hep-th/0606196,
math.QA/0606506 [this talk]
AM Semikhatov LCFT and representation theory
![Page 25: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/25.jpg)
MotivationRepresentation theory and CFT
Quantum groups
And more recently
PA Pearce, J Rasmussen, and JB Zuber (“Temperly–Lieb” approach)Pierce and Rasmussen (dense polymers)M Jeng, G Piroux, P Ruelle (sand-pile model)F Lesage, P Mathieu, J Rasmussen, H Saleur (WZW models)vertex-operator algebras with nonsemisimple representation categories:YZ Huang, J Lepowsky, and L Zhang;J Fuchs; M Miyamoto; A Milas;Flohr, N CarquevilleV Schomerus and H Saleur (supergeometry =⇒ logs)M Gaberdiel and I Runkel (boundary logarithmic theories)Flohr and Gaberdiel (torus amplitudes)H Eberle and Flohr (fusion)not to mention S Hwang, J Fuchs, Semikhatov, I Tipunin and B Feigin,A Gainutdinov, Semikhatov, Tipuninhep-th/0306274, hep-th/0504093, math.QA/0512621, hep-th/0606196,
math.QA/0606506 [this talk]
AM Semikhatov LCFT and representation theory
![Page 26: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/26.jpg)
MotivationRepresentation theory and CFT
Quantum groups
And more recently
PA Pearce, J Rasmussen, and JB Zuber (“Temperly–Lieb” approach)Pierce and Rasmussen (dense polymers)M Jeng, G Piroux, P Ruelle (sand-pile model)F Lesage, P Mathieu, J Rasmussen, H Saleur (WZW models)vertex-operator algebras with nonsemisimple representation categories:YZ Huang, J Lepowsky, and L Zhang;J Fuchs; M Miyamoto; A Milas;Flohr, N CarquevilleV Schomerus and H Saleur (supergeometry =⇒ logs)M Gaberdiel and I Runkel (boundary logarithmic theories)Flohr and Gaberdiel (torus amplitudes)H Eberle and Flohr (fusion)not to mention S Hwang, J Fuchs, Semikhatov, I Tipunin and B Feigin,A Gainutdinov, Semikhatov, Tipuninhep-th/0306274, hep-th/0504093, math.QA/0512621, hep-th/0606196,
math.QA/0606506 [this talk]
AM Semikhatov LCFT and representation theory
![Page 27: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/27.jpg)
MotivationRepresentation theory and CFT
Quantum groups
And more recently
PA Pearce, J Rasmussen, and JB Zuber (“Temperly–Lieb” approach)Pierce and Rasmussen (dense polymers)M Jeng, G Piroux, P Ruelle (sand-pile model)F Lesage, P Mathieu, J Rasmussen, H Saleur (WZW models)vertex-operator algebras with nonsemisimple representation categories:YZ Huang, J Lepowsky, and L Zhang;J Fuchs; M Miyamoto; A Milas;Flohr, N CarquevilleV Schomerus and H Saleur (supergeometry =⇒ logs)M Gaberdiel and I Runkel (boundary logarithmic theories)Flohr and Gaberdiel (torus amplitudes)H Eberle and Flohr (fusion)not to mention S Hwang, J Fuchs, Semikhatov, I Tipunin and B Feigin,A Gainutdinov, Semikhatov, Tipuninhep-th/0306274, hep-th/0504093, math.QA/0512621, hep-th/0606196,
math.QA/0606506 [this talk]
AM Semikhatov LCFT and representation theory
![Page 28: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/28.jpg)
MotivationRepresentation theory and CFT
Quantum groups
And more recently
PA Pearce, J Rasmussen, and JB Zuber (“Temperly–Lieb” approach)Pierce and Rasmussen (dense polymers)M Jeng, G Piroux, P Ruelle (sand-pile model)F Lesage, P Mathieu, J Rasmussen, H Saleur (WZW models)vertex-operator algebras with nonsemisimple representation categories:YZ Huang, J Lepowsky, and L Zhang;J Fuchs; M Miyamoto; A Milas;Flohr, N CarquevilleV Schomerus and H Saleur (supergeometry =⇒ logs)M Gaberdiel and I Runkel (boundary logarithmic theories)Flohr and Gaberdiel (torus amplitudes)H Eberle and Flohr (fusion)not to mention S Hwang, J Fuchs, Semikhatov, I Tipunin and B Feigin,A Gainutdinov, Semikhatov, Tipuninhep-th/0306274, hep-th/0504093, math.QA/0512621, hep-th/0606196,
math.QA/0606506 [this talk]
AM Semikhatov LCFT and representation theory
![Page 29: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/29.jpg)
MotivationRepresentation theory and CFT
Quantum groups
And more recently
PA Pearce, J Rasmussen, and JB Zuber (“Temperly–Lieb” approach)Pierce and Rasmussen (dense polymers)M Jeng, G Piroux, P Ruelle (sand-pile model)F Lesage, P Mathieu, J Rasmussen, H Saleur (WZW models)vertex-operator algebras with nonsemisimple representation categories:YZ Huang, J Lepowsky, and L Zhang;J Fuchs; M Miyamoto; A Milas;Flohr, N CarquevilleV Schomerus and H Saleur (supergeometry =⇒ logs)M Gaberdiel and I Runkel (boundary logarithmic theories)Flohr and Gaberdiel (torus amplitudes)H Eberle and Flohr (fusion)not to mention S Hwang, J Fuchs, Semikhatov, I Tipunin and B Feigin,A Gainutdinov, Semikhatov, Tipuninhep-th/0306274, hep-th/0504093, math.QA/0512621, hep-th/0606196,
math.QA/0606506 [this talk]
AM Semikhatov LCFT and representation theory
![Page 30: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/30.jpg)
MotivationRepresentation theory and CFT
Quantum groups
And more recently
PA Pearce, J Rasmussen, and JB Zuber (“Temperly–Lieb” approach)Pierce and Rasmussen (dense polymers)M Jeng, G Piroux, P Ruelle (sand-pile model)F Lesage, P Mathieu, J Rasmussen, H Saleur (WZW models)vertex-operator algebras with nonsemisimple representation categories:YZ Huang, J Lepowsky, and L Zhang;J Fuchs; M Miyamoto; A Milas;Flohr, N CarquevilleV Schomerus and H Saleur (supergeometry =⇒ logs)M Gaberdiel and I Runkel (boundary logarithmic theories)Flohr and Gaberdiel (torus amplitudes)H Eberle and Flohr (fusion)not to mention S Hwang, J Fuchs, Semikhatov, I Tipunin and B Feigin,A Gainutdinov, Semikhatov, Tipuninhep-th/0306274, hep-th/0504093, math.QA/0512621, hep-th/0606196,
math.QA/0606506 [this talk]
AM Semikhatov LCFT and representation theory
![Page 31: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/31.jpg)
MotivationRepresentation theory and CFT
Quantum groups
And more recently
PA Pearce, J Rasmussen, and JB Zuber (“Temperly–Lieb” approach)Pierce and Rasmussen (dense polymers)M Jeng, G Piroux, P Ruelle (sand-pile model)F Lesage, P Mathieu, J Rasmussen, H Saleur (WZW models)vertex-operator algebras with nonsemisimple representation categories:YZ Huang, J Lepowsky, and L Zhang;J Fuchs; M Miyamoto; A Milas;Flohr, N CarquevilleV Schomerus and H Saleur (supergeometry =⇒ logs)M Gaberdiel and I Runkel (boundary logarithmic theories)Flohr and Gaberdiel (torus amplitudes)H Eberle and Flohr (fusion)not to mention S Hwang, J Fuchs, Semikhatov, I Tipunin and B Feigin,A Gainutdinov, Semikhatov, Tipuninhep-th/0306274, hep-th/0504093, math.QA/0512621, hep-th/0606196,
math.QA/0606506 [this talk]
AM Semikhatov LCFT and representation theory
![Page 32: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/32.jpg)
MotivationRepresentation theory and CFT
Quantum groups
And yet more recently
T Creutzig, T Quella, and V Schomerus (boundary)N Read and H Saleur ×2D Adamovic and A Milas (Logarithmic intertwiners and W -algebras)Semikhatov (s`(2) model). . .
AM Semikhatov LCFT and representation theory
![Page 33: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/33.jpg)
MotivationRepresentation theory and CFT
Quantum groups
And yet more recently
T Creutzig, T Quella, and V Schomerus (boundary)N Read and H Saleur ×2D Adamovic and A Milas (Logarithmic intertwiners and W -algebras)Semikhatov (s`(2) model). . .
AM Semikhatov LCFT and representation theory
![Page 34: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/34.jpg)
MotivationRepresentation theory and CFT
Quantum groups
And yet more recently
T Creutzig, T Quella, and V Schomerus (boundary)N Read and H Saleur ×2D Adamovic and A Milas (Logarithmic intertwiners and W -algebras)Semikhatov (s`(2) model). . .
AM Semikhatov LCFT and representation theory
![Page 35: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/35.jpg)
MotivationRepresentation theory and CFT
Quantum groups
And yet more recently
T Creutzig, T Quella, and V Schomerus (boundary)N Read and H Saleur ×2D Adamovic and A Milas (Logarithmic intertwiners and W -algebras)Semikhatov (s`(2) model). . .
AM Semikhatov LCFT and representation theory
![Page 36: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/36.jpg)
MotivationRepresentation theory and CFT
Quantum groups
And yet more recently
T Creutzig, T Quella, and V Schomerus (boundary)N Read and H Saleur ×2D Adamovic and A Milas (Logarithmic intertwiners and W -algebras)Semikhatov (s`(2) model). . .
AM Semikhatov LCFT and representation theory
![Page 37: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/37.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Logarithms:
Logarithmic Conformal Field Theory:
nondiagonalizable action of a number of operators of the type of aHamiltonian
AM Semikhatov LCFT and representation theory
![Page 38: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/38.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Logarithms:
Logarithmic Conformal Field Theory:
nondiagonalizable action of a number of operators of the type of aHamiltonian
“Nonunitary evolution” etH =⇒ applications to models with disorder,systems with transient and recurrent states (sand-pile model), percolation,. . .
AM Semikhatov LCFT and representation theory
![Page 39: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/39.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Logarithms:
Logarithmic Conformal Field Theory:
nondiagonalizable action of a number of operators of the type of aHamiltonian
“Nonunitary evolution” etH =⇒ applications to models with disorder,systems with transient and recurrent states (sand-pile model), percolation,. . .
AM Semikhatov LCFT and representation theory
![Page 40: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/40.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Logarithms:
Logarithmic Conformal Field Theory:
nondiagonalizable action of a number of operators of the type of aHamiltonian
log: whence comest thou?
Let L0 ∼ z∂
∂zact nondiagonally:
zg ′(z) = ∆g(z),
zh ′(z) = ∆h(z) + g(z).
Solution:g(x) = B x∆,
AM Semikhatov LCFT and representation theory
![Page 41: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/41.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Logarithms:
Logarithmic Conformal Field Theory:
nondiagonalizable action of a number of operators of the type of aHamiltonian
log: whence comest thou?
Let L0 ∼ z∂
∂zact nondiagonally:
zg ′(z) = ∆g(z),
zh ′(z) = ∆h(z) + g(z).
Solution:g(x) = B x∆,
AM Semikhatov LCFT and representation theory
![Page 42: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/42.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Logarithms:
Logarithmic Conformal Field Theory:
nondiagonalizable action of a number of operators of the type of aHamiltonian
log: whence comest thou?
Let L0 ∼ z∂
∂zact nondiagonally:
zg ′(z) = ∆g(z),
zh ′(z) = ∆h(z) + g(z).
Solution:g(x) = B x∆,
h(x) = A x∆ + B x∆ log(x).
AM Semikhatov LCFT and representation theory
![Page 43: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/43.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Logarithms:
Logarithmic Conformal Field Theory:
nondiagonalizable action of a number of operators of the type of aHamiltonian
Logarithmic/nonsemisimple theories:
Being logarithmic/nonsemisimple is a property of representations chosen(even though algebras often get extended)
AM Semikhatov LCFT and representation theory
![Page 44: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/44.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Logarithms:
Logarithmic Conformal Field Theory:
nondiagonalizable action of a number of operators of the type of aHamiltonian
Logarithmic/nonsemisimple theories:
Being logarithmic/nonsemisimple is a property of representations chosen(even though algebras often get extended)
AM Semikhatov LCFT and representation theory
![Page 45: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/45.jpg)
MotivationRepresentation theory and CFT
Quantum groups
1 Motivation
2 Representation theory and CFT
3 Quantum groups
AM Semikhatov LCFT and representation theory
![Page 46: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/46.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Rational models: basic representation-theory unput
Virasoro algebra [Lm, Ln] = (m − n)Lm+n + c12(m3 − m)δm+n,0
Highest-weight modules Ln > 1|∆〉 = 0, L0|∆〉 = ∆|∆〉Vermairreducible
Rational (p, p ′)-models at c = 13 − pp ′ −
p ′
p :
Kac table of “good” modules:12 (p − 1) × (p ′ − 1) nonisomorphicVirasoro irreps
p ′−1
. . .
. . . . . . . . . . . . . . .. . .. . .︸ ︷︷ ︸
p−1
These irreps have no extensions among themselves =⇒ semisimple(diagonalizable)
=⇒ chiral space of states =⊕
(irreps)
=⇒ numerous deep properties of rcft. . .
AM Semikhatov LCFT and representation theory
![Page 47: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/47.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Rational models: basic representation-theory unput
Virasoro algebra [Lm, Ln] = (m − n)Lm+n + c12(m3 − m)δm+n,0
Highest-weight modules Ln > 1|∆〉 = 0, L0|∆〉 = ∆|∆〉Vermairreducible
Rational (p, p ′)-models at c = 13 − pp ′ −
p ′
p :
Kac table of “good” modules:12 (p − 1) × (p ′ − 1) nonisomorphicVirasoro irreps
p ′−1
. . .
. . . . . . . . . . . . . . .. . .. . .︸ ︷︷ ︸
p−1
These irreps have no extensions among themselves =⇒ semisimple(diagonalizable)
=⇒ chiral space of states =⊕
(irreps)
=⇒ numerous deep properties of rcft. . .
AM Semikhatov LCFT and representation theory
![Page 48: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/48.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Rational models: basic representation-theory unput
Virasoro algebra [Lm, Ln] = (m − n)Lm+n + c12(m3 − m)δm+n,0
Highest-weight modules Ln > 1|∆〉 = 0, L0|∆〉 = ∆|∆〉Vermairreducible
Rational (p, p ′)-models at c = 13 − pp ′ −
p ′
p :
Kac table of “good” modules:12 (p − 1) × (p ′ − 1) nonisomorphicVirasoro irreps
p ′−1
. . .
. . . . . . . . . . . . . . .. . .. . .︸ ︷︷ ︸
p−1
These irreps have no extensions among themselves =⇒ semisimple(diagonalizable)
=⇒ chiral space of states =⊕
(irreps)
=⇒ numerous deep properties of rcft. . .
AM Semikhatov LCFT and representation theory
![Page 49: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/49.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Rational models: basic representation-theory unput
Virasoro algebra [Lm, Ln] = (m − n)Lm+n + c12(m3 − m)δm+n,0
Highest-weight modules Ln > 1|∆〉 = 0, L0|∆〉 = ∆|∆〉Vermairreducible
Rational (p, p ′)-models at c = 13 − pp ′ −
p ′
p :
Kac table of “good” modules:12 (p − 1) × (p ′ − 1) nonisomorphicVirasoro irreps
p ′−1
. . .
. . . . . . . . . . . . . . .. . .. . .︸ ︷︷ ︸
p−1
These irreps have no extensions among themselves =⇒ semisimple(diagonalizable)
=⇒ chiral space of states =⊕
(irreps)
=⇒ numerous deep properties of rcft. . .
