Duality in Condensed Matter Physics and Field Theory · Duality in Statistical Mechanics • 2D...

20
Duality in Condensed Matter Physics and Field Theory Eduardo Fradkin Department of Physics and Institute for Condensed Matter Theory University of Illinois, Urbana, Illinois, USA Talk at the Rudolf Peierls Symposium for Theoretical Physics, Oxford University, July 5-6, 2018 Hart Goldman and Eduardo Fradkin, Phys. Rev. B 97, 195112 (2018) 1

Transcript of Duality in Condensed Matter Physics and Field Theory · Duality in Statistical Mechanics • 2D...

Page 1: Duality in Condensed Matter Physics and Field Theory · Duality in Statistical Mechanics • 2D Ising Model: Kramers- Wannier duality • Partition function as a sum over closed domain

Duality in Condensed Matter Physics and Field Theory

Eduardo FradkinDepartment of Physics and Institute for Condensed Matter Theory

University of Illinois, Urbana, Illinois, USA

Talk at the Rudolf Peierls Symposium for Theoretical Physics, Oxford University, July 5-6, 2018

Hart Goldman and Eduardo Fradkin, Phys. Rev. B 97, 195112 (2018)

�1

Page 2: Duality in Condensed Matter Physics and Field Theory · Duality in Statistical Mechanics • 2D Ising Model: Kramers- Wannier duality • Partition function as a sum over closed domain

Motivation• Recent interest in dualities in CM and QFT

• Particle-Vortex duality and its applications to the Fractional Quantum Hall Effect

• Conjectured dualities, bosonization and fermionization

• Can this “web of dualities” be “derived”?

• Loop models: flux attachment, duality and periodicity

• Periodicity vs Fractional Spin

• Fractional Spin and a web of dualities for loop models

!2

Page 3: Duality in Condensed Matter Physics and Field Theory · Duality in Statistical Mechanics • 2D Ising Model: Kramers- Wannier duality • Partition function as a sum over closed domain

Electromagnetic Duality

• EM duality: E ⟺ B

• Duality of forms (similar to Cauchy-Riemann in 2D)

• electric charge e ⟺ magnetic monopole m

• Dirac quantization em=2𝜋

!3

r · E =⇢, r · B = 0

r⇥B � 1c

@E

@t= j, r⇥E +

@B

@t= 0

<latexit sha1_base64="Se1SZXrBm73Pzlbk/Ef1+UDWYWM=">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</latexit><latexit sha1_base64="Se1SZXrBm73Pzlbk/Ef1+UDWYWM=">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</latexit><latexit sha1_base64="Se1SZXrBm73Pzlbk/Ef1+UDWYWM=">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</latexit><latexit sha1_base64="Se1SZXrBm73Pzlbk/Ef1+UDWYWM=">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</latexit><latexit sha1_base64="kO59cm/W//LWDO0rGNfdeSQGxWA=">AAAB9HicbVDLSsNAFL2prxqr1rWbwSK4KokbXQpuXFawD2hLmUxu2qGTSZi5EUrpyp1bv8Kd+Dd+hP/g9LFJ64ELh3Mu93GiXElLQfDjVfb2Dw6Pqsf+Sc0/PTuv1zo2K4zAtshUZnoRt6ikxjZJUtjLDfI0UtiNpo9Lv/uKxspMv9Asx2HKx1omUnByUmtUbwTNYAW2S8INacAGo/rvIM5EkaImobi1/TDIaTjnhqRQuPAHhcWciykf43x13IJdOylmSWZcaWIrtdTHU2tnaeQ6U04Tu+0txX+9KC3t6xeU3A/nUucFoRbrxUmhGGVs+TmLpUFBauYIF0a6i5mYcMMFuXxKk617bYLxwnfxhNth7JLObTMMmuFzAFW4hCu4gRDu4AGeoAVtEBDDO3x4b96n97WOseJt8ryAErzvP39qlUc=</latexit><latexit sha1_base64="nHlwgC/Z4tbhf1htHvaVp1xaJto=">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</latexit><latexit sha1_base64="aitEh1sEZmbDR09nHBDQirs0Uac=">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</latexit><latexit sha1_base64="JfdnustHQl5nF6V2ZtniepgNp/E=">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</latexit><latexit sha1_base64="Se1SZXrBm73Pzlbk/Ef1+UDWYWM=">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</latexit><latexit sha1_base64="Se1SZXrBm73Pzlbk/Ef1+UDWYWM=">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</latexit><latexit sha1_base64="Se1SZXrBm73Pzlbk/Ef1+UDWYWM=">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</latexit><latexit sha1_base64="Se1SZXrBm73Pzlbk/Ef1+UDWYWM=">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</latexit><latexit sha1_base64="Se1SZXrBm73Pzlbk/Ef1+UDWYWM=">AAAC8nicdVLLbhMxFPUMrxJeAZZsLCIqJErkqcqji6CqqBLLIjVtpTqKPB5PY2qPp/adSmHkFXwFO8SWH+Ij4BtwJonooPRKlq7PPT735bRU0gEhv6L42vUbN2+t3e7cuXvv/oPuw0eHzlSWiyE3ytjjlDmhZCGGIEGJ49IKplMljtKz97P40YWwTpriAKalGGl2WshccgYBGnf/1DTVmBYsVcxjyjMDuIH2/GCd2onZwPT8vGIZvoq46weENgxMaafNAqmFW9Je0twyXie+5iGUNRdaMguSqWVO/w8Bvz6Yw5/8xooaLovv+RcrBXfbggMy7vZInzSGLzmvSLL9OsHJAumhhe2Pu79pZnilRQFcMedOElLCqJ5JciV8h1ZOlIyfsVNRN9vw+FmAMpwbG04BuEFbPKadm+o0MDWDifs/NgNXxlLdyndSQf52VMuirEAUfJ44r0KjBs9WjTNpBQc1DQ7jVoaKMZ+wMCUIH6Kl7EJrE5H5TpjPcgj4audws5+QfvJxq7fzbjGpNfQEPUXPUYLeoB30Ae2jIeLRQfQ5+hJ9jSH+Fn+Pf8ypcbR48xi1LP75FxlM7po=</latexit><latexit sha1_base64="Se1SZXrBm73Pzlbk/Ef1+UDWYWM=">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</latexit>

Asymmetry of Maxwell’s equations

Page 4: Duality in Condensed Matter Physics and Field Theory · Duality in Statistical Mechanics • 2D Ising Model: Kramers- Wannier duality • Partition function as a sum over closed domain

Duality in Statistical Mechanics

• 2D Ising Model: Kramers-Wannier duality

• Partition function as a sum over closed domain walls

• Partition function as a sum over loops of the high T expansion

• high T ⟺ low T, order ⟺ disorder

• 3D: Duality of the ℤ2 gauge theory ⟺ Ising model

• loops dual to surfaces

• order ⟺ confinement

!4

Zdomains[exp(�2/T )] = Zloops[tanh(1/T )]<latexit sha1_base64="XG28OeZv9Hy2vRzkqBdqTxL2uY4=">AAACRHicdVBNS1tBFL3Pfqip1tgu3QyVQlw0vif4tagE3Li0YFSahDBv3o0ZnI/HzH1ieLy/05/hsqtuK/Q/1JW4LZ0kFkxpDwwczjkzd+5JcyU9xfGPaO7Z8xcv5xcWa6+Wll+v1FffnHpbOIFtYZV15yn3qKTBNklSeJ475DpVeJZeHo79syt0XlpzQqMce5pfGDmQglOQ+vXW537ZJbwmp8vMai6Nr6pOF6/zxoetzZON3scnAWVtPrGJmyFrJGO/X1+Pm/EE7AnZjpP9nYQlj8p6Ky5vGAAc9+t33cyKQqMhobj3nSTOqVdyR1IorGrdwmPOxSW/wHKyX8XeByljA+vCMcQm6kyOa+9HOg1JzWno//bG4j+9VM/M6xQ02OuV0uQFoRHTwYNCMbJsXB7LpENBahQIF06GHzMx5I4LChXPvOzDakPMqlro508J7P/kdKuZxM3kUyjqAKZYgDV4Bw1IYBdacATH0AYBX+AbfIfb6Gv0M7qPHqbRuejxzluYQfTrN6k4tEs=</latexit><latexit sha1_base64="HTfC/67isQZy1Vj392LsZMZcovY=">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</latexit><latexit sha1_base64="HTfC/67isQZy1Vj392LsZMZcovY=">AAACRHicdVBLS1tBGJ3rqza+ooVuuhkUIS6M9wr1sbAEuulSwaiYhDB37hczOI/LzHfFcLl/x5/h0pVbBTfdddeuitvSSaJgRA8MHM45M998J06lcBiGD8HY+MTk1Ifpj6WZ2bn5hfLi0pEzmeVQ50YaexIzB1JoqKNACSepBaZiCcfx+fe+f3wB1gmjD7GXQkuxMy06gjP0UrtcO23nTYRLtCpPjGJCu6JoNOEyraxvbhyutfZeBKQx6cBGpru0EvX9dnklrIYD0BfkaxjtbkU0elJWamF+vfzr5+f9dvlPMzE8U6CRS+ZcIwpTbOXMouASilIzc5Ayfs7OIB/sV9BVLyW0Y6w/GulAHckx5VxPxT6pGHbda68vvunFamReI8POTisXOs0QNB8O7mSSoqH98mgiLHCUPU8Yt8L/mPIus4yjr3jkZedX60JSlHw/zyXQ98nRZjUKq9GBL+obGWKafCHLpEIisk1q5AfZJ3XCyRW5JXfkPrgJfgd/g8dhdCx4uvOJjCD49x9W47Ym</latexit><latexit sha1_base64="HTfC/67isQZy1Vj392LsZMZcovY=">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</latexit><latexit sha1_base64="kO59cm/W//LWDO0rGNfdeSQGxWA=">AAAB9HicbVDLSsNAFL2prxqr1rWbwSK4KokbXQpuXFawD2hLmUxu2qGTSZi5EUrpyp1bv8Kd+Dd+hP/g9LFJ64ELh3Mu93GiXElLQfDjVfb2Dw6Pqsf+Sc0/PTuv1zo2K4zAtshUZnoRt6ikxjZJUtjLDfI0UtiNpo9Lv/uKxspMv9Asx2HKx1omUnByUmtUbwTNYAW2S8INacAGo/rvIM5EkaImobi1/TDIaTjnhqRQuPAHhcWciykf43x13IJdOylmSWZcaWIrtdTHU2tnaeQ6U04Tu+0txX+9KC3t6xeU3A/nUucFoRbrxUmhGGVs+TmLpUFBauYIF0a6i5mYcMMFuXxKk617bYLxwnfxhNth7JLObTMMmuFzAFW4hCu4gRDu4AGeoAVtEBDDO3x4b96n97WOseJt8ryAErzvP39qlUc=</latexit><latexit sha1_base64="yb8+4weCBoJNUkHosYR05ZJS2sk=">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</latexit><latexit sha1_base64="0g/j/xucKYx8puhJdfsGi5mgFTY=">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</latexit><latexit sha1_base64="uBu8Qp+NreH5heyAdS5HrnUxTTk=">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</latexit><latexit sha1_base64="uBu8Qp+NreH5heyAdS5HrnUxTTk=">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</latexit><latexit sha1_base64="HTfC/67isQZy1Vj392LsZMZcovY=">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</latexit><latexit sha1_base64="HTfC/67isQZy1Vj392LsZMZcovY=">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</latexit><latexit sha1_base64="HTfC/67isQZy1Vj392LsZMZcovY=">AAACRHicdVBLS1tBGJ3rqza+ooVuuhkUIS6M9wr1sbAEuulSwaiYhDB37hczOI/LzHfFcLl/x5/h0pVbBTfdddeuitvSSaJgRA8MHM45M998J06lcBiGD8HY+MTk1Ifpj6WZ2bn5hfLi0pEzmeVQ50YaexIzB1JoqKNACSepBaZiCcfx+fe+f3wB1gmjD7GXQkuxMy06gjP0UrtcO23nTYRLtCpPjGJCu6JoNOEyraxvbhyutfZeBKQx6cBGpru0EvX9dnklrIYD0BfkaxjtbkU0elJWamF+vfzr5+f9dvlPMzE8U6CRS+ZcIwpTbOXMouASilIzc5Ayfs7OIB/sV9BVLyW0Y6w/GulAHckx5VxPxT6pGHbda68vvunFamReI8POTisXOs0QNB8O7mSSoqH98mgiLHCUPU8Yt8L/mPIus4yjr3jkZedX60JSlHw/zyXQ98nRZjUKq9GBL+obGWKafCHLpEIisk1q5AfZJ3XCyRW5JXfkPrgJfgd/g8dhdCx4uvOJjCD49x9W47Ym</latexit><latexit sha1_base64="HTfC/67isQZy1Vj392LsZMZcovY=">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</latexit><latexit sha1_base64="uBu8Qp+NreH5heyAdS5HrnUxTTk=">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</latexit>

Page 5: Duality in Condensed Matter Physics and Field Theory · Duality in Statistical Mechanics • 2D Ising Model: Kramers- Wannier duality • Partition function as a sum over closed domain

3D Particle-Vortex Duality

• Theories with a global U(1) symmetry, e.g. the 3D XY model

• High T expansion loop gas: equivalent to histories of charged particles with short-range interactions

• Low T expansion: closed vortex loops with Biot-Savart interactions

• Third dimension = imaginary time; loops as worldlines

• Particle-Vortex duality: electric charge ⟺ vortex (magnetic charge)

• Mappings between phases of matter, most often between different theories

• Conjectured web of dualities between CFTs in 2+1 dimensions

• What is the relation between these different dualities?

!5

Page 6: Duality in Condensed Matter Physics and Field Theory · Duality in Statistical Mechanics • 2D Ising Model: Kramers- Wannier duality • Partition function as a sum over closed domain

Fractional Quantum Hall Effect

• Zeros of the wave function interpreted as the attachment of m-1 fluxes to each electron

• Landau-Ginzburg theory (Zhang, Hansson and Kivelson): Non-Relativistic gauged abelian-Higgs model with a Chern-Simons term: Electrons are “composite” bosons coupled to m fluxes

• Fermionic flux attachment (Jain, López and Fradkin, Halperin, Lee and Read)

• FQH plateau: composite bosons condense; the excitations are anyonic vortices

• FQH insulator: bosons are uncondensed and the gapped excitations are fermions

• Universal phase diagram for the FQH states based on particle-vortex duality (Kivelson, Lee and Zhang) with “super-universal” transitions

!6

m(z1, . . . , zN ) =Y

i<j

(zi � zj)m e�

PNi=1

|zi|2

4`20

<latexit sha1_base64="WEu5b/PmEcBqUr3HhOwCWgmpkRE=">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</latexit><latexit sha1_base64="MTUm3RT8mR81/J+g7aFBDrdzQE0=">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</latexit><latexit sha1_base64="MTUm3RT8mR81/J+g7aFBDrdzQE0=">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</latexit><latexit sha1_base64="MTUm3RT8mR81/J+g7aFBDrdzQE0=">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</latexit><latexit sha1_base64="kO59cm/W//LWDO0rGNfdeSQGxWA=">AAAB9HicbVDLSsNAFL2prxqr1rWbwSK4KokbXQpuXFawD2hLmUxu2qGTSZi5EUrpyp1bv8Kd+Dd+hP/g9LFJ64ELh3Mu93GiXElLQfDjVfb2Dw6Pqsf+Sc0/PTuv1zo2K4zAtshUZnoRt6ikxjZJUtjLDfI0UtiNpo9Lv/uKxspMv9Asx2HKx1omUnByUmtUbwTNYAW2S8INacAGo/rvIM5EkaImobi1/TDIaTjnhqRQuPAHhcWciykf43x13IJdOylmSWZcaWIrtdTHU2tnaeQ6U04Tu+0txX+9KC3t6xeU3A/nUucFoRbrxUmhGGVs+TmLpUFBauYIF0a6i5mYcMMFuXxKk617bYLxwnfxhNth7JLObTMMmuFzAFW4hCu4gRDu4AGeoAVtEBDDO3x4b96n97WOseJt8ryAErzvP39qlUc=</latexit><latexit sha1_base64="BTrjubiiNQAoK9NLMwKh7h1mFcw=">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</latexit><latexit sha1_base64="T/BdR14GV7iljS0v6Dm0LdY4mwc=">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</latexit><latexit sha1_base64="gAz8teB0+MVP3a4zA2q9ic3Etns=">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</latexit><latexit sha1_base64="gAz8teB0+MVP3a4zA2q9ic3Etns=">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</latexit><latexit sha1_base64="MTUm3RT8mR81/J+g7aFBDrdzQE0=">AAACZHicdVFdaxQxFM1O/ahrtVuLT4IEi9BCu0yK1RasFETwqazgtoVmN2Yyd7ppk8mYZAq70/kD/gV/jG+++uKLf8AnfdRns7sVXNELgcM5J/fmniSFks7H8ZdGNHfl6rXr8zeaNxdu3V5sLd05cKa0ArrCKGOPEu5AyRy6XnoFR4UFrhMFh8nZ87F+eA7WSZO/9sMCepqf5DKTgvtAsdYb2nGS6dURI+tUpca79RHbX9ulhTUpq+TT0xoHUW6M2OlaX2P6tuQphn61QV2pg2GX1P19TDPLRXURjBf9zbp6REEpFgdYs9ZK3I4nhf8AWzHZeUwwuWRW9l503r3/9PlDh7W+09SIUkPuheLOHZO48L2KWy+FgrpJSwcFF2f8BKrJ/jV+GKgUZ8aGk3s8YWd8XDs31Elwau4H7m9tTP5TS/TMvOPSZ9u9SuZF6SEX08FZqbA3eBwuTqUF4dUwAC6sDC/GYsBDND58wUxnF1YbQFo3Qz6/Q8D/BwebbRK3yasQ1DM0rXl0Dz1Aq4igJ2gPvUQd1EUCfUTf0A/0s/E1WoiWo7tTa9S4vLOMZiq6/ws+J77N</latexit><latexit sha1_base64="MTUm3RT8mR81/J+g7aFBDrdzQE0=">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</latexit><latexit sha1_base64="MTUm3RT8mR81/J+g7aFBDrdzQE0=">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</latexit><latexit sha1_base64="MTUm3RT8mR81/J+g7aFBDrdzQE0=">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</latexit><latexit sha1_base64="gAz8teB0+MVP3a4zA2q9ic3Etns=">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</latexit>

I

V

QH insulator

QH plateauQCP

Self-duality at the plateau transitions (I ⟷ V) (Shimshoni, Sondhi and Shahar)usually driven by disorder (Chalker-Coddington)

Two dimensional electron fluids in a large magnetic field at filling fraction 𝜈=1/m

Page 7: Duality in Condensed Matter Physics and Field Theory · Duality in Statistical Mechanics • 2D Ising Model: Kramers- Wannier duality • Partition function as a sum over closed domain

Bosonic Particle-Vortex Duality• Particle-vortex duality of 3D XY model (Peskin, Stone, Halperin-Dasgupta)

• U(1) broken symmetry phase (m2 < 0); excitations: closed quantized vortex lines with long range interactions

• U(1) unbroken symmetry phase (m2 > 0); excitations: massive charged bosons, closed worldlines with short-range repulsive interactions

• Duality: particles ⟺ vortices, high T ⟺ low T, strong coupling ⟺ weak coupling

• The 3D XY model as a loop model: lattice partition function sums over configurations of closed loops with short-range repulsive interactions

• Phase diagram for FQH fluids (Kivelson, Lee and Zhang) and plateau transitions

• 2+1-dimensional boson-boson complex scalar field theory mapping (A external field)

