AdS3/CFT2 and Integrability - fileRiccardo Borsato AdS 3/CFT 2 and Integrability Munich, 9 November...
Transcript of AdS3/CFT2 and Integrability - fileRiccardo Borsato AdS 3/CFT 2 and Integrability Munich, 9 November...
Riccardo Borsato
AdS3/CFT2 and Integrability
Munich, 9 November 2013.
AdS3/CFT2 and Integrability Riccardo Borsato
• Strings
• Integrability
• AdS3 × S3 × T 4
Based on 1303.5995, 1306.2512in collaboration with O. Ohlsson Sax, A. Sfondrini, B. Stefanski jr.& A. Torrielli
AdS3/CFT2 and Integrability Riccardo Borsato
Strings
AdS3/CFT2 and Integrability Riccardo Borsato
Strings
S = −g
2
∫dτdσ γαβ∂αX
M∂βXNGMN
Use uniform light-cone gauge in first order formalism
S =
∫dτdσ
(pµx
µ −H+γ01
γ00C1 +
1
2gγ00C2
)H = −p−
AdS3/CFT2 and Integrability Riccardo Borsato
Strings
Solve Virasoro
C1 = 0, C2 = 0,−→ x ′− = · · · , p− = · · ·
H = H2 +1
gH4 + · · ·
Free hamiltonian + higher order correctionsg as loop counting parameter
AdS3/CFT2 and Integrability Riccardo Borsato
Strings
Solve Virasoro
C1 = 0, C2 = 0,−→ x ′− = · · · , p− = · · ·
H = H2 +1
gH4 + · · ·
Free hamiltonian + higher order correctionsg as loop counting parameter
AdS3/CFT2 and Integrability Riccardo Borsato
Strings
Quantization
x(σ, τ) ∼∫
dp
2√ωp
(e ipσa(p, τ) + e−ipσa†(p, τ)
)
H2 −→ free harmonic oscillators
H4 −→ interaction term for tree-level scattering
· · · higher orders
AdS3/CFT2 and Integrability Riccardo Borsato
Strings
Quantization
x(σ, τ) ∼∫
dp
2√ωp
(e ipσa(p, τ) + e−ipσa†(p, τ)
)
H2 −→ free harmonic oscillators
H4 −→ interaction term for tree-level scattering
· · · higher orders
AdS3/CFT2 and Integrability Riccardo Borsato
Integrability
AdS3/CFT2 and Integrability Riccardo Borsato
Integrability
Classical and quantum integrability [2 × Zamolodchikov, ‘78]
Enough conserved quantities such that: in a N-particle scatteringthe number of particles and the set of momenta {pi , i = 1, · · · ,N}are conserved
N-particle scattering factorised as a sequence of 2-particlescatterings
2-body S-matrix as the fundamental object
S |X (in)p Y(in)
q 〉 = |Y(out)q X (out)
p 〉
AdS3/CFT2 and Integrability Riccardo Borsato
Integrability
Classical and quantum integrability [2 × Zamolodchikov, ‘78]
Enough conserved quantities such that: in a N-particle scatteringthe number of particles and the set of momenta {pi , i = 1, · · · ,N}are conserved
N-particle scattering factorised as a sequence of 2-particlescatterings
2-body S-matrix as the fundamental object
S |X (in)p Y(in)
q 〉 = |Y(out)q X (out)
p 〉
AdS3/CFT2 and Integrability Riccardo Borsato
Integrability
• No particle production
• Permutation of flavors
AdS3/CFT2 and Integrability Riccardo Borsato
Integrability
• No particle production
• Permutation of flavors
AdS3/CFT2 and Integrability Riccardo Borsato