AM Semikhatov LCFT and representation theory
![Page 50: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/50.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Rational models: basic representation-theory unput
Virasoro algebra [Lm, Ln] = (m − n)Lm+n + c12(m3 − m)δm+n,0
Highest-weight modules Ln > 1|∆〉 = 0, L0|∆〉 = ∆|∆〉Vermairreducibleindecomposable
Rational (p, p ′)-models at c = 13 − pp ′ −
p ′
p :
Kac table of “good” modules:12 (p − 1) × (p ′ − 1) nonisomorphicVirasoro irreps
p ′−1
. . .
. . . . . . . . . . . . . . .. . .. . .︸ ︷︷ ︸
p−1
These irreps have no extensions among themselves =⇒ semisimple(diagonalizable)
=⇒ chiral space of states =⊕
(irreps)
=⇒ numerous deep properties of rcft. . .
AM Semikhatov LCFT and representation theory
![Page 51: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/51.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Rational models: basic representation-theory unput
Virasoro algebra [Lm, Ln] = (m − n)Lm+n + c12(m3 − m)δm+n,0
Highest-weight modules Ln > 1|∆〉 = 0, L0|∆〉 = ∆|∆〉Vermairreducible
Rational (p, p ′)-models at c = 13 − pp ′ −
p ′
p :
Kac table of “good” modules:12 (p − 1) × (p ′ − 1) nonisomorphicVirasoro irreps
p ′−1
. . .
. . . . . . . . . . . . . . .. . .. . .︸ ︷︷ ︸
p−1
These irreps have no extensions among themselves =⇒ semisimple(diagonalizable)
=⇒ chiral space of states =⊕
(irreps)
=⇒ numerous deep properties of rcft. . .
AM Semikhatov LCFT and representation theory
![Page 52: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/52.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Rational models: basic representation-theory unput
Virasoro algebra [Lm, Ln] = (m − n)Lm+n + c12(m3 − m)δm+n,0
Highest-weight modules Ln > 1|∆〉 = 0, L0|∆〉 = ∆|∆〉Vermairreducible
Rational (p, p ′)-models at c = 13 − pp ′ −
p ′
p :
Kac table of “good” modules:12 (p − 1) × (p ′ − 1) nonisomorphicVirasoro irreps
p ′−1
. . .
. . . . . . . . . . . . . . .. . .. . .︸ ︷︷ ︸
p−1
These irreps have no extensions among themselves =⇒ semisimple(diagonalizable)
=⇒ chiral space of states =⊕
(irreps)
=⇒ numerous deep properties of rcft. . .
AM Semikhatov LCFT and representation theory
![Page 53: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/53.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Rational models: basic representation-theory unput
Virasoro algebra [Lm, Ln] = (m − n)Lm+n + c12(m3 − m)δm+n,0
Highest-weight modules Ln > 1|∆〉 = 0, L0|∆〉 = ∆|∆〉Vermairreducible
Rational (p, p ′)-models at c = 13 − pp ′ −
p ′
p :
Kac table of “good” modules:12 (p − 1) × (p ′ − 1) nonisomorphicVirasoro irreps
p ′−1
. . .
. . . . . . . . . . . . . . .. . .. . .︸ ︷︷ ︸
p−1
These irreps have no extensions among themselves =⇒ semisimple(diagonalizable)
=⇒ chiral space of states =⊕
(irreps)
=⇒ numerous deep properties of rcft. . .
AM Semikhatov LCFT and representation theory
![Page 54: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/54.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Rational models: basic representation-theory unput
Virasoro algebra [Lm, Ln] = (m − n)Lm+n + c12(m3 − m)δm+n,0
Highest-weight modules Ln > 1|∆〉 = 0, L0|∆〉 = ∆|∆〉Vermairreducible
Rational (p, p ′)-models at c = 13 − pp ′ −
p ′
p :
Kac table of “good” modules:12 (p − 1) × (p ′ − 1) nonisomorphicVirasoro irreps
p ′−1
. . .
. . . . . . . . . . . . . . .. . .. . .︸ ︷︷ ︸
p−1
These irreps have no extensions among themselves =⇒ semisimple(diagonalizable)
=⇒ chiral space of states =⊕
(irreps)
=⇒ numerous deep properties of rcft. . .
AM Semikhatov LCFT and representation theory
![Page 55: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/55.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFT: “minimal” extension
adding 1 row and 1 column:
p ′−1
. . .
. . . . . . . . . . . . . .. . .. . .︸ ︷︷ ︸
p−1
Also, a new possibility: (p, 1) modelswith the extended Kac table
. . .︸ ︷︷ ︸p
Drastic consequences
Representations admit indecomposable extensions
0→ X→ A→ Y→ 0, orY• −→ X•
=⇒ chiral space of states =⊕
(projective modules)
The symmetry extends from Virasoro to a larger W -algebra
AM Semikhatov LCFT and representation theory
![Page 56: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/56.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFT: “minimal” extension
adding 1 row and 1 column:“only” p + p ′ − 1 new boxes
p ′−1
. . .. . .
. . . . . . . . . . . . . .. . .. . .. . .︸ ︷︷ ︸
p−1
Also, a new possibility: (p, 1) modelswith the extended Kac table
. . .︸ ︷︷ ︸p
Drastic consequences
Representations admit indecomposable extensions
0→ X→ A→ Y→ 0, orY• −→ X•
=⇒ chiral space of states =⊕
(projective modules)
The symmetry extends from Virasoro to a larger W -algebra
AM Semikhatov LCFT and representation theory
![Page 57: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/57.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFT: “minimal” extension
adding 1 row and 1 column:“only” p + p ′ − 1 new boxes
p ′−1
. . .. . .
. . . . . . . . . . . . . .. . .. . .. . .︸ ︷︷ ︸
p−1
Also, a new possibility: (p, 1) modelswith the extended Kac table
. . .︸ ︷︷ ︸p
Drastic consequences
Representations admit indecomposable extensions
0→ X→ A→ Y→ 0, orY• −→ X•
=⇒ chiral space of states =⊕
(projective modules)
The symmetry extends from Virasoro to a larger W -algebra
AM Semikhatov LCFT and representation theory
![Page 58: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/58.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFT: “minimal” extension
adding 1 row and 1 column:“only” p + p ′ − 1 new boxes
p ′−1
. . .. . .
. . . . . . . . . . . . . .. . .. . .. . .︸ ︷︷ ︸
p−1
Also, a new possibility: (p, 1) modelswith the extended Kac table
. . .︸ ︷︷ ︸p
Nonlogarithmic content: void
Drastic consequences
Representations admit indecomposable extensions
0→ X→ A→ Y→ 0, orY• −→ X•
=⇒ chiral space of states =⊕
(projective modules)
The symmetry extends from Virasoro to a larger W -algebra
AM Semikhatov LCFT and representation theory
![Page 59: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/59.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFT: “minimal” extension
adding 1 row and 1 column:“only” p + p ′ − 1 new boxes
p ′−1
. . .. . .
. . . . . . . . . . . . . .. . .. . .. . .︸ ︷︷ ︸
p−1
Also, a new possibility: (p, 1) modelswith the extended Kac table
. . .︸ ︷︷ ︸p
Drastic consequences
Representations admit indecomposable extensions
0→ X→ A→ Y→ 0, orY• −→ X•
=⇒ chiral space of states =⊕
(projective modules)
The symmetry extends from Virasoro to a larger W -algebra
AM Semikhatov LCFT and representation theory
![Page 60: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/60.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFT: “minimal” extension
adding 1 row and 1 column:“only” p + p ′ − 1 new boxes
p ′−1
. . .. . .
. . . . . . . . . . . . . .. . .. . .. . .︸ ︷︷ ︸
p−1
Also, a new possibility: (p, 1) modelswith the extended Kac table
. . .︸ ︷︷ ︸p
Drastic consequences:
Representations admit indecomposable extensions
0→ X→ A→ Y→ 0, orY• −→ X•
=⇒ chiral space of states =⊕
(projective modules)
The symmetry extends from Virasoro to a larger W -algebra
AM Semikhatov LCFT and representation theory
![Page 61: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/61.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFT: “minimal” extension
adding 1 row and 1 column:“only” p + p ′ − 1 new boxes
p ′−1
. . .. . .
. . . . . . . . . . . . . .. . .. . .. . .︸ ︷︷ ︸
p−1
Also, a new possibility: (p, 1) modelswith the extended Kac table
. . .︸ ︷︷ ︸p
Drastic consequences:
Representations admit indecomposable extensions
0→ X→ A→ Y→ 0, orY• −→ X•
=⇒ chiral space of states =⊕
(projective modules)
The symmetry extends from Virasoro to a larger W -algebra
AM Semikhatov LCFT and representation theory
![Page 62: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/62.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFT: “minimal” extension
adding 1 row and 1 column:“only” p + p ′ − 1 new boxes
p ′−1
. . .. . .
. . . . . . . . . . . . . .. . .. . .. . .︸ ︷︷ ︸
p−1
Also, a new possibility: (p, 1) modelswith the extended Kac table
. . .︸ ︷︷ ︸p
Drastic consequences:
Representations admit indecomposable extensions
0→ X→ A→ Y→ 0, orY• −→ X•
=⇒ chiral space of states =⊕
(projective modules)
Projective modules: “maximally indecomposable”
The symmetry extends from Virasoro to a larger W -algebra
AM Semikhatov LCFT and representation theory
![Page 63: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/63.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFT: “minimal” extension
adding 1 row and 1 column:“only” p + p ′ − 1 new boxes
p ′−1
. . .. . .
. . . . . . . . . . . . . .. . .. . .. . .︸ ︷︷ ︸
p−1
Also, a new possibility: (p, 1) modelswith the extended Kac table
. . .︸ ︷︷ ︸p
Drastic consequences:
Representations admit indecomposable extensions
0→ X→ A→ Y→ 0, orY• −→ X•
=⇒ chiral space of states =⊕
(projective modules)
Projective modules: “maximally indecomposable”They are home for logarithmic partners
The symmetry extends from Virasoro to a larger W -algebra
AM Semikhatov LCFT and representation theory
![Page 64: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/64.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFT: “minimal” extension: New Hope?
adding 1 row and 1 column:“only” p + p ′ − 1 new boxes
p ′−1
. . .. . .
. . . . . . . . . . . . . .. . .. . .. . .︸ ︷︷ ︸
p−1
Also, a new possibility: (p, 1) modelswith the extended Kac table
. . .︸ ︷︷ ︸p
Drastic consequences:
Representations admit indecomposable extensions
0→ X→ A→ Y→ 0, orY• −→ X•
=⇒ chiral space of states =⊕
(projective modules)
The symmetry extends from Virasoro to a larger W -algebra
AM Semikhatov LCFT and representation theory
![Page 65: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/65.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFT: “minimal” extension: New Hope?
adding 1 row and 1 column:“only” p + p ′ − 1 new boxes
p ′−1
. . .. . .
. . . . . . . . . . . . . .. . .. . .. . .︸ ︷︷ ︸
p−1
Also, a new possibility: (p, 1) modelswith the extended Kac table
. . .︸ ︷︷ ︸p
Drastic consequences:
Representations admit indecomposable extensions
0→ X→ A→ Y→ 0, orY• −→ X•
=⇒ chiral space of states =⊕
(W -algebra projective modules)
The symmetry extends from Virasoro to a larger W -algebra
AM Semikhatov LCFT and representation theory
![Page 66: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/66.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFT: “minimal” extension: New Hope?
adding 1 row and 1 column:“only” p + p ′ − 1 new boxes
p ′−1
. . .. . .
. . . . . . . . . . . . . .. . .. . .. . .︸ ︷︷ ︸
p−1
Also, a new possibility: (p, 1) modelswith the extended Kac table
. . .︸ ︷︷ ︸p
Drastic consequences:
Representations admit indecomposable extensions
0→ X→ A→ Y→ 0, orY• −→ X•
=⇒ chiral space of states =⊕
(W -algebra projective modules)
The symmetry extends from Virasoro to a larger W -algebra(triplet W -algebra [(p, 1): Kausch, Kausch and Gaberdiel])
AM Semikhatov LCFT and representation theory
![Page 67: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/67.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Basic problem:
Virtually nothing is known about projective modules of Virasoro and“larger” algebras.
So —
in contrast to the rational case, representation theory fails?!
AM Semikhatov LCFT and representation theory
![Page 68: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/68.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Basic problem:
Virtually nothing is known about projective modules of Virasoro and“larger” algebras.
For the (p, 1) triplet W -algebra, projectivemodules must have the structure
Xas•
zz $$X−a
p−s◦��
X−ap−s◦
��Xa
s•
but none of theconstruction detailsare worked out
So —
in contrast to the rational case, representation theory fails?!
AM Semikhatov LCFT and representation theory
![Page 69: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/69.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Basic problem:
Virtually nothing is known about projective modules of Virasoro and“larger” algebras.
For the (p, 1) triplet W -algebra, projectivemodules must have the structure
Xas•
zz $$X−a
p−s◦��
X−ap−s◦
��Xa
s•
but none of theconstruction detailsare worked out
So —
in contrast to the rational case, representation theory fails?!
AM Semikhatov LCFT and representation theory
![Page 70: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/70.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Basic problem:
Virtually nothing is known about projective modules of Virasoro and“larger” algebras.
For the (p, p ′) triplet W -algebra, even the more complicated structure♣
vv
bdegikm ~~ ((
\ Z Y W U S Q ♦
�� ��
N >/
))
♥roljgqq ec
N >��/ ))
♥--
LO R T W Y [
��
p�
� ((
♦��
ss
`aabbccddeeffgg uu♠�� %%
[ Y W T R O L
♣�� ))
♠))vv
��p
♠uu ((
/>
N
♣((
/>
Nss
`aabbccddeeffgg
♠��yy
cegjlor♦
((
\ Z Y W U S Q
♥
♥~~
♦vv
bdegikm♣
although involved in the true projective module, is insufficient.
So —
in contrast to the rational case, representation theory fails?!AM Semikhatov LCFT and representation theory
![Page 71: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/71.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Basic problem:
Virtually nothing is known about projective modules of Virasoro and“larger” algebras.
So —
in contrast to the rational case, representation theory fails?!
AM Semikhatov LCFT and representation theory
![Page 72: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/72.jpg)
MotivationRepresentation theory and CFT
Quantum groups
The Indecomposables Strike Back
Basic problem:
Virtually nothing is known about projective modules of Virasoro and“larger” algebras.
So —
in contrast to the rational case, representation theory fails?!
AM Semikhatov LCFT and representation theory
![Page 73: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/73.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Return of the FreeField approach
Basic problem:
Virtually nothing is known about projective modules of Virasoro and“larger” algebras.
So —
in contrast to the rational case, representation theory fails?!
Resort to:
1 Free-field construction
2 Kazhdan–Lusztig correspondence
AM Semikhatov LCFT and representation theory
![Page 74: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/74.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Return of the FreeField approach
Basic problem:
Virtually nothing is known about projective modules of Virasoro and“larger” algebras.
So —
in contrast to the rational case, representation theory fails?!