!7

|D(A)�|2 �m2|�|2 � �|�|4 $ |D(a)'|2 + m2|'|2 � �|'|4 +12⇡

✏µ⌫�Aµ@⌫a�<latexit sha1_base64="nPgQosvncRPrR88eC9SCbc1lnNc=">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</latexit><latexit sha1_base64="/KtNUB9g5kKY/3UxoTx06ENPLkM=">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</latexit><latexit sha1_base64="/KtNUB9g5kKY/3UxoTx06ENPLkM=">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</latexit><latexit sha1_base64="/KtNUB9g5kKY/3UxoTx06ENPLkM=">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</latexit><latexit sha1_base64="Gr0oojzxuaZD/NhgaNUSxCLD73M=">AAAB3nicbVDLSgNBEOyNr7hGjWcvg0HwFHa9qDfBi8cIrgkkIczOziZD5rHM9AZCyA941ZP4Yfo3Th6XJBY0FFUNXV1pIYXDKPoNKnv7B4dH1ePwpBaenp3Xa2/OlJbxhBlpbCeljkuheYICJe8UllOVSt5Ox08Lvz3h1gmjX3Fa8L6iQy1ywSh6qTWoN6JmtATZJfGaNGCNQf2nlxlWKq6RSepcN44K7M+oRcEkn4e90vGCsjEd8tky3JxceykjubF+NJKlurFHlXNTlfpNRXHktr2F+K+Xqo173RLz+/5M6KJErtnqcF5KgoYsPieZsJyhnHpCmRU+MWEjailD30/om4i3/94lyW3zoRm/RFCFS7iCG4jhDh7hGVqQAIMM3uEj0MFn8LUqrBKsm7uADQTff2/UjFg=</latexit><latexit sha1_base64="sQyz2qE4E5kGNvqiNmsBuPzaqQg=">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</latexit><latexit sha1_base64="soOb74Rbwr3xjKh8Fq6sytbpGMI=">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</latexit><latexit sha1_base64="7MrbVN7f0NXUXOxvfQiTZ9iTHvM=">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</latexit><latexit sha1_base64="7MrbVN7f0NXUXOxvfQiTZ9iTHvM=">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</latexit><latexit sha1_base64="/KtNUB9g5kKY/3UxoTx06ENPLkM=">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</latexit><latexit sha1_base64="/KtNUB9g5kKY/3UxoTx06ENPLkM=">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</latexit><latexit sha1_base64="/KtNUB9g5kKY/3UxoTx06ENPLkM=">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</latexit><latexit sha1_base64="/KtNUB9g5kKY/3UxoTx06ENPLkM=">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</latexit><latexit sha1_base64="7MrbVN7f0NXUXOxvfQiTZ9iTHvM=">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</latexit>

jµ = 12⇡ ✏µ⌫�@⌫a�

<latexit sha1_base64="wqnbk+Qx8J0ADe9DMkVjSU33hSs=">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</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">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</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">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</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">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</latexit><latexit sha1_base64="kO59cm/W//LWDO0rGNfdeSQGxWA=">AAAB9HicbVDLSsNAFL2prxqr1rWbwSK4KokbXQpuXFawD2hLmUxu2qGTSZi5EUrpyp1bv8Kd+Dd+hP/g9LFJ64ELh3Mu93GiXElLQfDjVfb2Dw6Pqsf+Sc0/PTuv1zo2K4zAtshUZnoRt6ikxjZJUtjLDfI0UtiNpo9Lv/uKxspMv9Asx2HKx1omUnByUmtUbwTNYAW2S8INacAGo/rvIM5EkaImobi1/TDIaTjnhqRQuPAHhcWciykf43x13IJdOylmSWZcaWIrtdTHU2tnaeQ6U04Tu+0txX+9KC3t6xeU3A/nUucFoRbrxUmhGGVs+TmLpUFBauYIF0a6i5mYcMMFuXxKk617bYLxwnfxhNth7JLObTMMmuFzAFW4hCu4gRDu4AGeoAVtEBDDO3x4b96n97WOseJt8ryAErzvP39qlUc=</latexit><latexit sha1_base64="L6GplW3kQOgOv+eySGMp2oTTLjY=">AAACP3icbZDNSsNAFIVv/K21anXrZrAIrkrSjW4EQRCXFewPNG2ZTCbt2EkyzEyEEvJGPoord6Kv4MaduHHSZpPWCxcO5wz3zv08wZnStv1ubWxube/sVvaq+7WDw6P6ca2r4kQS2iExj2Xfw4pyFtGOZprTvpAUhx6nPW92m+e9ZyoVi6NHPRd0GOJJxAJGsDbWuH73NHbD5NoNJCapk6UtV7AMuVQoxk2emhC5kWluZvrYRAJLzTAf5S4eFf643rCb9qLQunAK0YCi2uP6l+vHJAlppAnHSg0cW+hhms8mnGZVN1FUYDLDE5ourszQubF8FMTSdKTRwi29w6FS89AzL0Osp2o1y81/My8s7RskOrgapiwSiaYRWS4OEo50jHKEyGeSEs3nRmAimfkxIlNsAGoDujRZmdOm1M+qho+zSmNddFtNx246DzZU4BTO4AIcuIQbuIc2dIDAC7zBB3xar9a39bMkuWEVSE+gVNbvH7bgs4k=</latexit><latexit sha1_base64="BGOhKL+5uwy1QBJ92RSd4nCUGZI=">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</latexit><latexit sha1_base64="tos+MjP3U3SUyLjDSTXdlGjEmTo=">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</latexit><latexit sha1_base64="Wy0JzcTrIquw6y+T1PmjcpwngPg=">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</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">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</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">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</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">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</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">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</latexit><latexit sha1_base64="kO59cm/W//LWDO0rGNfdeSQGxWA=">AAAB9HicbVDLSsNAFL2prxqr1rWbwSK4KokbXQpuXFawD2hLmUxu2qGTSZi5EUrpyp1bv8Kd+Dd+hP/g9LFJ64ELh3Mu93GiXElLQfDjVfb2Dw6Pqsf+Sc0/PTuv1zo2K4zAtshUZnoRt6ikxjZJUtjLDfI0UtiNpo9Lv/uKxspMv9Asx2HKx1omUnByUmtUbwTNYAW2S8INacAGo/rvIM5EkaImobi1/TDIaTjnhqRQuPAHhcWciykf43x13IJdOylmSWZcaWIrtdTHU2tnaeQ6U04Tu+0txX+9KC3t6xeU3A/nUucFoRbrxUmhGGVs+TmLpUFBauYIF0a6i5mYcMMFuXxKk617bYLxwnfxhNth7JLObTMMmuFzAFW4hCu4gRDu4AGeoAVtEBDDO3x4b96n97WOseJt8ryAErzvP39qlUc=</latexit><latexit sha1_base64="L6GplW3kQOgOv+eySGMp2oTTLjY=">AAACP3icbZDNSsNAFIVv/K21anXrZrAIrkrSjW4EQRCXFewPNG2ZTCbt2EkyzEyEEvJGPoord6Kv4MaduHHSZpPWCxcO5wz3zv08wZnStv1ubWxube/sVvaq+7WDw6P6ca2r4kQS2iExj2Xfw4pyFtGOZprTvpAUhx6nPW92m+e9ZyoVi6NHPRd0GOJJxAJGsDbWuH73NHbD5NoNJCapk6UtV7AMuVQoxk2emhC5kWluZvrYRAJLzTAf5S4eFf643rCb9qLQunAK0YCi2uP6l+vHJAlppAnHSg0cW+hhms8mnGZVN1FUYDLDE5ourszQubF8FMTSdKTRwi29w6FS89AzL0Osp2o1y81/My8s7RskOrgapiwSiaYRWS4OEo50jHKEyGeSEs3nRmAimfkxIlNsAGoDujRZmdOm1M+qho+zSmNddFtNx246DzZU4BTO4AIcuIQbuIc2dIDAC7zBB3xar9a39bMkuWEVSE+gVNbvH7bgs4k=</latexit><latexit sha1_base64="BGOhKL+5uwy1QBJ92RSd4nCUGZI=">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</latexit><latexit sha1_base64="tos+MjP3U3SUyLjDSTXdlGjEmTo=">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</latexit><latexit sha1_base64="Wy0JzcTrIquw6y+T1PmjcpwngPg=">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</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">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</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">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</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">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</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">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</latexit><latexit sha1_base64="Wy0JzcTrIquw6y+T1PmjcpwngPg=">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</latexit>

Page 8: Duality in Condensed Matter Physics and Field Theory · Duality in Statistical Mechanics • 2D Ising Model: Kramers- Wannier duality • Partition function as a sum over closed domain

Web of DualitiesRecently conjectured dualities between fixed points (relativistic CFTs)

Son’s conjecture for 2DEG at 𝜈=1/2

Fermionic particle-vortex duality (“QED3”) (Metlitski-Vishwanath) at the surface of

!8

Bosonization duality (Seiberg, Senthil, Wang and Witten):

i /D(A) � 1

8⇡AdA ! |D(a)�|2 � |�|4 + 1

4⇡ada+

1

2⇡adA

i /D(A) � 1

8⇡AdA ! i� /D(a)�� 1

4⇡adA� 1

8⇡AdA

Proven for SU(N)k gauge fields in the ’t Hooft limit: N, k ⟷∞, with N/k fixed(Jain, Mandlik, Minwalla, Takimi, Wadia, Tokoyama; Aharony, Gur-Ari, Yacoby)

Maps the Dirac fermion to a magnetic monopole bound to 𝜙jµ = 1

2⇡ ✏µ⌫�@⌫a�<latexit sha1_base64="wqnbk+Qx8J0ADe9DMkVjSU33hSs=">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</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">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</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">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</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">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</latexit><latexit sha1_base64="kO59cm/W//LWDO0rGNfdeSQGxWA=">AAAB9HicbVDLSsNAFL2prxqr1rWbwSK4KokbXQpuXFawD2hLmUxu2qGTSZi5EUrpyp1bv8Kd+Dd+hP/g9LFJ64ELh3Mu93GiXElLQfDjVfb2Dw6Pqsf+Sc0/PTuv1zo2K4zAtshUZnoRt6ikxjZJUtjLDfI0UtiNpo9Lv/uKxspMv9Asx2HKx1omUnByUmtUbwTNYAW2S8INacAGo/rvIM5EkaImobi1/TDIaTjnhqRQuPAHhcWciykf43x13IJdOylmSWZcaWIrtdTHU2tnaeQ6U04Tu+0txX+9KC3t6xeU3A/nUucFoRbrxUmhGGVs+TmLpUFBauYIF0a6i5mYcMMFuXxKk617bYLxwnfxhNth7JLObTMMmuFzAFW4hCu4gRDu4AGeoAVtEBDDO3x4b96n97WOseJt8ryAErzvP39qlUc=</latexit><latexit sha1_base64="L6GplW3kQOgOv+eySGMp2oTTLjY=">AAACP3icbZDNSsNAFIVv/K21anXrZrAIrkrSjW4EQRCXFewPNG2ZTCbt2EkyzEyEEvJGPoord6Kv4MaduHHSZpPWCxcO5wz3zv08wZnStv1ubWxube/sVvaq+7WDw6P6ca2r4kQS2iExj2Xfw4pyFtGOZprTvpAUhx6nPW92m+e9ZyoVi6NHPRd0GOJJxAJGsDbWuH73NHbD5NoNJCapk6UtV7AMuVQoxk2emhC5kWluZvrYRAJLzTAf5S4eFf643rCb9qLQunAK0YCi2uP6l+vHJAlppAnHSg0cW+hhms8mnGZVN1FUYDLDE5ourszQubF8FMTSdKTRwi29w6FS89AzL0Osp2o1y81/My8s7RskOrgapiwSiaYRWS4OEo50jHKEyGeSEs3nRmAimfkxIlNsAGoDujRZmdOm1M+qho+zSmNddFtNx246DzZU4BTO4AIcuIQbuIc2dIDAC7zBB3xar9a39bMkuWEVSE+gVNbvH7bgs4k=</latexit><latexit sha1_base64="BGOhKL+5uwy1QBJ92RSd4nCUGZI=">AAACP3icdZDLSsNAFIZP6q3WqtWtm8EiuCpJwdtCEARxWcFeoGnLZDJpx06SYWYilJA38lFcuRN9BTfuxI2TtoIVPXDg8P+Hc/k8wZnStv1sFZaWV1bXiuuljfLm1nZlp9xScSIJbZKYx7LjYUU5i2hTM81pR0iKQ4/Ttje+zP32PZWKxdGtngjaC/EwYgEjWBtpULm6G7hhcu4GEpPUydK6K1iGXCoU48ZPjYncyCQ3M31sLIGlZpj3cxX35/qgUrVr9jTQj+LIds6OHeTMlSrMozGovLl+TJKQRppwrFTXsYXupflswmlWchNFBSZjPKTp9MsMHRjJR0EsTUYaTdWFPhwqNQk90xliPVK/vVz80/PChX3dRAenvZRFItE0IrPFQcKRjlGOEPlMUqL5xBSYSGYuRmSEDUBtQC9MVua1EfWzkuHzDQH9X7TqNceuOTc2FGEP9uEQHDiBC7iGBjSBwAM8wQu8Wo/Wu/UxI1mw5kh3YSGszy/XTrOf</latexit><latexit sha1_base64="tos+MjP3U3SUyLjDSTXdlGjEmTo=">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</latexit><latexit sha1_base64="Wy0JzcTrIquw6y+T1PmjcpwngPg=">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</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">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</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">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</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">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</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">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</latexit><latexit sha1_base64="kO59cm/W//LWDO0rGNfdeSQGxWA=">AAAB9HicbVDLSsNAFL2prxqr1rWbwSK4KokbXQpuXFawD2hLmUxu2qGTSZi5EUrpyp1bv8Kd+Dd+hP/g9LFJ64ELh3Mu93GiXElLQfDjVfb2Dw6Pqsf+Sc0/PTuv1zo2K4zAtshUZnoRt6ikxjZJUtjLDfI0UtiNpo9Lv/uKxspMv9Asx2HKx1omUnByUmtUbwTNYAW2S8INacAGo/rvIM5EkaImobi1/TDIaTjnhqRQuPAHhcWciykf43x13IJdOylmSWZcaWIrtdTHU2tnaeQ6U04Tu+0txX+9KC3t6xeU3A/nUucFoRbrxUmhGGVs+TmLpUFBauYIF0a6i5mYcMMFuXxKk617bYLxwnfxhNth7JLObTMMmuFzAFW4hCu4gRDu4AGeoAVtEBDDO3x4b96n97WOseJt8ryAErzvP39qlUc=</latexit><latexit sha1_base64="L6GplW3kQOgOv+eySGMp2oTTLjY=">AAACP3icbZDNSsNAFIVv/K21anXrZrAIrkrSjW4EQRCXFewPNG2ZTCbt2EkyzEyEEvJGPoord6Kv4MaduHHSZpPWCxcO5wz3zv08wZnStv1ubWxube/sVvaq+7WDw6P6ca2r4kQS2iExj2Xfw4pyFtGOZprTvpAUhx6nPW92m+e9ZyoVi6NHPRd0GOJJxAJGsDbWuH73NHbD5NoNJCapk6UtV7AMuVQoxk2emhC5kWluZvrYRAJLzTAf5S4eFf643rCb9qLQunAK0YCi2uP6l+vHJAlppAnHSg0cW+hhms8mnGZVN1FUYDLDE5ourszQubF8FMTSdKTRwi29w6FS89AzL0Osp2o1y81/My8s7RskOrgapiwSiaYRWS4OEo50jHKEyGeSEs3nRmAimfkxIlNsAGoDujRZmdOm1M+qho+zSmNddFtNx246DzZU4BTO4AIcuIQbuIc2dIDAC7zBB3xar9a39bMkuWEVSE+gVNbvH7bgs4k=</latexit><latexit sha1_base64="BGOhKL+5uwy1QBJ92RSd4nCUGZI=">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</latexit><latexit sha1_base64="tos+MjP3U3SUyLjDSTXdlGjEmTo=">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</latexit><latexit sha1_base64="Wy0JzcTrIquw6y+T1PmjcpwngPg=">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</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">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</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">AAACSnicdZDNSgMxEMeztX7Vr6pHL8EqeCqbgl8HpSCIRwWrhW5bstlsG5vdDUlWKMu+gk/iiwievIm+ghdvIqLZtoIVHRgYfv/JTObvCs6Utu1HKzeRn5yanpktzM0vLC4Vl1cuVBRLQmsk4pGsu1hRzkJa00xzWheS4sDl9NLtHWX65TWVikXhue4L2gxwJ2Q+I1gb1C4eX7WdID5wfIlJgtKk4giWQocKxbjREyNCJzTJzUwPG0lgqRnmrYzi1oi3iyW7bA8C/ii2bbS/gyAakVJ14+PmrvGZP20XXxwvInFAQ004VqqBbKGbSTabcJoWnFhRgUkPd2gyuDKFmwZ50I+kyVDDAR3rw4FS/cA1nQHWXfVby+CfmhuM7WvE2t9rJiwUsaYhGS72Yw51BDMLocckJZr3TYGJZObHkHSxMVAbo8cmK3Nal3ppwfjzbQL8v7iolJFdRmeoVD0Ew5gBa2AdbAEEdkEVnIBTUAME3IIH8ASerXvr1Xqz3oetOWv0ZhWMRS7/BYSzuGQ=</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">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</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">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</latexit><latexit sha1_base64="kO59cm/W//LWDO0rGNfdeSQGxWA=">AAAB9HicbVDLSsNAFL2prxqr1rWbwSK4KokbXQpuXFawD2hLmUxu2qGTSZi5EUrpyp1bv8Kd+Dd+hP/g9LFJ64ELh3Mu93GiXElLQfDjVfb2Dw6Pqsf+Sc0/PTuv1zo2K4zAtshUZnoRt6ikxjZJUtjLDfI0UtiNpo9Lv/uKxspMv9Asx2HKx1omUnByUmtUbwTNYAW2S8INacAGo/rvIM5EkaImobi1/TDIaTjnhqRQuPAHhcWciykf43x13IJdOylmSWZcaWIrtdTHU2tnaeQ6U04Tu+0txX+9KC3t6xeU3A/nUucFoRbrxUmhGGVs+TmLpUFBauYIF0a6i5mYcMMFuXxKk617bYLxwnfxhNth7JLObTMMmuFzAFW4hCu4gRDu4AGeoAVtEBDDO3x4b96n97WOseJt8ryAErzvP39qlUc=</latexit><latexit sha1_base64="L6GplW3kQOgOv+eySGMp2oTTLjY=">AAACP3icbZDNSsNAFIVv/K21anXrZrAIrkrSjW4EQRCXFewPNG2ZTCbt2EkyzEyEEvJGPoord6Kv4MaduHHSZpPWCxcO5wz3zv08wZnStv1ubWxube/sVvaq+7WDw6P6ca2r4kQS2iExj2Xfw4pyFtGOZprTvpAUhx6nPW92m+e9ZyoVi6NHPRd0GOJJxAJGsDbWuH73NHbD5NoNJCapk6UtV7AMuVQoxk2emhC5kWluZvrYRAJLzTAf5S4eFf643rCb9qLQunAK0YCi2uP6l+vHJAlppAnHSg0cW+hhms8mnGZVN1FUYDLDE5ourszQubF8FMTSdKTRwi29w6FS89AzL0Osp2o1y81/My8s7RskOrgapiwSiaYRWS4OEo50jHKEyGeSEs3nRmAimfkxIlNsAGoDujRZmdOm1M+qho+zSmNddFtNx246DzZU4BTO4AIcuIQbuIc2dIDAC7zBB3xar9a39bMkuWEVSE+gVNbvH7bgs4k=</latexit><latexit sha1_base64="BGOhKL+5uwy1QBJ92RSd4nCUGZI=">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</latexit><latexit sha1_base64="tos+MjP3U3SUyLjDSTXdlGjEmTo=">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</latexit><latexit sha1_base64="Wy0JzcTrIquw6y+T1PmjcpwngPg=">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</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">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</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">AAACSnicdZDNSgMxEMeztX7Vr6pHL8EqeCqbgl8HpSCIRwWrhW5bstlsG5vdDUlWKMu+gk/iiwievIm+ghdvIqLZtoIVHRgYfv/JTObvCs6Utu1HKzeRn5yanpktzM0vLC4Vl1cuVBRLQmsk4pGsu1hRzkJa00xzWheS4sDl9NLtHWX65TWVikXhue4L2gxwJ2Q+I1gb1C4eX7WdID5wfIlJgtKk4giWQocKxbjREyNCJzTJzUwPG0lgqRnmrYzi1oi3iyW7bA8C/ii2bbS/gyAakVJ14+PmrvGZP20XXxwvInFAQ004VqqBbKGbSTabcJoWnFhRgUkPd2gyuDKFmwZ50I+kyVDDAR3rw4FS/cA1nQHWXfVby+CfmhuM7WvE2t9rJiwUsaYhGS72Yw51BDMLocckJZr3TYGJZObHkHSxMVAbo8cmK3Nal3ppwfjzbQL8v7iolJFdRmeoVD0Ew5gBa2AdbAEEdkEVnIBTUAME3IIH8ASerXvr1Xqz3oetOWv0ZhWMRS7/BYSzuGQ=</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">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</latexit><latexit sha1_base64="RKAD/Ab0tmYfTRY6DDh5/sAQ9L0=">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</latexit><latexit sha1_base64="Wy0JzcTrIquw6y+T1PmjcpwngPg=">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</latexit>

Page 9: Duality in Condensed Matter Physics and Field Theory · Duality in Statistical Mechanics • 2D Ising Model: Kramers- Wannier duality • Partition function as a sum over closed domain

Strategy

• “Derive” this web of dualities using quantum loop models near criticality, but still in the gapped phases.

• These models are related to modular invariant models originally introduced by Kivelson and myself, but we will see that modular invariance cannot be kept close to the CFT.

• The “fractional spin” responsible for breaking modular invariance gives rise to Dirac fermions, leading to loop model based “proofs” of the CFT duality web.