Integrability
• No particle production
• Permutation of flavors
AdS3/CFT2 and Integrability Riccardo Borsato
Integrability
Braiding (and Physical) Unitarity
S12 S21 = 1
AdS3/CFT2 and Integrability Riccardo Borsato
Integrability
• Factorization of scattering: Yang-Baxter equation
S23 S13 S12 = S12 S13 S23
AdS3/CFT2 and Integrability Riccardo Borsato
Integrability
Crossing symmetry
AdS3/CFT2 and Integrability Riccardo Borsato
Integrability
Compatibility with symmetries
[J,S] = 0, J ∈ A
Hamiltonian H is central element of A
Non-relativistic dispersion relation
E =
√m2 + 16g2 sin2 p
2
AdS3/CFT2 and Integrability Riccardo Borsato
Integrability
Compatibility with symmetries
[J,S] = 0, J ∈ A
Hamiltonian H is central element of A
Non-relativistic dispersion relation
E =
√m2 + 16g2 sin2 p
2
AdS3/CFT2 and Integrability Riccardo Borsato
Integrability
Two new central elements C,C†
C |{pj}〉 = g(e i∑
j pj − 1) |{pj}〉
C |· · ·〉phys = 0 (closed strings)
Off-shell statesC |· · ·〉 6= 0
Q |p, q〉 = (Q(p)× 1 + e ip × Q(q)) |p, q〉
Spin-chain interpretation
AdS3/CFT2 and Integrability Riccardo Borsato
Integrability
Two new central elements C,C†
C |{pj}〉 = g(e i∑
j pj − 1) |{pj}〉
C |· · ·〉phys = 0 (closed strings)
Off-shell statesC |· · ·〉 6= 0
Q |p, q〉 = (Q(p)× 1 + e ip × Q(q)) |p, q〉
Spin-chain interpretation
AdS3/CFT2 and Integrability Riccardo Borsato
Integrability
Two new central elements C,C†
C |{pj}〉 = g(e i∑
j pj − 1) |{pj}〉
C |· · ·〉phys = 0 (closed strings)
Off-shell statesC |· · ·〉 6= 0
Q |p, q〉 = (Q(p)× 1 + e ip × Q(q)) |p, q〉
Spin-chain interpretation
AdS3/CFT2 and Integrability Riccardo Borsato
Integrability
Two new central elements C,C†
C |{pj}〉 = g(e i∑
j pj − 1) |{pj}〉
C |· · ·〉phys = 0 (closed strings)
Off-shell statesC |· · ·〉 6= 0
Q |p, q〉 = (Q(p)× 1 + e ip × Q(q)) |p, q〉
Spin-chain interpretation
AdS3/CFT2 and Integrability Riccardo Borsato
AdS3 × S3 × T 4
For AdS5 × S5 see:
[Arutyunov, Frolov, ‘09]
[Beisert et al., ‘11]
AdS3/CFT2 and Integrability Riccardo Borsato
AdS3 × S3 × T 4
Type IIB, pure RR flux
Classically integrable [Babichenko, Stefanski, Zarembo, ‘10] [Zarembo, ‘10]
[Sundin, Wulff, ‘11]
Massive sector: [1303.5995]
4 bosons + 4 fermions,m=1
AdS3/CFT2 and Integrability Riccardo Borsato
AdS3 × S3 × T 4
Type IIB, pure RR flux
Classically integrable [Babichenko, Stefanski, Zarembo, ‘10] [Zarembo, ‘10]
[Sundin, Wulff, ‘11]
Massive sector: [1303.5995]
4 bosons + 4 fermions,m=1
AdS3/CFT2 and Integrability Riccardo Borsato
AdS3 × S3 × T 4
A = centrally extended psu(1|1)4
4 copies of psu(1|1)
{Q,S} = H
QLi ,S
Li ,Q
Ri ,S
Ri , i = 1, 2
{QL,Ri ,SL,R
j } = δijHL,R , H = HL + HR
{QLi ,Q
Rj } = δijC, {SL
i ,SRj } = δijC
†
AdS3/CFT2 and Integrability Riccardo Borsato
AdS3 × S3 × T 4
A = centrally extended psu(1|1)4