Resort to:
1 Free-field construction
2 Kazhdan–Lusztig correspondence
AM Semikhatov LCFT and representation theory
![Page 75: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/75.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFTs in terms of free fields
1 Take screenings in a free-field realization: e.g., S±=
∫eα±ϕ(z)dz
2 rational models are the cohomology of (the differential associated with)screenings
3 Take the kernel of the screenings
4 The kernel is a representation space of a W -algebra— the maximum local algebra acting in the kernel.
AM Semikhatov LCFT and representation theory
![Page 76: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/76.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFTs in terms of free fields
1 Take screenings in a free-field realization: e.g., S±=
∫eα±ϕ(z)dz
2 rational models are the cohomology of (the differential associated with)screenings
3 Take the kernel of the screenings
4 The kernel is a representation space of a W -algebra— the maximum local algebra acting in the kernel.
AM Semikhatov LCFT and representation theory
![Page 77: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/77.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFTs in terms of free fields
1 Take screenings in a free-field realization: e.g., S±=
∫eα±ϕ(z)dz
2 rational models are the cohomology of (the differential associated with)screenings
3 Take the kernel of the screenings
4 The kernel is a representation space of a W -algebra— the maximum local algebra acting in the kernel.
AM Semikhatov LCFT and representation theory
![Page 78: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/78.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFTs in terms of free fields
1 Take screenings in a free-field realization: e.g., S±=
∫eα±ϕ(z)dz
2 rational models are the cohomology of (the differential associated with)screenings
3 Take the kernel of the screenings
4 The kernel is a representation space of a W -algebra— the maximum local algebra acting in the kernel.
AM Semikhatov LCFT and representation theory
![Page 79: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/79.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFTs in terms of free fields
1 Take screenings in a free-field realization: e.g., S±=
∫eα±ϕ(z)dz
2 rational models are the cohomology of (the differential associated with)screenings
3 Take the kernel of the screenings (“kernel > cohomology”)
4 The kernel is a representation space of a W -algebra— the maximum local algebra acting in the kernel.
AM Semikhatov LCFT and representation theory
![Page 80: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/80.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFTs in terms of free fields
1 Take screenings in a free-field realization: e.g., S±=
∫eα±ϕ(z)dz
2 rational models are the cohomology of (the differential associated with)screenings
3 Take the kernel of the screenings
4 The kernel is a representation space of a W -algebra— the maximum local algebra acting in the kernel.
AM Semikhatov LCFT and representation theory
![Page 81: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/81.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFTs in terms of free fields
1 Take screenings in a free-field realization: e.g., S±=
∫eα±ϕ(z)dz
2 rational models are the cohomology of (the differential associated with)screenings
3 Take the kernel of the screenings
4 The kernel is a representation space of a W -algebra— the maximum local algebra acting in the kernel.
AM Semikhatov LCFT and representation theory
![Page 82: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/82.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFTs in terms of free fields
1 Take screenings in a free-field realization: e.g., S±=
∫eα±ϕ(z)dz
2 rational models are the cohomology of (the differential associated with)screenings
3 Take the kernel of the screenings
4 The kernel is a representation space of a W -algebra— the maximum local algebra acting in the kernel.×
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AM Semikhatov LCFT and representation theory
![Page 83: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/83.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFTs in terms of free fields
1 Take screenings in a free-field realization: e.g., S±=
∫eα±ϕ(z)dz
2 rational models are the cohomology of (the differential associated with)screenings
3 Take the kernel of the screenings
4 The kernel is a representation space of a W -algebra— the maximum local algebra acting in the kernel.W -algebra generators for (3, 2)
W + =(
3527
(∂
4ϕ)2
+ 5627∂
5ϕ∂
3ϕ+ 28
27∂
6ϕ∂
2ϕ+ 8
27∂
7ϕ∂ϕ− 280
9√
3
(∂
3ϕ)2∂
2ϕ
− 70
3√
3∂
4ϕ(∂
2ϕ)2
− 280
9√
3∂
4ϕ∂
3ϕ∂ϕ− 56
3√
3∂
5ϕ∂
2ϕ∂ϕ− 28
9√
3∂
6ϕ(∂ϕ)2
+ 353
(∂
2ϕ)4 + 280
3∂
3ϕ(∂
2ϕ)2∂ϕ+ 280
9
(∂
3ϕ)2(
∂ϕ)2
+ 1403∂
4ϕ∂
2ϕ(∂ϕ
)2+ 56
9∂
5ϕ(∂ϕ)3
− 140√3
(∂
2ϕ)3(∂ϕ)2
− 560
3√
3∂
3ϕ∂
2ϕ(∂ϕ)2
− 70
3√
3∂
4ϕ(∂ϕ)4
+ 70(∂
2ϕ)2(∂ϕ)4
+ 563∂
3ϕ(∂ϕ)5 − 28√
3∂
2ϕ(∂ϕ)6
+(∂ϕ)8 − 1
27√
3∂
8ϕ)e2√
3ϕ,
AM Semikhatov LCFT and representation theory
![Page 84: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/84.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFTs in terms of free fields
1 Take screenings in a free-field realization: e.g., S±=
∫eα±ϕ(z)dz
2 rational models are the cohomology of (the differential associated with)screenings
3 Take the kernel of the screenings
4 The kernel is a representation space of a W -algebra— the maximum local algebra acting in the kernel.W -algebra generators for (3, 2)
W −=
(217192
(∂
5ϕ
)2− 2653
3456∂
6ϕ∂
4ϕ− 23
384∂
7ϕ∂
3ϕ− 11
1152∂
8ϕ∂
2ϕ− 1
768∂
9ϕ∂ϕ− 1225
64√
3∂
4ϕ
(∂
3ϕ
)2
− 13475576√
3
(∂
4ϕ
)2∂
2ϕ+ 2695
64√
3∂
5ϕ∂
3ϕ∂
2ϕ+ 2555
192√
3∂
5ϕ∂
4ϕ∂ϕ− 2891
576√
3∂
6ϕ
(∂
2ϕ
)2− 1351
192√
3∂
6ϕ∂
3ϕ∂ϕ
− 103192√
3∂
7ϕ∂
2ϕ∂ϕ− 13
384√
3∂
8ϕ
(∂ϕ
)2+ 3535
32
(∂
3ϕ
)2(∂
2ϕ
)2− 735
16
(∂
3ϕ
)3∂ϕ− 3395
54∂
4ϕ
(∂
2ϕ
)3
+ 24524∂
4ϕ∂
3ϕ∂
2ϕ∂ϕ+ 12635
576
(∂
4ϕ
)2(∂ϕ
)2+ 245
12∂
5ϕ
(∂
2ϕ
)2∂ϕ+ 105
32∂
5ϕ∂
3ϕ
(∂ϕ
)2
− 2443288
∂6ϕ∂
2ϕ
(∂ϕ
)2− 19
96∂
7ϕ
(∂ϕ
)3− 13405
144√
3
(∂
2ϕ
)5+ 8225
24√
3∂
3ϕ
(∂
2ϕ)
3∂ϕ− 105
√3
4
(∂
3ϕ
)2∂
2ϕ
(∂ϕ
)2
+ 66524√
3∂
4ϕ
(∂
2ϕ
)2(∂ϕ
)2+ 245
2√
3∂
4ϕ∂
3ϕ
(∂ϕ
)3− 245
8√
3∂
5ϕ∂
2ϕ
(∂ϕ
)3− 91
24√
3∂
6ϕ
(∂ϕ
)4+ 16205
144
(∂
2ϕ
)4(∂ϕ
)2
+ 3854∂
3ϕ
(∂
2ϕ
)2(∂ϕ
)3+ 525
8
(∂
3ϕ
)2(∂ϕ
)4+ 35
3∂
4ϕ∂
2ϕ
(∂ϕ
)4− 7∂5
ϕ(∂ϕ
)5+ 665
3√
3
(∂
2ϕ
)3(∂ϕ
)4
+ 105√
32
∂3ϕ∂
2ϕ
(∂ϕ
)5− 35
3√
3∂
4ϕ
(∂ϕ
)6+ 455
6
(∂
2ϕ
)2(∂ϕ
)6+ 5∂3
ϕ(∂ϕ
)7+ 25√
3∂
2ϕ
(∂ϕ
)8
+(∂ϕ
)10− 1
13824√
3∂
10ϕ
)e−2√
3ϕ,
AM Semikhatov LCFT and representation theory
![Page 85: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/85.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFTs in terms of free fields
1 Take screenings in a free-field realization: e.g., S±=
∫eα±ϕ(z)dz
2 rational models are the cohomology of (the differential associated with)screenings
3 Take the kernel of the screenings
4 The kernel is a representation space of a W -algebra— the maximum local algebra acting in the kernel.
Key differences from the rational case
The LCFT model may be dependent on the free-field representation taken,on the screenings chosen, etc.
The symmetry algebra of a LCFT model is larger than the “naive” algebra(e.g., Virasoro).
The space of states in a LCFT is not the direct sum of irreduciblerepresentations but the sum of all (finitely many) projective modules
P =⊕ι
Pι.
AM Semikhatov LCFT and representation theory
![Page 86: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/86.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFTs in terms of free fields
1 Take screenings in a free-field realization: e.g., S±=
∫eα±ϕ(z)dz
2 rational models are the cohomology of (the differential associated with)screenings
3 Take the kernel of the screenings
4 The kernel is a representation space of a W -algebra— the maximum local algebra acting in the kernel.
Key differences from the rational case
The LCFT model may be dependent on the free-field representation taken,on the screenings chosen, etc.
The symmetry algebra of a LCFT model is larger than the “naive” algebra(e.g., Virasoro).
The space of states in a LCFT is not the direct sum of irreduciblerepresentations but the sum of all (finitely many) projective modules
P =⊕ι
Pι.
AM Semikhatov LCFT and representation theory
![Page 87: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/87.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFTs in terms of free fields
1 Take screenings in a free-field realization: e.g., S±=
∫eα±ϕ(z)dz
2 rational models are the cohomology of (the differential associated with)screenings
3 Take the kernel of the screenings
4 The kernel is a representation space of a W -algebra— the maximum local algebra acting in the kernel.
Key differences from the rational case
The LCFT model may be dependent on the free-field representation taken,on the screenings chosen, etc.
The symmetry algebra of a LCFT model is larger than the “naive” algebra(e.g., Virasoro).
The space of states in a LCFT is not the direct sum of irreduciblerepresentations but the sum of all (finitely many) projective modules
P =⊕ι
Pι.
AM Semikhatov LCFT and representation theory
![Page 88: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/88.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFTs in terms of free fields
1 Take screenings in a free-field realization: e.g., S±=
∫eα±ϕ(z)dz
2 rational models are the cohomology of (the differential associated with)screenings
3 Take the kernel of the screenings
4 The kernel is a representation space of a W -algebra— the maximum local algebra acting in the kernel.
Key differences from the rational case
The LCFT model may be dependent on the free-field representation taken,on the screenings chosen, etc.
The symmetry algebra of a LCFT model is larger than the “naive” algebra(e.g., Virasoro).
The space of states in a LCFT is not the direct sum of irreduciblerepresentations but the sum of all (finitely many) projective modules
P =⊕ι
Pι.
AM Semikhatov LCFT and representation theory
![Page 89: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/89.jpg)
MotivationRepresentation theory and CFT
Quantum groups
LCFTs in terms of free fields
1 Take screenings in a free-field realization: e.g., S±=
∫eα±ϕ(z)dz
2 rational models are the cohomology of (the differential associated with)screenings
3 Take the kernel of the screenings
4 The kernel is a representation space of a W -algebra— the maximum local algebra acting in the kernel.
Key differences from the rational case
The LCFT model may be dependent on the free-field representation taken,on the screenings chosen, etc.
The symmetry algebra of a LCFT model is larger than the “naive” algebra(e.g., Virasoro).
The space of states in a LCFT is not the direct sum of irreduciblerepresentations but the sum of all (finitely many) projective modules
P =⊕ι
Pι.
AM Semikhatov LCFT and representation theory
![Page 90: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/90.jpg)
MotivationRepresentation theory and CFT
Quantum groups
W-algebras and their representations
(p, 1) models: the triplet W -algebra Wp
has 2p irreps X±r , r = 1, . . . , p.
∆X+(r) =(p − r)2
4p+
c − 1
24, ∆X−(r) =
(2p − r)2
4p+
c − 1
24.
(p, p ′) models: 2pp ′ irreps of the corresponding Wp,p ′ :X±r ,r ′ , r = 1, . . . , p, r ′ = 1, . . . , p ′,∆X+
r,r ′= ∆r,p ′−r ′;1, ∆X−
r,r ′= ∆p−r,r ′;−2,
∆r,r ′;n =(pr ′−p ′r+pp ′n)2 − (p−p ′)2
4pp ′.
PLUS the 12(p − 1)(p ′ − 1) representations from the Virasoro
minimal model.
kernel of the screenings =
N⊕A
XA
— finite sum of irreducible representatons
AM Semikhatov LCFT and representation theory
![Page 91: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/91.jpg)
MotivationRepresentation theory and CFT
Quantum groups
W-algebras and their representations
(p, 1) models: the triplet W -algebra Wp
has 2p irreps X±r , r = 1, . . . , p.
∆X+(r) =(p − r)2
4p+
c − 1
24, ∆X−(r) =
(2p − r)2
4p+
c − 1
24.
(p, p ′) models: 2pp ′ irreps of the corresponding Wp,p ′ :X±r ,r ′ , r = 1, . . . , p, r ′ = 1, . . . , p ′,∆X+
r,r ′= ∆r,p ′−r ′;1, ∆X−
r,r ′= ∆p−r,r ′;−2,
∆r,r ′;n =(pr ′−p ′r+pp ′n)2 − (p−p ′)2
4pp ′.
PLUS the 12(p − 1)(p ′ − 1) representations from the Virasoro
minimal model.
kernel of the screenings =
N⊕A
XA
— finite sum of irreducible representatons
AM Semikhatov LCFT and representation theory
![Page 92: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/92.jpg)
MotivationRepresentation theory and CFT
Quantum groups
W-algebras and their representations
(p, 1) models: the triplet W -algebra Wp
has 2p irreps X±r , r = 1, . . . , p.
∆X+(r) =(p − r)2
4p+
c − 1
24, ∆X−(r) =
(2p − r)2
4p+
c − 1
24.
(p, p ′) models: 2pp ′ irreps of the corresponding Wp,p ′ :X±r ,r ′ , r = 1, . . . , p, r ′ = 1, . . . , p ′,∆X+
r,r ′= ∆r,p ′−r ′;1, ∆X−
r,r ′= ∆p−r,r ′;−2,
∆r,r ′;n =(pr ′−p ′r+pp ′n)2 − (p−p ′)2
4pp ′.
PLUS the 12(p − 1)(p ′ − 1) representations from the Virasoro
minimal model.
kernel of the screenings =
N⊕A
XA
— finite sum of irreducible representatons
AM Semikhatov LCFT and representation theory
![Page 93: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/93.jpg)
MotivationRepresentation theory and CFT
Quantum groups
W-algebras and their representations
(p, 1) models: the triplet W -algebra Wp
has 2p irreps X±r , r = 1, . . . , p.
∆X+(r) =(p − r)2
4p+
c − 1
24, ∆X−(r) =
(2p − r)2
4p+
c − 1
24.
(p, p ′) models: 2pp ′ irreps of the corresponding Wp,p ′ :X±r ,r ′ , r = 1, . . . , p, r ′ = 1, . . . , p ′,∆X+
r,r ′= ∆r,p ′−r ′;1, ∆X−
r,r ′= ∆p−r,r ′;−2,
∆r,r ′;n =(pr ′−p ′r+pp ′n)2 − (p−p ′)2
4pp ′.
PLUS the 12(p − 1)(p ′ − 1) representations from the Virasoro
minimal model.
kernel of the screenings =
N⊕A
XA
— finite sum of irreducible representatons
AM Semikhatov LCFT and representation theory
![Page 94: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/94.jpg)
MotivationRepresentation theory and CFT
Quantum groups
W-algebras and their representations
(p, 1) models: the triplet W -algebra Wp
has 2p irreps X±r , r = 1, . . . , p.
∆X+(r) =(p − r)2
4p+
c − 1
24, ∆X−(r) =
(2p − r)2
4p+
c − 1
24.
(p, p ′) models: 2pp ′ irreps of the corresponding Wp,p ′ :X±r ,r ′ , r = 1, . . . , p, r ′ = 1, . . . , p ′,∆X+
r,r ′= ∆r,p ′−r ′;1, ∆X−
r,r ′= ∆p−r,r ′;−2,
∆r,r ′;n =(pr ′−p ′r+pp ′n)2 − (p−p ′)2
4pp ′.
