Conformal sigma models on supergroups

Raphael BenichouVUB, Brussels01/02/2011

Fusion of line operators

Quantum integrability

and

in

Based on arXiv:1011.3158

[hep-th]

In this talk, we will consider the following model:

This model admits a one-parameter family of flat connections:

SWZ = − i

24π

Bd3yαβγTr(g−1∂αgg

−1∂βgg−1∂γg)

Skin =1

16π

d2zTr[−∂µg−1∂µg]

S =1

f2Skin + kSWZg

SupergroupPSl(n|n)

∀α, dA(α) +A(α) ∧A(α) = 0

Ω(α) = P exp

−

A(α)

Consequently the monodromy matrix Ω codes an infinite number of conserved charges, and the model is classically integrable.

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Plan1. Motivations

2. The quantum current algebra

3. Line operators and UV divergences

4. Fusion of line operators

5. Derivation of the Hirota equation

6. Conclusion

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Part 1:Motivations

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1. Integrability in AdS/CFT

Type IIB string theory on AdS5×S5

N=4 SYM

Conformal dimensions

Energy of string states

In the classical string theory limit ⇔ planar gauge theory limit, integrable structures appear.

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Y-system

T-system, or Hirota equation

A system of equations has been proposed to solve the spectrum problem:

Each string state corresponds to a solution of the Y-system with specific analytic properties.

Ta,s(u+ 1)Ta,s(u− 1) = Ta+1,s(u+ 1)Ta−1,s(u− 1) + Ta,s+1(u− 1)Ta,s−1(u+ 1)

Gromov, Kazakov & Vieira, 2009

⇔This approach relies on some crucial assumptions:

• Quantum integrability• String hypothesis.

Gromov, Kazakov, Kozak & Vieira, 2009

Bombardelli, Fioravanti & Tateo, 2009

Arutyunov & Frolov, 2009

The AdS/CFT Y- and T-systems

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The Y/T-systems can be derived using the Thermodynamic Bethe Ansatz machinery.

In this talk we will derive the T-system in a ‟toy model’’: a sigma-model on a supergroup.

The T-functions are believed to be the transfer matrices of the worldsheet theory. The T-system can presumably be derived from the computation of the fusion of transfer matrices.

Pure spinor string on AdS5xS5

Sigma model on PSU(2, 2|4)

SO(5)× SO(4, 1)

+ ghostsPSl(n|n)

Sigma model on ⇔

First-principles derivation of the T-systemTa,s(u+ 1)Ta,s(u− 1) = Ta+1,s(u+ 1)Ta−1,s(u− 1) + Ta,s+1(u− 1)Ta,s−1(u+ 1)

The string worldsheet theory is a sigma-model on a supercoset coupled to ghosts.

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2. Superstrings in RR backgroundsIn type II string theory, several fields can take a non-zero expectation value in the vacuum: metric, dilaton... and RR-fluxes.

We don’t know how to quantize string theory when RR fluxes are present.

Type II string theory vacua

Small curvature: Supergravity

No RR fluxes:RNS formalism ???

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Some ideas have been proposed to quantize string theory in RR backgrounds. Spacetime is embedded in a superspace. The worldsheet theory is a sigma model on a superspace coupled to a ghost system.

‟Pure spinor formalism’’, ‟Hybrid formalism’’...

None of these formalisms has been applied with success yet. Sigma models on superspaces need to be understood better. Sigma models on PSl(n|n) are a good starting point.

Berkovits et al.

These models also play a role in non-supersymmetric condensed matter system: Quantum Hall effect, disordered fermions...

Berkovits, Vafa & Witten, 1999

Zirnbauer, 1999 Guruswamy, LeClair & Ludwig, 1999

Hybrid string on AdS3xS3

Sigma model on

+ ghosts

⇔ PSU(1, 1|2)

Quantization of strings in RR backgrounds

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For instance:

Part 2:Current algebra

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The action is invariant under:

The left-current is:

The current is conserved and satisfies the Maurer-Cartan equation:

This implies that the following connection is flat:

S =1

f2Skin +

2η − 1

f2SWZ

Global symmetry and classical integrability

g(z, z) → hL g(z, z)hR hL,R ∈ PSl(n|n)

WZW model

jz = −η∂gg−1 jz = −(1− η)∂gg−1

∂jz + ∂jz = 0 d(dgg−1) = dgg−1dgg−1

A(α; z) =2

1 + αjz(z)dz +

2

1− αjz(z)dz

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η = 0, η = 1 :1

f2Radius of

target space⇔

,

∀α, dA(α) +A(α) ∧A(α) = 0

Current-current OPEs

The ellipses contain an infinite series of subleading singular terms, for instance:

The structure of the current-current OPEs is the following:

The current take value in the Lie super-algebra:

jz,z = jaz,ztaGenerator of psl(n|n)

jaz,z(z)jbz,z(0) = κab(2nd− order pole) + fab

c jcz,z(1st− order pole) + ...