!9

Page 10: Duality in Condensed Matter Physics and Field Theory · Duality in Statistical Mechanics • 2D Ising Model: Kramers- Wannier duality • Partition function as a sum over closed domain

Quantum Loop Models and Duality(EF and Kivelson 1996)

• Non-intersecting linked loops [J𝜇] in 3D Euclidean space-time (with no spin) with exact particle-hole symmetry

• flux attachment with fractional statistics 𝜃, long ranged interactions with coupling g, and short-range repulsion (to avoid crossings)

• The imaginary part of the action is given in terms of the loops linking number

!10

L = |Dµ�|2 �m2|�|2 � �|�|4 � 14g2

fµ⌫1p�@2

fµ⌫ +k

4⇡✏µ⌫�aµ@⌫a�

<latexit sha1_base64="Fr0N2Zi//hU6FajDIYESkpYgktE=">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</latexit><latexit sha1_base64="HUJsyb6AlbPs9XcF8HAXsBZQh+s=">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</latexit><latexit sha1_base64="HUJsyb6AlbPs9XcF8HAXsBZQh+s=">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</latexit><latexit sha1_base64="HUJsyb6AlbPs9XcF8HAXsBZQh+s=">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</latexit><latexit sha1_base64="Gr0oojzxuaZD/NhgaNUSxCLD73M=">AAAB3nicbVDLSgNBEOyNr7hGjWcvg0HwFHa9qDfBi8cIrgkkIczOziZD5rHM9AZCyA941ZP4Yfo3Th6XJBY0FFUNXV1pIYXDKPoNKnv7B4dH1ePwpBaenp3Xa2/OlJbxhBlpbCeljkuheYICJe8UllOVSt5Ox08Lvz3h1gmjX3Fa8L6iQy1ywSh6qTWoN6JmtATZJfGaNGCNQf2nlxlWKq6RSepcN44K7M+oRcEkn4e90vGCsjEd8tky3JxceykjubF+NJKlurFHlXNTlfpNRXHktr2F+K+Xqo173RLz+/5M6KJErtnqcF5KgoYsPieZsJyhnHpCmRU+MWEjailD30/om4i3/94lyW3zoRm/RFCFS7iCG4jhDh7hGVqQAIMM3uEj0MFn8LUqrBKsm7uADQTff2/UjFg=</latexit><latexit sha1_base64="zqm5dtcQTnK5+uFkizb0HuiT6ps=">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</latexit><latexit sha1_base64="cPMkI8AWp8zn9H2KvCLUbgCxMtQ=">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</latexit><latexit sha1_base64="cbQdYyTjBefMGoc1EkjzdiA2Sck=">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</latexit><latexit sha1_base64="cbQdYyTjBefMGoc1EkjzdiA2Sck=">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</latexit><latexit sha1_base64="HUJsyb6AlbPs9XcF8HAXsBZQh+s=">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</latexit><latexit sha1_base64="HUJsyb6AlbPs9XcF8HAXsBZQh+s=">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</latexit><latexit sha1_base64="HUJsyb6AlbPs9XcF8HAXsBZQh+s=">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</latexit><latexit sha1_base64="HUJsyb6AlbPs9XcF8HAXsBZQh+s=">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</latexit><latexit sha1_base64="cbQdYyTjBefMGoc1EkjzdiA2Sck=">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</latexit>

Field theory picture: 2+1 D complex scalar field coupled to 3+1 D Maxwell field with a 𝜃 term

Z[g, ✓] =X

{Jµ}2Z�(�µJ

µ)e�S[Jµ]

S[Jµ] =g2

2

X

x,y

Jµ(x)Gµ⌫(x� y)J⌫(y) + i✓X

x,y

Jµ(x)Kµ⌫(x� y)J⌫(y)

Gµ⌫(p) =1pp2

✓�µ⌫ � pµp⌫

p2

◆, Kµ⌫(p) = i✏µ⌫�

p�p2

long ranged interactionslinking number =𝜃𝛷[J]

Page 11: Duality in Condensed Matter Physics and Field Theory · Duality in Statistical Mechanics • 2D Ising Model: Kramers- Wannier duality • Partition function as a sum over closed domain

Self-Duality and Modular Invariance

• The partition functions of loop models regularized without self-linking (fractional spin) have the symmetries

• 𝑆: duality: Z[τ]=Z[-1/τ], and 𝒯: Periodicity: Z[τ]=Z[τ+1]

• 𝑆 and 𝒯 generate the modular group PSL(2,ℤ)

• The partition function is self dual at the fixed points of the modular group

• Two types of PSL(2,ℤ) fixed points: “bosonic” and “fermionic”

• FK showed that the finite modular fixed points are quantum critical points with 𝜎xx ≠ 0 and 𝜎xy = 0

• Problem: the predicted conductivities are different in the FK loop models and the web of dualities

⌧ =✓

⇡+ i

g2

2⇡<latexit sha1_base64="+xZmN/rH5CN1YEiI03mMKSNgazQ=">AAACGXicdZDJSgNBEIZr4hbjFvXgwUtjEIRAmAm45KAEvHhUMCo4MfR0epImPQvdNUIY5iG8+x5e1ZN49aIv4TPYWQQjWtBQfP/fVNXvxVJotO0PKzc1PTM7l58vLCwuLa8UV9cudJQoxhsskpG68qjmUoS8gQIlv4oVp4En+aXXOx7ol7dcaRGF59iPeTOgnVD4glE0qFUsu0iTQ7ftK8pSF7scaZa6scjKYgw7N9UsrQ5Qq1iyK/awyI9m13Zqew5xxqRU3/j8JABw2iq+u+2IJQEPkUmq9bVjx9hMqULBJM8KbqJ5TFmPdng6PCUj2wa1iR8p80IkQzrho4HW/cAzzoBiV//WBvBPzQsm5l0n6B80UxHGCfKQjQb7iSQYkUFOpC0UZyj7pqFMCbMxYV1qAkGTZsFE8X0v+b9pVCu1inPmlOpHMKo8bMIW7IAD+1CHEziFBjC4gwd4hCfr3nq2XqzXkTVnjf+sw0RZb18DIKP0</latexit><latexit sha1_base64="twnFONC12P6IFr2ePzcAYxPpNrI=">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</latexit><latexit sha1_base64="twnFONC12P6IFr2ePzcAYxPpNrI=">AAACGXicdZC7SgNBFIZnvcZ4SdTCwmYwCEIg7Aa8BFQCNpYRjAlkY5idzCZDZi/MnBXCsg9h71tY2KqV2NroA9jmGZxcBCN6YODw/f9wzvmdUHAFpvlhzMzOzS8sppbSyyura5ns+saVCiJJWZUGIpB1hygmuM+qwEGweigZ8RzBak7vbKjXbphUPPAvoR+ypkc6Pnc5JaBRK5u3gUQndtuVhMY2dBmQJLZDnuT5BHaui0lcHKJWNmcWzFHhH82+aZUOLGxNSK68NRhkju8/K63su90OaOQxH6ggSjUsM4RmTCRwKliStiPFQkJ7pMPi0SkJ3tWojd1A6ucDHtEpH/GU6nuOdnoEuuq3NoR/ao43Na8RgXvUjLkfRsB8Oh7sRgJDgIc54TaXjILo64ZQyfXGmHaJDgR0mmkdxfe9+P+mWiyUCtaFlSufonGl0DbaQXvIQoeojM5RBVURRbfoAT2iJ+POeDZejNexdcaY/NlEU2W8fQFlxKW3</latexit><latexit sha1_base64="twnFONC12P6IFr2ePzcAYxPpNrI=">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</latexit><latexit sha1_base64="Gr0oojzxuaZD/NhgaNUSxCLD73M=">AAAB3nicbVDLSgNBEOyNr7hGjWcvg0HwFHa9qDfBi8cIrgkkIczOziZD5rHM9AZCyA941ZP4Yfo3Th6XJBY0FFUNXV1pIYXDKPoNKnv7B4dH1ePwpBaenp3Xa2/OlJbxhBlpbCeljkuheYICJe8UllOVSt5Ox08Lvz3h1gmjX3Fa8L6iQy1ywSh6qTWoN6JmtATZJfGaNGCNQf2nlxlWKq6RSepcN44K7M+oRcEkn4e90vGCsjEd8tky3JxceykjubF+NJKlurFHlXNTlfpNRXHktr2F+K+Xqo173RLz+/5M6KJErtnqcF5KgoYsPieZsJyhnHpCmRU+MWEjailD30/om4i3/94lyW3zoRm/RFCFS7iCG4jhDh7hGVqQAIMM3uEj0MFn8LUqrBKsm7uADQTff2/UjFg=</latexit><latexit sha1_base64="hFfHjjbHxjGqke4yFQf2OmcV70s=">AAACDnicbZDNSsNAFIVv6l+tVaNbN8EiCIWSdKMuBMGNywrGFppaJtNJO3Tyw8yNUEIewr3v4VZdiU+gb+OkzaatBwYO37lw5x4/EVyhbf8alY3Nre2d6m5tr75/cGge1R9VnErKXBqLWPZ8opjgEXORo2C9RDIS+oJ1/eltkXefmVQ8jh5wlrBBSMYRDzglqNHQbHpI0mtvFEhCMw8nDEmeeQnPm7yE46d2nrULNDQbdsuey1o3TmkaUKozNH+8UUzTkEVIBVGq79gJDjIikVPB8pqXKpYQOiVjls1Pya0zjUZWEEv9IrTmdGmOhErNQl9PhgQnajUr4L+ZHy7t66cYXA4yHiUpsoguFgepsDC2ip6sEZeMophpQ6jk+scWnRBdCOo2a7oKZ/XwdeO2W1ct596GKpzAKZyDAxdwA3fQARcovMAbvMOH8Wp8Gl+LzipGWd4xLMn4/gPHW6AO</latexit><latexit sha1_base64="070UOSgO5j6NuPKpemygNALAcOo=">AAACDnicdZDNSgMxFIXv+Ftr1erWTbAIQqHMFPzpQhDcuKxgbaFTSybNtKGZH5I7QhnmIdz7Hm7VlfgE+jZm2gpW9EDg8J0LN/d4sRQabfvTWlpeWV1bL2wUN0tb2zvl3dKtjhLFeItFMlIdj2ouRchbKFDyTqw4DTzJ2974Ms/b91xpEYU3OIl5L6DDUPiCUTSoX666SJNzd+ArylIXRxxplrqxyKpiDod39Syt56hfrtg1eyrywxzbTuPEIc6cVGCuZr/84Q4ilgQ8RCap1l3HjrGXUoWCSZ4V3UTzmLIxHfJ0ekpGDg0aED9S5oVIpnRhjgZaTwLPTAYUR/p3lsM/My9Y2NdN0D/rpSKME+Qhmy32E0kwInlPZCAUZygnxlCmhPkxYSNqCkHTZtFU8X0v+d+06rVGzbm2oQD7cABH4MApXMAVNKEFDB7gCZ7hxXq0Xq23WWdL1ry8PViQ9f4F5eWgJA==</latexit><latexit sha1_base64="/GHWNy+y4doLhcLQTQDMdPUlniI=">AAACGXicdZDLSsNAFIYn9VbrrerSzWARhEJJCl66UApuXFYwttDUMplM2qGTCzMnQgl5CPe+h1t1JW7d6Ns4bSNY0QMDh+//h3PO78aCKzDNT6OwsLi0vFJcLa2tb2xulbd3blSUSMpsGolIdlyimOAhs4GDYJ1YMhK4grXd0cVEb98xqXgUXsM4Zr2ADELuc0pAo3656gBJzhzPl4SmDgwZkCx1Yp5VeQ4Ht/UsrU9Qv1wxa+a08I/myLQaxxa2clJBebX65Q/Hi2gSsBCoIEp1LTOGXkokcCpYVnISxWJCR2TA0ukpGT7QyMN+JPULAU/pnI8ESo0DVzsDAkP1W5vAPzU3mJvXTcA/7aU8jBNgIZ0N9hOBIcKTnLDHJaMgxrohVHK9MaZDogMBnWZJR/F9L/6/seu1Rs26MivN8zyTItpD++gQWegENdElaiEbUXSPHtETejYejBfj1XibWQtG/mcXzZXx/gXTXqGg</latexit><latexit sha1_base64="UjsMBaZRDAZiUVyJuCcjj1TFyfY=">AAACGXicdZDLSsNAFIYn9VbrrerSzWARhEJJCl66UApuXFYwttDUMplM2qGTCzMnQgl5CPe+h1t1JW7d6Ns4bSNY0QMDh+//h3PO78aCKzDNT6OwsLi0vFJcLa2tb2xulbd3blSUSMpsGolIdlyimOAhs4GDYJ1YMhK4grXd0cVEb98xqXgUXsM4Zr2ADELuc0pAo3656gBJzhzPl4SmDgwZkCx1Yp5VeQ4Ht/UsrU9Qv1wxa+a08I/myLQaxxa2clJBebX65Q/Hi2gSsBCoIEp1LTOGXkokcCpYVnISxWJCR2TA0ukpGT7QyMN+JPULAU/pnI8ESo0DVzsDAkP1W5vAPzU3mJvXTcA/7aU8jBNgIZ0N9hOBIcKTnLDHJaMgxrohVHK9MaZDogMBnWZJR/F9L/6/seu1Rs26sirN8zyTItpD++gQWegENdElaiEbUXSPHtETejYejBfj1XibWQtG/mcXzZXx/gXTrqGh</latexit><latexit sha1_base64="twnFONC12P6IFr2ePzcAYxPpNrI=">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</latexit><latexit sha1_base64="twnFONC12P6IFr2ePzcAYxPpNrI=">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</latexit><latexit sha1_base64="twnFONC12P6IFr2ePzcAYxPpNrI=">AAACGXicdZC7SgNBFIZnvcZ4SdTCwmYwCEIg7Aa8BFQCNpYRjAlkY5idzCZDZi/MnBXCsg9h71tY2KqV2NroA9jmGZxcBCN6YODw/f9wzvmdUHAFpvlhzMzOzS8sppbSyyura5ns+saVCiJJWZUGIpB1hygmuM+qwEGweigZ8RzBak7vbKjXbphUPPAvoR+ypkc6Pnc5JaBRK5u3gUQndtuVhMY2dBmQJLZDnuT5BHaui0lcHKJWNmcWzFHhH82+aZUOLGxNSK68NRhkju8/K63su90OaOQxH6ggSjUsM4RmTCRwKliStiPFQkJ7pMPi0SkJ3tWojd1A6ucDHtEpH/GU6nuOdnoEuuq3NoR/ao43Na8RgXvUjLkfRsB8Oh7sRgJDgIc54TaXjILo64ZQyfXGmHaJDgR0mmkdxfe9+P+mWiyUCtaFlSufonGl0DbaQXvIQoeojM5RBVURRbfoAT2iJ+POeDZejNexdcaY/NlEU2W8fQFlxKW3</latexit><latexit sha1_base64="twnFONC12P6IFr2ePzcAYxPpNrI=">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</latexit><latexit sha1_base64="UjsMBaZRDAZiUVyJuCcjj1TFyfY=">AAACGXicdZDLSsNAFIYn9VbrrerSzWARhEJJCl66UApuXFYwttDUMplM2qGTCzMnQgl5CPe+h1t1JW7d6Ns4bSNY0QMDh+//h3PO78aCKzDNT6OwsLi0vFJcLa2tb2xulbd3blSUSMpsGolIdlyimOAhs4GDYJ1YMhK4grXd0cVEb98xqXgUXsM4Zr2ADELuc0pAo3656gBJzhzPl4SmDgwZkCx1Yp5VeQ4Ht/UsrU9Qv1wxa+a08I/myLQaxxa2clJBebX65Q/Hi2gSsBCoIEp1LTOGXkokcCpYVnISxWJCR2TA0ukpGT7QyMN+JPULAU/pnI8ESo0DVzsDAkP1W5vAPzU3mJvXTcA/7aU8jBNgIZ0N9hOBIcKTnLDHJaMgxrohVHK9MaZDogMBnWZJR/F9L/6/seu1Rs26sirN8zyTItpD++gQWegENdElaiEbUXSPHtETejYejBfj1XibWQtG/mcXzZXx/gXTrqGh</latexit>

Modular parameter:

!11

Page 12: Duality in Condensed Matter Physics and Field Theory · Duality in Statistical Mechanics • 2D Ising Model: Kramers- Wannier duality • Partition function as a sum over closed domain

The Role of Fractional Spin

• The linking number of two separate loops 𝑙1 and 𝑙2 is

• Witten: point-split the loops into ribbons so that the writhe is a frame-dependent topological invariant W[𝑙] = SL[𝑙] = integer. Only consistent deep in the topological phase, not as the critical point is approached.

• Polyakov: no-point splitting and W[𝑙] = SL[𝑙] - T[𝑙] (writhe = self-linking - twist)

T[𝑙] is a Berry phase (fractional spin) and e is the tangent vector to the loop. The twist T[𝑙] is not quantized and depends on the metric.

!12

T [`] =12⇡

Z L

0ds

Z 1

0due · @se⇥ @ue

<latexit sha1_base64="Xen2ybEi/VP87JtJZIbs9yQxVr0=">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</latexit><latexit sha1_base64="vuO5CJW/+PP7vg7wRD98hIFrvR8=">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</latexit><latexit sha1_base64="vuO5CJW/+PP7vg7wRD98hIFrvR8=">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</latexit><latexit sha1_base64="vuO5CJW/+PP7vg7wRD98hIFrvR8=">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</latexit><latexit sha1_base64="cSEpZM4nr16xSbsp5iCCuNGq5wo=">AAAB83icbZDNSsNAFIVv6l+NVevazWARXJXEjboT3LisYGyhLWUyuWmHTn6YuRFK6cqdW5/Clfg4PoTv4KTtJq0XBg7nXJhzvzBX0pDn/Ti1nd29/YP6oXvUcI9PTpuNF5MVWmAgMpXpXsgNKpliQJIU9nKNPAkVdsPpQ5l3X1EbmaXPNMtxmPBxKmMpOFmrM2q2vLa3HLYt/LVowXpGzd9BlIkiwZSE4sb0fS+n4ZxrkkLhwh0UBnMupnyM82W5Bbu0VsTiTNuXElu6lT2eGDNLQruZcJqYzaw0/83CpPqfsYUmGFXMfkHx7XAu07wgTMWqTVwoRhkrcbBIahSkZlZwoaU9g4kJ11yQheZaPP4mjG0RXLfv2v6TB3U4hwu4Ah9u4B4eoQMBCIjgHT6cN+fT+VpRrDlrnGdQGef7D/wclRw=</latexit><latexit sha1_base64="nf2UCLYfyS8gp3AAt0pLTyLT2eY=">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</latexit><latexit sha1_base64="cfLe7/Sv7+GF5n0eAlZBRS5hbjk=">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</latexit><latexit sha1_base64="d0jJS0XbcAGw2i769xGAx6DugRg=">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</latexit><latexit sha1_base64="d0jJS0XbcAGw2i769xGAx6DugRg=">AAACaXicdZHLSgMxFIbT8VbrrdaN6CZYBFdlRrwuBMGNCxcKVoVOHTKZMzY0mRmSM0IZ+oa+gO+gG7e6MK31UtEDgZ/vP0lO/oSZFAZd97HkTExOTc+UZytz8wuLS9Xl2pVJc82hyVOZ6puQGZAigSYKlHCTaWAqlHAddk8G/vU9aCPS5BJ7GbQVu0tELDhDi4JqfNnyQcr2kR9rxguvX2z7mehTXyQYuLdnkfmUXpTTwg8VBevyKEXqZ0yjYDIwXxyFAvNtfO0IqnW34Q6L/hC7rne451FvROpkVOdB9dmPUp4rSJBLZkzLczNsF4NzuYR+xc8NZIx32R0UwxT6dNOiiMaptitBOqRjfUwZ01Oh7VQMO+a3N4B/eqEav8/YgToQjcFWjvFBuxBJliMk/GOaOJcUUzrInUZCA0fZs4JxLewzKO8wmzna36nYfD5DoP+L5nbjsOFduPXjnVFQZbJONsgW8cg+OSan5Jw0CScP5IW8krfSk1NzVp21j1anNNqzQsbKqb8DXdS9GA==</latexit><latexit sha1_base64="vuO5CJW/+PP7vg7wRD98hIFrvR8=">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</latexit><latexit sha1_base64="vuO5CJW/+PP7vg7wRD98hIFrvR8=">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</latexit><latexit sha1_base64="vuO5CJW/+PP7vg7wRD98hIFrvR8=">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</latexit><latexit sha1_base64="vuO5CJW/+PP7vg7wRD98hIFrvR8=">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</latexit><latexit sha1_base64="d0jJS0XbcAGw2i769xGAx6DugRg=">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</latexit>

�[J = `1 + `2] = 2⇥ (Linking number of `1 with `2) +W [`1] +W [`2]

“Writhe.” Associated with self linking. Not necessarily a topological invariant.

Page 13: Duality in Condensed Matter Physics and Field Theory · Duality in Statistical Mechanics • 2D Ising Model: Kramers- Wannier duality • Partition function as a sum over closed domain

Fractional Spin: Periodicity Lost, 3D Bosonization Regained

!13

L[J]: length of loop, 𝛷[J]: linking number (including the spin factor)

• T[𝑙] is not quantized: Duality 𝑆 remains a symmetry, but periodicity 𝒯 is lost

• Polyakov: fractional spin leads to the (IR) duality between a complex massive scalar with CS at k = 1 and a massive Dirac spinor (with a parity anomaly)

• Loop model representation

Zfermion = det[i/@ �M ] =ZDJ �(@µJµ) e�|m|L[J]�isign(M)⇡�[J]