PLUS the 12(p − 1)(p ′ − 1) representations from the Virasoro
minimal model.
kernel of the screenings =
N⊕A
XA
— finite sum of irreducible representatons
AM Semikhatov LCFT and representation theory
![Page 95: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/95.jpg)
MotivationRepresentation theory and CFT
Quantum groups
W-algebra characters
(p, 1): The irreducible W -representation characters are given by
χ+r (q) =
1
η(q)
(r
pθp−r ,p(q) +
2
pθ ′p−r ,p(q)
),
χ−r (q) =
1
η(q)
(r
pθr ,p(q) −
2
pθ ′r ,p(q)
),
1 6 r 6 p.
AM Semikhatov LCFT and representation theory
![Page 96: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/96.jpg)
MotivationRepresentation theory and CFT
Quantum groups
W-algebra characters
(p, p ′): The irreducible W -representation characters are given by
χr ,r ′(q) =1
η(q)(θpr ′−p ′r ,pp ′(q) − θpr ′+p ′r ,pp ′(q)), (r , r ′) ∈ I1,
χ+r ,r ′ =
1
(pp ′)2η
(θ ′′pr ′+p ′r − θ ′′pr ′−p ′r
− (pr ′+p ′r)θ ′pr ′+p ′r + (pr ′−p ′r)θ ′pr ′−p ′r
+(pr ′+p ′r)2
4θpr ′+p ′r −
(pr ′−p ′r)2
4θpr ′−p ′r
), 1 6 r 6 p, 1 6 r ′6 p ′,
χ−r ,r ′ =
1
(pp ′)2η
(θ ′′pp ′−pr ′−p ′r − θ ′′pp ′+pr ′−p ′r
+ (pr ′+p ′r)θ ′pp ′−pr ′−p ′r + (pr ′−p ′r)θ ′pp ′+pr ′−p ′r
+(pr ′ + p ′r)2−(pp ′)2
4θpp ′−pr ′−p ′r
−(pr ′ − p ′r)2−(pp ′)2
4θpp ′+pr ′−p ′r
), 1 6 r 6 p, 1 6 r ′6 p ′.
AM Semikhatov LCFT and representation theory
![Page 97: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/97.jpg)
MotivationRepresentation theory and CFT
Quantum groups
W-algebra characters
(p, p ′): The irreducible W -representation characters are given by
χr ,r ′(q) =1
η(q)(θpr ′−p ′r ,pp ′(q) − θpr ′+p ′r ,pp ′(q)), (r , r ′) ∈ I1,
χ+r ,r ′ =
1
(pp ′)2η
(θ ′′pr ′+p ′r − θ ′′pr ′−p ′r
− (pr ′+p ′r)θ ′pr ′+p ′r + (pr ′−p ′r)θ ′pr ′−p ′r
+(pr ′+p ′r)2
4θpr ′+p ′r −
(pr ′−p ′r)2
4θpr ′−p ′r
), 1 6 r 6 p, 1 6 r ′6 p ′,
χ−r ,r ′ =
1
(pp ′)2η
(θ ′′pp ′−pr ′−p ′r − θ ′′pp ′+pr ′−p ′r
+ (pr ′+p ′r)θ ′pp ′−pr ′−p ′r + (pr ′−p ′r)θ ′pp ′+pr ′−p ′r
+(pr ′ + p ′r)2−(pp ′)2
4θpp ′−pr ′−p ′r
−(pr ′ − p ′r)2−(pp ′)2
4θpp ′+pr ′−p ′r
), 1 6 r 6 p, 1 6 r ′6 p ′.
AM Semikhatov LCFT and representation theory
![Page 98: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/98.jpg)
MotivationRepresentation theory and CFT
Quantum groups
W-algebra characters
(p, p ′): The irreducible W -representation characters are given by
χr ,r ′(q) =1
η(q)(θpr ′−p ′r ,pp ′(q) − θpr ′+p ′r ,pp ′(q)), (r , r ′) ∈ I1,
χ+r ,r ′ =
1
(pp ′)2η
(θ ′′pr ′+p ′r − θ ′′pr ′−p ′r
− (pr ′+p ′r)θ ′pr ′+p ′r + (pr ′−p ′r)θ ′pr ′−p ′r
+(pr ′+p ′r)2
4θpr ′+p ′r −
(pr ′−p ′r)2
4θpr ′−p ′r
), 1 6 r 6 p, 1 6 r ′6 p ′,
χ−r ,r ′ =
1
(pp ′)2η
(θ ′′pp ′−pr ′−p ′r − θ ′′pp ′+pr ′−p ′r
+ (pr ′+p ′r)θ ′pp ′−pr ′−p ′r + (pr ′−p ′r)θ ′pp ′+pr ′−p ′r
+(pr ′ + p ′r)2−(pp ′)2
4θpp ′−pr ′−p ′r
−(pr ′ − p ′r)2−(pp ′)2
4θpp ′+pr ′−p ′r
), 1 6 r 6 p, 1 6 r ′6 p ′.
AM Semikhatov LCFT and representation theory
![Page 99: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/99.jpg)
MotivationRepresentation theory and CFT
Quantum groups
W-algebra characters
(p, p ′): The irreducible W -representation characters are given by
χr ,r ′(q) =1
η(q)(θpr ′−p ′r ,pp ′(q) − θpr ′+p ′r ,pp ′(q)), (r , r ′) ∈ I1,
χ+r ,r ′ =
1
(pp ′)2η
(θ ′′pr ′+p ′r − θ ′′pr ′−p ′r
− (pr ′+p ′r)θ ′pr ′+p ′r + (pr ′−p ′r)θ ′pr ′−p ′r
+(pr ′+p ′r)2
4θpr ′+p ′r −
(pr ′−p ′r)2
4θpr ′−p ′r
), 1 6 r 6 p, 1 6 r ′6 p ′,
χ−r ,r ′ =
1
(pp ′)2η
(θ ′′pp ′−pr ′−p ′r − θ ′′pp ′+pr ′−p ′r
+ (pr ′+p ′r)θ ′pp ′−pr ′−p ′r + (pr ′−p ′r)θ ′pp ′+pr ′−p ′r
+(pr ′ + p ′r)2−(pp ′)2
4θpp ′−pr ′−p ′r
−(pr ′ − p ′r)2−(pp ′)2
4θpp ′+pr ′−p ′r
), 1 6 r 6 p, 1 6 r ′6 p ′.
AM Semikhatov LCFT and representation theory
![Page 100: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/100.jpg)
MotivationRepresentation theory and CFT
Quantum groups
W-Characters =⇒ Modular Group Representation
The need for generalized characters:
In LCFT, characters alone are not closed under SL(2,Z) action
(p, 1) models: The 2p characters give rise to a (3p − 1)-dimensionalSL(2,Z)-representation.
(p, p ′) models: The 2pp ′ + 12(p − 1)(p ′ − 1) characters give rise to
a 12(3p − 1)(3p ′ − 1)-dimensional SL(2,Z)-representation.
AM Semikhatov LCFT and representation theory
![Page 101: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/101.jpg)
MotivationRepresentation theory and CFT
Quantum groups
W-Characters =⇒ Modular Group Representation
(p, 1) models: The 2p characters give rise to a (3p − 1)-dimensionalSL(2,Z)-representation.
(p, p ′) models: The 2pp ′ + 12(p − 1)(p ′ − 1) characters give rise to
a 12(3p − 1)(3p ′ − 1)-dimensional SL(2,Z)-representation.
AM Semikhatov LCFT and representation theory
![Page 102: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/102.jpg)
MotivationRepresentation theory and CFT
Quantum groups
W-Characters =⇒ Modular Group Representation
(p, 1) models: The 2p characters give rise to a (3p − 1)-dimensionalSL(2,Z)-representation.
(p, p ′) models: The 2pp ′ + 12(p − 1)(p ′ − 1) characters give rise to
a 12(3p − 1)(3p ′ − 1)-dimensional SL(2,Z)-representation.
AM Semikhatov LCFT and representation theory
![Page 103: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/103.jpg)
MotivationRepresentation theory and CFT
Quantum groups
W-Characters =⇒ Modular Group Representation
(p, 1) models: The 2p characters give rise to a (3p − 1)-dimensionalSL(2,Z)-representation.
(p, p ′) models: The 2pp ′ + 12(p − 1)(p ′ − 1) characters give rise to
a 12(3p − 1)(3p ′ − 1)-dimensional SL(2,Z)-representation.
It is highly probable that these dimensions 3p − 1 and 12 (3p − 1)(3p ′ − 1) are the
dimensions of the spaces of torus amplitudes.
AM Semikhatov LCFT and representation theory
![Page 104: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/104.jpg)
MotivationRepresentation theory and CFT
Quantum groups
W-Characters =⇒ Modular Group Representation
(p, 1) models: The 2p characters give rise to a (3p − 1)-dimensionalSL(2,Z)-representation.
(p, p ′) models: The 2pp ′ + 12(p − 1)(p ′ − 1) characters give rise to
a 12(3p − 1)(3p ′ − 1)-dimensional SL(2,Z)-representation.
Theorem
The (3p − 1)-dimension SL(2,Z)-representation Zcft has the structure
Zcft = Rp+1 ⊕ C2 ⊗Rp−1,
Rp−1 is the [“sinπrsp ”] SL(2,Z)-representation realized in the s`(2)p−2
minimal model, Rp+1 is a [“cosπrsp ”] SL(2,Z)-representations of dimension
p + 1, and C2 is the defining two-dimensional representation of SL(2,Z).
AM Semikhatov LCFT and representation theory
![Page 105: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/105.jpg)
MotivationRepresentation theory and CFT
Quantum groups
W-Characters =⇒ Modular Group Representation
(p, 1) models: The 2p characters give rise to a (3p − 1)-dimensionalSL(2,Z)-representation.
(p, p ′) models: The 2pp ′ + 12(p − 1)(p ′ − 1) characters give rise to
a 12(3p − 1)(3p ′ − 1)-dimensional SL(2,Z)-representation.
Theorem
The (3p − 1)-dimension SL(2,Z)-representation Zcft has the structure
Zcft = Rp+1 ⊕ C2 ⊗Rp−1,
Rp−1 is the [“sinπrsp ”] SL(2,Z)-representation realized in the s`(2)p−2
minimal model, Rp+1 is a [“cosπrsp ”] SL(2,Z)-representations of dimension
p + 1, and C2 is the defining two-dimensional representation of SL(2,Z).
AM Semikhatov LCFT and representation theory
![Page 106: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/106.jpg)
MotivationRepresentation theory and CFT
Quantum groups
W-Characters =⇒ Modular Group Representation
(p, 1) models: The 2p characters give rise to a (3p − 1)-dimensionalSL(2,Z)-representation.
(p, p ′) models: The 2pp ′ + 12(p − 1)(p ′ − 1) characters give rise to
a 12(3p − 1)(3p ′ − 1)-dimensional SL(2,Z)-representation.
Theorem
The (3p − 1)-dimension SL(2,Z)-representation Zcft has the structure
Zcft = Rp+1 ⊕ C2 ⊗Rp−1,
Rp−1 is the [“sinπrsp ”] SL(2,Z)-representation realized in the s`(2)p−2
minimal model, Rp+1 is a [“cosπrsp ”] SL(2,Z)-representations of dimension
p + 1, and C2 is the defining two-dimensional representation of SL(2,Z).
AM Semikhatov LCFT and representation theory
![Page 107: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/107.jpg)
MotivationRepresentation theory and CFT
Quantum groups
W-Characters =⇒ Modular Group Representation
(p, 1) models: The 2p characters give rise to a (3p − 1)-dimensionalSL(2,Z)-representation.
(p, p ′) models: The 2pp ′ + 12(p − 1)(p ′ − 1) characters give rise to
a 12(3p − 1)(3p ′ − 1)-dimensional SL(2,Z)-representation.
Theorem
The (3p − 1)-dimension SL(2,Z)-representation Zcft has the structure
Zcft = Rp+1 ⊕ C2 ⊗Rp−1,
Rp−1 is the [“sinπrsp ”] SL(2,Z)-representation realized in the s`(2)p−2
minimal model, Rp+1 is a [“cosπrsp ”] SL(2,Z)-representations of dimension
p + 1, and C2 is the defining two-dimensional representation of SL(2,Z).
AM Semikhatov LCFT and representation theory
![Page 108: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/108.jpg)
MotivationRepresentation theory and CFT
Quantum groups
W-Characters =⇒ Modular Group Representation
(p, 1) models: The 2p characters give rise to a (3p − 1)-dimensionalSL(2,Z)-representation.
(p, p ′) models: The 2pp ′ + 12(p − 1)(p ′ − 1) characters give rise to
a 12(3p − 1)(3p ′ − 1)-dimensional SL(2,Z)-representation.
Theorem
The 12(3p − 1)(3p ′ − 1)-dimensional SL(2,Z)-representation Zcft has the
structure
Zcft = Rmin ⊕ Rproj ⊕ C2 ⊗ (R� ⊕ R�)⊕ C3 ⊗ Rmin,
Rmin is the 12 (p − 1)(p ′− 1)-dimensional SL(2,Z)-representation on the
characters of the (p, p ′) Virasoro minimal model, C3 ∼= S2(C2), and Rproj, R�,and R� are SL(2,Z)-representations of the respective dimensions12 (p + 1)(p ′+ 1), 1
2 (p − 1)(p ′+ 1), and 12 (p + 1)(p ′− 1).
AM Semikhatov LCFT and representation theory
![Page 109: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/109.jpg)
MotivationRepresentation theory and CFT
Quantum groups
W-Characters =⇒ Modular Group Representation
(p, 1) models: The 2p characters give rise to a (3p − 1)-dimensionalSL(2,Z)-representation.
(p, p ′) models: The 2pp ′ + 12(p − 1)(p ′ − 1) characters give rise to
a 12(3p − 1)(3p ′ − 1)-dimensional SL(2,Z)-representation.
Theorem
The 12(3p − 1)(3p ′ − 1)-dimensional SL(2,Z)-representation Zcft has the
structure
Zcft = Rmin ⊕ Rproj ⊕ C2 ⊗ (R� ⊕ R�)⊕ C3 ⊗ Rmin,
Rmin is the 12 (p − 1)(p ′− 1)-dimensional SL(2,Z)-representation on the
characters of the (p, p ′) Virasoro minimal model, C3 ∼= S2(C2), and Rproj, R�,and R� are SL(2,Z)-representations of the respective dimensions12 (p + 1)(p ′+ 1), 1
2 (p − 1)(p ′+ 1), and 12 (p + 1)(p ′− 1).
AM Semikhatov LCFT and representation theory
![Page 110: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/110.jpg)
MotivationRepresentation theory and CFT
Quantum groups
W-Characters =⇒ Modular Group Representation
(p, 1) models: The 2p characters give rise to a (3p − 1)-dimensionalSL(2,Z)-representation.
(p, p ′) models: The 2pp ′ + 12(p − 1)(p ′ − 1) characters give rise to
a 12(3p − 1)(3p ′ − 1)-dimensional SL(2,Z)-representation.
Theorem
The 12(3p − 1)(3p ′ − 1)-dimensional SL(2,Z)-representation Zcft has the
structure
Zcft = Rmin ⊕ Rproj ⊕ C2 ⊗ (R� ⊕ R�)⊕ C3 ⊗ Rmin,
Rmin is the 12 (p − 1)(p ′− 1)-dimensional SL(2,Z)-representation on the
characters of the (p, p ′) Virasoro minimal model, C3 ∼= S2(C2), and Rproj, R�,and R� are SL(2,Z)-representations of the respective dimensions12 (p + 1)(p ′+ 1), 1
2 (p − 1)(p ′+ 1), and 12 (p + 1)(p ′− 1).
AM Semikhatov LCFT and representation theory
![Page 111: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/111.jpg)
MotivationRepresentation theory and CFT
Quantum groups
W-Characters =⇒ Modular Group Representation
(p, 1) models: The 2p characters give rise to a (3p − 1)-dimensionalSL(2,Z)-representation.
(p, p ′) models: The 2pp ′ + 12(p − 1)(p ′ − 1) characters give rise to
a 12(3p − 1)(3p ′ − 1)-dimensional SL(2,Z)-representation.