[ta, tb] = ifabct

cNon-degenerate metricStructure constant

faecf

bed : jcz j

dz :

z

z

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Computation of the current algebra

The coefficients of the 2nd- and 1st-order poles follow respectively from the 2- and 3-points functions for the currents.

They can be computed to all orders in perturbation theory

⇒ Many loops vanish.

jaz,z(z)jbz,z(0) = κab(2nd− order pole) + fab

c jcz,z(1st− order pole) + ...

jaz,z(z)j

bz,z(w)

jaz,z(z)j

bz,z(w)j

cz,z(x)

g = exp(Xata)We write the group element as:

Then all interaction vertices are proportional to structure constants.

facdf

bcd = 0 Vanishing of the dual Coxeter number

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For PSl(n|n):

The current algebra encodes the Virasoro algebra:

All terms are of order

jaz (z)jbz(0) = f2η2

κab

z2+ f2η(2− η)

ifabc

z

jcz(z) + jcz(0)

2+ f2η2

ifabcz

z2jcz(z) + jcz(0)

2+ ...

jaz (z)jbz(0) = f2(1− η)2

κab

z2+ f2(1− η)(1 + η)

ifabc

z

jcz(z) + jcz(0)

2+ f2(1− η)2

ifabcz

z2jcz(z) + jcz(0)

2+ ...

jaz (z)jbz(0) = f2(1− η)2

ifabcj

cz(z)

z+ f2η2

ifabcj

cz(0)

z+ ...

The quantum current algebra

f2 Computation at p-th order in ⇔ Perform p OPEs.

f2

Ashok, Benichou & Troost, 2009

T =1

2f2η2κab : j

az j

bz :

c = −2with

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T (z)T (0) =c

2z4+

2T (0)

z2+

∂T (0)

z

The current algebra is compatible with the Maurer-Cartan equation:

Actually demanding the Maurer-equation to hold in the quantum theory fixes completely all OPEs of the form current-operator:

The Maurer-Cartan equation

jz,z(z)MC(0) = 0 MC = (1− η)∂jaz ta − η∂jaz ta + ifabc : j

czj

bz : tawith

MC(z)O(0) = 0 =⇒ jz,z(z)O(0)

This is enough to compute the conformal dimension of any operator. This method is very efficient for small (Konishi-like) operators. For instance:

Benichou & Troost, 2010

O(0) = jzΦRR : ∆ = 1 + c(2)R f2

η − 1

2

+ c(2)

Rf2 (1− η) +O(f4)

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The equal-time commutator is defined as:

From the current-current OPEs, we deduce the equal-time commutator of two connections:

We recognize a (r,s) Maillet system with:

This commutator is exact to all orders in perturbation theory.

OPEs and commutators

[A(σ, τ), B(σ, τ)] = lim→0+

(A(σ, τ + )B(σ, τ)−B(σ, τ + )A(σ, τ))

[AR(α;σ1), AR(β;σ2)] = 2sδ(σ1 − σ2)

+ [AR(α;σ1) +AR(β;σ2), r] δ(σ1 − σ2)

+ [AR(α;σ1)−AR(β;σ2), s] δ(σ1 − σ2)

AR(α; z) =2

1 + αjaz (z)t

(R)a dz +

2

1− αjaz (z)t

(R)a dz

r = πif2 2

α− β

(1 + β − 2η)2

(1 + β)(1− β)+

(1 + α− 2η)2

(1 + α)(1− α)

κabta,R ⊗ tb,R

s = πif2

4

(1 + α)(1 + β)η2 − 4

(1− α)(1− β)(1− η)2

κabta,R ⊗ tb,RMaillet, 1985 Maillet, 1986

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Part 3:Divergences in line

operators

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Line operators: definitions

• Transition matrix:

• Monodromy matrix:

• Transfer matrix:

Flat connection⇒ contour

independence

T b,aR (α) = P exp

− b

aAR(α)

ΩR(α) = P exp

−

AR(α)

TR(α) = STr ΩR(α)

a b

For simplicity, we only consider constant - t ime contours.