<latexit sha1_base64="JOz3VvnM9X5V21YRnTQnea2nCzc=">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</latexit><latexit sha1_base64="gvk0163/DjASSvEOn5Kq+epoyvs=">AAACkHicbVHdahNBFJ5d/2r8aVpvBG/GFiGFJuwKYisUK3ohwUIE0wYz6TI7ezYZOrO7zJwVw3bfwPfwmXwCr7zxBXQ2iRdpPTDDx/edM+fMd+JCSYtB8MPzb9y8dfvOxt3WvfsPHm62t7ZPbV4aAUORq9yMYm5ByQyGKFHBqDDAdazgLL542+hnX8BYmWefcF7ARPNpJlMpODoqan/7HFVMc5wZXaVgtCPr+oglgGPJrOJ2BknFCm5QclXTLj2ZtI6YzHBRJLiq3tV9tu8KFPLOv8SI6ZL2z929x/YpnFfdS335YdyfdKlkCF/RdbNymtWdkz1WSDaYSSfWUXs36AWLoNdBuAK7xy++Bz//jF4NovYvluSi1JChcLPacRgUOKmaIYSCusVKCwUXF3wK1cKrmj5zVELT3LiTIV2wa3lcWzvXsctsfmivag35Xy3Wa/3GJaYHk0pmRYmQiWXjtFQUc9osgibSgEA1d4ALI93EVMy44QLdutZeXq2hbjl/wqtuXAenz3th0As/OqMOyTI2yBOyQzokJC/JMXlPBmRIBPntPfaeejv+tn/gv/bfLFN9b1XziKyF3/8LALDNhQ==</latexit><latexit sha1_base64="gvk0163/DjASSvEOn5Kq+epoyvs=">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</latexit><latexit sha1_base64="gvk0163/DjASSvEOn5Kq+epoyvs=">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</latexit><latexit sha1_base64="kO59cm/W//LWDO0rGNfdeSQGxWA=">AAAB9HicbVDLSsNAFL2prxqr1rWbwSK4KokbXQpuXFawD2hLmUxu2qGTSZi5EUrpyp1bv8Kd+Dd+hP/g9LFJ64ELh3Mu93GiXElLQfDjVfb2Dw6Pqsf+Sc0/PTuv1zo2K4zAtshUZnoRt6ikxjZJUtjLDfI0UtiNpo9Lv/uKxspMv9Asx2HKx1omUnByUmtUbwTNYAW2S8INacAGo/rvIM5EkaImobi1/TDIaTjnhqRQuPAHhcWciykf43x13IJdOylmSWZcaWIrtdTHU2tnaeQ6U04Tu+0txX+9KC3t6xeU3A/nUucFoRbrxUmhGGVs+TmLpUFBauYIF0a6i5mYcMMFuXxKk617bYLxwnfxhNth7JLObTMMmuFzAFW4hCu4gRDu4AGeoAVtEBDDO3x4b96n97WOseJt8ryAErzvP39qlUc=</latexit><latexit sha1_base64="K6Kha57V+VVxgeGVc3hs11SnD78=">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</latexit><latexit sha1_base64="K6Kha57V+VVxgeGVc3hs11SnD78=">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</latexit><latexit sha1_base64="0Lvim4iMEFaRbGy+ivNoSlwUGZQ=">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</latexit><latexit sha1_base64="0Lvim4iMEFaRbGy+ivNoSlwUGZQ=">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</latexit><latexit sha1_base64="gvk0163/DjASSvEOn5Kq+epoyvs=">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</latexit><latexit sha1_base64="gvk0163/DjASSvEOn5Kq+epoyvs=">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</latexit><latexit sha1_base64="gvk0163/DjASSvEOn5Kq+epoyvs=">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</latexit><latexit sha1_base64="gvk0163/DjASSvEOn5Kq+epoyvs=">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</latexit><latexit sha1_base64="kO59cm/W//LWDO0rGNfdeSQGxWA=">AAAB9HicbVDLSsNAFL2prxqr1rWbwSK4KokbXQpuXFawD2hLmUxu2qGTSZi5EUrpyp1bv8Kd+Dd+hP/g9LFJ64ELh3Mu93GiXElLQfDjVfb2Dw6Pqsf+Sc0/PTuv1zo2K4zAtshUZnoRt6ikxjZJUtjLDfI0UtiNpo9Lv/uKxspMv9Asx2HKx1omUnByUmtUbwTNYAW2S8INacAGo/rvIM5EkaImobi1/TDIaTjnhqRQuPAHhcWciykf43x13IJdOylmSWZcaWIrtdTHU2tnaeQ6U04Tu+0txX+9KC3t6xeU3A/nUucFoRbrxUmhGGVs+TmLpUFBauYIF0a6i5mYcMMFuXxKk617bYLxwnfxhNth7JLObTMMmuFzAFW4hCu4gRDu4AGeoAVtEBDDO3x4b96n97WOseJt8ryAErzvP39qlUc=</latexit><latexit sha1_base64="K6Kha57V+VVxgeGVc3hs11SnD78=">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</latexit><latexit sha1_base64="K6Kha57V+VVxgeGVc3hs11SnD78=">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</latexit><latexit sha1_base64="0Lvim4iMEFaRbGy+ivNoSlwUGZQ=">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</latexit><latexit sha1_base64="0Lvim4iMEFaRbGy+ivNoSlwUGZQ=">AAACkHicbVFNb9QwEHXCV1m+tuWCxMV0hbSVuquEC+2hoggOaEWlRWLbinUaOc5k16qdRPYEsUrzD/iD/Agu/AKcbThsy0i2nt6b8YzfJKWSFoPgl+ffuXvv/oOth71Hj588fdbf3jm1RWUEzEShCnOecAtK5jBDiQrOSwNcJwrOkssPrX72HYyVRf4VVyVEmi9ymUnB0VFx/+e3uGaa49LoOgOjHdk0RywFnEtmFbdLSGtWcoOSq4aO6EnUO2Iyx3WR4Kr+2EzYvitQyIf/EmOmKzq5cPce26dwUY+u9NXn+SQaUckQfqDrZuUib4Yne6yUbLqUTmzi/iAYB+ugt0HYgQHpYhr3f7O0EJWGHIWb1c7DoMSobocQCpoeqyyUXFzyBdRrrxr62lEpzQrjTo50zW7kcW3tSicus/2hvam15H+1RG/0m1eYHUS1zMsKIRfXjbNKUSxouwiaSgMC1coBLox0E1Ox5IYLdOvaeLlbQ9Nz/oQ33bgNTt+Mw2AcfgkGx4edU1vkJdklQxKSt+SYfCJTMiOC/PFeeK+8XX/HP/Df+e+vU32vq3lONsKf/AWKisnB</latexit><latexit sha1_base64="gvk0163/DjASSvEOn5Kq+epoyvs=">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</latexit><latexit sha1_base64="gvk0163/DjASSvEOn5Kq+epoyvs=">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</latexit><latexit sha1_base64="gvk0163/DjASSvEOn5Kq+epoyvs=">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</latexit><latexit sha1_base64="gvk0163/DjASSvEOn5Kq+epoyvs=">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</latexit><latexit sha1_base64="kO59cm/W//LWDO0rGNfdeSQGxWA=">AAAB9HicbVDLSsNAFL2prxqr1rWbwSK4KokbXQpuXFawD2hLmUxu2qGTSZi5EUrpyp1bv8Kd+Dd+hP/g9LFJ64ELh3Mu93GiXElLQfDjVfb2Dw6Pqsf+Sc0/PTuv1zo2K4zAtshUZnoRt6ikxjZJUtjLDfI0UtiNpo9Lv/uKxspMv9Asx2HKx1omUnByUmtUbwTNYAW2S8INacAGo/rvIM5EkaImobi1/TDIaTjnhqRQuPAHhcWciykf43x13IJdOylmSWZcaWIrtdTHU2tnaeQ6U04Tu+0txX+9KC3t6xeU3A/nUucFoRbrxUmhGGVs+TmLpUFBauYIF0a6i5mYcMMFuXxKk617bYLxwnfxhNth7JLObTMMmuFzAFW4hCu4gRDu4AGeoAVtEBDDO3x4b96n97WOseJt8ryAErzvP39qlUc=</latexit><latexit sha1_base64="K6Kha57V+VVxgeGVc3hs11SnD78=">AAAChXicbVFNb9QwEHXCV1kKbHtC4mJaIW2l7irhQi9VQXBAKyotEttWrNPIcSa7Vu0ksieIVep/wB/kR3DhF+Bsl8O2jGTr6b0Zz/hNVitpMYp+BeG9+w8ePtp63Huy/fTZ8/7O9pmtGiNgKipVmYuMW1CyhClKVHBRG+A6U3CeXX3o9PPvYKysyq+4rCHRfF7KQgqOnkr7P7+lLdMcF0a3BRjtSeeOWQ44k8wqbheQt6zmBiVXjg7padI7ZrLEVZHgqv3oxuzQFyjkg3+JKdMNHV/6+4AdUrhsh9f6+vNsnAypZAg/0Hezcl66wekBqyWbLKQXXdrfj0bRKuhdEK/BPlnHJO3/ZnklGg0lCj+rncVRjUnbDSEUuB5rLNRcXPE5tCuvHH3tqZwWlfGnRLpiN/K4tnapM5/Z/dDe1jryv1qmN/rNGiyOklaWdYNQipvGRaMoVrRbBM2lAYFq6QEXRvqJqVhwwwX6dW28vF6D63l/4ttu3AVnb0ZxNIq/RGSLvCR7ZEBi8pa8I5/IhEyJIH+CF8GrYC/cDY/Ckxsnw2Bt6S7ZiPD9X8w0yNI=</latexit><latexit sha1_base64="K6Kha57V+VVxgeGVc3hs11SnD78=">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</latexit><latexit sha1_base64="0Lvim4iMEFaRbGy+ivNoSlwUGZQ=">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</latexit><latexit sha1_base64="0Lvim4iMEFaRbGy+ivNoSlwUGZQ=">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</latexit><latexit sha1_base64="gvk0163/DjASSvEOn5Kq+epoyvs=">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</latexit><latexit sha1_base64="gvk0163/DjASSvEOn5Kq+epoyvs=">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</latexit><latexit sha1_base64="gvk0163/DjASSvEOn5Kq+epoyvs=">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</latexit><latexit sha1_base64="gvk0163/DjASSvEOn5Kq+epoyvs=">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</latexit><latexit sha1_base64="0Lvim4iMEFaRbGy+ivNoSlwUGZQ=">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</latexit>

For general statistical angle 𝜃 we have the loop model

Z =ZDJ �(@µJµ) e�|m|L[J]+i✓�[J]

<latexit sha1_base64="Wyt8fh4+ETwFxGOCGyAZYleWfvM=">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</latexit><latexit sha1_base64="DiCTbb9zTh5GHAf3pcmTq2MyODY=">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</latexit><latexit sha1_base64="DiCTbb9zTh5GHAf3pcmTq2MyODY=">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</latexit><latexit sha1_base64="DiCTbb9zTh5GHAf3pcmTq2MyODY=">AAACT3icbVBNbxMxEPWmBdrw0VCOXKxWSEWUaBcJ8SFVqgQHFDgEibQRcRrNeiddq/buyp6tFG3393Dmp3BC3ODUU8/cAG+aS1pGsvX85mme58WFVo7C8EfQWlm9cfPW2nr79p279zY69zcPXF5aiQOZ69wOY3CoVYYDUqRxWFgEE2s8jE/eNP3DU7RO5dknmhU4NnCcqamSQJ6adN5/3hMqI2GAUgm6elv3xK5IUBPsiAIsKdATYUreO/L3Y7GLR9XTM3P2YdQbP1GCUiQQ/VT5Zz3pbIfdcF78OogWYHv/+Zfw/O/wdX/SuRBJLkuDGUkNzo2isKBx1bhKjXVblA4LkCdwjNV81Zo/8lTCp7n1JyM+Z5d0YJybmdgrm5Xc1V5D/rcXmyW/UUnTl+NKZUVJmMlL42mpOeW8yZEnyqIkPfMApFX+x1ymYEGST3tpsvOrpZjUbZ9PdDWN6+DgWTcKu9FHH9Qrdllr7CHbYjssYi/YPnvH+mzAJPvKvrOf7FfwLfgd/GktpK1gAR6wpWqt/wPIQbhJ</latexit><latexit sha1_base64="kO59cm/W//LWDO0rGNfdeSQGxWA=">AAAB9HicbVDLSsNAFL2prxqr1rWbwSK4KokbXQpuXFawD2hLmUxu2qGTSZi5EUrpyp1bv8Kd+Dd+hP/g9LFJ64ELh3Mu93GiXElLQfDjVfb2Dw6Pqsf+Sc0/PTuv1zo2K4zAtshUZnoRt6ikxjZJUtjLDfI0UtiNpo9Lv/uKxspMv9Asx2HKx1omUnByUmtUbwTNYAW2S8INacAGo/rvIM5EkaImobi1/TDIaTjnhqRQuPAHhcWciykf43x13IJdOylmSWZcaWIrtdTHU2tnaeQ6U04Tu+0txX+9KC3t6xeU3A/nUucFoRbrxUmhGGVs+TmLpUFBauYIF0a6i5mYcMMFuXxKk617bYLxwnfxhNth7JLObTMMmuFzAFW4hCu4gRDu4AGeoAVtEBDDO3x4b96n97WOseJt8ryAErzvP39qlUc=</latexit><latexit sha1_base64="Orxe9gaFOMEWQSS5PAjnF7nPijs=">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</latexit><latexit sha1_base64="Orxe9gaFOMEWQSS5PAjnF7nPijs=">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</latexit><latexit sha1_base64="Qiq0WfVOVWMu0+Vj4Dw86Ffrqqk=">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</latexit><latexit sha1_base64="Qiq0WfVOVWMu0+Vj4Dw86Ffrqqk=">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</latexit><latexit sha1_base64="DiCTbb9zTh5GHAf3pcmTq2MyODY=">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</latexit><latexit sha1_base64="DiCTbb9zTh5GHAf3pcmTq2MyODY=">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</latexit><latexit sha1_base64="DiCTbb9zTh5GHAf3pcmTq2MyODY=">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</latexit><latexit sha1_base64="DiCTbb9zTh5GHAf3pcmTq2MyODY=">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</latexit><latexit sha1_base64="Qiq0WfVOVWMu0+Vj4Dw86Ffrqqk=">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</latexit>

Page 14: Duality in Condensed Matter Physics and Field Theory · Duality in Statistical Mechanics • 2D Ising Model: Kramers- Wannier duality • Partition function as a sum over closed domain

!14

• Can we use this to “derive” the web of dualities? Yes!

• First step: We introduce background fields to the boson side of Polyakov’s duality in the unbroken phase

Exact rewriting as loop model coupled to gauge fields

Fractional Spin: Periodicity Lost, 3D Bosonization Regained

LB = |D[a]�|2 �m20|�|2 � |�|4 + 1

4⇡ ada + 12⇡ adA

<latexit sha1_base64="hILbSeSFbGohM/sLvyVLhFybNak=">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</latexit><latexit sha1_base64="HjMjbjNHVz0YR9DhbkFAs6ghPCg=">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</latexit><latexit sha1_base64="HjMjbjNHVz0YR9DhbkFAs6ghPCg=">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</latexit><latexit sha1_base64="HjMjbjNHVz0YR9DhbkFAs6ghPCg=">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</latexit><latexit sha1_base64="kO59cm/W//LWDO0rGNfdeSQGxWA=">AAAB9HicbVDLSsNAFL2prxqr1rWbwSK4KokbXQpuXFawD2hLmUxu2qGTSZi5EUrpyp1bv8Kd+Dd+hP/g9LFJ64ELh3Mu93GiXElLQfDjVfb2Dw6Pqsf+Sc0/PTuv1zo2K4zAtshUZnoRt6ikxjZJUtjLDfI0UtiNpo9Lv/uKxspMv9Asx2HKx1omUnByUmtUbwTNYAW2S8INacAGo/rvIM5EkaImobi1/TDIaTjnhqRQuPAHhcWciykf43x13IJdOylmSWZcaWIrtdTHU2tnaeQ6U04Tu+0txX+9KC3t6xeU3A/nUucFoRbrxUmhGGVs+TmLpUFBauYIF0a6i5mYcMMFuXxKk617bYLxwnfxhNth7JLObTMMmuFzAFW4hCu4gRDu4AGeoAVtEBDDO3x4b96n97WOseJt8ryAErzvP39qlUc=</latexit><latexit sha1_base64="we/ArsJfNZ3PbSpruEGlqxbA1bg=">AAACTnicbZDLSsNAFIZP4q3WW3XrJlgEQSxJEXQjeFu4cFHBVqGp4WQysYOTCzMToaR5Mp9EXLlVfAenbURaPTDwz38Oc+b//JQzqWz71TDn5hcWlyrL1ZXVtfWN2uZqRyaZILRNEp6Iex8l5SymbcUUp/epoBj5nN75Txej/t0zFZIl8a0apLQX4WPMQkZQacurtd0IVZ8gz68L7/xkeNnFnpv22fCheRB59kNz+HMrxeG+GwokuVPkh27KCgzw12lOnDOvVrcb9risv8IpRR3Kanm1LzdISBbRWBGOUnYdO1W9HIVihNOi6maSpkie8JHm49CFtautwAoToU+srLE7NYeRlIPI15OjiHK2NzL/7fnR1L5upsLjXs7iNFM0JpPFYcYtlVgjolbABCWKD7RAIpj+sUX6qIkozX3qZamj9WlQVDUfZ5bGX9FpNhy74dzYUIFt2IE9cOAITuEKWtAGAi/wDh/wabyZhlmZkDSNEukWTJVZ/QbsCLX2</latexit><latexit sha1_base64="we/ArsJfNZ3PbSpruEGlqxbA1bg=">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</latexit><latexit sha1_base64="Qltk4uqIRkqPygqs+jKczgPM/MU=">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</latexit><latexit sha1_base64="tMDV4J1yV+PkpGa1UpTmgYpXIdE=">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</latexit><latexit sha1_base64="HjMjbjNHVz0YR9DhbkFAs6ghPCg=">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</latexit><latexit sha1_base64="HjMjbjNHVz0YR9DhbkFAs6ghPCg=">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</latexit><latexit sha1_base64="HjMjbjNHVz0YR9DhbkFAs6ghPCg=">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</latexit><latexit sha1_base64="HjMjbjNHVz0YR9DhbkFAs6ghPCg=">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</latexit><latexit sha1_base64="tMDV4J1yV+PkpGa1UpTmgYpXIdE=">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</latexit>

Z[A] =ZDJDa �(@µJµ) e�|m|L[J]+iS[J,a,A]

<latexit sha1_base64="CPjsxTJveirAfGTcDa0eLAC6mqc=">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</latexit><latexit sha1_base64="5YjxSwFWYXfDAKqcOyrkQmZ8brc=">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</latexit><latexit sha1_base64="5YjxSwFWYXfDAKqcOyrkQmZ8brc=">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</latexit><latexit sha1_base64="5YjxSwFWYXfDAKqcOyrkQmZ8brc=">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</latexit><latexit sha1_base64="kO59cm/W//LWDO0rGNfdeSQGxWA=">AAAB9HicbVDLSsNAFL2prxqr1rWbwSK4KokbXQpuXFawD2hLmUxu2qGTSZi5EUrpyp1bv8Kd+Dd+hP/g9LFJ64ELh3Mu93GiXElLQfDjVfb2Dw6Pqsf+Sc0/PTuv1zo2K4zAtshUZnoRt6ikxjZJUtjLDfI0UtiNpo9Lv/uKxspMv9Asx2HKx1omUnByUmtUbwTNYAW2S8INacAGo/rvIM5EkaImobi1/TDIaTjnhqRQuPAHhcWciykf43x13IJdOylmSWZcaWIrtdTHU2tnaeQ6U04Tu+0txX+9KC3t6xeU3A/nUucFoRbrxUmhGGVs+TmLpUFBauYIF0a6i5mYcMMFuXxKk617bYLxwnfxhNth7JLObTMMmuFzAFW4hCu4gRDu4AGeoAVtEBDDO3x4b96n97WOseJt8ryAErzvP39qlUc=</latexit><latexit sha1_base64="Kls3leiFpZG6kC0YA1MEN/Drc2A=">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</latexit><latexit sha1_base64="Kls3leiFpZG6kC0YA1MEN/Drc2A=">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</latexit><latexit sha1_base64="SlFdnclzFOGAiX7I4l8ZhfhSdoY=">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</latexit><latexit sha1_base64="SlFdnclzFOGAiX7I4l8ZhfhSdoY=">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</latexit><latexit sha1_base64="5YjxSwFWYXfDAKqcOyrkQmZ8brc=">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</latexit><latexit sha1_base64="5YjxSwFWYXfDAKqcOyrkQmZ8brc=">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</latexit><latexit sha1_base64="5YjxSwFWYXfDAKqcOyrkQmZ8brc=">AAACWXicbVBdaxNBFJ2sXzF+Rfvoy2ARKsawK0irILTogwQfKpo2uLMNd2dvmqEzu8vMXSFs9/f4I/wl4pNPguJv0NmkD6b1wgyHc++dM+ekpVaOwvBrJ7h0+crVa93rvRs3b92+079778AVlZU4loUu7CQFh1rlOCZFGielRTCpxsP05FXbP/yE1qki/0CLEhMDx7maKQnkqWl//DHeS14KlZMwQHMJun7djP7BIAYiQ02wJUqwpEBPhan46Mjfj8QAj+onp+b0bTxKHqv38WgAg72kmfY3w2G4LH4RRGdgc/fZ5/DHn8mL/Wn/t8gKWRnMSWpwLo7CkpK6FZQam56oHJYgT+AY66Xphj/0VMZnhfUnJ75k1+bAOLcwqZ9s3bjzvZb8by81a3pxRbOdpFZ5WRHmciU8qzSngreJ8kxZlKQXHoC0yv+YyzlYkORzX3vZeWtzzJqezyc6n8ZFcPB0GIXD6J0P6jlbVZfdZw/YFovYNttlb9g+GzPJvrDv7Cf71fkWdIJu0FuNBp2znQ22VsHGX0xCudk=</latexit><latexit sha1_base64="5YjxSwFWYXfDAKqcOyrkQmZ8brc=">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</latexit><latexit sha1_base64="SlFdnclzFOGAiX7I4l8ZhfhSdoY=">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</latexit>

S[J, a,A] =Z

d3x

J(a�A) +

14⇡

ada� 14⇡

AdA + . . .