Theorem
The 12(3p − 1)(3p ′ − 1)-dimensional SL(2,Z)-representation Zcft has the
structure
Zcft = Rmin ⊕ Rproj ⊕ C2 ⊗ (R� ⊕ R�)⊕ C3 ⊗ Rmin,
Rmin is the 12 (p − 1)(p ′− 1)-dimensional SL(2,Z)-representation on the
characters of the (p, p ′) Virasoro minimal model, C3 ∼= S2(C2), and Rproj, R�,and R� are SL(2,Z)-representations of the respective dimensions12 (p + 1)(p ′+ 1), 1
2 (p − 1)(p ′+ 1), and 12 (p + 1)(p ′− 1).
AM Semikhatov LCFT and representation theory
![Page 112: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/112.jpg)
MotivationRepresentation theory and CFT
Quantum groups
W-Characters =⇒ Modular Group Representation
(p, 1) models: The 2p characters give rise to a (3p − 1)-dimensionalSL(2,Z)-representation.
(p, p ′) models: The 2pp ′ + 12(p − 1)(p ′ − 1) characters give rise to
a 12(3p − 1)(3p ′ − 1)-dimensional SL(2,Z)-representation.
Theorem
The 12(3p − 1)(3p ′ − 1)-dimensional SL(2,Z)-representation Zcft has the
structure
Zcft = Rmin ⊕ Rproj ⊕ C2 ⊗ (R� ⊕ R�)⊕ C3 ⊗ Rmin,
Rmin is the 12 (p − 1)(p ′− 1)-dimensional SL(2,Z)-representation on the
characters of the (p, p ′) Virasoro minimal model, C3 ∼= S2(C2), and Rproj, R�,and R� are SL(2,Z)-representations of the respective dimensions12 (p + 1)(p ′+ 1), 1
2 (p − 1)(p ′+ 1), and 12 (p + 1)(p ′− 1).
AM Semikhatov LCFT and representation theory
![Page 113: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/113.jpg)
MotivationRepresentation theory and CFT
Quantum groups
W-Characters =⇒ Modular Group Representation
(p, 1) models: The 2p characters give rise to a (3p − 1)-dimensionalSL(2,Z)-representation.
(p, p ′) models: The 2pp ′ + 12(p − 1)(p ′ − 1) characters give rise to
a 12(3p − 1)(3p ′ − 1)-dimensional SL(2,Z)-representation.
It is highly probable that these dimensions 3p − 1 and 12 (3p − 1)(3p ′ − 1) are
the dimensions of the spaces of torus amplitudes.
AM Semikhatov LCFT and representation theory
![Page 114: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/114.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Some details for (p, p ′)Generalized characters:
subrep. dimension basis
Rmin12(p − 1)(p ′− 1) χr,r ′ , (r , r ′) ∈ I1
Rproj12(p + 1)(p ′+ 1) κr,r ′ , (r , r ′) ∈ I0
C2 ⊗ R� 2 · 12(p − 1)(p ′+ 1) ρ�
r,r ′ , ϕ�r,r ′ , (r , r ′) ∈ I�
C2 ⊗ R� 2 · 12(p + 1)(p ′− 1) ρ�
r,r ′ , ϕ�r,r ′ , (r , r ′) ∈ I�
C3 ⊗ Rmin 3 · 12(p − 1)(p ′− 1) ρr,r ′ , ψr,r ′ , ϕr,r ′ , (r , r ′) ∈ I1
AM Semikhatov LCFT and representation theory
![Page 115: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/115.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Some details for (p, p ′)Generalized characters:
subrep. dimension basis
Rmin12(p − 1)(p ′− 1) χr,r ′ , (r , r ′) ∈ I1
Rproj12(p + 1)(p ′+ 1) κr,r ′ , (r , r ′) ∈ I0
C2 ⊗ R� 2 · 12(p − 1)(p ′+ 1) ρ�
r,r ′ , ϕ�r,r ′ , (r , r ′) ∈ I�
C2 ⊗ R� 2 · 12(p + 1)(p ′− 1) ρ�
r,r ′ , ϕ�r,r ′ , (r , r ′) ∈ I�
C3 ⊗ Rmin 3 · 12(p − 1)(p ′− 1) ρr,r ′ , ψr,r ′ , ϕr,r ′ , (r , r ′) ∈ I1
κr,r ′ = χr,r ′ + 2χ+r,r ′ + 2χ−
r,p ′−r ′ + 2χ−p−r,r ′ + 2χ+
p−r,p ′−r ′ , (r , r ′) ∈ I1,
κ0,r ′ = 2χ+p,p ′−r ′ + 2χ−
p,r ′ , 1 6 r ′6 p ′− 1,
κr,0 = 2χ+p−r,p ′ + 2χ−
r,p ′ , 1 6 r 6 p − 1,
κ0,0 = 2χ+p,p ′ ,
κp,0 = 2χ−p,p ′
AM Semikhatov LCFT and representation theory
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MotivationRepresentation theory and CFT
Quantum groups
Some details for (p, p ′)Generalized characters:
subrep. dimension basis
Rmin12(p − 1)(p ′− 1) χr,r ′ , (r , r ′) ∈ I1
Rproj12(p + 1)(p ′+ 1) κr,r ′ , (r , r ′) ∈ I0
C2 ⊗ R� 2 · 12(p − 1)(p ′+ 1) ρ�
r,r ′ , ϕ�r,r ′ , (r , r ′) ∈ I�
C2 ⊗ R� 2 · 12(p + 1)(p ′− 1) ρ�
r,r ′ , ϕ�r,r ′ , (r , r ′) ∈ I�
C3 ⊗ Rmin 3 · 12(p − 1)(p ′− 1) ρr,r ′ , ψr,r ′ , ϕr,r ′ , (r , r ′) ∈ I1
ρ�r,r ′(τ) = p ′r−pr ′
2 χr,r ′(τ) + p ′(r − p)(χ+r,r ′(τ) + χ−
r,p ′−r ′(τ))
+ p ′r(χ+p−r,p ′−r ′(τ) + χ−
p−r,r ′(τ)), (r , r ′) ∈ I1,
ρ�r,0(τ) = p ′(rχ+
p−r,p ′(τ) − (p−r)χ−r,p ′(τ)), 1 6 r 6 p−1,
ϕ�r,r ′(τ) = τρ�
r,r ′(τ), (r , r ′) ∈ I�,
AM Semikhatov LCFT and representation theory
![Page 117: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/117.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Some details for (p, p ′)Generalized characters:
subrep. dimension basis
Rmin12(p − 1)(p ′− 1) χr,r ′ , (r , r ′) ∈ I1
Rproj12(p + 1)(p ′+ 1) κr,r ′ , (r , r ′) ∈ I0
C2 ⊗ R� 2 · 12(p − 1)(p ′+ 1) ρ�
r,r ′ , ϕ�r,r ′ , (r , r ′) ∈ I�
C2 ⊗ R� 2 · 12(p + 1)(p ′− 1) ρ�
r,r ′ , ϕ�r,r ′ , (r , r ′) ∈ I�
C3 ⊗ Rmin 3 · 12(p − 1)(p ′− 1) ρr,r ′ , ψr,r ′ , ϕr,r ′ , (r , r ′) ∈ I1
ρ�r,r ′(τ) = pr ′−p ′r
2 χr,r ′(τ) − p(p ′−r ′)(χ+r,r ′(τ) + χ−
p−r,r ′(τ))
+ pr ′(χ+p−r,p ′−r ′(τ) + χ−
r,p ′−r ′(τ)), (r , r ′) ∈ I1,
ρ�0,r ′(τ) = p(r ′χ+
p,p ′−r ′(τ) − (p ′−r ′)χ−p,r ′(τ)), 1 6 r ′6 p ′−1,
ϕ�r,r ′(τ) = τρ�
r,r ′(τ), (r , r ′) ∈ I�,
AM Semikhatov LCFT and representation theory
![Page 118: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/118.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Some details for (p, p ′)Generalized characters:
subrep. dimension basis
Rmin12(p − 1)(p ′− 1) χr,r ′ , (r , r ′) ∈ I1
Rproj12(p + 1)(p ′+ 1) κr,r ′ , (r , r ′) ∈ I0
C2 ⊗ R� 2 · 12(p − 1)(p ′+ 1) ρ�
r,r ′ , ϕ�r,r ′ , (r , r ′) ∈ I�
C2 ⊗ R� 2 · 12(p + 1)(p ′− 1) ρ�
r,r ′ , ϕ�r,r ′ , (r , r ′) ∈ I�
C3 ⊗ Rmin 3 · 12(p − 1)(p ′− 1) ρr,r ′ , ψr,r ′ , ϕr,r ′ , (r , r ′) ∈ I1
ρr,r ′(τ) = pp ′((p−r)(p ′−r ′)χ+
r,r ′(τ) + rr ′χ+p−r,p ′−r ′(τ) −
(pr ′−p ′r)2
4pp ′ χr,r ′(τ)
− (p−r)r ′χ−r,p ′−r ′(τ) − r(p ′−r ′)χ−
p−r,r ′(τ)), (r , r ′) ∈ I1
ψr,r ′(τ) = 2τρr,r ′(τ) + iπpp ′χr,r ′(τ), (r , r ′) ∈ I1,
ϕr,r ′(τ) = τ2ρr,r ′(τ) + iπpp ′τχr,r ′(τ), (r , r ′) ∈ I1.
AM Semikhatov LCFT and representation theory
![Page 119: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/119.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Corollary: Several modular invariants
involving τ explicitly:
ρ�(τ, τ) =
p−1∑r=1
im τ |ρ�r,0(τ)|
2 + 2∑
(r,r ′)∈I1
im τ |ρ�r,r ′(τ)|
2,
ρ(τ, τ) =∑
(r,r ′)∈I1
ρr,r ′(τ)(8(im τ)2ρr,r ′(τ) + 4pp ′πim τ χr,r ′(τ))
+ χr,r ′(τ)(4pp ′πim τ ρr,r ′(τ) + (πpp ′)2χr,r ′(τ)).
A-series:κ[A](τ, τ) =
= |κ0,0(τ)|2 + |κp,0(τ)|
2 + 2
p−1∑r=1
|κr,0(τ)|2 + 2
p ′−1∑r ′=1
|κ0,r ′(τ)|2 + 4
∑(r,r ′)∈I1
|κr,r ′(τ)|2
D-series (in the case p ′ ≡ 0 mod 4):
κ[D](τ, τ) = |κ0,0(τ) + κp,0(τ)|2 +
p−1∑r=1
|κr,0(τ) + κp−r,0(τ)|2
+∑
2 6 r ′ 6 p ′−1r ′ even
|κ0,r ′(τ) + κ0,p ′−r ′(τ)|2 +
∑(r,r ′)∈I1
r ′ even
2 |κr,r ′(τ) + κr,p ′−r ′(τ)|2.
E6-type invariant for (p, p ′) = (5, 12):
κ[E6](τ, τ) = |κ0,1(τ) − κ0,7(τ)|2 + |κ0,2(τ) − κ0,10(τ)|
2 + |κ0,5(τ) − κ0,11(τ)|2
+ 2|κ1,1(τ) − κ1,7(τ)|2 + 2|κ2,1(τ) − κ2,7(τ)|
2 + 2|κ2,5(τ) − κ3,1(τ)|2
+ 2|κ2,2(τ) − κ3,2(τ)|2 + 2|κ1,5(τ) − κ4,1(τ)|
2 + 2|κ1,2(τ) − κ4,2(τ)|2.
AM Semikhatov LCFT and representation theory
![Page 120: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/120.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Corollary: Several modular invariants
A-series:κ[A](τ, τ) =
= |κ0,0(τ)|2 + |κp,0(τ)|
2 + 2
p−1∑r=1
|κr,0(τ)|2 + 2
p ′−1∑r ′=1
|κ0,r ′(τ)|2 + 4
∑(r,r ′)∈I1
|κr,r ′(τ)|2
D-series (in the case p ′ ≡ 0 mod 4):
κ[D](τ, τ) = |κ0,0(τ) + κp,0(τ)|2 +
p−1∑r=1
|κr,0(τ) + κp−r,0(τ)|2
+∑
2 6 r ′ 6 p ′−1r ′ even
|κ0,r ′(τ) + κ0,p ′−r ′(τ)|2 +
∑(r,r ′)∈I1
r ′ even
2 |κr,r ′(τ) + κr,p ′−r ′(τ)|2.
E6-type invariant for (p, p ′) = (5, 12):
κ[E6](τ, τ) = |κ0,1(τ) − κ0,7(τ)|2 + |κ0,2(τ) − κ0,10(τ)|
2 + |κ0,5(τ) − κ0,11(τ)|2
+ 2|κ1,1(τ) − κ1,7(τ)|2 + 2|κ2,1(τ) − κ2,7(τ)|
2 + 2|κ2,5(τ) − κ3,1(τ)|2
+ 2|κ2,2(τ) − κ3,2(τ)|2 + 2|κ1,5(τ) − κ4,1(τ)|
2 + 2|κ1,2(τ) − κ4,2(τ)|2.
AM Semikhatov LCFT and representation theory
![Page 121: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/121.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Corollary: Several modular invariants
A-series:κ[A](τ, τ) =
= |κ0,0(τ)|2 + |κp,0(τ)|
2 + 2
p−1∑r=1
|κr,0(τ)|2 + 2
p ′−1∑r ′=1
|κ0,r ′(τ)|2 + 4
∑(r,r ′)∈I1
|κr,r ′(τ)|2
D-series (in the case p ′ ≡ 0 mod 4):
κ[D](τ, τ) = |κ0,0(τ) + κp,0(τ)|2 +
p−1∑r=1
|κr,0(τ) + κp−r,0(τ)|2
+∑
2 6 r ′ 6 p ′−1r ′ even
|κ0,r ′(τ) + κ0,p ′−r ′(τ)|2 +
∑(r,r ′)∈I1
r ′ even
2 |κr,r ′(τ) + κr,p ′−r ′(τ)|2.
E6-type invariant for (p, p ′) = (5, 12):
κ[E6](τ, τ) = |κ0,1(τ) − κ0,7(τ)|2 + |κ0,2(τ) − κ0,10(τ)|
2 + |κ0,5(τ) − κ0,11(τ)|2
+ 2|κ1,1(τ) − κ1,7(τ)|2 + 2|κ2,1(τ) − κ2,7(τ)|
2 + 2|κ2,5(τ) − κ3,1(τ)|2
+ 2|κ2,2(τ) − κ3,2(τ)|2 + 2|κ1,5(τ) − κ4,1(τ)|
2 + 2|κ1,2(τ) − κ4,2(τ)|2.
AM Semikhatov LCFT and representation theory
![Page 122: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/122.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Corollary: Several modular invariants
A-series:κ[A](τ, τ) =
= |κ0,0(τ)|2 + |κp,0(τ)|
2 + 2
p−1∑r=1
|κr,0(τ)|2 + 2
p ′−1∑r ′=1
|κ0,r ′(τ)|2 + 4
∑(r,r ′)∈I1
|κr,r ′(τ)|2
D-series (in the case p ′ ≡ 0 mod 4):
κ[D](τ, τ) = |κ0,0(τ) + κp,0(τ)|2 +
p−1∑r=1
|κr,0(τ) + κp−r,0(τ)|2
+∑
2 6 r ′ 6 p ′−1r ′ even
|κ0,r ′(τ) + κ0,p ′−r ′(τ)|2 +
∑(r,r ′)∈I1
r ′ even
2 |κr,r ′(τ) + κr,p ′−r ′(τ)|2.
E6-type invariant for (p, p ′) = (5, 12):
κ[E6](τ, τ) = |κ0,1(τ) − κ0,7(τ)|2 + |κ0,2(τ) − κ0,10(τ)|
2 + |κ0,5(τ) − κ0,11(τ)|2
+ 2|κ1,1(τ) − κ1,7(τ)|2 + 2|κ2,1(τ) − κ2,7(τ)|
2 + 2|κ2,5(τ) − κ3,1(τ)|2
+ 2|κ2,2(τ) − κ3,2(τ)|2 + 2|κ1,5(τ) − κ4,1(τ)|
2 + 2|κ1,2(τ) − κ4,2(τ)|2.