4 copies of psu(1|1)
{Q,S} = H
QLi ,S
Li ,Q
Ri ,S
Ri , i = 1, 2
{QL,Ri ,SL,R
j } = δijHL,R , H = HL + HR
{QLi ,Q
Rj } = δijC, {SL
i ,SRj } = δijC
†
AdS3/CFT2 and Integrability Riccardo Borsato
AdS3 × S3 × T 4
A = centrally extended psu(1|1)4
4 copies of psu(1|1)
{Q,S} = H
QLi ,S
Li ,Q
Ri ,S
Ri , i = 1, 2
{QL,Ri ,SL,R
j } = δijHL,R , H = HL + HR
{QLi ,Q
Rj } = δijC, {SL
i ,SRj } = δijC
†
AdS3/CFT2 and Integrability Riccardo Borsato
AdS3 × S3 × T 4
Action of left supercharges on left excitations
|Φ++〉
|Φ−+〉 |Φ+−〉
|Φ−−〉
+QL1
+SL1
+QL1
+SL1
+QL2
+SL2
−QL2
−SL2
C |Φ〉 6= 0 −→ QR |Φ〉 6= 0
AdS3/CFT2 and Integrability Riccardo Borsato
AdS3 × S3 × T 4
Action of left supercharges on left excitations
|Φ++〉
|Φ−+〉 |Φ+−〉
|Φ−−〉
+QL1
+SL1
+QL1
+SL1
+QL2
+SL2
−QL2
−SL2
C |Φ〉 6= 0 −→ QR |Φ〉 6= 0
AdS3/CFT2 and Integrability Riccardo Borsato
AdS3 × S3 × T 4
Impose symmetries, unitarity and LR-symmetryS-matrix decomposed in block form: LL, RR, LR
Yang-Baxter satisfied!
Completely fixed up to two scalar factors σpq, σpqConstraint from crossing symmetry [1306.2512]
σpqσpq = fpq, σpqσpq = gpq
Comparison to perturbative results[Sundin, Rughoonauth, Wulff, ‘12], [Sundin, Wulff ,‘13],
[Beccaria,Levkovich-Maslyuk,Macorini,Tseytlin, ‘12],[Abbott, ‘13],
[Engelund, McKeown, Roiban, ‘13]
AdS3/CFT2 and Integrability Riccardo Borsato
AdS3 × S3 × T 4
Impose symmetries, unitarity and LR-symmetryS-matrix decomposed in block form: LL, RR, LR
Yang-Baxter satisfied!
Completely fixed up to two scalar factors σpq, σpqConstraint from crossing symmetry [1306.2512]
σpqσpq = fpq, σpqσpq = gpq
Comparison to perturbative results[Sundin, Rughoonauth, Wulff, ‘12], [Sundin, Wulff ,‘13],
[Beccaria,Levkovich-Maslyuk,Macorini,Tseytlin, ‘12],[Abbott, ‘13],
[Engelund, McKeown, Roiban, ‘13]
AdS3/CFT2 and Integrability Riccardo Borsato
AdS3 × S3 × T 4
Impose symmetries, unitarity and LR-symmetryS-matrix decomposed in block form: LL, RR, LR
Yang-Baxter satisfied!
Completely fixed up to two scalar factors σpq, σpqConstraint from crossing symmetry [1306.2512]
σpqσpq = fpq, σpqσpq = gpq
Comparison to perturbative results[Sundin, Rughoonauth, Wulff, ‘12], [Sundin, Wulff ,‘13],
[Beccaria,Levkovich-Maslyuk,Macorini,Tseytlin, ‘12],[Abbott, ‘13],
[Engelund, McKeown, Roiban, ‘13]
AdS3/CFT2 and Integrability Riccardo Borsato
Conclusions
When possible, use Integrability!
Open questions
• include massless excitations (work in progress)
• consider wrapping corrections
• spin-chain interpretation for the dual CFT2
• understand the limit AdS3 × S3 × S3 × S1 →AdS3 × S3 × T 4
• extend the full description to mixed R-R and NS-NS fluxes[Hoare, Tseytlin ‘13], [Hoare, Stepanchuk, Tseytlin ‘13]
• · · ·
AdS3/CFT2 and Integrability Riccardo Borsato
Thank you!
AdS3/CFT2 and Integrability Riccardo Borsato