<latexit sha1_base64="EfV7VHOncwSPhT7IetPseGN4oR4=">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</latexit><latexit sha1_base64="EbJ2wzRDRZ2cCybuOwJzY0eKDxA=">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</latexit><latexit sha1_base64="EbJ2wzRDRZ2cCybuOwJzY0eKDxA=">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</latexit><latexit sha1_base64="EbJ2wzRDRZ2cCybuOwJzY0eKDxA=">AAACVHicbVDLSsNAFJ3Gd31VXboJiqCoJVFBXQgWN+KqolWhieVmMmmHTh7M3Igl5D/8EPFTBMGFG935By6ctm6iXhg4nHPmPo6XCK7Qsl5Kxsjo2PjE5FR5emZ2br6ysHil4lRS1qCxiOWNB4oJHrEGchTsJpEMQk+wa6970tev75hUPI4usZcwN4R2xANOATXVqtQvmmdbsFVzjxweoenf7t47ggXYPFuH7drGphNIoJmdZ3tOwnPwYbvI1PzapiP8GJUjebuDbquyalWtQZl/gf0DVo/335Ye6MFjvVX5dPyYpiGLkApQqmlbCboZSORUsLzspIolQLvQZtng3txc05RvBrHUTy89YAs+CJXqhZ52hoAd9Vvrk/9qXliY10wxOHAzHiUpsogOBwepMDE2+2GaPpeMouhpAFRyvbFJO6DzQR15obPSp3WYn5d1PvbvNP6Cq52qbVXtcx3UIRnWJFkmK2Sd2GSfHJNTUicNQskTeSXv5KP0XPoyRoyxodUo/fxZIoUy5r4B9vq3tA==</latexit><latexit sha1_base64="kO59cm/W//LWDO0rGNfdeSQGxWA=">AAAB9HicbVDLSsNAFL2prxqr1rWbwSK4KokbXQpuXFawD2hLmUxu2qGTSZi5EUrpyp1bv8Kd+Dd+hP/g9LFJ64ELh3Mu93GiXElLQfDjVfb2Dw6Pqsf+Sc0/PTuv1zo2K4zAtshUZnoRt6ikxjZJUtjLDfI0UtiNpo9Lv/uKxspMv9Asx2HKx1omUnByUmtUbwTNYAW2S8INacAGo/rvIM5EkaImobi1/TDIaTjnhqRQuPAHhcWciykf43x13IJdOylmSWZcaWIrtdTHU2tnaeQ6U04Tu+0txX+9KC3t6xeU3A/nUucFoRbrxUmhGGVs+TmLpUFBauYIF0a6i5mYcMMFuXxKk617bYLxwnfxhNth7JLObTMMmuFzAFW4hCu4gRDu4AGeoAVtEBDDO3x4b96n97WOseJt8ryAErzvP39qlUc=</latexit><latexit sha1_base64="JCENDJlJDXWtfXgDePlefqIFsxU=">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</latexit><latexit sha1_base64="JCENDJlJDXWtfXgDePlefqIFsxU=">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</latexit><latexit sha1_base64="foVBj+YSuRovTdmN8T+5rfo22rs=">AAACVHicbVDLSsNAFJ3Gd31VXboJFkFRS6KCuhBa3EhXFa0KTSw3k0k7dPJg5kYsIb/lpwiCW935By6c1m6iXhg4nHPmPo6XCK7Qsl5LxtT0zOzc/EJ5cWl5ZbWytn6r4lRS1qaxiOW9B4oJHrE2chTsPpEMQk+wO29wMdLvHplUPI5ucJgwN4RexANOATXVrbSuO8192G+45w6P0PQfjp4cwQLsNHfgoLG75wQSaGbn2bGT8Bx8OCgyDb+x5wg/RuVI3uuj261UrZo1LvMvsCegSibV6lY+HT+macgipAKU6thWgm4GEjkVLC87qWIJ0AH0WDa+Nze3NeWbQSz100uP2YIPQqWGoaedIWBf/dZG5L+aFxbmdVIMTt2MR0mKLKI/g4NUmBibozBNn0tGUQw1ACq53tikfdD5oI680Fnp0/rMz8s6H/t3Gn/B7WHNtmr2lVWtn02SmiebZIvsEJuckDq5JC3SJpQ8kzfyTj5KL6UvY8qY+bEapcmfDVIoY+UbJ9K0Yw==</latexit><latexit sha1_base64="foVBj+YSuRovTdmN8T+5rfo22rs=">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</latexit><latexit sha1_base64="EbJ2wzRDRZ2cCybuOwJzY0eKDxA=">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</latexit><latexit sha1_base64="EbJ2wzRDRZ2cCybuOwJzY0eKDxA=">AAACVHicbVDLSsNAFJ3Gd31VXboJiqCoJVFBXQgWN+KqolWhieVmMmmHTh7M3Igl5D/8EPFTBMGFG935By6ctm6iXhg4nHPmPo6XCK7Qsl5Kxsjo2PjE5FR5emZ2br6ysHil4lRS1qCxiOWNB4oJHrEGchTsJpEMQk+wa6970tev75hUPI4usZcwN4R2xANOATXVqtQvmmdbsFVzjxweoenf7t47ggXYPFuH7drGphNIoJmdZ3tOwnPwYbvI1PzapiP8GJUjebuDbquyalWtQZl/gf0DVo/335Ye6MFjvVX5dPyYpiGLkApQqmlbCboZSORUsLzspIolQLvQZtng3txc05RvBrHUTy89YAs+CJXqhZ52hoAd9Vvrk/9qXliY10wxOHAzHiUpsogOBwepMDE2+2GaPpeMouhpAFRyvbFJO6DzQR15obPSp3WYn5d1PvbvNP6Cq52qbVXtcx3UIRnWJFkmK2Sd2GSfHJNTUicNQskTeSXv5KP0XPoyRoyxodUo/fxZIoUy5r4B9vq3tA==</latexit><latexit sha1_base64="EbJ2wzRDRZ2cCybuOwJzY0eKDxA=">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</latexit><latexit sha1_base64="EbJ2wzRDRZ2cCybuOwJzY0eKDxA=">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</latexit><latexit sha1_base64="foVBj+YSuRovTdmN8T+5rfo22rs=">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</latexit>

Integrating out a results in a term involving the linking number and the spin factor

�⇡ �[J ] +Z

d3x

JA� 1

4⇡AdA

<latexit sha1_base64="bva+fZ2mdxDOsUi5iJ9tw1iD+ZE=">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</latexit><latexit sha1_base64="i4yWxsEM9D9qVeevIM3peeAY7vg=">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</latexit><latexit sha1_base64="i4yWxsEM9D9qVeevIM3peeAY7vg=">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</latexit><latexit sha1_base64="i4yWxsEM9D9qVeevIM3peeAY7vg=">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</latexit><latexit sha1_base64="kO59cm/W//LWDO0rGNfdeSQGxWA=">AAAB9HicbVDLSsNAFL2prxqr1rWbwSK4KokbXQpuXFawD2hLmUxu2qGTSZi5EUrpyp1bv8Kd+Dd+hP/g9LFJ64ELh3Mu93GiXElLQfDjVfb2Dw6Pqsf+Sc0/PTuv1zo2K4zAtshUZnoRt6ikxjZJUtjLDfI0UtiNpo9Lv/uKxspMv9Asx2HKx1omUnByUmtUbwTNYAW2S8INacAGo/rvIM5EkaImobi1/TDIaTjnhqRQuPAHhcWciykf43x13IJdOylmSWZcaWIrtdTHU2tnaeQ6U04Tu+0txX+9KC3t6xeU3A/nUucFoRbrxUmhGGVs+TmLpUFBauYIF0a6i5mYcMMFuXxKk617bYLxwnfxhNth7JLObTMMmuFzAFW4hCu4gRDu4AGeoAVtEBDDO3x4b96n97WOseJt8ryAErzvP39qlUc=</latexit><latexit sha1_base64="YsfvLkgAkF8VIw/S0SlEGS/QBXI=">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</latexit><latexit sha1_base64="YsfvLkgAkF8VIw/S0SlEGS/QBXI=">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</latexit><latexit sha1_base64="H1D6l5HTNtfw1BbgqkurC1C7H6g=">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</latexit><latexit sha1_base64="H1D6l5HTNtfw1BbgqkurC1C7H6g=">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</latexit><latexit sha1_base64="i4yWxsEM9D9qVeevIM3peeAY7vg=">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</latexit><latexit sha1_base64="i4yWxsEM9D9qVeevIM3peeAY7vg=">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</latexit><latexit sha1_base64="i4yWxsEM9D9qVeevIM3peeAY7vg=">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</latexit><latexit sha1_base64="i4yWxsEM9D9qVeevIM3peeAY7vg=">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</latexit><latexit sha1_base64="H1D6l5HTNtfw1BbgqkurC1C7H6g=">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</latexit>

Page 15: Duality in Condensed Matter Physics and Field Theory · Duality in Statistical Mechanics • 2D Ising Model: Kramers- Wannier duality • Partition function as a sum over closed domain

!15

LF = (i /D[A]�M) � 18⇡

AdA<latexit sha1_base64="2/wHMU+syZJylZ+AYhVs772int4=">AAACRXicbZDLSsNAFIZPvNZ6q4orN8Ei1EVL4sa6EFqU4kKhgr1AU8pkMmmHTi7MTIQS8jy+hjtXbhUfojtxq5O2m7YeGPj5z5n553x2yKiQhvGlrayurW9sZray2zu7e/u5g8OmCCKOSQMHLOBtGwnCqE8akkpG2iEnyLMZadnDm7TfeiZc0MB/kqOQdD3U96lLMZLK6uWqlofkACMW3ye92rVlIx5bdUGTArUEQ2JAnPg26VS7xYfz1C9aLkc4NpO4bIU0qTrVXi5vlIxJ6cvCnIl8xawdvwJAvZcbW06AI4/4EqsE0TGNUHZjxCXFjCRZKxIkRHiI+iSeLJjoZ8pydDfg6vhSn7hzc8gTYuTZajJdRyz2UvPfnu3N5XUi6Za7MfXDSBIfT4PdiOky0FN6ukM5wZKNlECYU/VjHQ+QIiIV47mXZ/CSrOJjLtJYFs2LkmmUzEcF6gqmlYETOIUCmHAJFbiDOjQAwwu8wwd8am/aWPvWfqajK9rszhHMlfb7B3ZCtBU=</latexit><latexit sha1_base64="WoH4dG7fn95Lz7+OXy2gYAbF11A=">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</latexit><latexit sha1_base64="WoH4dG7fn95Lz7+OXy2gYAbF11A=">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</latexit><latexit sha1_base64="WoH4dG7fn95Lz7+OXy2gYAbF11A=">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</latexit><latexit sha1_base64="kO59cm/W//LWDO0rGNfdeSQGxWA=">AAAB9HicbVDLSsNAFL2prxqr1rWbwSK4KokbXQpuXFawD2hLmUxu2qGTSZi5EUrpyp1bv8Kd+Dd+hP/g9LFJ64ELh3Mu93GiXElLQfDjVfb2Dw6Pqsf+Sc0/PTuv1zo2K4zAtshUZnoRt6ikxjZJUtjLDfI0UtiNpo9Lv/uKxspMv9Asx2HKx1omUnByUmtUbwTNYAW2S8INacAGo/rvIM5EkaImobi1/TDIaTjnhqRQuPAHhcWciykf43x13IJdOylmSWZcaWIrtdTHU2tnaeQ6U04Tu+0txX+9KC3t6xeU3A/nUucFoRbrxUmhGGVs+TmLpUFBauYIF0a6i5mYcMMFuXxKk617bYLxwnfxhNth7JLObTMMmuFzAFW4hCu4gRDu4AGeoAVtEBDDO3x4b96n97WOseJt8ryAErzvP39qlUc=</latexit><latexit sha1_base64="cvKv5pbdOjuDirfUo+Bl02UnqUI=">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</latexit><latexit sha1_base64="cvKv5pbdOjuDirfUo+Bl02UnqUI=">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</latexit><latexit sha1_base64="UC/C0GWeDI3jX9lUmP1hXZdX7F8=">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</latexit><latexit sha1_base64="UC/C0GWeDI3jX9lUmP1hXZdX7F8=">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</latexit><latexit sha1_base64="WoH4dG7fn95Lz7+OXy2gYAbF11A=">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</latexit><latexit sha1_base64="WoH4dG7fn95Lz7+OXy2gYAbF11A=">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</latexit><latexit sha1_base64="WoH4dG7fn95Lz7+OXy2gYAbF11A=">AAACRXicbZDLSsNAFIYn9VbrrSqu3ASLUBctiRvrQmhQiguFCvYCTSiTyaQdOrkwMxFKyJu4FX0Fn8CVW8WH6E7c6qTtpq0HBn7+c2b+OZ8dUsKFpn0pmaXlldW17HpuY3Nreye/u9fkQcQQbqCABqxtQ44p8XFDEEFxO2QYejbFLXtwmfZbD5hxEvj3Yhhiy4M9n7gEQSGtbt4wPSj6CNL4JunWLkwbstisc5IUickp5H3sxFdJx7BKtyepXzJdBlGsJ3HFDEliOEY3X9DK2rjURaFPRaGq1w5en14e6938yHQCFHnYF0gm8I6uhcKKIRMEUZzkzIjjEKIB7OF4vGCiHkvLUd2AyeMLdezOzEGP86Fny8l0HT7fS81/e7Y3k9eJhFuxYuKHkcA+mgS7EVVFoKb0VIcwjAQdSgERI/LHKupDSURIxjMvT+ElOclHn6exKJqnZV0r63cS1DmYVBYcgiNQBDo4A1VwDeqgARB4Bu/gA3wqb8pI+VZ+JqMZZXpnH8yU8vsHLSy19w==</latexit><latexit sha1_base64="WoH4dG7fn95Lz7+OXy2gYAbF11A=">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</latexit><latexit sha1_base64="UC/C0GWeDI3jX9lUmP1hXZdX7F8=">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</latexit>

with M<0

Zfermion[A;M < 0] e�i CS[A]/2 =ZDJ �(@µJµ) e�|m|L[J]+iSfermion[J,A;M<0]e�i CS[A]/2

<latexit sha1_base64="PWqXNbeYQTo9yxO44PSxw17o42g=">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</latexit><latexit sha1_base64="tfA9oVZMf4sB7ugMNC7uu26kf0A=">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</latexit><latexit sha1_base64="tfA9oVZMf4sB7ugMNC7uu26kf0A=">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</latexit><latexit sha1_base64="tfA9oVZMf4sB7ugMNC7uu26kf0A=">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</latexit><latexit sha1_base64="kO59cm/W//LWDO0rGNfdeSQGxWA=">AAAB9HicbVDLSsNAFL2prxqr1rWbwSK4KokbXQpuXFawD2hLmUxu2qGTSZi5EUrpyp1bv8Kd+Dd+hP/g9LFJ64ELh3Mu93GiXElLQfDjVfb2Dw6Pqsf+Sc0/PTuv1zo2K4zAtshUZnoRt6ikxjZJUtjLDfI0UtiNpo9Lv/uKxspMv9Asx2HKx1omUnByUmtUbwTNYAW2S8INacAGo/rvIM5EkaImobi1/TDIaTjnhqRQuPAHhcWciykf43x13IJdOylmSWZcaWIrtdTHU2tnaeQ6U04Tu+0txX+9KC3t6xeU3A/nUucFoRbrxUmhGGVs+TmLpUFBauYIF0a6i5mYcMMFuXxKk617bYLxwnfxhNth7JLObTMMmuFzAFW4hCu4gRDu4AGeoAVtEBDDO3x4b96n97WOseJt8ryAErzvP39qlUc=</latexit><latexit sha1_base64="Mw7RAdetF6nFyOQ5su9nIc3n2lQ=">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</latexit><latexit sha1_base64="Mw7RAdetF6nFyOQ5su9nIc3n2lQ=">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</latexit><latexit sha1_base64="oiGQyIAkNOXeGAFMfVRdDFizj5k=">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</latexit><latexit sha1_base64="oiGQyIAkNOXeGAFMfVRdDFizj5k=">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</latexit><latexit sha1_base64="tfA9oVZMf4sB7ugMNC7uu26kf0A=">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</latexit><latexit sha1_base64="tfA9oVZMf4sB7ugMNC7uu26kf0A=">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</latexit><latexit sha1_base64="tfA9oVZMf4sB7ugMNC7uu26kf0A=">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</latexit><latexit sha1_base64="tfA9oVZMf4sB7ugMNC7uu26kf0A=">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</latexit><latexit sha1_base64="oiGQyIAkNOXeGAFMfVRdDFizj5k=">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</latexit>

Loop model representation

Sfermion[J,A;M < 0] =Z

d3x

✓JA� 1

8⇡AdA

◆� ⇡�[J ]

<latexit sha1_base64="LLHDwE5TTf3cMPYeN4PADuk60N0=">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</latexit><latexit sha1_base64="NISuBviYLWUtHMjp7VIQUdrB1MA=">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</latexit><latexit sha1_base64="NISuBviYLWUtHMjp7VIQUdrB1MA=">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</latexit><latexit sha1_base64="NISuBviYLWUtHMjp7VIQUdrB1MA=">AAACW3icbVBdSxtBFJ2s/bCprWmlT74MlYIFDbvtgylVMPhSAkJKGyNktmF29m52cGZ3mblbDMv+D3+Ir/6QPhSEvup/cJL4Eu2FgcM5d+6950SFkhZ9/0/DW3ny9Nnz1RfNl2uvXq+33rw9sXlpBAxErnJzGnELSmYwQIkKTgsDXEcKhtHZ0Uwf/gZjZZ79xGkBoeaTTCZScHTUuDX8Ma6Y5pgaXSVgtCPretTb6X493vfDAyYzpPGvz+dshylIcLvX3WWJ4aIK6qrDClnTbtxlRk5S/Eh3HcH6qRz1wnFry2/786KPQXAPtg73rjcuROeyP27dsjgXpYYMheLWjgK/wLDiBqVQUDdZaaHg4oxPoJrbrukHR8U0yY177sw5u9THtbVTHbnOmUP7UJuR/9UivbRvVGLSCSuZFSVCJhaLk1JRzOksUxpLAwLV1AEujHQXU5FylxK65JcmW2cthbhuunyCh2k8Bief2oHfDr67oL6QRa2STfKebJOA7JFD8o30yYAIckX+kRty2/jrrXhNb23R6jXu/2yQpfLe3QHyxrlt</latexit><latexit sha1_base64="kO59cm/W//LWDO0rGNfdeSQGxWA=">AAAB9HicbVDLSsNAFL2prxqr1rWbwSK4KokbXQpuXFawD2hLmUxu2qGTSZi5EUrpyp1bv8Kd+Dd+hP/g9LFJ64ELh3Mu93GiXElLQfDjVfb2Dw6Pqsf+Sc0/PTuv1zo2K4zAtshUZnoRt6ikxjZJUtjLDfI0UtiNpo9Lv/uKxspMv9Asx2HKx1omUnByUmtUbwTNYAW2S8INacAGo/rvIM5EkaImobi1/TDIaTjnhqRQuPAHhcWciykf43x13IJdOylmSWZcaWIrtdTHU2tnaeQ6U04Tu+0txX+9KC3t6xeU3A/nUucFoRbrxUmhGGVs+TmLpUFBauYIF0a6i5mYcMMFuXxKk617bYLxwnfxhNth7JLObTMMmuFzAFW4hCu4gRDu4AGeoAVtEBDDO3x4b96n97WOseJt8ryAErzvP39qlUc=</latexit><latexit sha1_base64="rK0plCabwUyllqSHnIPIIUbxFXA=">AAACUHicbZDPattAEMbHSv+krts6vfayJBRSSIzUHhpoAjW9FEPBoXUc8KpmtRpZS3YlsTsKNUKP1gfpoadem3fo2vHFSQcWPr4Zdub7JZVWjsLwVyfYefDw0ePdJ92nvWfPX/T3eheurK3EiSx1aS8T4VCrAiekSONlZVGYROM0ufq06k+v0TpVFt9oWWFsxKJQmZKCvDXvT7/OG24E5dY0GVrjzbadjY6GH76chvEZVwWx9Pu7H/yIa8zocDQ85pkVsona5oRXqmXDdMitWuT0hh17g49zNRvF8/5BOAjXxe6LaCMOYFPjef+Gp6WsDRYktXBuFoUVxY2wpKTGtstrh5WQV2KBzTp2y157K2VZaf3zZ67drTlhnFuaxE+uErq7vZX5315itvbNaspO4kYVVU1YyNvFWa0ZlWzFlKXKoiS99EJIq/zFTObCUyJPfutn56PlmLZdzye6S+O+uHg7iMJBdB7CLryCfTiECN7DR/gMY5iAhJ/wB/7CTed3sBN0b0kGnQ3Sl7BVQe8fV8S1Yg==</latexit><latexit sha1_base64="rK0plCabwUyllqSHnIPIIUbxFXA=">AAACUHicbZDPattAEMbHSv+krts6vfayJBRSSIzUHhpoAjW9FEPBoXUc8KpmtRpZS3YlsTsKNUKP1gfpoadem3fo2vHFSQcWPr4Zdub7JZVWjsLwVyfYefDw0ePdJ92nvWfPX/T3eheurK3EiSx1aS8T4VCrAiekSONlZVGYROM0ufq06k+v0TpVFt9oWWFsxKJQmZKCvDXvT7/OG24E5dY0GVrjzbadjY6GH76chvEZVwWx9Pu7H/yIa8zocDQ85pkVsona5oRXqmXDdMitWuT0hh17g49zNRvF8/5BOAjXxe6LaCMOYFPjef+Gp6WsDRYktXBuFoUVxY2wpKTGtstrh5WQV2KBzTp2y157K2VZaf3zZ67drTlhnFuaxE+uErq7vZX5315itvbNaspO4kYVVU1YyNvFWa0ZlWzFlKXKoiS99EJIq/zFTObCUyJPfutn56PlmLZdzye6S+O+uHg7iMJBdB7CLryCfTiECN7DR/gMY5iAhJ/wB/7CTed3sBN0b0kGnQ3Sl7BVQe8fV8S1Yg==</latexit><latexit sha1_base64="sqG3pu0Xkql/zCANqSS06A2JO4c=">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</latexit><latexit sha1_base64="sqG3pu0Xkql/zCANqSS06A2JO4c=">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</latexit><latexit sha1_base64="NISuBviYLWUtHMjp7VIQUdrB1MA=">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</latexit><latexit sha1_base64="NISuBviYLWUtHMjp7VIQUdrB1MA=">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</latexit><latexit sha1_base64="NISuBviYLWUtHMjp7VIQUdrB1MA=">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</latexit><latexit sha1_base64="NISuBviYLWUtHMjp7VIQUdrB1MA=">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</latexit><latexit sha1_base64="sqG3pu0Xkql/zCANqSS06A2JO4c=">AAACW3icbVDRShtBFJ2sWm1q22jpU1+GBkFBw259UNGCoS8lUIi0MUJmDbOzd7ODM7vLzF0xLPtpfogPfepr/YdOYl6ivTBwOOfOvfecqFDSou8/NLyV1bVX6xuvm282375739ravrR5aQQMRK5ycxVxC0pmMECJCq4KA1xHCobRzbeZPrwFY2We/cJpAaHmk0wmUnB01Lg1/DmumOaYGl0lYLQj63rU2++e/jjzw69MZkjj68M7ts8UJLjb6x6wxHBRBXV1zApZ027cZUZOUtyjB45g/VSOeuG41fY7/rzoSxAsQJssqj9uPbI4F6WGDIXi1o4Cv8Cw4galUFA3WWmh4OKGT6Ca267pjqNimuTGPXfmnF3q49raqY5c58yhfa7NyP9qkV7aNyoxOQ4rmRUlQiaeFielopjTWaY0lgYEqqkDXBjpLqYi5S4ldMkvTbbOWgpx3XT5BM/TeAkuv3QCvxNc+O3zk0VSG+QT+Ux2SUCOyDn5TvpkQAS5J3/IX/LY+O2teE1v86nVayz+fCBL5X38ByOethw=</latexit>

The bosonization identity in the phase with broken time reversal, M>0, is obtained by a particle vortex duality in the bosonic theory

• In the fermionic theory, changing the sign of the mass term drives a transition from a trivial state with 𝜎xy =0 to a topological state with 𝜎xy=e^2/h

• This the integer QH transition• In the bosonic theory this is a transition between the unbroken state to the

broken state

Page 16: Duality in Condensed Matter Physics and Field Theory · Duality in Statistical Mechanics • 2D Ising Model: Kramers- Wannier duality • Partition function as a sum over closed domain

Loop Models: Tools for Deriving Dualities

!16

1. Start with a proposed duality and write down boson loop models for each theory using Polyakov’s duality.

2. Use path integral manipulations to equate the two loop model partition functions.

3. Match both sides of the critical point using bosonic particle-vortex duality. Relates superfluid of particles to an insulator of vortices.

