Fusion of line Quantum and operators Conformal sigma models on

of 46 /46
Conformal sigma models on supergroups Raphael Benichou VUB, Brussels 01/02/2011 Fusion of line operators Quantum integrability and in Based on arXiv:1011.3158 [hep-th]

Embed Size (px)

Transcript of Fusion of line Quantum and operators Conformal sigma models on

Raphael Benichou VUB, Brussels01/02/2011
Fusion of line operators
SWZ = − i
−1∂βgg −1∂γg)
(α) = P exp

Consequently the monodromy matrix Ω codes an infinite number of conserved charges, and the model is classically integrable.
1/44
3. Line operators and UV divergences
4. Fusion of line operators
5. Derivation of the Hirota equation
6. Conclusion
N=4 SYM
Energy of string states
In the classical string theory limit ⇔ planar gauge theory limit, integrable structures appear.
4/44
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
• Quantum integrability • String hypothesis.
Bombardelli, Fioravanti & Tateo, 2009
5/44
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)
+ ghosts PSl(n|n)
Sigma model on ⇔
First-principles derivation of the T-system 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)
The string worldsheet theory is a sigma- model on a supercoset coupled to ghosts.
6/44
2. Superstrings in RR backgrounds In 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
7/44
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
Hybrid string on AdS3xS3
8/44
The left-current is:
This implies that the following connection is flat:
S = 1
f2 Skin +
2η − 1
f2 SWZ
g(z, z) → hL g(z, z)hR hL,R ∈ PSl(n|n)
WZW model
∂jz + ∂jz = 0 d(dgg−1) = dgg−1dgg−1
A(α; z) = 2
1 + α jz(z)dz +
f2 Radius of
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,zta Generator of psl(n|n)
jaz,z(z)j b z,z(0) = κab(2nd− order pole) + fab
c j c z,z(1st− order pole) + ...
[ta, tb] = ifab ct
cNon-degenerate metric Structure constant
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.
c j c z,z(1st− order pole) + ...
jaz,z(z)j
Then all interaction vertices are proportional to structure constants.
fa cdf
12/44
All terms are of order
jaz (z)j b z(0) = f2η2
κab
κab
ifab cj
c z(z)
z + f2η2
ifab cj
c z(0)
f2
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 + ifa bc : j
c zj
b z : 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
η − 1
14/44
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)−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
κabta,R ⊗ tb,RMaillet, 1985 Maillet, 1986
15/44
operators
16/44
− b
17/44
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,a R (α) = P exp
− b
aN (α;σN )ta1 ...taNwith:
18/44
2nd-order poles
These divergences are cancelled by a scalar wave-function renormalization of the transition matrix:
a b...A(σi) A(σi+1)...
= c(2)
∝ (N − 1)κabtatb log T b,a N−2
∝ −(N − 2)κabtatb log T b,a N−2
19/44
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)...
∝ − log κabtaN−2tbA(σ1) A(σN )
20/44
• The two OPEs can be taken between distinct pairs of connections:
Transition matrix: Divergences at second order
a b...A(σi) A(σi+1)...
a b
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
∝ (−faiaj cf
cf cai
caj dtaitai+1 = 0
∝ fai+1aj cκ
caitaitai+1 = 0
+ a b...... ...A(σj)A(σi+2)
cai = 0
Renorm.(α) = (α) + # log κab(tatb(α)− ta(α)tb)
• Two OPEs taken between distinct pairs of connections:
+ 1
• The triple OPEs still do not contribute:
A(σ1) A(σN ) A(σi)... A(σ1) A(σN ) A(σi)...+
∝ faNai cκ
23/44
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
25/44
Collisions of integrated connections induce quantum corrections. We write the singular terms in the OPEs as:
a b
c d
R (β) = P exp
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

+
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.
27/44
AR(σi)

a bc d
The net result for the fusion of transition matrices at first order is:
lim →0+
R (β) = T b,a R (α)T d,c
R (β)
r + s
R (β)− χ(a; c, d)T b,a R (α)T d,a
R (β) r + s
+ χ(d; a, b)T b,d R (α)
r − s
R (β)− χ(c; a, b)T b,c R (α)T d,c
R (β) r − s
+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
AR(σi)
2. “Triple collisions”.
The second-order corrections contribute to the symmetric product of the transition matrices.
29/44
Fusion at second order: first contribution We start from the first-order result:
= a bc d a bc d
r − s
+
1st order
We pull the contours away and perform a second OPE: a b
c d
r − s
e− r+s 2 e
r−s 2
30/44
AR(σi)
... jz,z
We have to compute the additional corrections coming from: AR(σi+1)
jz,z
=
gh k


β − α
r−s 2
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:
∅ ??
2
d
d
= −1
2 Maillet, 1985 Maillet, 1986
• In the OPE formalism: the distance between the contours provides a natural regularization:
lim →0+
→0+
1
π
arctan

+ +
= 3
Contributions:
33/44
34/44
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: generalities 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)
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
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
TR(α) = χR(g(α))
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
Derivation of the Hirota equation I
We look for δ such that the previous quantity vanishes. We assume: d=f2 We perform an expansion in powers of f^2:

corrections from fusion
+Of^4 Previously we identified two kinds of quantum corrections:
3

β − α


etc.
37/44
j j =− f4 fef


R,R
=
= −f4 4π 2
= −4π2 f 4
38/44
From the derivative expansion
From the triple collisions
1− α2
− 4π2 f 4
If we choose:
We obtain:
We have derived from first principles the Hirota equation up to first non-trivial order in perturvation theory.
39/44
δ = −2πf2 (1 + α− 2η)2
1− α2 We have shown: for
We perform a change of variables such that the Hirota equation takes its usual form:
u = 1
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
40/44
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
No hypothesis
42/44
Superstrings in AdS3×S3 and integrability Strings in AdS3×S3 with RR fluxes can be described in the hybrid formalism:
Berkovits, Vafa & Witten, 1999
Hybrid string on AdS3xS3xT4
⇔ 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

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