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Line operators and UV divergences

We expand the line operators:

Collisions of integrated operators lead to divergences. We need to regularize, and potentially renormalize the line operators. We use a “principal value” regularization scheme:

T b,aR (α) = P exp

− b

aAR(α)

=

∞

N=0

(−1)NT b,aN (α)

T b,aN (α) =

b>σ1>...>σN>adσ1...dσNAa1(α;σ1)...A

aN (α;σN )ta1 ...taNwith:

1

σ−→ P.V.

1

σ=

1

2

1

σ + i+

1

σ − i

=

σ

σ2 + 2

1

σ2−→ P.V.

1

σ2=

1

2

1

(σ + i)2+

1

(σ − i)2

=

σ2 − 2

(σ2 + 2)2

a bA(σ1)A(σ2)A(σN ) ...T b,aN :

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Transition matrix: Divergences at first order1st-order poles

2nd-order poles

These divergences are cancelled by a scalar wave-function renormalization of the transition matrix:

a b...A(σi) A(σi+1)...

jz,z

∝ fabctatb ∝ fab

cfabdtd = 0

a b...A(σi) A(σi+1)...

a b...A(σi) A(σi+1)... A(σi+2)

= c(2)

T b,aRenorm.(α) = (1 + #c(2) log )T b,a(α)

∝ (N − 1)κabtatb log Tb,aN−2

∝ −(N − 2)κabtatb log Tb,aN−2

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Monodromy matrix: Divergences at first order

Transfer matrix: No divergence at first order

...A(σi) A(σi+1)... A(σi+2)

There is a new source of divergence in the monodromy matrix:

...A(σi) A(σi+1)...

∝ −(N − 2)κabtatb log ΩN−2

∝ (N − 1)κabtatb log ΩN−2

∝ − log κabtaΩN−2tbA(σ1) A(σN )

ΩRenorm.(α) = Ω(α) + # log κab(tatbΩ(α)− taΩ(α)tb)

The renormalized monodromy matrix reads:

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• The two OPEs can be taken between distinct pairs of connections:

Transition matrix: Divergences at second order

a b...A(σi) A(σi+1)...

jz,z jz,z

...A(σj) A(σj+1)

etc.

T b,aRenorm.(α) =

1 + #c(2) log +

1

2(#c(2) log )2

T b,a(α)

• We also have to consider “triple OPEs”:

a b

jz,z

A A A

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This leads to a straightforward generalization of the previous result:

a b...A(σi) A(σi+1)... ...A(σj)

All divergences coming from the triple OPEs cancel:

a b...... ...+

jz,z jz,z

∝ (−faiajcf

cai+1d + fai+1aj

cfcai

d)taitai+1 = fai+1aicf

cajdtaitai+1 = 0

a b...A(σi) A(σi+1)... ...A(σj)

∝ fai+1ajcκ

caitaitai+1 = 0

a b...A(σi)A(σi+1)... ...A(σj)A(σi+2)

+a b...... ...A(σj)A(σi+2)

∝ faiajcκ

cai+2 + fai+2ajcκ

cai = 0

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A(σi) A(σi+1) A(σj)

A(σi)A(σi+1)

Monodromy matrix: Divergences at second order

ΩRenorm.(α) = Ω(α) + # log κab(tatbΩ(α)− taΩ(α)tb)

• Two OPEs taken between distinct pairs of connections:

+1

2(# log )2κabκcd(tatbtctdΩ(α)− 2tatbtcΩ(α)td + tatcΩ(α)tdtb)

• The triple OPEs still do not contribute:

A(σ1) A(σN ) A(σi)... A(σ1) A(σN ) A(σi)...+

∝ faNaicκ

ca1 + fa1aicκ

caN = 0

Transfer matrix: No divergence up to second order

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Renormalization of line operators: Summary

Up to second order in perturbation theory:

• Divergences in the transition and monodromy matrices are cancelled by a wave-function renormalization that depends on the representation.

• The transfer matrix is free of divergences.

Essentially these nice properties follow from the vanishing of the dual Coxeter number of PSl(n|n). For generic group the sprectral parameter has to be renormalized.