AM Semikhatov LCFT and representation theory
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MotivationRepresentation theory and CFT
Quantum groups
1 Motivation
2 Representation theory and CFT
3 Quantum groups
AM Semikhatov LCFT and representation theory
![Page 124: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/124.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Quantum groups: Kazhdan–Lusztig correspondence
THE SAME SL(2,Z) REPRESENTATIONS ARE REALIZED ONCENTERS OF THE CORRESPONDING QUANTUM GROUPS
AM Semikhatov LCFT and representation theory
![Page 125: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/125.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Quantum groups: Kazhdan–Lusztig correspondence
THE SAME SL(2,Z) REPRESENTATIONS ARE REALIZED ONCENTERS OF THE CORRESPONDING QUANTUM GROUPS
Screenings =⇒ quantum group g (“Kazhdan–Lusztig-dual”)
At a root of unity =⇒ finite-dimensional
Center Z
Quantum group g is ribbon and factorizable
AM Semikhatov LCFT and representation theory
![Page 126: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/126.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Quantum groups: Kazhdan–Lusztig correspondence
THE SAME SL(2,Z) REPRESENTATIONS ARE REALIZED ONCENTERS OF THE CORRESPONDING QUANTUM GROUPS
Screenings =⇒ quantum group g (“Kazhdan–Lusztig-dual”)
At a root of unity =⇒ finite-dimensional
Center Z
Quantum group g is ribbon and factorizable
AM Semikhatov LCFT and representation theory
![Page 127: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/127.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Quantum groups: Kazhdan–Lusztig correspondence
THE SAME SL(2,Z) REPRESENTATIONS ARE REALIZED ONCENTERS OF THE CORRESPONDING QUANTUM GROUPS
Screenings =⇒ quantum group g (“Kazhdan–Lusztig-dual”)
At a root of unity =⇒ finite-dimensional (q2p = 1, dim g = 2p3 andq2pp ′ = 1, dim g = 2p3p ′3)
Center Z
Quantum group g is ribbon and factorizable
AM Semikhatov LCFT and representation theory
![Page 128: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/128.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Quantum groups: Kazhdan–Lusztig correspondence
THE SAME SL(2,Z) REPRESENTATIONS ARE REALIZED ONCENTERS OF THE CORRESPONDING QUANTUM GROUPS
Screenings =⇒ quantum group g (“Kazhdan–Lusztig-dual”)
At a root of unity =⇒ finite-dimensional (q2p = 1, dim g = 2p3 andq2pp ′ = 1, dim g = 2p3p ′3)
Center Z
Quantum group g is ribbon and factorizable
AM Semikhatov LCFT and representation theory
![Page 129: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/129.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Quantum groups: Kazhdan–Lusztig correspondence
THE SAME SL(2,Z) REPRESENTATIONS ARE REALIZED ONCENTERS OF THE CORRESPONDING QUANTUM GROUPS
Screenings =⇒ quantum group g (“Kazhdan–Lusztig-dual”)
At a root of unity =⇒ finite-dimensional (q2p = 1, dim g = 2p3 andq2pp ′ = 1, dim g = 2p3p ′3)
Center Z: dim Z = 3p − 1 and dim Z = 12(3p − 1)(3p ′ − 1)
Quantum group g is ribbon and factorizable
AM Semikhatov LCFT and representation theory
![Page 130: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/130.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Quantum groups: Kazhdan–Lusztig correspondence
THE SAME SL(2,Z) REPRESENTATIONS ARE REALIZED ONCENTERS OF THE CORRESPONDING QUANTUM GROUPS
Screenings =⇒ quantum group g (“Kazhdan–Lusztig-dual”)
At a root of unity =⇒ finite-dimensional (q2p = 1, dim g = 2p3 andq2pp ′ = 1, dim g = 2p3p ′3)
Center Z: dim Z = 3p − 1 and dim Z = 12(3p − 1)(3p ′ − 1)
Quantum group g is ribbon and factorizable =⇒ its center carries anSL(2,Z) representation
AM Semikhatov LCFT and representation theory
![Page 131: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/131.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Quantum groups: Kazhdan–Lusztig correspondence
THE SAME SL(2,Z) REPRESENTATIONS ARE REALIZED ONCENTERS OF THE CORRESPONDING QUANTUM GROUPS
Screenings =⇒ quantum group g (“Kazhdan–Lusztig-dual”)
At a root of unity =⇒ finite-dimensional (q2p = 1, dim g = 2p3 andq2pp ′ = 1, dim g = 2p3p ′3)
Center Z: dim Z = 3p − 1 and dim Z = 12(3p − 1)(3p ′ − 1)
Quantum group g is ribbon and factorizable =⇒ its center carries anSL(2,Z) representation [Lyubashenko, Turaev, Kerler]
AM Semikhatov LCFT and representation theory
![Page 132: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/132.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Quantum groups: Kazhdan–Lusztig correspondence
THE SAME SL(2,Z) REPRESENTATIONS ARE REALIZED ONCENTERS OF THE CORRESPONDING QUANTUM GROUPS
Screenings =⇒ quantum group g (“Kazhdan–Lusztig-dual”)
At a root of unity =⇒ finite-dimensional (q2p = 1, dim g = 2p3 andq2pp ′ = 1, dim g = 2p3p ′3)
Center Z: dim Z = 3p − 1 and dim Z = 12(3p − 1)(3p ′ − 1)
Quantum group g is ribbon and factorizable =⇒ its center carries anSL(2,Z) representation
Theorem
This SL(2,Z)-representation on Z coincides with theSL(2,Z)-representation generated by the LCFT characters
AM Semikhatov LCFT and representation theory
![Page 133: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/133.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Quantum groups: Kazhdan–Lusztig correspondence
THE SAME SL(2,Z) REPRESENTATIONS ARE REALIZED ONCENTERS OF THE CORRESPONDING QUANTUM GROUPS
Screenings =⇒ quantum group g (“Kazhdan–Lusztig-dual”)
At a root of unity =⇒ finite-dimensional (q2p = 1, dim g = 2p3 andq2pp ′ = 1, dim g = 2p3p ′3)
Center Z: dim Z = 3p − 1 and dim Z = 12(3p − 1)(3p ′ − 1)
Quantum group g is ribbon and factorizable =⇒ its center carries anSL(2,Z) representation
The quantum group knows surprisingly much about the LCFT
AM Semikhatov LCFT and representation theory
![Page 134: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/134.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Quantum groups: Kazhdan–Lusztig correspondence
THE SAME SL(2,Z) REPRESENTATIONS ARE REALIZED ONCENTERS OF THE CORRESPONDING QUANTUM GROUPS
Screenings =⇒ quantum group g (“Kazhdan–Lusztig-dual”)
At a root of unity =⇒ finite-dimensional (q2p = 1, dim g = 2p3 andq2pp ′ = 1, dim g = 2p3p ′3)
Center Z: dim Z = 3p − 1 and dim Z = 12(3p − 1)(3p ′ − 1)
Quantum group g is ribbon and factorizable =⇒ its center carries anSL(2,Z) representation
The quantum group knows surprisingly much about the LCFT
Anything else?
AM Semikhatov LCFT and representation theory
![Page 135: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/135.jpg)
MotivationRepresentation theory and CFT
Quantum groups
More on KL: Grothendieck rings/Fusion
The “(p, 1)” quantum group has 2p irreps X±r , 1 6 r 6 p. Grothendieck ring:
Xαr Xα′
s =
r+s−1∑t=|r−s|+1
step=2
Xαα′
t ,
Xαr =
{Xαr , 1 6 r 6 p,
Xα2p−r + 2X−αr−p, p + 1 6 r 6 2p − 1.
AM Semikhatov LCFT and representation theory
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MotivationRepresentation theory and CFT
Quantum groups
More on KL: Grothendieck rings/Fusion
The “(p, 1)” quantum group has 2p irreps X±r , 1 6 r 6 p. Grothendieck ring:
Xαr Xα′
s =
r+s−1∑t=|r−s|+1
step=2
Xαα′
t ,
Xαr =
{Xαr , 1 6 r 6 p,
Xα2p−r + 2X−αr−p, p + 1 6 r 6 2p − 1.
AM Semikhatov LCFT and representation theory
![Page 137: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/137.jpg)
MotivationRepresentation theory and CFT
Quantum groups
More on KL: Grothendieck rings/Fusion
The “(p, 1)” quantum group has 2p irreps X±r , 1 6 r 6 p. Grothendieck ring:
Xαr Xα′
s =
r+s−1∑t=|r−s|+1
step=2
Xαα′
t ,
Xαr =
{Xαr , 1 6 r 6 p,
Xα2p−r + 2X−αr−p, p + 1 6 r 6 2p − 1.
This nonsemisimple algebra G2p contains the ideal Vp+1 of projective modules;the quotient G2p/Vp+1 is a semisimple fusion algebra — the fusion of the unitary
s`(2) representations of level k = p − 2:
Xr Xs =
p−1−|p−r−s|∑t=|r−s|+1
step=2
Xt , r , s = 1, . . . , p − 1.
AM Semikhatov LCFT and representation theory
![Page 138: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/138.jpg)
MotivationRepresentation theory and CFT
Quantum groups
More on KL: Grothendieck rings/Fusion
The “(p, 1)” quantum group has 2p irreps X±r , 1 6 r 6 p. Grothendieck ring:
Xαr Xα′
s =
r+s−1∑t=|r−s|+1
step=2
Xαα′
t ,
Xαr =
{Xαr , 1 6 r 6 p,
Xα2p−r + 2X−αr−p, p + 1 6 r 6 2p − 1.
The triplet (p, 1) W -algebra has just 2p irreps
AM Semikhatov LCFT and representation theory
![Page 139: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/139.jpg)
MotivationRepresentation theory and CFT
Quantum groups
More on KL: Grothendieck rings/Fusion
The “(p, 1)” quantum group has 2p irreps X±r , 1 6 r 6 p. Grothendieck ring:
Xαr Xα′
s =
r+s−1∑t=|r−s|+1
step=2
Xαα′
t ,
Xαr =
{Xαr , 1 6 r 6 p,
Xα2p−r + 2X−αr−p, p + 1 6 r 6 2p − 1.
The triplet (p, 1) W -algebra has just 2p irreps
This “(p, 1)”-quantum-group Grothendieck ring IS A CANDIDATE FORFUSION in the (p, 1) LCFT model
AM Semikhatov LCFT and representation theory
![Page 140: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/140.jpg)
MotivationRepresentation theory and CFT
Quantum groups
More on KL: Grothendieck rings/Fusion
The “(p, 1)” quantum group has 2p irreps X±r , 1 6 r 6 p. Grothendieck ring:
Xαr Xα′
s =
r+s−1∑t=|r−s|+1
step=2
Xαα′
t , CORROBORATED BY a derivationfrom characters [FHST (2003)]
Xαr =
{Xαr , 1 6 r 6 p,
Xα2p−r + 2X−αr−p, p + 1 6 r 6 2p − 1.
The triplet (p, 1) W -algebra has just 2p irreps
This “(p, 1)”-quantum-group Grothendieck ring IS A CANDIDATE FORFUSION in the (p, 1) LCFT model
AM Semikhatov LCFT and representation theory
![Page 141: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/141.jpg)
MotivationRepresentation theory and CFT
Quantum groups
More on KL: Grothendieck rings/Fusion
The “(p, p ′)” quantum group has 2pp ′ irreps X±r,r ′ , 1 6 r 6 p, 1 6 r ′6 p ′.Grothendieck ring:
Xαr,r ′X
βs,s ′ =
r+s−1∑u=|r−s|+1
step=2
r ′+s ′−1∑u ′=|r ′−s ′|+1
step=2
Xαβu,u ′ ,
Xαr,r ′ =
Xαr,r ′ , 1 6 r 6 p, 1 6 r ′6 p ′,
Xα2p−r,r ′ + 2X−αr−p,r ′ , p+1 6 r 6 2p−1, 1 6 r ′6 p ′,
Xαr,2p ′−r ′ + 2X−αr,r ′−p ′ , 1 6 r 6 p, p ′+1 6 r ′6 2p ′−1,
Xα2p−r,2p ′−r ′ + 2X
−α2p−r,r ′−p ′
+ 2X−αr−p,2p ′−r ′ + 4X
αr−p,r ′−p ′ , p+1 6 r 6 2p−1, p ′+1 6 r ′6 2p ′−1.
AM Semikhatov LCFT and representation theory
![Page 142: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/142.jpg)
MotivationRepresentation theory and CFT
Quantum groups
More on KL: Grothendieck rings/Fusion
The “(p, p ′)” quantum group has 2pp ′ irreps X±r,r ′ , 1 6 r 6 p, 1 6 r ′6 p ′.Grothendieck ring:
Xαr,r ′X
βs,s ′ =
r+s−1∑u=|r−s|+1
step=2
r ′+s ′−1∑u ′=|r ′−s ′|+1
step=2
Xαβu,u ′ ,
Xαr,r ′ =
Xαr,r ′ , 1 6 r 6 p, 1 6 r ′6 p ′,
Xα2p−r,r ′ + 2X−αr−p,r ′ , p+1 6 r 6 2p−1, 1 6 r ′6 p ′,
Xαr,2p ′−r ′ + 2X−αr,r ′−p ′ , 1 6 r 6 p, p ′+1 6 r ′6 2p ′−1,
Xα2p−r,2p ′−r ′ + 2X
−α2p−r,r ′−p ′
+ 2X−αr−p,2p ′−r ′ + 4X
αr−p,r ′−p ′ , p+1 6 r 6 2p−1, p ′+1 6 r ′6 2p ′−1.
AM Semikhatov LCFT and representation theory
![Page 143: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/143.jpg)
MotivationRepresentation theory and CFT
Quantum groups
More on KL: Grothendieck rings/Fusion
The “(p, p ′)” quantum group has 2pp ′ irreps X±r,r ′ , 1 6 r 6 p, 1 6 r ′6 p ′.Grothendieck ring:
Xαr,r ′X
βs,s ′ =
r+s−1∑u=|r−s|+1
step=2
r ′+s ′−1∑u ′=|r ′−s ′|+1
step=2
Xαβu,u ′ ,
Xαr,r ′ =
Xαr,r ′ , 1 6 r 6 p, 1 6 r ′6 p ′,
Xα2p−r,r ′ + 2X−αr−p,r ′ , p+1 6 r 6 2p−1, 1 6 r ′6 p ′,
Xαr,2p ′−r ′ + 2X−αr,r ′−p ′ , 1 6 r 6 p, p ′+1 6 r ′6 2p ′−1,
Xα2p−r,2p ′−r ′ + 2X
−α2p−r,r ′−p ′
+ 2X−αr−p,2p ′−r ′ + 4X
αr−p,r ′−p ′ , p+1 6 r 6 2p−1, p ′+1 6 r ′6 2p ′−1.
1 This algebra is generated by two elements X+1,2 and X+
2,1;
2 its radical is generated by the algebra action on X+p,p ′ ; the quotient over the
radical coincides with the fusion of the (p, p ′) Virasoro minimal models;
3 X+1,1 is the identity;
4 X−1,1 acts as a simple current, X−
1,1Xαr,r ′ = X−α
r,r ′ .
