Fusion of line Quantum and operators Conformal sigma models on
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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.
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