Bachas & Gaberdiel, 2004

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Part 4:Fusion of line operators

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Fusion of line operators

Collisions of integrated connections induce quantum corrections. We write the singular terms in the OPEs as:

a b

c d

a bc dFusion

Classically: lim→0+

T b+i,a+iR (α)T d,c

R (β) = P exp

− b

aAR(α)−

d

cAR(β)

τ = 0

τ = τ = 0

1

σ + i− σ =1

2

1

σ + i− σ +1

σ − i− σ

+

1

2

1

σ + i− σ −1

σ − i− σ

= P.V.1

σ − σ − i

(σ − σ)2 + 2

AR(α;σ + i)AR(β;σ) →

Divergence in the fused operator of the type studied previously.

Regularization of the delta function. Gives a fi n i t e c o r r e c t i o n associated with fusion.Mikhailov & Schafer-

Nameki, 2007

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AR(σi)

Fusion at first order

The relevant OPE reads:

a b

c d

We expand the line operators: =

M,N

a b

c d

AR(σ1)

AR(σ1)AR(σN )

AR(σj)

AR(σM ) ...

...

We compute:

M,N

i,j

a b

c d

...

...

...

...OPE

(1− P.V.)AR(α;σ + i)AR(β;σ) = sδ(σ − σ)

+

AR(α;σ),

r + s

2

δ(σ − σ) +

AR(β;σ),

r − s

2

δ(σ − σ)

This is nothing but the commutator: [AR(α;σ), AR(β;σ)]

The first-order corrections only contribute to the commutator of the transition matrices.

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AR(σi)

Fusion at first order

AR(σj)

With some efforts we can resum the series to get:

M,N

i,j

a b

c d

...

...

...

...OPE

lim→0+ =

a bc d

The net result for the fusion of transition matrices at first order is:

lim→0+

T b+i,a+iR (α)T d,c

R (β) = T b,aR (α)T d,c

R (β)

+ χ(b; c, d)T d,bR (β)

r + s

2T b,aR (α)T b,c

R (β)− χ(a; c, d)T b,aR (α)T d,a

R (β)r + s

2T b,cR (β)

+ χ(d; a, b)T b,dR (α)

r − s

2T d,aR (α)T d,c

R (β)− χ(c; a, b)T b,cR (α)T d,c

R (β)r − s

2T c,aR (α)

+O(f4)

This agrees with the result of Maillet for the commutator of transition matrices, derived in the Hamiltonian formalism. Maillet, 1986

a bc d

r − s

2r + s

2

-

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Fusion at second order

We have two different kinds of contributions:

AR(σi)

AR(σj)

a b

c d

...

...

...

...

AR(σk)

AR(σl)...

...

AR(σi)

AR(σj)

a b

c d

...

...

...

...

AR(σk) ...jz,z

1. Two OPEs between two distinct pairs of connections.

2. “Triple collisions”.

The second-order corrections contribute to the symmetric product of the transition matrices.

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Fusion at second order: first contribution We start from the first-order result:

=a bc d a bc d

r − s

2

r + s

2

-a b

c da bc d

+

1st order

We pull the contours away and perform a second OPE:a b

cd

r − s

2

AR

ARa b

cd

AR

AR a b

cd

AR

AR

r + s

2

→ + -

We obtain a natural generalization of the first-order result:

e−r+s2 e

r−s2

=

a b

c d

a bc d

+ + O(f6)

Additional corrections from triple collisions

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Fusion at second order: triple collisions

AR(σi)

AR(σj)

a b

cd

...

... ...

...jz,z

We have to compute the additional corrections coming from:AR(σi+1)

jz,z

AR(σi+2)AR(σi)

AR(σj)

a b

c d

...

... ...

...jz,z

AR(σi+1)

We obtain the following result:

=

a b

c d

a bc d

+ +O(f6)a bc d

with: j = −f4 fefgf

ghk

h1t

R(et

Rh) ⊗ tR

f + h2tRe ⊗ tR

(h tR

f)

jkz (σ) +

h1t

R(et

Rh) ⊗ tR

f + h2tRe ⊗ tR

(h tR

f)

jkz (σ)

d

aj

h1 = π2

1

2

2

1 + α

2

α− β

(1 + β − 2η)2

(1 + β)(1− β)+ (1− η)24

β − α

(1− α2)(1− β2)

2

1 + α

(1 + α− 2η)2

1− α2etc.