AM Semikhatov LCFT and representation theory
![Page 144: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/144.jpg)
MotivationRepresentation theory and CFT
Quantum groups
More on KL: Grothendieck rings/Fusion
The “(p, p ′)” quantum group has 2pp ′ irreps X±r,r ′ , 1 6 r 6 p, 1 6 r ′6 p ′.Grothendieck ring:
Xαr,r ′X
βs,s ′ =
r+s−1∑u=|r−s|+1
step=2
r ′+s ′−1∑u ′=|r ′−s ′|+1
step=2
Xαβu,u ′ ,
Xαr,r ′ =
Xαr,r ′ , 1 6 r 6 p, 1 6 r ′6 p ′,
Xα2p−r,r ′ + 2X−αr−p,r ′ , p+1 6 r 6 2p−1, 1 6 r ′6 p ′,
Xαr,2p ′−r ′ + 2X−αr,r ′−p ′ , 1 6 r 6 p, p ′+1 6 r ′6 2p ′−1,
Xα2p−r,2p ′−r ′ + 2X
−α2p−r,r ′−p ′
+ 2X−αr−p,2p ′−r ′ + 4X
αr−p,r ′−p ′ , p+1 6 r 6 2p−1, p ′+1 6 r ′6 2p ′−1.
1 This algebra is generated by two elements X+1,2 and X+
2,1;
2 its radical is generated by the algebra action on X+p,p ′ ; the quotient over the
radical coincides with the fusion of the (p, p ′) Virasoro minimal models;
3 X+1,1 is the identity;
4 X−1,1 acts as a simple current, X−
1,1Xαr,r ′ = X−α
r,r ′ .
AM Semikhatov LCFT and representation theory
![Page 145: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/145.jpg)
MotivationRepresentation theory and CFT
Quantum groups
More on KL: Grothendieck rings/Fusion
The “(p, p ′)” quantum group has 2pp ′ irreps X±r,r ′ , 1 6 r 6 p, 1 6 r ′6 p ′.Grothendieck ring:
Xαr,r ′X
βs,s ′ =
r+s−1∑u=|r−s|+1
step=2
r ′+s ′−1∑u ′=|r ′−s ′|+1
step=2
Xαβu,u ′ ,
Xαr,r ′ =
Xαr,r ′ , 1 6 r 6 p, 1 6 r ′6 p ′,
Xα2p−r,r ′ + 2X−αr−p,r ′ , p+1 6 r 6 2p−1, 1 6 r ′6 p ′,
Xαr,2p ′−r ′ + 2X−αr,r ′−p ′ , 1 6 r 6 p, p ′+1 6 r ′6 2p ′−1,
Xα2p−r,2p ′−r ′ + 2X
−α2p−r,r ′−p ′
+ 2X−αr−p,2p ′−r ′ + 4X
αr−p,r ′−p ′ , p+1 6 r 6 2p−1, p ′+1 6 r ′6 2p ′−1.
1 This algebra is generated by two elements X+1,2 and X+
2,1;
2 its radical is generated by the algebra action on X+p,p ′ ; the quotient over the
radical coincides with the fusion of the (p, p ′) Virasoro minimal models;
3 X+1,1 is the identity;
4 X−1,1 acts as a simple current, X−
1,1Xαr,r ′ = X−α
r,r ′ .
AM Semikhatov LCFT and representation theory
![Page 146: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/146.jpg)
MotivationRepresentation theory and CFT
Quantum groups
More on KL: Grothendieck rings/Fusion
The “(p, p ′)” quantum group has 2pp ′ irreps X±r,r ′ , 1 6 r 6 p, 1 6 r ′6 p ′.Grothendieck ring:
Xαr,r ′X
βs,s ′ =
r+s−1∑u=|r−s|+1
step=2
r ′+s ′−1∑u ′=|r ′−s ′|+1
step=2
Xαβu,u ′ ,
Xαr,r ′ =
Xαr,r ′ , 1 6 r 6 p, 1 6 r ′6 p ′,
Xα2p−r,r ′ + 2X−αr−p,r ′ , p+1 6 r 6 2p−1, 1 6 r ′6 p ′,
Xαr,2p ′−r ′ + 2X−αr,r ′−p ′ , 1 6 r 6 p, p ′+1 6 r ′6 2p ′−1,
Xα2p−r,2p ′−r ′ + 2X
−α2p−r,r ′−p ′
+ 2X−αr−p,2p ′−r ′ + 4X
αr−p,r ′−p ′ , p+1 6 r 6 2p−1, p ′+1 6 r ′6 2p ′−1.
1 This algebra is generated by two elements X+1,2 and X+
2,1;
2 its radical is generated by the algebra action on X+p,p ′ ; the quotient over the
radical coincides with the fusion of the (p, p ′) Virasoro minimal models;
3 X+1,1 is the identity;
4 X−1,1 acts as a simple current, X−
1,1Xαr,r ′ = X−α
r,r ′ .
AM Semikhatov LCFT and representation theory
![Page 147: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/147.jpg)
MotivationRepresentation theory and CFT
Quantum groups
More on KL: Grothendieck rings/Fusion
The “(p, p ′)” quantum group has 2pp ′ irreps X±r,r ′ , 1 6 r 6 p, 1 6 r ′6 p ′.Grothendieck ring:
Xαr,r ′X
βs,s ′ =
r+s−1∑u=|r−s|+1
step=2
r ′+s ′−1∑u ′=|r ′−s ′|+1
step=2
Xαβu,u ′ ,
Xαr,r ′ =
Xαr,r ′ , 1 6 r 6 p, 1 6 r ′6 p ′,
Xα2p−r,r ′ + 2X−αr−p,r ′ , p+1 6 r 6 2p−1, 1 6 r ′6 p ′,
Xαr,2p ′−r ′ + 2X−αr,r ′−p ′ , 1 6 r 6 p, p ′+1 6 r ′6 2p ′−1,
Xα2p−r,2p ′−r ′ + 2X
−α2p−r,r ′−p ′
+ 2X−αr−p,2p ′−r ′ + 4X
αr−p,r ′−p ′ , p+1 6 r 6 2p−1, p ′+1 6 r ′6 2p ′−1.
Related to fusion in (p, p ′) LCFT models?!
AM Semikhatov LCFT and representation theory
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MotivationRepresentation theory and CFT
Quantum groups
More on KL: Grothendieck rings/Fusion
The “(p, p ′)” quantum group has 2pp ′ irreps X±r,r ′ , 1 6 r 6 p, 1 6 r ′6 p ′.Grothendieck ring:
Xαr,r ′X
βs,s ′ =
r+s−1∑u=|r−s|+1
step=2
r ′+s ′−1∑u ′=|r ′−s ′|+1
step=2
Xαβu,u ′ , CORROBORATED BY computer
calculations [EF (2006)]
Xαr,r ′ =
Xαr,r ′ , 1 6 r 6 p, 1 6 r ′6 p ′,
Xα2p−r,r ′ + 2X−αr−p,r ′ , p+1 6 r 6 2p−1, 1 6 r ′6 p ′,
Xαr,2p ′−r ′ + 2X−αr,r ′−p ′ , 1 6 r 6 p, p ′+1 6 r ′6 2p ′−1,
Xα2p−r,2p ′−r ′ + 2X
−α2p−r,r ′−p ′
+ 2X−αr−p,2p ′−r ′ + 4X
αr−p,r ′−p ′ , p+1 6 r 6 2p−1, p ′+1 6 r ′6 2p ′−1.
Related to fusion in (p, p ′) LCFT models?!
AM Semikhatov LCFT and representation theory
![Page 149: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/149.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Example: (2, 3) model
The 2pp ′ = 12 representations:
X+1,1 (2), X−
1,1 (7) X+2,1 (1), X−
2,1 (5) X+3,1 ( 1
3), X−
3,1 ( 103)
X+1,2 ( 5
8), X−
1,2 ( 338) X+
2,2 ( 18), X−
2,2 ( 218) X+
3,2 (− 124
), X−3,2 ( 35
24)
X+1,2X
+1,2 = 2X
−1,1 + 2X
+1,1, X
+1,2X
+2,1 = X
+2,2, X
+1,2X
+2,2 = 2X
−2,1 + 2X
+2,1,
X+1,2X
+3,1 = X
+3,2, X
+1,2X
+3,2 = 2X
−3,1 + 2X
+3,1,
X+2,1X
+2,1 = X
+1,1 + X
+3,1, X
+2,1X
+2,2 = X
+1,2 + X
+3,2, X
+2,1X
+3,1 = 2X
−1,1 + 2X
+2,1,
X+2,1X
+3,2 = 2X
−1,2 + 2X
+2,2,
X+2,2X
+2,2 = 2X
−1,1 + 2X
−3,1 + 2X
+1,1 + 2X
+3,1, X
+2,2X
+3,1 = 2X
−1,2 + 2X
+2,2,
X+2,2X
+3,2 = 4X
−1,1 + 4X
−2,1 + 4X
+1,1 + 4X
+2,1,
X+3,1X
+3,1 = 2X
−2,1 + 2X
+1,1 + X
+3,1, X
+3,1X
+3,2 = 2X
−2,2 + 2X
+1,2 + X
+3,2,
X+3,2X
+3,2 = 4X
−1,1 + 4X
−2,1 + 2X
−3,1 + 4X
+1,1 + 4X
+2,1 + 2X
+3,1.
X−1,1 acts as a simple current, X−
1,1X±
r,r ′ = X∓r,r ′ .
AM Semikhatov LCFT and representation theory
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MotivationRepresentation theory and CFT
Quantum groups
Example: (2, 3) model
The 2pp ′ = 12 representations:
X+1,1 (2), X−
1,1 (7) X+2,1 (1), X−
2,1 (5) X+3,1 ( 1
3), X−
3,1 ( 103)
X+1,2 ( 5
8), X−
1,2 ( 338) X+
2,2 ( 18), X−
2,2 ( 218) X+
3,2 (− 124
), X−3,2 ( 35
24)
X+1,2X
+1,2 = 2X
−1,1 + 2X
+1,1, X
+1,2X
+2,1 = X
+2,2, X
+1,2X
+2,2 = 2X
−2,1 + 2X
+2,1,
X+1,2X
+3,1 = X
+3,2, X
+1,2X
+3,2 = 2X
−3,1 + 2X
+3,1,
X+2,1X
+2,1 = X
+1,1 + X
+3,1, X
+2,1X
+2,2 = X
+1,2 + X
+3,2, X
+2,1X
+3,1 = 2X
−1,1 + 2X
+2,1,
X+2,1X
+3,2 = 2X
−1,2 + 2X
+2,2,
X+2,2X
+2,2 = 2X
−1,1 + 2X
−3,1 + 2X
+1,1 + 2X
+3,1, X
+2,2X
+3,1 = 2X
−1,2 + 2X
+2,2,
X+2,2X
+3,2 = 4X
−1,1 + 4X
−2,1 + 4X
+1,1 + 4X
+2,1,
X+3,1X
+3,1 = 2X
−2,1 + 2X
+1,1 + X
+3,1, X
+3,1X
+3,2 = 2X
−2,2 + 2X
+1,2 + X
+3,2,
X+3,2X
+3,2 = 4X
−1,1 + 4X
−2,1 + 2X
−3,1 + 4X
+1,1 + 4X
+2,1 + 2X
+3,1.
X−1,1 acts as a simple current, X−
1,1X±
r,r ′ = X∓r,r ′ .
AM Semikhatov LCFT and representation theory
![Page 151: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/151.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Beyond the Grothendieck ringThe TRUE tensor algebra of Uqs`(2)-representations [K Erdmann et al]:r + s − p 6 1, then
Xαr ⊗ Xβs =
r+s−1⊕t=|r−s |+1
step=2
Xαβt (min(r , s) terms in the sum).
r + s − p > 2, even: r + s − p = 2n with n > 1, then
Xαr ⊗ Xβs =
2p−r−s−1⊕t=|r−s |+1
step=2
Xαβt ⊕
n⊕a=1
Pαβp+1−2a.
r + s − p > 3, odd: r + s − p = 2n + 1 with n > 1, then
Xαr ⊗ Xβs =
2p−r−s−1⊕t=|r−s |+1
step=2
Xαβt ⊕
n⊕a=0
Pαβp−2a.
The same structure of “indecomposable-aware” fusion in (p, 1) LCFT?
AM Semikhatov LCFT and representation theory
![Page 152: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/152.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Beyond the Grothendieck ringThe TRUE tensor algebra of Uqs`(2)-representations [K Erdmann et al]:r + s − p 6 1, then
Xαr ⊗ Xβs =
r+s−1⊕t=|r−s |+1
step=2
Xαβt (min(r , s) terms in the sum).
r + s − p > 2, even: r + s − p = 2n with n > 1, then
Xαr ⊗ Xβs =
2p−r−s−1⊕t=|r−s |+1
step=2
Xαβt ⊕
n⊕a=1
Pαβp+1−2a.
r + s − p > 3, odd: r + s − p = 2n + 1 with n > 1, then
Xαr ⊗ Xβs =
2p−r−s−1⊕t=|r−s |+1
step=2
Xαβt ⊕
n⊕a=0
Pαβp−2a.
The same structure of “indecomposable-aware” fusion in (p, 1) LCFT?
AM Semikhatov LCFT and representation theory
![Page 153: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/153.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Beyond the Grothendieck ringThe TRUE tensor algebra of Uqs`(2)-representations [K Erdmann et al]:r + s − p 6 1, then
Xαr ⊗ Xβs =
r+s−1⊕t=|r−s |+1
step=2
Xαβt (min(r , s) terms in the sum).
r + s − p > 2, even: r + s − p = 2n with n > 1, then
Xαr ⊗ Xβs =
2p−r−s−1⊕t=|r−s |+1
step=2
Xαβt ⊕
n⊕a=1
Pαβp+1−2a.
r + s − p > 3, odd: r + s − p = 2n + 1 with n > 1, then
Xαr ⊗ Xβs =
2p−r−s−1⊕t=|r−s |+1
step=2
Xαβt ⊕
n⊕a=0
Pαβp−2a.
The same structure of “indecomposable-aware” fusion in (p, 1) LCFT?
AM Semikhatov LCFT and representation theory
![Page 154: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/154.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Beyond the Grothendieck ringThe TRUE tensor algebra of Uqs`(2)-representations [K Erdmann et al]:r + s − p 6 1, then
Xαr ⊗ Xβs =
r+s−1⊕t=|r−s |+1
step=2
Xαβt (min(r , s) terms in the sum).
r + s − p > 2, even: r + s − p = 2n with n > 1, then
Xαr ⊗ Xβs =
2p−r−s−1⊕t=|r−s |+1
step=2
Xαβt ⊕
n⊕a=1
Pαβp+1−2a.
r + s − p > 3, odd: r + s − p = 2n + 1 with n > 1, then
Xαr ⊗ Xβs =
2p−r−s−1⊕t=|r−s |+1
step=2
Xαβt ⊕
n⊕a=0
Pαβp−2a.
The same structure of “indecomposable-aware” fusion in (p, 1) LCFT?
AM Semikhatov LCFT and representation theory
![Page 155: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/155.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Beyond the Grothendieck ringThe TRUE tensor algebra of Uqs`(2)-representations [K Erdmann et al]:r + s − p 6 1, then
Xαr ⊗ Xβs =
r+s−1⊕t=|r−s |+1
step=2
Xαβt (min(r , s) terms in the sum).
r + s − p > 2, even: r + s − p = 2n with n > 1, then
Xαr ⊗ Xβs =
2p−r−s−1⊕t=|r−s |+1
step=2
Xαβt ⊕
n⊕a=1
Pαβp+1−2a.
r + s − p > 3, odd: r + s − p = 2n + 1 with n > 1, then
Xαr ⊗ Xβs =
2p−r−s−1⊕t=|r−s |+1
step=2
Xαβt ⊕
n⊕a=0
Pαβp−2a.
The same structure of “indecomposable-aware” fusion in (p, 1) LCFT?