4. The dualities are IR identities

5. In the bosonic theories the short-distance repulsion between loops become the 𝜙4 coupling, which in the massless limit flow in the IR into the WF fixed point

Page 17: Duality in Condensed Matter Physics and Field Theory · Duality in Statistical Mechanics • 2D Ising Model: Kramers- Wannier duality • Partition function as a sum over closed domain

Example: Fermion particle-vortex duality

!17

• Case for opposite mass signs (QH phase) follows from the same logic

• Current mapping also natural upon integrating out b:

Use loop models to derive the duality between free Dirac fermion and QED3 with (quantized) Chern-Simons terms

Zd3xJµA

µ + ⇡�[J ] �⇡�[J ] +

Zd3x

Jµa

µ � 1

2⇡adb+

2

4⇡bdb� 1

2⇡bdA

Integrate out a, b

�µ ! 1

4⇡✏µ⌫�@

⌫a�

i /D[A] � 18⇡ AdA$ i /D[a] + 1

8⇡ ada� 12⇡ adb + 2

4⇡ bdb� 12⇡ bdA

<latexit sha1_base64="3RzxxGEgKE5wxCo0uYSvDhylCXI=">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</latexit><latexit sha1_base64="YTrqZHspCP3Cv2b7SiyBRV3N8qM=">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</latexit><latexit sha1_base64="YTrqZHspCP3Cv2b7SiyBRV3N8qM=">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</latexit><latexit sha1_base64="YTrqZHspCP3Cv2b7SiyBRV3N8qM=">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</latexit><latexit sha1_base64="kO59cm/W//LWDO0rGNfdeSQGxWA=">AAAB9HicbVDLSsNAFL2prxqr1rWbwSK4KokbXQpuXFawD2hLmUxu2qGTSZi5EUrpyp1bv8Kd+Dd+hP/g9LFJ64ELh3Mu93GiXElLQfDjVfb2Dw6Pqsf+Sc0/PTuv1zo2K4zAtshUZnoRt6ikxjZJUtjLDfI0UtiNpo9Lv/uKxspMv9Asx2HKx1omUnByUmtUbwTNYAW2S8INacAGo/rvIM5EkaImobi1/TDIaTjnhqRQuPAHhcWciykf43x13IJdOylmSWZcaWIrtdTHU2tnaeQ6U04Tu+0txX+9KC3t6xeU3A/nUucFoRbrxUmhGGVs+TmLpUFBauYIF0a6i5mYcMMFuXxKk617bYLxwnfxhNth7JLObTMMmuFzAFW4hCu4gRDu4AGeoAVtEBDDO3x4b96n97WOseJt8ryAErzvP39qlUc=</latexit><latexit sha1_base64="17ycB8zRvHs0vkak+Rl5i3yfpjE=">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</latexit><latexit sha1_base64="17ycB8zRvHs0vkak+Rl5i3yfpjE=">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</latexit><latexit sha1_base64="YfP3vBFDPO2ZLH+KVueCknUOwcE=">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</latexit><latexit sha1_base64="TNbpz7gO0VgFm30lvaD0qYR3lb0=">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</latexit><latexit sha1_base64="YTrqZHspCP3Cv2b7SiyBRV3N8qM=">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</latexit><latexit sha1_base64="YTrqZHspCP3Cv2b7SiyBRV3N8qM=">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</latexit><latexit sha1_base64="YTrqZHspCP3Cv2b7SiyBRV3N8qM=">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</latexit><latexit sha1_base64="YTrqZHspCP3Cv2b7SiyBRV3N8qM=">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</latexit><latexit sha1_base64="TNbpz7gO0VgFm30lvaD0qYR3lb0=">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</latexit>

�M 0 , M 0 > 0<latexit sha1_base64="2awZGIlUH8pyqdPHTzojSEocQjg=">AAACJHicbVC7TsMwFL0pr1JeBQYGFosKlQGqhAWmqhILS6Ui0YfUVJXjOK2F88B2kKooK5/CxApfwYYYWPgD+g24jyUtR7J1dM69vtfHiTiTyjS/jdzK6tr6Rn6zsLW9s7tX3D9oyTAWhDZJyEPRcbCknAW0qZjitBMJin2H07bzcDPx209USBYG92oU0Z6PBwHzGMFKS/0iuqiXUWI7WCA7kiyd3ufIfoyxi+rlqtkvlsyKOQVaJtaclGpH4zECgEa/+Gu7IYl9GijCsZRdy4xUL8FCMcJpWrBjSSNMHvCAJtP9U3SqJRd5odAnUGiqZuqwL+XId3Slj9VQLnoT8V/P8TPzurHyrnsJC6JY0YDMBnsxRypEk3CQywQlio80wUQwvTEiQywwUTrCzMtSf21I3bSg87EW01gmrcuKZVasO6tUq8IMeTiGEzgDC66gBrfQgCYQeIZXeIN348X4MD6Nr1lpzpj3HEIGxs8fjzSlkA==</latexit><latexit sha1_base64="FD3a24PjZwoNDm9mkCrUc5MnjUk=">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</latexit><latexit sha1_base64="FD3a24PjZwoNDm9mkCrUc5MnjUk=">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</latexit><latexit sha1_base64="FD3a24PjZwoNDm9mkCrUc5MnjUk=">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</latexit><latexit sha1_base64="kO59cm/W//LWDO0rGNfdeSQGxWA=">AAAB9HicbVDLSsNAFL2prxqr1rWbwSK4KokbXQpuXFawD2hLmUxu2qGTSZi5EUrpyp1bv8Kd+Dd+hP/g9LFJ64ELh3Mu93GiXElLQfDjVfb2Dw6Pqsf+Sc0/PTuv1zo2K4zAtshUZnoRt6ikxjZJUtjLDfI0UtiNpo9Lv/uKxspMv9Asx2HKx1omUnByUmtUbwTNYAW2S8INacAGo/rvIM5EkaImobi1/TDIaTjnhqRQuPAHhcWciykf43x13IJdOylmSWZcaWIrtdTHU2tnaeQ6U04Tu+0txX+9KC3t6xeU3A/nUucFoRbrxUmhGGVs+TmLpUFBauYIF0a6i5mYcMMFuXxKk617bYLxwnfxhNth7JLObTMMmuFzAFW4hCu4gRDu4AGeoAVtEBDDO3x4b96n97WOseJt8ryAErzvP39qlUc=</latexit><latexit sha1_base64="KyJXSjqnWQDHolm4ZvwiW1QUWcc=">AAACGXicbZDLSgMxFIbPeK216ujWTbBIXWjJuNGVCG7cFCrYC3RKyWQybWjmYpIRyjBbH8WVW30Kd+LCh/AdTC+baT2Q8PP/h5yTz0sEVxrjH2ttfWNza7u0U96t7O0f2IeVtopTSVmLxiKWXY8oJnjEWpprwbqJZCT0BOt447tp3nlmUvE4etSThPVDMox4wCnRxhrY6KJRQ5nrEYncRPF8dp8j9yklPmrUbvDAruI6nhVaFc5CVGFRzYH96/oxTUMWaSqIUj0HJ7qfEak5FSwvu6liCaFjMmTZbP8cnRrLR0EszYk0mrmFPhIqNQk90xkSPVLL2dT8N/PCwrxeqoPrfsajJNUsovPBQSqQjtEUDvK5ZFSLiRGESm42RnREJKHaICy8rMzXRszPy4aPs0xjVbQv6w6uOw8YSnAMJ3AGDlzBLdxDE1pA4QXe4B0+rFfr0/qak1yzFkiPoFDW9x8sAqG1</latexit><latexit sha1_base64="KyJXSjqnWQDHolm4ZvwiW1QUWcc=">AAACGXicbZDLSgMxFIbPeK216ujWTbBIXWjJuNGVCG7cFCrYC3RKyWQybWjmYpIRyjBbH8WVW30Kd+LCh/AdTC+baT2Q8PP/h5yTz0sEVxrjH2ttfWNza7u0U96t7O0f2IeVtopTSVmLxiKWXY8oJnjEWpprwbqJZCT0BOt447tp3nlmUvE4etSThPVDMox4wCnRxhrY6KJRQ5nrEYncRPF8dp8j9yklPmrUbvDAruI6nhVaFc5CVGFRzYH96/oxTUMWaSqIUj0HJ7qfEak5FSwvu6liCaFjMmTZbP8cnRrLR0EszYk0mrmFPhIqNQk90xkSPVLL2dT8N/PCwrxeqoPrfsajJNUsovPBQSqQjtEUDvK5ZFSLiRGESm42RnREJKHaICy8rMzXRszPy4aPs0xjVbQv6w6uOw8YSnAMJ3AGDlzBLdxDE1pA4QXe4B0+rFfr0/qak1yzFkiPoFDW9x8sAqG1</latexit><latexit sha1_base64="5RH89clWZnwuj9KdJ3stwsuDVW0=">AAACJHicbVC7TsMwFHXKq5RXgJHFokJlgMphgamqxMJSqUj0ITVV5ThOa9V5YDtIVZSVT2Fiha9gQwws/AH/gJtmScuRbB2dc6/v9XEizqRC6Nsora1vbG6Vtys7u3v7B+bhUVeGsSC0Q0Ieir6DJeUsoB3FFKf9SFDsO5z2nOnt3O89USFZGDyoWUSHPh4HzGMEKy2NTHjZqsHEdrCAdiRZmt0X0H6MsQtbtQYamVVURxngKrFyUgU52iPz13ZDEvs0UIRjKQcWitQwwUIxwmlasWNJI0ymeEyTbP8UnmnJhV4o9AkUzNRCHfalnPmOrvSxmshlby7+6zl+Yd4gVt7NMGFBFCsakMVgL+ZQhXAeDnSZoETxmSaYCKY3hmSCBSZKR1h4WeqvTaibVnQ+1nIaq6R7VbdQ3bpH1WYjT6oMTsApOAcWuAZNcAfaoAMIeAav4A28Gy/Gh/FpfC1KS0becwwKMH7+ACxfozw=</latexit><latexit sha1_base64="8e5MNiBOf+fAvEKF57BRljdatB8=">AAACJHicbVC7TsMwFL0pr1JeAUYWiwqVAaqEBaaqEgtLpSLRh9RUleM4rVXnge0gVVVWPoWJFb6CDTGw8Af8A27aJS1HsnV0zr2+18eNOZPKsr6Nwtr6xuZWcbu0s7u3f2AeHrVllAhCWyTikei6WFLOQtpSTHHajQXFgctpxx3fzvzOExWSReGDmsS0H+BhyHxGsNLSwESXjQqaOi4WyIklS7P7AjmPCfZQo1KzBmbZqloZ0CqxF6QMCzQH5q/jRSQJaKgIx1L2bCtW/SkWihFO05KTSBpjMsZDOs32T9GZljzkR0KfUKFMzdXhQMpJ4OrKAKuRXPZm4r+eG+Tm9RLl3/SnLIwTRUMyH+wnHKkIzcJBHhOUKD7RBBPB9MaIjLDAROkIcy9L/bUR9dKSzsdeTmOVtK+qtlW17+1yvbZIqggncArnYMM11OEOmtACAs/wCm/wbrwYH8an8TUvLRiLnmPIwfj5Ayyvoz0=</latexit><latexit sha1_base64="FD3a24PjZwoNDm9mkCrUc5MnjUk=">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</latexit><latexit sha1_base64="FD3a24PjZwoNDm9mkCrUc5MnjUk=">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</latexit><latexit sha1_base64="FD3a24PjZwoNDm9mkCrUc5MnjUk=">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</latexit><latexit sha1_base64="FD3a24PjZwoNDm9mkCrUc5MnjUk=">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</latexit><latexit sha1_base64="8e5MNiBOf+fAvEKF57BRljdatB8=">AAACJHicbVC7TsMwFL0pr1JeAUYWiwqVAaqEBaaqEgtLpSLRh9RUleM4rVXnge0gVVVWPoWJFb6CDTGw8Af8A27aJS1HsnV0zr2+18eNOZPKsr6Nwtr6xuZWcbu0s7u3f2AeHrVllAhCWyTikei6WFLOQtpSTHHajQXFgctpxx3fzvzOExWSReGDmsS0H+BhyHxGsNLSwESXjQqaOi4WyIklS7P7AjmPCfZQo1KzBmbZqloZ0CqxF6QMCzQH5q/jRSQJaKgIx1L2bCtW/SkWihFO05KTSBpjMsZDOs32T9GZljzkR0KfUKFMzdXhQMpJ4OrKAKuRXPZm4r+eG+Tm9RLl3/SnLIwTRUMyH+wnHKkIzcJBHhOUKD7RBBPB9MaIjLDAROkIcy9L/bUR9dKSzsdeTmOVtK+qtlW17+1yvbZIqggncArnYMM11OEOmtACAs/wCm/wbrwYH8an8TUvLRiLnmPIwfj5Ayyvoz0=</latexit>�M , M < 0

<latexit sha1_base64="G+FyB81MOKZaTd29FLABijV2+CI=">AAACInicbVDLSgMxFL1TX7W+Rl24ECRYBBdaZtzoQqTgxk2hgn1Ap5RMJm1DMw+TjFDK7PwUV271K9yJK8FfsN9gOu1mWg8kHM65N/fmuBFnUlnWt5FbWl5ZXcuvFzY2t7Z3zN29ugxjQWiNhDwUTRdLyllAa4opTpuRoNh3OW24g9uJ33iiQrIweFDDiLZ93AtYlxGstNQxj84raOS4WCCnKlmS3mfIeYyxhyrXVscsWiUrBVok9owUywfjMQKAasf8dbyQxD4NFOFYypZtRao9wkIxwmlScGJJI0wGuEdH6fYJOtGSh7qh0CdQKFUzddiXcui7utLHqi/nvYn4r+f6mXmtWHWv2iMWRLGiAZkO7sYcqRBNokEeE5QoPtQEE8H0xoj0scBE6QAzL0v9tT71koLOx55PY5HUL0q2VbLv7WL5BqbIwyEcwynYcAlluIMq1IDAM7zCG7wbL8aH8Wl8TUtzxqxnHzIwfv4ASFuk7A==</latexit><latexit sha1_base64="m84XkHYrK6cnsqiOw0elZhGkyaU=">AAACInicbVC7SgNBFJ2NrxgfWbWwEGQwCBYadm20CBKwsQlEMA/IhjA7O5sMmX04MyuEZTs/xco2foWdWAm2luYbnGzSbOKBGQ7n3Dv3zrFDRoU0jC8tt7K6tr6R3yxsbe/sFvW9/aYIIo5JAwcs4G0bCcKoTxqSSkbaISfIsxlp2cPbqd96IlzQwH+Qo5B0PdT3qUsxkkrq6ccXNRhbNuLQqguapPc5tB4j5MBaxejpJaNspIDLxJyTUvVwMilWxj/1nv5rOQGOPOJLzJAQHdMIZTdGXFLMSFKwIkFChIeoT+J0+wSeKsmBbsDV8SVM1Uwd8oQYebaq9JAciEVvKv7r2V5mXieS7nU3pn4YSeLj2WA3YlAGcBoNdCgnWLKRIghzqjaGeIA4wlIFmHlZqK8NiJMUVD7mYhrLpHlZNo2yeW+Wqjdghjw4AifgDJjgClTBHaiDBsDgGbyCMXjTXrR37UP7nJXmtHnPAchA+/4D0cGmrw==</latexit><latexit sha1_base64="m84XkHYrK6cnsqiOw0elZhGkyaU=">AAACInicbVC7SgNBFJ2NrxgfWbWwEGQwCBYadm20CBKwsQlEMA/IhjA7O5sMmX04MyuEZTs/xco2foWdWAm2luYbnGzSbOKBGQ7n3Dv3zrFDRoU0jC8tt7K6tr6R3yxsbe/sFvW9/aYIIo5JAwcs4G0bCcKoTxqSSkbaISfIsxlp2cPbqd96IlzQwH+Qo5B0PdT3qUsxkkrq6ccXNRhbNuLQqguapPc5tB4j5MBaxejpJaNspIDLxJyTUvVwMilWxj/1nv5rOQGOPOJLzJAQHdMIZTdGXFLMSFKwIkFChIeoT+J0+wSeKsmBbsDV8SVM1Uwd8oQYebaq9JAciEVvKv7r2V5mXieS7nU3pn4YSeLj2WA3YlAGcBoNdCgnWLKRIghzqjaGeIA4wlIFmHlZqK8NiJMUVD7mYhrLpHlZNo2yeW+Wqjdghjw4AifgDJjgClTBHaiDBsDgGbyCMXjTXrR37UP7nJXmtHnPAchA+/4D0cGmrw==</latexit><latexit sha1_base64="m84XkHYrK6cnsqiOw0elZhGkyaU=">AAACInicbVC7SgNBFJ2NrxgfWbWwEGQwCBYadm20CBKwsQlEMA/IhjA7O5sMmX04MyuEZTs/xco2foWdWAm2luYbnGzSbOKBGQ7n3Dv3zrFDRoU0jC8tt7K6tr6R3yxsbe/sFvW9/aYIIo5JAwcs4G0bCcKoTxqSSkbaISfIsxlp2cPbqd96IlzQwH+Qo5B0PdT3qUsxkkrq6ccXNRhbNuLQqguapPc5tB4j5MBaxejpJaNspIDLxJyTUvVwMilWxj/1nv5rOQGOPOJLzJAQHdMIZTdGXFLMSFKwIkFChIeoT+J0+wSeKsmBbsDV8SVM1Uwd8oQYebaq9JAciEVvKv7r2V5mXieS7nU3pn4YSeLj2WA3YlAGcBoNdCgnWLKRIghzqjaGeIA4wlIFmHlZqK8NiJMUVD7mYhrLpHlZNo2yeW+Wqjdghjw4AifgDJjgClTBHaiDBsDgGbyCMXjTXrR37UP7nJXmtHnPAchA+/4D0cGmrw==</latexit><latexit sha1_base64="kO59cm/W//LWDO0rGNfdeSQGxWA=">AAAB9HicbVDLSsNAFL2prxqr1rWbwSK4KokbXQpuXFawD2hLmUxu2qGTSZi5EUrpyp1bv8Kd+Dd+hP/g9LFJ64ELh3Mu93GiXElLQfDjVfb2Dw6Pqsf+Sc0/PTuv1zo2K4zAtshUZnoRt6ikxjZJUtjLDfI0UtiNpo9Lv/uKxspMv9Asx2HKx1omUnByUmtUbwTNYAW2S8INacAGo/rvIM5EkaImobi1/TDIaTjnhqRQuPAHhcWciykf43x13IJdOylmSWZcaWIrtdTHU2tnaeQ6U04Tu+0txX+9KC3t6xeU3A/nUucFoRbrxUmhGGVs+TmLpUFBauYIF0a6i5mYcMMFuXxKk617bYLxwnfxhNth7JLObTMMmuFzAFW4hCu4gRDu4AGeoAVtEBDDO3x4b96n97WOseJt8ryAErzvP39qlUc=</latexit><latexit sha1_base64="QsEHTMWENf6DTpSY5wWtc7VDBLs=">AAACF3icbZDLSgMxFIbP1FutVUe3ggSL4EJLxo0uXAhu3BQq2At0SslkMm1o5mKSEUqZnY/iyq0+hTtx5UP4DqbTbqb1QMLP/x9yTj4vEVxpjH+s0tr6xuZWebuyU93d27cPqm0Vp5KyFo1FLLseUUzwiLU014J1E8lI6AnW8cZ3s7zzzKTicfSoJwnrh2QY8YBToo01sI8vGmjqekQit6l4lt/nyH1KiY8aN3hg13Ad54VWhbMQNVhUc2D/un5M05BFmgqiVM/Bie5PidScCpZV3FSxhNAxGbJpvn2GTo3loyCW5kQa5W6hj4RKTULPdIZEj9RyNjP/zbywMK+X6uC6P+VRkmoW0fngIBVIx2iGBvlcMqrFxAhCJTcbIzoiklBtABZeVuZrI+ZnFcPHWaaxKtqXdQfXnQcMZTiCEzgDB67gFu6hCS2g8AJv8A4f1qv1aX3NSZasBdJDKJT1/QfryqET</latexit><latexit sha1_base64="QsEHTMWENf6DTpSY5wWtc7VDBLs=">AAACF3icbZDLSgMxFIbP1FutVUe3ggSL4EJLxo0uXAhu3BQq2At0SslkMm1o5mKSEUqZnY/iyq0+hTtx5UP4DqbTbqb1QMLP/x9yTj4vEVxpjH+s0tr6xuZWebuyU93d27cPqm0Vp5KyFo1FLLseUUzwiLU014J1E8lI6AnW8cZ3s7zzzKTicfSoJwnrh2QY8YBToo01sI8vGmjqekQit6l4lt/nyH1KiY8aN3hg13Ad54VWhbMQNVhUc2D/un5M05BFmgqiVM/Bie5PidScCpZV3FSxhNAxGbJpvn2GTo3loyCW5kQa5W6hj4RKTULPdIZEj9RyNjP/zbywMK+X6uC6P+VRkmoW0fngIBVIx2iGBvlcMqrFxAhCJTcbIzoiklBtABZeVuZrI+ZnFcPHWaaxKtqXdQfXnQcMZTiCEzgDB67gFu6hCS2g8AJv8A4f1qv1aX3NSZasBdJDKJT1/QfryqET</latexit><latexit sha1_base64="JyzXArxkPqTPHs1yOIAIt4OB6Xg=">AAACInicbVDLSgMxFL1TX7W+Rl0KEiyCCy0zbnQhUnDjplDBPqBTSiaTaUMzD5OMUMrs/BRXbvUr3IkrwV/wH0zH2UzrgYTDOffm3hw35kwqy/oySkvLK6tr5fXKxubW9o65u9eWUSIIbZGIR6LrYkk5C2lLMcVpNxYUBy6nHXd8M/M7j1RIFoX3ahLTfoCHIfMZwUpLA/PwrIGmjosFcpqSpdl9ipyHBHuocWUNzKpVszKgRWLnpAo5mgPzx/EikgQ0VIRjKXu2Fav+FAvFCKdpxUkkjTEZ4yGdZtun6FhLHvIjoU+oUKYW6nAg5SRwdWWA1UjOezPxX88NCvN6ifIv+1MWxomiIfkb7CccqQjNokEeE5QoPtEEE8H0xoiMsMBE6QALL0v9tRH10orOx55PY5G0z2u2VbPvrGr9Ok+qDAdwBCdgwwXU4Raa0AICT/ACr/BmPBvvxofx+VdaMvKefSjA+P4F5XeimA==</latexit><latexit sha1_base64="bvt+qBcHLCigQtt3swsFb2vQKu8=">AAACInicbVDLSgMxFL1TX7W+Rl0KEiyCCy0zbnQhUnDjplDBPqBTSiaTaUMzD5OMUMrs/BRXbvUr3IkrwV/wH0zH2UzrgYTDOffm3hw35kwqy/oySkvLK6tr5fXKxubW9o65u9eWUSIIbZGIR6LrYkk5C2lLMcVpNxYUBy6nHXd8M/M7j1RIFoX3ahLTfoCHIfMZwUpLA/PwrIGmjosFcpqSpdl9ipyHBHuocWUNzKpVszKgRWLnpAo5mgPzx/EikgQ0VIRjKXu2Fav+FAvFCKdpxUkkjTEZ4yGdZtun6FhLHvIjoU+oUKYW6nAg5SRwdWWA1UjOezPxX88NCvN6ifIv+1MWxomiIfkb7CccqQjNokEeE5QoPtEEE8H0xoiMsMBE6QALL0v9tRH10orOx55PY5G0z2u2VbPv7Gr9Ok+qDAdwBCdgwwXU4Raa0AICT/ACr/BmPBvvxofx+VdaMvKefSjA+P4F5ceimQ==</latexit><latexit sha1_base64="m84XkHYrK6cnsqiOw0elZhGkyaU=">AAACInicbVC7SgNBFJ2NrxgfWbWwEGQwCBYadm20CBKwsQlEMA/IhjA7O5sMmX04MyuEZTs/xco2foWdWAm2luYbnGzSbOKBGQ7n3Dv3zrFDRoU0jC8tt7K6tr6R3yxsbe/sFvW9/aYIIo5JAwcs4G0bCcKoTxqSSkbaISfIsxlp2cPbqd96IlzQwH+Qo5B0PdT3qUsxkkrq6ccXNRhbNuLQqguapPc5tB4j5MBaxejpJaNspIDLxJyTUvVwMilWxj/1nv5rOQGOPOJLzJAQHdMIZTdGXFLMSFKwIkFChIeoT+J0+wSeKsmBbsDV8SVM1Uwd8oQYebaq9JAciEVvKv7r2V5mXieS7nU3pn4YSeLj2WA3YlAGcBoNdCgnWLKRIghzqjaGeIA4wlIFmHlZqK8NiJMUVD7mYhrLpHlZNo2yeW+Wqjdghjw4AifgDJjgClTBHaiDBsDgGbyCMXjTXrR37UP7nJXmtHnPAchA+/4D0cGmrw==</latexit><latexit sha1_base64="m84XkHYrK6cnsqiOw0elZhGkyaU=">AAACInicbVC7SgNBFJ2NrxgfWbWwEGQwCBYadm20CBKwsQlEMA/IhjA7O5sMmX04MyuEZTs/xco2foWdWAm2luYbnGzSbOKBGQ7n3Dv3zrFDRoU0jC8tt7K6tr6R3yxsbe/sFvW9/aYIIo5JAwcs4G0bCcKoTxqSSkbaISfIsxlp2cPbqd96IlzQwH+Qo5B0PdT3qUsxkkrq6ccXNRhbNuLQqguapPc5tB4j5MBaxejpJaNspIDLxJyTUvVwMilWxj/1nv5rOQGOPOJLzJAQHdMIZTdGXFLMSFKwIkFChIeoT+J0+wSeKsmBbsDV8SVM1Uwd8oQYebaq9JAciEVvKv7r2V5mXieS7nU3pn4YSeLj2WA3YlAGcBoNdCgnWLKRIghzqjaGeIA4wlIFmHlZqK8NiJMUVD7mYhrLpHlZNo2yeW+Wqjdghjw4AifgDJjgClTBHaiDBsDgGbyCMXjTXrR37UP7nJXmtHnPAchA+/4D0cGmrw==</latexit><latexit sha1_base64="m84XkHYrK6cnsqiOw0elZhGkyaU=">AAACInicbVC7SgNBFJ2NrxgfWbWwEGQwCBYadm20CBKwsQlEMA/IhjA7O5sMmX04MyuEZTs/xco2foWdWAm2luYbnGzSbOKBGQ7n3Dv3zrFDRoU0jC8tt7K6tr6R3yxsbe/sFvW9/aYIIo5JAwcs4G0bCcKoTxqSSkbaISfIsxlp2cPbqd96IlzQwH+Qo5B0PdT3qUsxkkrq6ccXNRhbNuLQqguapPc5tB4j5MBaxejpJaNspIDLxJyTUvVwMilWxj/1nv5rOQGOPOJLzJAQHdMIZTdGXFLMSFKwIkFChIeoT+J0+wSeKsmBbsDV8SVM1Uwd8oQYebaq9JAciEVvKv7r2V5mXieS7nU3pn4YSeLj2WA3YlAGcBoNdCgnWLKRIghzqjaGeIA4wlIFmHlZqK8NiJMUVD7mYhrLpHlZNo2yeW+Wqjdghjw4AifgDJjgClTBHaiDBsDgGbyCMXjTXrR37UP7nJXmtHnPAchA+/4D0cGmrw==</latexit><latexit sha1_base64="m84XkHYrK6cnsqiOw0elZhGkyaU=">AAACInicbVC7SgNBFJ2NrxgfWbWwEGQwCBYadm20CBKwsQlEMA/IhjA7O5sMmX04MyuEZTs/xco2foWdWAm2luYbnGzSbOKBGQ7n3Dv3zrFDRoU0jC8tt7K6tr6R3yxsbe/sFvW9/aYIIo5JAwcs4G0bCcKoTxqSSkbaISfIsxlp2cPbqd96IlzQwH+Qo5B0PdT3qUsxkkrq6ccXNRhbNuLQqguapPc5tB4j5MBaxejpJaNspIDLxJyTUvVwMilWxj/1nv5rOQGOPOJLzJAQHdMIZTdGXFLMSFKwIkFChIeoT+J0+wSeKsmBbsDV8SVM1Uwd8oQYebaq9JAciEVvKv7r2V5mXieS7nU3pn4YSeLj2WA3YlAGcBoNdCgnWLKRIghzqjaGeIA4wlIFmHlZqK8NiJMUVD7mYhrLpHlZNo2yeW+Wqjdghjw4AifgDJjgClTBHaiDBsDgGbyCMXjTXrR37UP7nJXmtHnPAchA+/4D0cGmrw==</latexit><latexit sha1_base64="bvt+qBcHLCigQtt3swsFb2vQKu8=">AAACInicbVDLSgMxFL1TX7W+Rl0KEiyCCy0zbnQhUnDjplDBPqBTSiaTaUMzD5OMUMrs/BRXbvUr3IkrwV/wH0zH2UzrgYTDOffm3hw35kwqy/oySkvLK6tr5fXKxubW9o65u9eWUSIIbZGIR6LrYkk5C2lLMcVpNxYUBy6nHXd8M/M7j1RIFoX3ahLTfoCHIfMZwUpLA/PwrIGmjosFcpqSpdl9ipyHBHuocWUNzKpVszKgRWLnpAo5mgPzx/EikgQ0VIRjKXu2Fav+FAvFCKdpxUkkjTEZ4yGdZtun6FhLHvIjoU+oUKYW6nAg5SRwdWWA1UjOezPxX88NCvN6ifIv+1MWxomiIfkb7CccqQjNokEeE5QoPtEEE8H0xoiMsMBE6QALL0v9tRH10orOx55PY5G0z2u2VbPv7Gr9Ok+qDAdwBCdgwwXU4Raa0AICT/ACr/BmPBvvxofx+VdaMvKefSjA+P4F5ceimQ==</latexit>