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e−r+s2 e

r−s2

The two regularization schemes are equivalent.

Fusion of line operators with coinciding endpoints• In the Hamiltonian formalism: amiguities appear because of non-

utra local terms in commutators ⇔ 2nd order poles in the OPEs.

Consider for instance:

a b

d

A(σ)

A(σ)

d

adσ

b

adσδ(σ − σ) =

d

adσ(δ(b− σ)− δ(a− σ))

∅ ??

Maillet introduced the following regularization: 1

2

d

adσ

b

a+ηdσδ(σ − σ) +

d

adσ

b

a−ηdσδ(σ − σ)

= −1

2Maillet, 1985 Maillet, 1986

• In the OPE formalism: the distance between the contours provides a natural regularization:

lim→0+

d

adσ

b

adσδ(σ − σ) = lim

→0+

1

π

arctan

b− a

− arctan

d− a

+ arctan

d− b

= −1

2

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+

Fusion of monodromy and transfer matrices

⇔

We work on the universal cover of the cylinder:

+ +

We obtain for the fusion of transfer matrices:

=3

4s2

+

j

+ O(f6)

∅[TR(α), TR(β)] = 0 +O(f6)⇒

Contributions:

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Part 5: The Hirota equation

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The T-functions are interpreted as the transfer matrices of the underlying theory. Using the fusion of transfer matrices that we computed previously, we want to show that the Hirota equation holds in the sigma-models on PSl(n|n).

The integrer indices (a,s) label representation of PSl(n|n). They take value in a T-shaped lattice.

The Hirota equation: generalitiesTa,s(u+ 1)Ta,s(u− 1) = Ta+1,s(u+ 1)Ta−1,s(u− 1) + Ta,s+1(u− 1)Ta,s−1(u+ 1)

a

s

n

p n-p

The precise shape of the lattice depends on the real form one considers. For PSU(p,n-p|n):

Gromov, Kazakov & Tsuboi, 2010

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The transfer matrix is a super-character:

The shifts in the spectral parameter come from quantum effects associated with fusion. We will demonstrate that up to 1st non-trivial order.

The classical limit of the Hirota equation

TR(α) = STr P exp

−

AR(α)

⇒

Characters of PSl(n|n) satisfy:

TR(α) = χR(g(α))

= g(α) ∈ PSl(n|n)

χ(a,s)(g(α))χ(a,s)(g(α)) = χ(a,s+1)(g(α))χ(a,s−1)(g(α)) + χ(a+1,s)(g(α))χ(a+1,s)(g(α))

Ta,s(u+ 1)Ta,s(u− 1) = Ta+1,s(u+ 1)Ta−1,s(u− 1) + Ta,s+1(u− 1)Ta,s−1(u+ 1)

Kazakov & Vieira, 2007

u 1 Classical limit

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We evaluate:

Derivation of the Hirota equation I

We look for δ such that the previous quantity vanishes. We assume: d=f2We perform an expansion in powers of f^2:

Ta,s(α+ δ)Ta,s(α− δ)− Ta+1,s(α+ δ)Ta−1,s(α− δ)− Ta,s−1(α+ δ)Ta,s+1(α− δ)

≡

R,R

TR(α+ δ)TR(α− δ)

R,R

TR(α+ δ)TR(α− δ) =

R,R

TR(α)TR(α) + δ

R,R

(∂αTR(α)TR(α)− TR(α)∂αTR(α))+Quantum

corrections from fusion

+Of^4Previously we identified two kinds of quantum corrections:

3

4s2TR(α)TR(β)

s ∝ f2

4

(1 + α)(1 + β)η2 − 4

(1− α)(1− β)(1− η)2

j

⇒ s2 = O(f4)

h1 = π2

1

2

2

1 + α

2

α− β

(1 + β − 2η)2

(1 + β)(1− β)+ (1− η)24

β − α

(1− α2)(1− β2)

2

1 + α

(1 + α− 2η)2

1− α2

j = −f4 fefgf

ghk

h1t

R(et

Rh) ⊗ tR

f + h2tRe ⊗ tR

(h tR

f)

jkz (σ) +

h1t

R(et

Rh) ⊗ tR

f + h2tRe ⊗ tR

(h tR

f)

jkz (σ)

α− β = O(f2) ⇒ j = O(f2)

etc.

