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

Strings

S = −g

∫dτdσ γαβ∂αX

M∂βXNGMN

Use uniform light-cone gauge in first order formalism

∫dτdσ

µ −H+γ01

γ00C1 +

2gγ00C2

)H = −p−

Strings

Solve Virasoro

C1 = 0, C2 = 0,−→ x ′− = · · · , p− = · · ·

H = H2 +1

gH4 + · · ·

Free hamiltonian + higher order correctionsg as loop counting parameter

Strings

Solve Virasoro

C1 = 0, C2 = 0,−→ x ′− = · · · , p− = · · ·

H = H2 +1

gH4 + · · ·

Free hamiltonian + higher order correctionsg as loop counting parameter

Strings

Quantization

x(σ, τ) ∼∫

2√ωp

(e ipσa(p, τ) + e−ipσa†(p, τ)

H2 −→ free harmonic oscillators

H4 −→ interaction term for tree-level scattering

· · · higher orders

Strings

Quantization

x(σ, τ) ∼∫

2√ωp

(e ipσa(p, τ) + e−ipσa†(p, τ)

H2 −→ free harmonic oscillators

H4 −→ interaction term for tree-level scattering

· · · higher orders

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)

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)

Integrability

• No particle production

• Permutation of flavors

Integrability

Braiding (and Physical) Unitarity

S12 S21 = 1

Integrability

• Factorization of scattering: Yang-Baxter equation

S23 S13 S12 = S12 S13 S23

Integrability

Crossing symmetry

Integrability

Compatibility with symmetries

[J,S] = 0, J ∈ A

Hamiltonian H is central element of A

Non-relativistic dispersion relation

√m2 + 16g2 sin2 p

Integrability

Compatibility with symmetries

[J,S] = 0, J ∈ A

Hamiltonian H is central element of A

Non-relativistic dispersion relation

√m2 + 16g2 sin2 p

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

Integrability

C |{pj}〉 = g(e i∑

j pj − 1) |{pj}〉

Q |p, q〉 = (Q(p)× 1 + e ip × Q(q)) |p, q〉

Integrability

C |{pj}〉 = g(e i∑

j pj − 1) |{pj}〉

Q |p, q〉 = (Q(p)× 1 + e ip × Q(q)) |p, q〉

Integrability

C |{pj}〉 = g(e i∑

j pj − 1) |{pj}〉

Q |p, q〉 = (Q(p)× 1 + e ip × Q(q)) |p, q〉

AdS3 × S3 × T 4

For AdS5 × S5 see:

[Arutyunov, Frolov, ‘09]

[Beisert et al., ‘11]

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 × 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 × S3 × T 4

A = centrally extended psu(1|1)4

4 copies of psu(1|1)

{Q,S} = H

QLi ,S

Ri , i = 1, 2

{QL,Ri ,SL,R

j } = δijHL,R , H = HL + HR

{QLi ,Q

Rj } = δijC, {SL

i ,SRj } = δijC

AdS3 × S3 × T 4

{Q,S} = H

QLi ,S

Ri , i = 1, 2

{QL,Ri ,SL,R

{QLi ,Q

Rj } = δijC, {SL

i ,SRj } = δijC

AdS3 × S3 × T 4

{Q,S} = H

QLi ,S

Ri , i = 1, 2

{QL,Ri ,SL,R

{QLi ,Q

Rj } = δijC, {SL

i ,SRj } = δijC

AdS3 × S3 × T 4

Action of left supercharges on left excitations

|Φ++〉

|Φ−+〉 |Φ+−〉

|Φ−−〉

−QL2

−SL2

C |Φ〉 6= 0 −→ QR |Φ〉 6= 0

AdS3 × S3 × T 4

Action of left supercharges on left excitations

|Φ++〉

|Φ−+〉 |Φ+−〉

|Φ−−〉

−QL2

−SL2

C |Φ〉 6= 0 −→ QR |Φ〉 6= 0

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 × S3 × T 4

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]

• · · ·

Thank you!

AdS3/CFT2 and Integrability - fileRiccardo Borsato AdS 3/CFT 2 and Integrability Munich, 9 November...

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Transcript of AdS3/CFT2 and Integrability - fileRiccardo Borsato AdS 3/CFT 2 and Integrability Munich, 9 November...