ZF[A;M < 0] = ZQED3[A;M 0 > 0], ZF[A;M > 0] = ZQED3

[A;M 0 < 0]<latexit sha1_base64="qvL9zDBEJQSGGWmm9xjTNbnFRK4=">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</latexit><latexit sha1_base64="nJThqOqfKQ2Vw/aTHZ/uBuiZVaM=">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</latexit><latexit sha1_base64="nJThqOqfKQ2Vw/aTHZ/uBuiZVaM=">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</latexit><latexit sha1_base64="nJThqOqfKQ2Vw/aTHZ/uBuiZVaM=">AAACZnicdVHLSgMxFM2Mr1q1VkVcuIkW0YWUjC5U1FLxgRtRwfpqhyGTpm1o5kGSEcowX+CP+Qv+gRvd6tJM68K2eiFwOOc+T9yQM6kQejXMkdGx8YnMZHZqeiY3m5+bv5VBJAitkIAH4t7FknLm04piitP7UFDsuZzeue3jVL97okKywL9RnZDaHm76rMEIVppy8u6jE9c8rFrCi8+SpHq0f3GA7MNf7PXpSeJsd5X1ErI34XBF6f8K3czJF1ARdQMOA+sHFMorOWRcPj9cOfn3Wj0gkUd9RTiWsmqhUNkxFooRTpNsLZI0xKSNmzTuOpDANU3VYSMQ+vkKdtm+POxJ2fFcnZkuKQe1lPxTc72+edVINXbtmPlhpKhPeoMbEYcqgKm9sM4EJYp3NMBEML0xJC0sMFH6E/o6S31ai9aTrPbHGnRjGNxuFS1UtK6tQnkP9CIDlsEq2AAW2AFlcA6uQAUQ8AI+wCf4Mt7MnLloLvVSTeOnZgH0hQm/AXD6u9w=</latexit><latexit sha1_base64="kO59cm/W//LWDO0rGNfdeSQGxWA=">AAAB9HicbVDLSsNAFL2prxqr1rWbwSK4KokbXQpuXFawD2hLmUxu2qGTSZi5EUrpyp1bv8Kd+Dd+hP/g9LFJ64ELh3Mu93GiXElLQfDjVfb2Dw6Pqsf+Sc0/PTuv1zo2K4zAtshUZnoRt6ikxjZJUtjLDfI0UtiNpo9Lv/uKxspMv9Asx2HKx1omUnByUmtUbwTNYAW2S8INacAGo/rvIM5EkaImobi1/TDIaTjnhqRQuPAHhcWciykf43x13IJdOylmSWZcaWIrtdTHU2tnaeQ6U04Tu+0txX+9KC3t6xeU3A/nUucFoRbrxUmhGGVs+TmLpUFBauYIF0a6i5mYcMMFuXxKk617bYLxwnfxhNth7JLObTMMmuFzAFW4hCu4gRDu4AGeoAVtEBDDO3x4b96n97WOseJt8ryAErzvP39qlUc=</latexit><latexit sha1_base64="K5gYZbRxP3BidBChdWurDfQg+qQ=">AAACW3icdVHLSgMxFL0dX7XWWgVx4SZYRBdSMrpQUIviAzeCBatiOwyZNGODmQdJRijD/J+/4B+40a0uzdQurNULgcM593nixYIrjfFLwZqYnJqeKc6W5srzlYXqYvlGRYmkrEUjEck7jygmeMhammvB7mLJSOAJdus9nuT67ROTikfhte7HzAnIQ8h9Tok2lFv17t20ExDdk0F6nmXt4/3LA+wc/mCbZ6eZuzNQNhrY2ULjFY3/K0wzt1rDdTwINA7sIajBMK7c6lunG9EkYKGmgijVtnGsnZRIzalgWamTKBYT+kgeWDpwIEPrhuoiP5LmhRoN2JE8EijVDzyTmS+pfms5+afmBSPz2on295yUh3GiWUi/B/uJQDpCub2oyyWjWvQNIFRyszGiPSIJ1eYTRjorc1qPdbOS8cf+7cY4uNmu27huNzEUYRXWYBNs2IUjuIAraAGFZ3iHD/gsvFoVa/nbSaswtHQJRsJa+QLXcblD</latexit><latexit sha1_base64="K5gYZbRxP3BidBChdWurDfQg+qQ=">AAACW3icdVHLSgMxFL0dX7XWWgVx4SZYRBdSMrpQUIviAzeCBatiOwyZNGODmQdJRijD/J+/4B+40a0uzdQurNULgcM593nixYIrjfFLwZqYnJqeKc6W5srzlYXqYvlGRYmkrEUjEck7jygmeMhammvB7mLJSOAJdus9nuT67ROTikfhte7HzAnIQ8h9Tok2lFv17t20ExDdk0F6nmXt4/3LA+wc/mCbZ6eZuzNQNhrY2ULjFY3/K0wzt1rDdTwINA7sIajBMK7c6lunG9EkYKGmgijVtnGsnZRIzalgWamTKBYT+kgeWDpwIEPrhuoiP5LmhRoN2JE8EijVDzyTmS+pfms5+afmBSPz2on295yUh3GiWUi/B/uJQDpCub2oyyWjWvQNIFRyszGiPSIJ1eYTRjorc1qPdbOS8cf+7cY4uNmu27huNzEUYRXWYBNs2IUjuIAraAGFZ3iHD/gsvFoVa/nbSaswtHQJRsJa+QLXcblD</latexit><latexit sha1_base64="oj8fC9MafGZHqiSn0HWZpINqNeM=">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</latexit><latexit sha1_base64="vg6auApc7aPCkU2pYfV9Tzd8EKQ=">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</latexit><latexit sha1_base64="nJThqOqfKQ2Vw/aTHZ/uBuiZVaM=">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</latexit><latexit sha1_base64="nJThqOqfKQ2Vw/aTHZ/uBuiZVaM=">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</latexit><latexit sha1_base64="nJThqOqfKQ2Vw/aTHZ/uBuiZVaM=">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</latexit><latexit sha1_base64="nJThqOqfKQ2Vw/aTHZ/uBuiZVaM=">AAACZnicdVHLSgMxFM2Mr1q1VkVcuIkW0YWUjC5U1FLxgRtRwfpqhyGTpm1o5kGSEcowX+CP+Qv+gRvd6tJM68K2eiFwOOc+T9yQM6kQejXMkdGx8YnMZHZqeiY3m5+bv5VBJAitkIAH4t7FknLm04piitP7UFDsuZzeue3jVL97okKywL9RnZDaHm76rMEIVppy8u6jE9c8rFrCi8+SpHq0f3GA7MNf7PXpSeJsd5X1ErI34XBF6f8K3czJF1ARdQMOA+sHFMorOWRcPj9cOfn3Wj0gkUd9RTiWsmqhUNkxFooRTpNsLZI0xKSNmzTuOpDANU3VYSMQ+vkKdtm+POxJ2fFcnZkuKQe1lPxTc72+edVINXbtmPlhpKhPeoMbEYcqgKm9sM4EJYp3NMBEML0xJC0sMFH6E/o6S31ai9aTrPbHGnRjGNxuFS1UtK6tQnkP9CIDlsEq2AAW2AFlcA6uQAUQ8AI+wCf4Mt7MnLloLvVSTeOnZgH0hQm/AXD6u9w=</latexit><latexit sha1_base64="vg6auApc7aPCkU2pYfV9Tzd8EKQ=">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</latexit>

Page 18: Duality in Condensed Matter Physics and Field Theory · Duality in Statistical Mechanics • 2D Ising Model: Kramers- Wannier duality • Partition function as a sum over closed domain

General Bosonization Dualities

!18

L = |Da�|2 � |�|4 +k�

4⇡ada +

12⇡

adA<latexit sha1_base64="haUhz8WUbRUgg+IDkZmBaGQggCw=">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</latexit><latexit sha1_base64="h40nlEQsbCta6zWv7i51zqZNbys=">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</latexit><latexit sha1_base64="h40nlEQsbCta6zWv7i51zqZNbys=">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</latexit><latexit sha1_base64="h40nlEQsbCta6zWv7i51zqZNbys=">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</latexit><latexit sha1_base64="cSEpZM4nr16xSbsp5iCCuNGq5wo=">AAAB83icbZDNSsNAFIVv6l+NVevazWARXJXEjboT3LisYGyhLWUyuWmHTn6YuRFK6cqdW5/Clfg4PoTv4KTtJq0XBg7nXJhzvzBX0pDn/Ti1nd29/YP6oXvUcI9PTpuNF5MVWmAgMpXpXsgNKpliQJIU9nKNPAkVdsPpQ5l3X1EbmaXPNMtxmPBxKmMpOFmrM2q2vLa3HLYt/LVowXpGzd9BlIkiwZSE4sb0fS+n4ZxrkkLhwh0UBnMupnyM82W5Bbu0VsTiTNuXElu6lT2eGDNLQruZcJqYzaw0/83CpPqfsYUmGFXMfkHx7XAu07wgTMWqTVwoRhkrcbBIahSkZlZwoaU9g4kJ11yQheZaPP4mjG0RXLfv2v6TB3U4hwu4Ah9u4B4eoQMBCIjgHT6cN+fT+VpRrDlrnGdQGef7D/wclRw=</latexit><latexit sha1_base64="ThLjdt4a+eM5ISqpbS6Ct9JX6kg=">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</latexit><latexit sha1_base64="InoZHk8LpxqQ/ngTEgwBmGPvKbQ=">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</latexit><latexit sha1_base64="4z+qmLHNsOrgOU1pPcvyj2owXgE=">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</latexit><latexit sha1_base64="4z+qmLHNsOrgOU1pPcvyj2owXgE=">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</latexit><latexit sha1_base64="h40nlEQsbCta6zWv7i51zqZNbys=">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</latexit><latexit sha1_base64="h40nlEQsbCta6zWv7i51zqZNbys=">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</latexit><latexit sha1_base64="h40nlEQsbCta6zWv7i51zqZNbys=">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</latexit><latexit sha1_base64="h40nlEQsbCta6zWv7i51zqZNbys=">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</latexit><latexit sha1_base64="4z+qmLHNsOrgOU1pPcvyj2owXgE=">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</latexit>

|Da�|2�|�|4+k�

4⇡ada+

12⇡

adA$ i /Db +18⇡

bdb+12⇡

ad(b+A)+k� + 1

4⇡ada

<latexit sha1_base64="K6hCt5p6ijynOPZBng7Km7BquRg=">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</latexit><latexit sha1_base64="kN6eW36QjfYMOBToiCbelaeFWlk=">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</latexit><latexit sha1_base64="kN6eW36QjfYMOBToiCbelaeFWlk=">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</latexit><latexit sha1_base64="kN6eW36QjfYMOBToiCbelaeFWlk=">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</latexit><latexit sha1_base64="cSEpZM4nr16xSbsp5iCCuNGq5wo=">AAAB83icbZDNSsNAFIVv6l+NVevazWARXJXEjboT3LisYGyhLWUyuWmHTn6YuRFK6cqdW5/Clfg4PoTv4KTtJq0XBg7nXJhzvzBX0pDn/Ti1nd29/YP6oXvUcI9PTpuNF5MVWmAgMpXpXsgNKpliQJIU9nKNPAkVdsPpQ5l3X1EbmaXPNMtxmPBxKmMpOFmrM2q2vLa3HLYt/LVowXpGzd9BlIkiwZSE4sb0fS+n4ZxrkkLhwh0UBnMupnyM82W5Bbu0VsTiTNuXElu6lT2eGDNLQruZcJqYzaw0/83CpPqfsYUmGFXMfkHx7XAu07wgTMWqTVwoRhkrcbBIahSkZlZwoaU9g4kJ11yQheZaPP4mjG0RXLfv2v6TB3U4hwu4Ah9u4B4eoQMBCIjgHT6cN+fT+VpRrDlrnGdQGef7D/wclRw=</latexit><latexit sha1_base64="FxL4Waon1VHlWuDfYRymTbajhfI=">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</latexit><latexit sha1_base64="9IZJRHD4uEAPjctLKZUycygc9h8=">AAACoXicdZHNbtQwEMedlI8SCl165WJRIRWtWJJVC+0FqOiBY5EIrbReorHjbKx1Est2QKtsHohH4iF4B5zdoBI+RrI0/s2MZ/wfqqQwNgy/e/7Ordt37u7eC+7vPXi4P3q098lUtWY8ZpWs9DUFw6UoeWyFlfxaaQ4FlfyKLt918asvXBtRlR/tSvF5AYtSZIKBdSgZfVtfJEBULtafp8/XW+d4TDINrFkm3b1tjokSLaTQ46htpltyHhDJM6vFIregdfU1EISCJsoIYiSYnKfNRZvQDtwUn3bFNKV/PXdEx+fPBr3H0U33ZHQYTsKN4d+ckzA6exnhqCeHqLfLZPSDpBWrC15a5oYxsyhUdt6AtoJJ3gakNlwBW8KCNxshW/zUoRRnlXantHhDB3lQGLMqqMsswObmz1gH/xmjxbBfr84Azmqbnc4bUara8pJtp8lqiW2Fu9XhVGjOrFw5B5gW7huY5eDUsm7BgdPnlwj4/048nZxNog8h2kWP0RN0hCL0Cr1F79ElihHz9r0T77X3xn/hx/5sK6Tv9YoeoIH585/3edQ5</latexit><latexit sha1_base64="wTRMLt36fx7Cx5mVw73tAzaOwZE=">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</latexit><latexit sha1_base64="wTRMLt36fx7Cx5mVw73tAzaOwZE=">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</latexit><latexit sha1_base64="kN6eW36QjfYMOBToiCbelaeFWlk=">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</latexit><latexit sha1_base64="kN6eW36QjfYMOBToiCbelaeFWlk=">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</latexit><latexit sha1_base64="kN6eW36QjfYMOBToiCbelaeFWlk=">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</latexit><latexit sha1_base64="kN6eW36QjfYMOBToiCbelaeFWlk=">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</latexit><latexit sha1_base64="wTRMLt36fx7Cx5mVw73tAzaOwZE=">AAACrHicdZHbbtQwEIadcGgJhy5w2RuLFVLRiiVZtdC9Ka3oBZdFIrTSZonGjrOx1kks26FaZfNAPBIP0XfA2Q0q4TCSpfE3M57xP0QKro3v/3DcO3fv3d/ZfeA9fPT4yd7g6bMvuqwUZSEtRamuCGgmeMFCw41gV1IxyIlgl2T5oY1ffmNK87L4bFaSzXNYFDzlFIxF8eD7+jyGSGZ8/XXyer11DkdRqoDWy7i9N/VhJHkDCXQ4aOrJlpx5kWCpUXyRGVCqvPZ4REBFUvNIC9AZS+rzJiYtuC0+botJQv567oCMzl71eo+C2+7xYOiP/Y3h35wjP5i+DXDQkSHq7CIe3ERJSaucFYbaYfQs8KWZ16AMp4I1XlRpJoEuYcHqjZANfmlRgtNS2VMYvKG9PMi1XuXEZuZgMv1nrIX/jJG8369TpwdnlUmP5zUvZGVYQbfTpJXApsTt6nDCFaNGrKwDVHH7DUwzsGoZu2DP6vNLBPx/J5yMp+Pgkz88nXZC7aJ99AIdoAC9Q6foI7pAIaLOnnPknDjv3Tdu6M7c+TbVdbqa56hnbvoTODjVRA==</latexit>