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We consider:

Derivation of the Hirota equation II

j j =− f4 fef

gfgh

k

h1t

R(et

Rh) ⊗ tR

f + h2tRe ⊗ tR

(h tR

f)

jkz (σ)

+h1t

R(et

Rh) ⊗ tR

f + h2tRe ⊗ tR

(h tR

f)

jkz (σ)

R,R

We use the character identities:

R,R

STr(tatbg ⊗ tcg)fcb

dfda

e =

R,R

STr(teg ⊗ g)−

R,R

STr(g ⊗ teg)

R,R

STr(tag ⊗ tbtcg)fcdef

dba = −

R,R

STr(teg ⊗ g) +

R,R

STr(g ⊗ teg)Kazakov & Vieira, 2007

=

R,R

j j = f4((h1 + h2)j

ez + (h1 + h2)j

ez)(te ⊗ 1− 1⊗ te)

= −f4 4π2

δ

(1 + α− 2η)4

(1− α2)2∂αA

e(α)(te ⊗ 1− 1⊗ te) +O(f4)

= −4π2 f4

δ

(1 + α− 2η)4

1− α2

R,R

(∂αTR(α)TR(α)− TR(α)∂αTR(α)) +O(f4)

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R,R

TR(α+ δ)TR(α− δ) = 0 +O(f4)

Derivation of the Hirota equation III

We obtain eventually: Classical Hirota equation ⇒∅

From the derivative expansion

From the triple collisions

δ = −2πf2 (1 + α− 2η)2

1− α2

R,R

TR(α+ δ)TR(α− δ) =

R,R

TR(α)TR(α) + δ

R,R

(∂αTR(α)TR(α)− TR(α)∂αTR(α))

− 4π2 f4

δ

(1 + α− 2η)4

(1− α2)2

R,R

(∂αTR(α)TR(α)− TR(α)∂αTR(α)) +O(f4)

If we choose:

We obtain:

We have derived from first principles the Hirota equation up to first non-trivial order in perturvation theory.

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R,R

TR(α+ δ)TR(α− δ) = 0 +O(f4)

Derivation of the Hirota equation IV

δ = −2πf2 (1 + α− 2η)2

1− α2We have shown: for

We perform a change of variables such that the Hirota equation takes its usual form:

u =1

2πf2

α+ 2(2η − 1) log(α+ 1− 2η) +

1− (1− 2η)2

α+ 1− 2η

Ta,s(u+ 1)Ta,s(u− 1) = Ta+1,s(u+ 1)Ta−1,s(u− 1) + Ta,s+1(u− 1)Ta,s−1(u+ 1)⇒ +O(f4)

Notice that for η =1

2, ie when there is no Wess-Zumino term in the action:

u =1

2πf2

α+

1

α

This is the famous Zhukowsky map.

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Part 6: Conclusions

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We studied line operators in the sigma-models on PSl(n|n).

• We studied UV divergences up to second order in perturbation theory.

• We computed the fusion of line operators up to second order in perturbation theory.

• We deduced a perturbative proof of the Hirota equation.

Summary of the results

Comparison with the Thermodynamic Bethe Anzatz approach:

No hypothesis

Perturbative

The two approaches are complementary.

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Superstrings in AdS3×S3 and integrabilityStrings in AdS3×S3 with RR fluxes can be described in the hybrid formalism:

Berkovits, Vafa & Witten, 1999

Hybrid string on AdS3xS3xT4

Sigma model on

+ ghosts

⇔ PSU(1, 1|2)

This theory admits a consistent expansion in powers of the ghosts eφ, eφ.

We have proven that string theory on AdS3×S3 is governed by the Hirota equation at zeroth-order in the ghost expansion and at first non-trivial order in the semi-classical expansion.

1

f2ηThe parameters of

the sigma model are: ⇔ Spacetime radius

Ratio of RR and NSNS fluxes

⇔

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Superstrings in AdS5×S5 and integrability

• The pure spinor string on AdS5×S5 realizes a (r,s) Maillet system.

• The transfer matrix is free of divergences.

...at least up to first order in perturbation theory.

Mikhailov & Schafer-Nameki, 2007

Mikhailov & Schafer-Nameki, 2007

Magro, 2008

There is good hope that the computation we presented straightforwardly generalizes to this case.

To be continued.

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Thank you.