AM Semikhatov LCFT and representation theory
![Page 156: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/156.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Indecomposable modules: Was(n) and Ma
s(n)
1 6 s 6 p−1, a =±, and n > 2, module Was (n):
Xas• xa
1
��
Xas•xa
2
��
xa1
��
. . .xa2
��
xa1
��
Xas•xa
2
��X−a
p−s• X−ap−s• . . . X−a
p−s•1 6 s 6 p−1, a =± , and n > 2, module Ma
s (n):
Xas•xa
2
��
xa1
��
. . .xa2
��
xa1
��
Xas•xa
2
��
xa1
��X−a
p−s• X−ap−s• . . . X−a
p−s• X−ap−s•
AM Semikhatov LCFT and representation theory
![Page 157: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/157.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Indecomposable modules: Was(n) and Ma
s(n)
1 6 s 6 p−1, a =±, and n > 2, module Was (n):
Xas• xa
1
��
Xas•xa
2
��
xa1
��
. . .xa2
��
xa1
��
Xas•xa
2
��X−a
p−s• X−ap−s• . . . X−a
p−s•1 6 s 6 p−1, a =± , and n > 2, module Ma
s (n):
Xas•xa
2
��
xa1
��
. . .xa2
��
xa1
��
Xas•xa
2
��
xa1
��X−a
p−s• X−ap−s• . . . X−a
p−s• X−ap−s•
AM Semikhatov LCFT and representation theory
![Page 158: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/158.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Indecomposable modules: Oas(n, z)
1 6 s 6 p − 1, a =±, n > 1, and z ∈ CP1, module Oas (n, z):
Xas•xa
2
��
xa1
��
Xas•xa
2
��
. . . Xas• xa
1
��X−a
p−s• X−ap−s• . . . X−a
p−s•
Xas•
z2xa2
>>
z1xa1
ee
CP1 3 z = (z1 : z2)
AM Semikhatov LCFT and representation theory
![Page 159: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/159.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Indecomposable modules: Oas(n, z)
1 6 s 6 p − 1, a =±, n > 1, and z ∈ CP1, module Oas (n, z):
Xas•xa
2
��
xa1
��
Xas•xa
2
��
. . . Xas• xa
1
��X−a
p−s• X−ap−s• . . . X−a
p−s•
Xas•
z2xa2
>>
z1xa1
ee
CP1 3 z = (z1 : z2)
AM Semikhatov LCFT and representation theory
![Page 160: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/160.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Summary
We know
extended symmetry of the model:W -algebra
W -algebra irreps
Quantum-group projective modules
Quantum-group Grothendieckring/“fusion”
ZCFT: Characters and generalizedcharacters, modulartransformations on ZCFT
Modular transformations onquantum-group center Z
(some) modular invariants
We don’t
structure of W -algebra singularvectors etc.
W -algebra projective modules
honest W -algebra fusion
honest W -algebra torus amplitudes
full partition function
correlation functions
AM Semikhatov LCFT and representation theory
![Page 161: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/161.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Summary
We know
extended symmetry of the model:W -algebra
W -algebra irreps
Quantum-group projective modules
Quantum-group Grothendieckring/“fusion”
ZCFT: Characters and generalizedcharacters, modulartransformations on ZCFT
Modular transformations onquantum-group center Z
(some) modular invariants
We don’t
structure of W -algebra singularvectors etc.
W -algebra projective modules
honest W -algebra fusion
honest W -algebra torus amplitudes
full partition function
correlation functions
AM Semikhatov LCFT and representation theory
![Page 162: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/162.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Summary
We know
extended symmetry of the model:W -algebra
W -algebra irreps
Quantum-group projective modules
Quantum-group Grothendieckring/“fusion”
ZCFT: Characters and generalizedcharacters, modulartransformations on ZCFT
Modular transformations onquantum-group center Z
(some) modular invariants
We don’t
structure of W -algebra singularvectors etc.
W -algebra projective modules
honest W -algebra fusion
honest W -algebra torus amplitudes
full partition function
correlation functions
AM Semikhatov LCFT and representation theory
![Page 163: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/163.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Summary
We know
extended symmetry of the model:W -algebra
W -algebra irreps
Quantum-group projective modules
Quantum-group Grothendieckring/“fusion”
ZCFT: Characters and generalizedcharacters, modulartransformations on ZCFT
Modular transformations onquantum-group center Z
(some) modular invariants
We don’t
structure of W -algebra singularvectors etc.
W -algebra projective modules
honest W -algebra fusion
honest W -algebra torus amplitudes
full partition function
correlation functions
AM Semikhatov LCFT and representation theory
![Page 164: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/164.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Summary
We know
extended symmetry of the model:W -algebra
W -algebra irreps
Quantum-group projective modules
Quantum-group Grothendieckring/“fusion”
ZCFT: Characters and generalizedcharacters, modulartransformations on ZCFT
Modular transformations onquantum-group center Z
(some) modular invariants
We don’t
structure of W -algebra singularvectors etc.
W -algebra projective modules
honest W -algebra fusion
honest W -algebra torus amplitudes
full partition function
correlation functions
AM Semikhatov LCFT and representation theory
![Page 165: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/165.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Summary
We know
extended symmetry of the model:W -algebra
W -algebra irreps
Quantum-group projective modules
Quantum-group Grothendieckring/“fusion”
ZCFT: Characters and generalizedcharacters, modulartransformations on ZCFT
Modular transformations onquantum-group center Z
(some) modular invariants
We don’t
structure of W -algebra singularvectors etc.
W -algebra projective modules
honest W -algebra fusion
honest W -algebra torus amplitudes
full partition function
correlation functions
AM Semikhatov LCFT and representation theory
![Page 166: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/166.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Summary
We know
extended symmetry of the model:W -algebra
W -algebra irreps
Quantum-group projective modules
Quantum-group Grothendieckring/“fusion”
ZCFT: Characters and generalizedcharacters, modulartransformations on ZCFT
Modular transformations onquantum-group center Z
(some) modular invariants
We don’t
structure of W -algebra singularvectors etc.
W -algebra projective modules
honest W -algebra fusion
honest W -algebra torus amplitudes
full partition function
correlation functions
AM Semikhatov LCFT and representation theory
![Page 167: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/167.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Summary
We know
extended symmetry of the model:W -algebra
W -algebra irreps
Quantum-group projective modules
Quantum-group Grothendieckring/“fusion”
ZCFT: Characters and generalizedcharacters, modulartransformations on ZCFT
Modular transformations onquantum-group center Z
(some) modular invariants
We don’t
structure of W -algebra singularvectors etc.
W -algebra projective modules
honest W -algebra fusion
honest W -algebra torus amplitudes
full partition function
correlation functions
AM Semikhatov LCFT and representation theory
![Page 168: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/168.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Summary
We know
extended symmetry of the model:W -algebra
W -algebra irreps
Quantum-group projective modules
Quantum-group Grothendieckring/“fusion”
ZCFT: Characters and generalizedcharacters, modulartransformations on ZCFT
Modular transformations onquantum-group center Z
(some) modular invariants
We don’t
structure of W -algebra singularvectors etc.
W -algebra projective modules
honest W -algebra fusion
honest W -algebra torus amplitudes
full partition function
correlation functions
AM Semikhatov LCFT and representation theory
![Page 169: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/169.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Summary
We know
extended symmetry of the model:W -algebra
W -algebra irreps
Quantum-group projective modules
Quantum-group Grothendieckring/“fusion”
ZCFT: Characters and generalizedcharacters, modulartransformations on ZCFT
Modular transformations onquantum-group center Z
(some) modular invariants
We don’t
structure of W -algebra singularvectors etc.
W -algebra projective modules
honest W -algebra fusion
honest W -algebra torus amplitudes
full partition function
correlation functions
AM Semikhatov LCFT and representation theory
![Page 170: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/170.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Summary
We know
extended symmetry of the model:W -algebra
W -algebra irreps
Quantum-group projective modules
Quantum-group Grothendieckring/“fusion”
ZCFT: Characters and generalizedcharacters, modulartransformations on ZCFT
Modular transformations onquantum-group center Z
(some) modular invariants
We don’t
structure of W -algebra singularvectors etc.
W -algebra projective modules
honest W -algebra fusion
honest W -algebra torus amplitudes
full partition function
correlation functions
AM Semikhatov LCFT and representation theory
![Page 171: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/171.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Summary
We know
extended symmetry of the model:W -algebra
W -algebra irreps
Quantum-group projective modules
Quantum-group Grothendieckring/“fusion”
ZCFT: Characters and generalizedcharacters, modulartransformations on ZCFT
Modular transformations onquantum-group center Z = ZCFT
(some) modular invariants
We don’t
structure of W -algebra singularvectors etc.
W -algebra projective modules
honest W -algebra fusion
honest W -algebra torus amplitudes
full partition function
correlation functions
AM Semikhatov LCFT and representation theory
![Page 172: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/172.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Summary
We know
extended symmetry of the model:W -algebra
W -algebra irreps
Quantum-group projective modules
Quantum-group Grothendieckring/“fusion”
ZCFT: Characters and generalizedcharacters, modulartransformations on ZCFT
Modular transformations onquantum-group center Z = ZCFT
(some) modular invariants
We don’t
structure of W -algebra singularvectors etc.
W -algebra projective modules
honest W -algebra fusion
honest W -algebra torus amplitudes
full partition function
correlation functions
AM Semikhatov LCFT and representation theory
![Page 173: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/173.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Summary
We know
extended symmetry of the model:W -algebra
W -algebra irreps
Quantum-group projective modules
Quantum-group Grothendieckring/“fusion”
ZCFT: Characters and generalizedcharacters, modulartransformations on ZCFT
Modular transformations onquantum-group center Z = ZCFT
(some) modular invariants
We don’t
structure of W -algebra singularvectors etc.
W -algebra projective modules
honest W -algebra fusion
honest W -algebra torus amplitudes
full partition function
correlation functions
AM Semikhatov LCFT and representation theory
![Page 174: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/174.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Summary
We know
extended symmetry of the model:W -algebra
W -algebra irreps
Quantum-group projective modules
Quantum-group Grothendieckring/“fusion”
ZCFT: Characters and generalizedcharacters, modulartransformations on ZCFT
Modular transformations onquantum-group center Z = ZCFT
(some) modular invariants
We don’t
structure of W -algebra singularvectors etc.
W -algebra projective modules
honest W -algebra fusion
honest W -algebra torus amplitudes
full partition function
correlation functions
AM Semikhatov LCFT and representation theory
![Page 175: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/175.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Summary
We know
extended symmetry of the model:W -algebra
W -algebra irreps
Quantum-group projective modules
Quantum-group Grothendieckring/“fusion”
ZCFT: Characters and generalizedcharacters, modulartransformations on ZCFT
Modular transformations onquantum-group center Z = ZCFT
(some) modular invariants
We don’t
structure of W -algebra singularvectors etc.
W -algebra projective modules
honest W -algebra fusion
honest W -algebra torus amplitudes
full partition function
correlation functions
More than fifty percent success:
AM Semikhatov LCFT and representation theory
![Page 176: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/176.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Summary
We know
extended symmetry of the model:W -algebra
W -algebra irreps
Quantum-group projective modules
Quantum-group Grothendieckring/“fusion”
ZCFT: Characters and generalizedcharacters, modulartransformations on ZCFT
Modular transformations onquantum-group center Z = ZCFT
(some) modular invariants
We don’t
structure of W -algebra singularvectors etc.
W -algebra projective modules
honest W -algebra fusion
honest W -algebra torus amplitudes
full partition function
correlation functions
More than fifty percent success: Passed!
AM Semikhatov LCFT and representation theory
![Page 177: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/177.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Summary
We know
extended symmetry of the model:W -algebra
W -algebra irreps
Quantum-group projective modules
Quantum-group Grothendieckring/“fusion”
ZCFT: Characters and generalizedcharacters, modulartransformations on ZCFT
Modular transformations onquantum-group center Z = ZCFT
(some) modular invariants
We don’t
structure of W -algebra singularvectors etc.
W -algebra projective modules
honest W -algebra fusion
honest W -algebra torus amplitudes
full partition function
correlation functions
EVEN THOUGH I WAS CHEATING !!
AM Semikhatov LCFT and representation theory
![Page 178: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/178.jpg)
MotivationRepresentation theory and CFT
Quantum groups
Thank You :)
AM Semikhatov LCFT and representation theory
![Page 179: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/179.jpg)
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AM Semikhatov LCFT and representation theory
![Page 180: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/180.jpg)
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4 More
AM Semikhatov LCFT and representation theory
![Page 181: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/181.jpg)
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From Free Fields to the Quantum Group
Consider the (p, 1) case. Screening: E =
∮e
−√
2pϕ
.
The Wp algebra:
W −(z) = e−√
2pϕ(z), W 0(z) = [S+,W−(z)], W +(z) = [S+,W
0(z)],
where S+ =
∮e√
2pϕ. The W±,0(z) are primary fields of dimension 2p − 1
with respect to the energy-momentum tensor
T (z) =1
2∂ϕ∂ϕ(z) +
(√2p −
√2
p
)∂2ϕ(z).
On a suitably defined free-field space F,
Ker E∣∣∣F=
p⊕r=1
X+r ⊕ X−
r ,
a sum of 2p Wp-representations.
AM Semikhatov LCFT and representation theory
![Page 182: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/182.jpg)
More
From Free Fields to the Quantum Group
Consider the (p, 1) case. Screening: E =
∮e
−√
2pϕ
.
The Wp algebra:
W −(z) = e−√
2pϕ(z), W 0(z) = [S+,W−(z)], W +(z) = [S+,W
0(z)],
where S+ =
∮e√
2pϕ. The W±,0(z) are primary fields of dimension 2p − 1
with respect to the energy-momentum tensor
T (z) =1
2∂ϕ∂ϕ(z) +
(√2p −
√2
p
)∂2ϕ(z).
On a suitably defined free-field space F,
Ker E∣∣∣F=
p⊕r=1
X+r ⊕ X−
r ,
a sum of 2p Wp-representations.
AM Semikhatov LCFT and representation theory
![Page 183: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/183.jpg)
More
From Free Fields to the Quantum Group
Consider the (p, 1) case. Screening: E =
∮e
−√
2pϕ
.
The Wp algebra:
W −(z) = e−√
2pϕ(z), W 0(z) = [S+,W−(z)], W +(z) = [S+,W
0(z)],
where S+ =
∮e√
2pϕ. The W±,0(z) are primary fields of dimension 2p − 1
with respect to the energy-momentum tensor
T (z) =1
2∂ϕ∂ϕ(z) +
(√2p −
√2
p
)∂2ϕ(z).
On a suitably defined free-field space F,
Ker E∣∣∣F=
p⊕r=1
X+r ⊕ X−
r ,
a sum of 2p Wp-representations.
AM Semikhatov LCFT and representation theory
![Page 184: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/184.jpg)
More
From Free Fields to the Quantum Group
Consider the (p, 1) case. Screening: E =
∮e
−√
2pϕ
.
The Wp algebra:
W −(z) = e−√
2pϕ(z), W 0(z) = [S+,W−(z)], W +(z) = [S+,W
0(z)],
where S+ =
∮e√
2pϕ. The W±,0(z) are primary fields of dimension 2p − 1
with respect to the energy-momentum tensor
T (z) =1
2∂ϕ∂ϕ(z) +
(√2p −
√2
p
)∂2ϕ(z).
On a suitably defined free-field space F,
Ker E∣∣∣F=
p⊕r=1
X+r ⊕ X−
r ,
a sum of 2p Wp-representations.
AM Semikhatov LCFT and representation theory
![Page 185: theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from](https://reader033.fdocuments.us/reader033/viewer/2022053014/5f12dd698a4b955c1538e59b/html5/thumbnails/185.jpg)
More
From Free Fields to the Quantum Group
Consider the (p, 1) case. Screening: E =
∮e
−√
2pϕ
.
The Wp algebra:
W −(z) = e−√
2pϕ(z), W 0(z) = [S+,W−(z)], W +(z) = [S+,W
0(z)],
where S+ =
∮e√
2pϕ. The W±,0(z) are primary fields of dimension 2p − 1
with respect to the energy-momentum tensor
T (z) =1
2∂ϕ∂ϕ(z) +
(√2p −
√2
p
)∂2ϕ(z).
On a suitably defined free-field space F,
Ker E∣∣∣F=
p⊕r=1
X+r ⊕ X−
r ,
a sum of 2p Wp-representations.
AM Semikhatov LCFT and representation theory