k =12

k� � 1k� + 1

<latexit sha1_base64="SeCtC356lzktwYCUw+h2EQJk6mM=">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</latexit><latexit sha1_base64="wsvqqq2CdDAQl5NXYJHwcUAeahk=">AAACLnicdZDLSsNAFIYn9VbrLerSzWARhNKSFNR2IRTcdCNUMLbQlDKZTtqhkwszE6GEPISP4kpwpU8hLsStG9/BSVPBiB4Y+Pn+M5xzfidkVEjDeNUKS8srq2vF9dLG5tb2jr67dyOCiGNi4YAFvOcgQRj1iSWpZKQXcoI8h5GuM71I/e4t4YIG/rWchWTgobFPXYqRVGioV6ZDOxT03HY5wrGZxPUkkymf0KoimaqYyVAvGzVjXvCHODHM5qkJzQUptxpe9fHSb3eG+qc9CnDkEV9ihoTom0YoBzHikmJGkpIdCRIiPEVjEs9PSeCRQiPoBlw9X8I5zfUhT4iZ56hOD8mJ+O2l8E/P8fLzhFpoQkY52I+k2xjE1A8jSXycbeNGDMoApuHBEeUESzZTAmFO1RkQT5CKS6qISyqf7xDg/8Kq15o180rl1ARZFcEBOATHwARnoAXaoAMsgMEdeABP4Fm71160N+09ay1oiz/7IFfaxxc5iKyZ</latexit><latexit sha1_base64="wsvqqq2CdDAQl5NXYJHwcUAeahk=">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</latexit><latexit sha1_base64="wsvqqq2CdDAQl5NXYJHwcUAeahk=">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</latexit><latexit sha1_base64="cSEpZM4nr16xSbsp5iCCuNGq5wo=">AAAB83icbZDNSsNAFIVv6l+NVevazWARXJXEjboT3LisYGyhLWUyuWmHTn6YuRFK6cqdW5/Clfg4PoTv4KTtJq0XBg7nXJhzvzBX0pDn/Ti1nd29/YP6oXvUcI9PTpuNF5MVWmAgMpXpXsgNKpliQJIU9nKNPAkVdsPpQ5l3X1EbmaXPNMtxmPBxKmMpOFmrM2q2vLa3HLYt/LVowXpGzd9BlIkiwZSE4sb0fS+n4ZxrkkLhwh0UBnMupnyM82W5Bbu0VsTiTNuXElu6lT2eGDNLQruZcJqYzaw0/83CpPqfsYUmGFXMfkHx7XAu07wgTMWqTVwoRhkrcbBIahSkZlZwoaU9g4kJ11yQheZaPP4mjG0RXLfv2v6TB3U4hwu4Ah9u4B4eoQMBCIjgHT6cN+fT+VpRrDlrnGdQGef7D/wclRw=</latexit><latexit sha1_base64="sQNeYj8Ee8dVEgHP9ji4VZ0lHuw=">AAACI3icbZDNSsNAEMcn9avWqtGrl2ARhGJJelEPguDFYwVjC00Jm82mWbr5YHcjlJCH8FE8edWnEA/iE/gObppe0jqw8Oc3Azvz81JGhTTNL62xsbm1vdPcbe219w8O9aP2k0gyjomNE5bwkYcEYTQmtqSSkVHKCYo8Robe7K7sD58JFzSJH+U8JZMITWMaUIykQq7enblOKuiNE3CEc6vI+0UVSx7SC0Wq1LUKV++YPXNRxnqwlqEDyxq4+q/jJziLSCwxQ0KMLTOVkxxxSTEjRcvJBEkRnqEpyRenFMaZQr4RJFy9WBoLWptDkRDzyFOTEZKhWO2V8N+eF9X/E2qhkPg1OM5kcDXJaZxmksS42ibImCETo5Rn+JQTLNlcBYQ5VWcYOERKl1SKW8qPtWpjPdj93nXPejChCSdwCudgwSXcwj0MwAYML/AG7/ChvWqf2nclsqEtjR5DrbSfP8PeqCg=</latexit><latexit sha1_base64="SUjxq31pjPMfUt1M8Ft+k1fHDyY=">AAACI3icdZDLSsNAFIZP6q3WqtWtm8EiCMWSFLx0IQhuXFYwtmBKmUwmzdDJhZmJUEIewkdx5VafQlyIT+A7OGkqWNEDAz/fOTDnfG7CmVSm+WZUlpZXVteq67WN+ubWdmOnfivjVBBqk5jHYuBiSTmLqK2Y4nSQCIpDl9O+O7ks+v17KiSLoxs1TegwxOOI+YxgpdGo0ZqMnESyc8cXmGRWnnXyMhY8YEealKll5aNG02ybs0I/wrFpdU8sZM1JE+bVGzU+HS8maUgjRTiW8s4yEzXMsFCMcJrXnFTSBJMJHtNsdkqODjTykB8L/SKFZnRhDodSTkNXT4ZYBfJ3r4B/9txw8T+pFwqotwDvUuWfDTMWJamiESm38VOOVIwKechjghLFpzpgIpg+A5EAa11KK65pP98S0P/B7rS7bevahCrswT4cggWncAFX0AMbCDzAEzzDi/FovBrvpciKMTe6CwtlfHwB5DaoPg==</latexit><latexit sha1_base64="hE+FkMMPg0NVhp7+9t/SbdQ1y+I=">AAACLnicdZDLSsNAFIYnXmu9RV26GSyCUCxJwUsXQsGNywrGFpoSJpNJO3RyYWYilJCH8FFcudWnEBfi1o3v4KSJYEQPDPx8/xnOOb8bMyqkYbxqC4tLyyurtbX6+sbm1ra+s3srooRjYuGIRXzgIkEYDYklqWRkEHOCApeRvju9zP3+HeGCRuGNnMVkFKBxSH2KkVTI0ZtTx44FvbB9jnBqZmk7K2TOJ/RYkUI1zczRG0bLmBf8IU4Ms3NqQrMkDVBWz9E/bS/CSUBCiRkSYmgasRyliEuKGcnqdiJIjPAUjUk6PyWDhwp50I+4eqGEc1rpQ4EQs8BVnQGSE/Hby+GfnhtU5wm10IR4FThMpH8+SmkYJ5KEuNjGTxiUEczDgx7lBEs2UwJhTtUZEE+QikuqiOsqn+8Q4P/Carc6LfPaaHQ7ZVA1sA8OwBEwwRnogivQAxbA4B48gifwrD1oL9qb9l60Lmjlnz1QKe3jCylwqco=</latexit><latexit sha1_base64="hE+FkMMPg0NVhp7+9t/SbdQ1y+I=">AAACLnicdZDLSsNAFIYnXmu9RV26GSyCUCxJwUsXQsGNywrGFpoSJpNJO3RyYWYilJCH8FFcudWnEBfi1o3v4KSJYEQPDPx8/xnOOb8bMyqkYbxqC4tLyyurtbX6+sbm1ra+s3srooRjYuGIRXzgIkEYDYklqWRkEHOCApeRvju9zP3+HeGCRuGNnMVkFKBxSH2KkVTI0ZtTx44FvbB9jnBqZmk7K2TOJ/RYkUI1zczRG0bLmBf8IU4Ms3NqQrMkDVBWz9E/bS/CSUBCiRkSYmgasRyliEuKGcnqdiJIjPAUjUk6PyWDhwp50I+4eqGEc1rpQ4EQs8BVnQGSE/Hby+GfnhtU5wm10IR4FThMpH8+SmkYJ5KEuNjGTxiUEczDgx7lBEs2UwJhTtUZEE+QikuqiOsqn+8Q4P/Carc6LfPaaHQ7ZVA1sA8OwBEwwRnogivQAxbA4B48gifwrD1oL9qb9l60Lmjlnz1QKe3jCylwqco=</latexit><latexit sha1_base64="wsvqqq2CdDAQl5NXYJHwcUAeahk=">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</latexit><latexit sha1_base64="wsvqqq2CdDAQl5NXYJHwcUAeahk=">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</latexit><latexit sha1_base64="wsvqqq2CdDAQl5NXYJHwcUAeahk=">AAACLnicdZDLSsNAFIYn9VbrLerSzWARhNKSFNR2IRTcdCNUMLbQlDKZTtqhkwszE6GEPISP4kpwpU8hLsStG9/BSVPBiB4Y+Pn+M5xzfidkVEjDeNUKS8srq2vF9dLG5tb2jr67dyOCiGNi4YAFvOcgQRj1iSWpZKQXcoI8h5GuM71I/e4t4YIG/rWchWTgobFPXYqRVGioV6ZDOxT03HY5wrGZxPUkkymf0KoimaqYyVAvGzVjXvCHODHM5qkJzQUptxpe9fHSb3eG+qc9CnDkEV9ihoTom0YoBzHikmJGkpIdCRIiPEVjEs9PSeCRQiPoBlw9X8I5zfUhT4iZ56hOD8mJ+O2l8E/P8fLzhFpoQkY52I+k2xjE1A8jSXycbeNGDMoApuHBEeUESzZTAmFO1RkQT5CKS6qISyqf7xDg/8Kq15o180rl1ARZFcEBOATHwARnoAXaoAMsgMEdeABP4Fm71160N+09ay1oiz/7IFfaxxc5iKyZ</latexit><latexit sha1_base64="wsvqqq2CdDAQl5NXYJHwcUAeahk=">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</latexit><latexit sha1_base64="hE+FkMMPg0NVhp7+9t/SbdQ1y+I=">AAACLnicdZDLSsNAFIYnXmu9RV26GSyCUCxJwUsXQsGNywrGFpoSJpNJO3RyYWYilJCH8FFcudWnEBfi1o3v4KSJYEQPDPx8/xnOOb8bMyqkYbxqC4tLyyurtbX6+sbm1ra+s3srooRjYuGIRXzgIkEYDYklqWRkEHOCApeRvju9zP3+HeGCRuGNnMVkFKBxSH2KkVTI0ZtTx44FvbB9jnBqZmk7K2TOJ/RYkUI1zczRG0bLmBf8IU4Ms3NqQrMkDVBWz9E/bS/CSUBCiRkSYmgasRyliEuKGcnqdiJIjPAUjUk6PyWDhwp50I+4eqGEc1rpQ4EQs8BVnQGSE/Hby+GfnhtU5wm10IR4FThMpH8+SmkYJ5KEuNjGTxiUEczDgx7lBEs2UwJhTtUZEE+QikuqiOsqn+8Q4P/Carc6LfPaaHQ7ZVA1sA8OwBEwwRnogivQAxbA4B48gifwrD1oL9qb9l60Lmjlnz1QKe3jCylwqco=</latexit>

⌧ =⌧� � 1⌧� + 1

<latexit sha1_base64="BMo88RtXwS1NZ3I9RZhUrBUY9Vw=">AAACLHicdVDLSgMxFL1TX7W+qi7dBIugSMuMoLYLoeDGZQVrC50yZDKZNjTzIMkIpcw3+Cmu3Ko/4ULErTv/wfQhOqIHAveecy6597gxZ1KZ5ouRm5tfWFzKLxdWVtfWN4qbW9cySgShTRLxSLRdLClnIW0qpjhtx4LiwOW05Q7Ox3rrhgrJovBKDWPaDXAvZD4jWGnKKR7YCieOHUt2ZvsCk9Gs77OylX43h1bqFEtmxZwA/SiOTat2YiFrxpTq1aD8BAANp/hhexFJAhoqwrGUHcuMVXeEhWKE07RgJ5LGmAxwj44mh6RoT1Me8iOhX6jQhM34cCDlMHC1M8CqL39rY/JPzQ2y/0m9UJ96GbKTKL/aHbEwThQNyXQbP+FIRWgcHfKYoETxoS4wEUyfgUgf69CUDrig8/kKAf1fNI8qtYp1qXOqwRR52IFd2AcLTqEOF9CAJhC4hXt4gEfjzng2Xo23qTVnzGa2IQPj/RNtwaru</latexit><latexit sha1_base64="LuVUE5T73tziQCRPRa0qAlMOmgs=">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</latexit><latexit sha1_base64="LuVUE5T73tziQCRPRa0qAlMOmgs=">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</latexit><latexit sha1_base64="LuVUE5T73tziQCRPRa0qAlMOmgs=">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</latexit><latexit sha1_base64="cSEpZM4nr16xSbsp5iCCuNGq5wo=">AAAB83icbZDNSsNAFIVv6l+NVevazWARXJXEjboT3LisYGyhLWUyuWmHTn6YuRFK6cqdW5/Clfg4PoTv4KTtJq0XBg7nXJhzvzBX0pDn/Ti1nd29/YP6oXvUcI9PTpuNF5MVWmAgMpXpXsgNKpliQJIU9nKNPAkVdsPpQ5l3X1EbmaXPNMtxmPBxKmMpOFmrM2q2vLa3HLYt/LVowXpGzd9BlIkiwZSE4sb0fS+n4ZxrkkLhwh0UBnMupnyM82W5Bbu0VsTiTNuXElu6lT2eGDNLQruZcJqYzaw0/83CpPqfsYUmGFXMfkHx7XAu07wgTMWqTVwoRhkrcbBIahSkZlZwoaU9g4kJ11yQheZaPP4mjG0RXLfv2v6TB3U4hwu4Ah9u4B4eoQMBCIjgHT6cN+fT+VpRrDlrnGdQGef7D/wclRw=</latexit><latexit sha1_base64="uQxkkXP2JrOiq/GrVStKt8z44rI=">AAACIXicbZDNSsNAFIVv6l+tVaNbN4NFUMSSuFEXguDGZQVjC00Jk8mkGTr5YWYilJJn8FFcudWncCHiI/gOTtuApPXAwLnnXph7Pz/jTCrL+jRqK6tr6xv1zcZWc3tn19xrPso0F4Q6JOWp6PlYUs4S6iimOO1lguLY57Trj26n/e4TFZKlyYMaZ3QQ42HCQkaw0pFnnrgK556bSXbthgKTSVlH7Mwu/opTu/DMltW2ZkLLxi5NC0p1PPPHDVKSxzRRhGMp+7aVqcEEC8UIp0XDzSXNMBnhIZ3MDinQkY4CFKZCv0ShWVqZw7GU49jXkzFWkVzsTcN/e35c/U/qhSIaVMJ+rsLLwYQlWa5oQubbhDlHKkVTdChgghLFx9pgIpg+A5EIa2hKA25oPvYijWXjnLev2va9BXU4gEM4Bhsu4AbuoAMOEHiGV3iDd+PF+DC+5iBrRkl0Hyoyvn8BavynfA==</latexit><latexit sha1_base64="jgWDAvzMX5+MZV5aiFUWoHL0Bt0=">AAACIXicdZDLSsNAFIZPvNZaNbp1M1gERSyJ4KULQXDjsoK1haaEyWTSDJ1cmJkIJeQZfBRXbvUpXIj4CL6D0zaiFf1h4Jz/PzDnfF7KmVSW9WrMzS8sLi1XVqqrtbX1DXOzdiuTTBDaJglPRNfDknIW07ZiitNuKiiOPE473vBynHfuqJAsiW/UKKX9CA9iFjCClbZcc99ROHOdVLJzJxCY5GUfskO7+G4O7MI161bDmgj9KI4tu3liI7t06lCq5Zofjp+QLKKxIhxL2bOtVPVzLBQjnBZVJ5M0xWSIBzSfHFKgXW35KEiEfrFCE3dmDkdSjiJPT0ZYhfJ3Njb/zLxo9j+pFwqpP2P2MhWc9XMWp5miMZluE2QcqQSN0SGfCUoUH+kCE8H0GYiEWENTGnBV8/mCgP4v2keNZsO+tqAC27ADe2DDKVzAFbSgDQTu4RGe4Nl4MF6MtynIOaMkugUzMt4/AYtUp5I=</latexit><latexit sha1_base64="jVacdh2lV3DspR02V6TpCd0gcrI=">AAACLHicdVDLSsNAFJ3UV62vqEs3g0VQxJIRfHQhFNy4rGCs0JQwmUzaoZMHMxOhhHyDn+LKrX6FCxG37vwHp21EI3pg4N5zzmXuPV7CmVSW9WJUZmbn5heqi7Wl5ZXVNXN941rGqSDUJjGPxY2HJeUsorZiitObRFAcepx2vOH5WO/cUiFZHF2pUUJ7Ie5HLGAEK0255p6jcOo6iWRnTiAwyYp+wA5Q/t3so9w161bDmgD+KI4s1DxGEBVMHRRou+aH48ckDWmkCMdSdpGVqF6GhWKE07zmpJImmAxxn2aTQ3K4oykfBrHQL1JwwpZ8OJRyFHraGWI1kL+1Mfmn5oXl/6ReaED9EtlNVXDay1iUpIpGZLpNkHKoYjiODvpMUKL4SBeYCKbPgGSAdWhKB1zT+XyFAP8v7MNGs4EurXqrWQRVBVtgG+wCBE5AC1yANrABAXfgATyCJ+PeeDZejbeptWIUM5ugBOP9E8lhqRw=</latexit><latexit sha1_base64="jVacdh2lV3DspR02V6TpCd0gcrI=">AAACLHicdVDLSsNAFJ3UV62vqEs3g0VQxJIRfHQhFNy4rGCs0JQwmUzaoZMHMxOhhHyDn+LKrX6FCxG37vwHp21EI3pg4N5zzmXuPV7CmVSW9WJUZmbn5heqi7Wl5ZXVNXN941rGqSDUJjGPxY2HJeUsorZiitObRFAcepx2vOH5WO/cUiFZHF2pUUJ7Ie5HLGAEK0255p6jcOo6iWRnTiAwyYp+wA5Q/t3so9w161bDmgD+KI4s1DxGEBVMHRRou+aH48ckDWmkCMdSdpGVqF6GhWKE07zmpJImmAxxn2aTQ3K4oykfBrHQL1JwwpZ8OJRyFHraGWI1kL+1Mfmn5oXl/6ReaED9EtlNVXDay1iUpIpGZLpNkHKoYjiODvpMUKL4SBeYCKbPgGSAdWhKB1zT+XyFAP8v7MNGs4EurXqrWQRVBVtgG+wCBE5AC1yANrABAXfgATyCJ+PeeDZejbeptWIUM5ugBOP9E8lhqRw=</latexit><latexit sha1_base64="LuVUE5T73tziQCRPRa0qAlMOmgs=">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</latexit><latexit sha1_base64="LuVUE5T73tziQCRPRa0qAlMOmgs=">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</latexit><latexit sha1_base64="LuVUE5T73tziQCRPRa0qAlMOmgs=">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</latexit><latexit sha1_base64="LuVUE5T73tziQCRPRa0qAlMOmgs=">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</latexit><latexit sha1_base64="jVacdh2lV3DspR02V6TpCd0gcrI=">AAACLHicdVDLSsNAFJ3UV62vqEs3g0VQxJIRfHQhFNy4rGCs0JQwmUzaoZMHMxOhhHyDn+LKrX6FCxG37vwHp21EI3pg4N5zzmXuPV7CmVSW9WJUZmbn5heqi7Wl5ZXVNXN941rGqSDUJjGPxY2HJeUsorZiitObRFAcepx2vOH5WO/cUiFZHF2pUUJ7Ie5HLGAEK0255p6jcOo6iWRnTiAwyYp+wA5Q/t3so9w161bDmgD+KI4s1DxGEBVMHRRou+aH48ckDWmkCMdSdpGVqF6GhWKE07zmpJImmAxxn2aTQ3K4oykfBrHQL1JwwpZ8OJRyFHraGWI1kL+1Mfmn5oXl/6ReaED9EtlNVXDay1iUpIpGZLpNkHKoYjiODvpMUKL4SBeYCKbPgGSAdWhKB1zT+XyFAP8v7MNGs4EurXqrWQRVBVtgG+wCBE5AC1yANrABAXfgATyCJ+PeeDZejbeptWIUM5ugBOP9E8lhqRw=</latexit>

No periodicity in flux attachment

The same result works for the nonlocal self-dual models with modular parameter

Loop models can also be used to fermionize theories of the form

One obtains, consistently with the duality web

Page 19: Duality in Condensed Matter Physics and Field Theory · Duality in Statistical Mechanics • 2D Ising Model: Kramers- Wannier duality • Partition function as a sum over closed domain

Conclusions

• Loop models offer a bridge to the “derivation” of the web of dualities for relativistic theories

• The loop models are always interacting

• The scalar fields are never free

• Fractional spin plays a key role

• It is always possible to find a dual theory

• Periodicity of flux attachment, and SL(2,ℤ), is not a symmetry for relativistic theories

!19

Page 20: Duality in Condensed Matter Physics and Field Theory · Duality in Statistical Mechanics • 2D Ising Model: Kramers- Wannier duality • Partition function as a sum over closed domain

Some References on Dualities• M. E. Peskin, Ann. Phys. 113, 122 (1978)

• C. Dasgupta and B. I. Halperin, Phys. Rev. Lett. 47, 1556 (1981)

• A. M. Polyakov, Mod. Phys. Lett. A 3, 325 (1988)

• E. Witten, Comm. Math. Phys. 121, 351 (1989)

• C. P. Burgess and F. Quevedo, Nucl. Phys. B 421, 373 (1993)

• E. Fradkin and F. A. Schaposnik, Phys. Lett. B 338, 253 (1994)

• E. Fradkin and S. A. Kivelson, Nucl. Phys. B 474, 543 (1996)

• D. T. Son, Phys. Rev. X 5, 031027 (2015)

• N. Seiberg, T. Senthil, C. Wang, and E. Witten, Ann. Phys. 374, 395 (2016)

• A. Karch and D. Tong, Phys. Rev. X 6, 031043 (2016)

• O. Aharony, JHEP 02, 093 (2016)

• M. Metlitski and A. Vishwanath, Phys. Rev. B 93, 255151 (2016)

!20