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A new class of integrable deformations of CFTs
Received: December
new class of integrable deformations of CFTs
Open Access 0 1
c The Authors. 0 1
Theory, Field Theories in Lower Dimensions
0 National and Kapodistrian University of Athens , Athens 15784 , Greece
1 Ag. Paraskevi, GR-15310 Athens , Greece
2 Department of Nuclear and Particle Physics, Faculty of Physics
3 Institute of Nuclear and Particle Physics, National Center for Scienti c Research Demokritos
We construct a new class of integrable -models based on current algebra theories for a general semisimple group G by utilizing a left-right asymmetric gauging. Their action can be thought of as the all-loop e ective action of two independent WZW models for G both at level k, perturbed by current bilinears mixing the di erent WZW models. A non-perturbative symmetry in the couplings parametric space is revealed. We perform the Hamiltonian analysis of the action and demonstrate integrability in several cases. We extend our construction to deformations of G=H CFTs and show integrability when G=H is a symmetric space. Our method resembles that used for constructing the -deformed integrable -models, but the results are distinct and novel.
Integrable Field Theories; Sigma Models; Conformal Field Models in String
-
Constructing the new models
Contents
1 Introduction
3 Integrability 3.1 3.2 An example
Conclusions
Introduction
Lax pairs
Canonical treatment and charges in involution
of two independent WZW models both at level k.
This action is given by
Sk(g; g~; A ) = Sk(g) +
A A+
Tr tag~ 1D+g~)EabTr(tbg~ 1D g~ :
The WZW action Sk(g) for a group element g 2 G is given by1
Sk(g) =
2 Tr @+g 1
1Our conventions are
= 2d2 ;
Sk(g1g2) = Sk(g1) + Sk(g2)
A g~. Both lines are separately
invariant and so is the total action, under the transformation
g = [g; u] ;
@ u + [A ; u] ;
-model action
where the matrix
Sk; (g) = Sk(g) +
= k(kI + E) 1
J+a =
iTr(ta@+gg 1);
a =
iTr(tag 1
Dab = Tr(tagtbg 1) ;
The simplest case in which the matrix
is proportional to the identity is integrable [1]. In
tions in [4]. The action (1.6) for small deformation parameters becomes
Sk; (g) = Sk(g) +
matrix elements
ab will run under the renormalization group (RG)
ow. Indeed, the
computation of the corresponding RG
ow equations using gravitational methods was
performed in [2, 3] and is in agreement with results from
eld theoretical methods [8{
isotropic case, i.e. exact in ab =
ab, but for k
1, of current and primary eld operators
and related recent works can be found in [30{36].
SU(2)k=U(1) exact CFT.
Constructing the new models
gauged WZW action with [37]
Sk(g; A ; B ) = Sk(g) +
where now we use two di erent algebra valued elds A
and B . The above action is not
gauge invariant, but instead, under the in nitesimal transformation
g = guR
@ uL + [A ; uL] ;
@ uR + [B ; uR] ; (2.2)
forms as
PCMs as follows
Sk(g; A ; B ) =
Tr (A+@ uL
A @+uL)
(B+@ uR
B @+uR) : (2.3)
1 Z
Tr tag~1 1D+g~1)E1abTr(tbg~1 1D g~1
Tr tag~2 1D+g~2)E2abTr(tbg~2 1D g~2 ;
where we notice the interchange of the A
and B in the two gauged WZW actions.
A g~1 and D g~2 =
B g~2. The matrices E1 and E2 parametrizing the couplings.
This action is invariant under the transformation
g1 = g1uR
@ uL + [A ; uL] ;
g2 =
g2 = g2uL
@ uR + [B ; uR] :
2 Tr (B+@ uR
B @+uR) (A+@ uL
A @+uL) = 0 :
own. The using (2.4) we have that
Sk(g1; g2; A ; B ) =
2 Tr (A+@ uL
A @+uL) (B+@ uR
B @+uR)
we introduce the de nition
i = k(kI + Ei) 1 ;
i = 1; 2 ;
action is given by
Integrating out the gauge elds we nd that
and that
2 Tr A @+g1g1 1
B+g1 1@ g1 + A g1B+g1 1
A+g2 1@ g2 + B g2A+g2 1
A+ = i I
= i I
1 1T J1+ + D1 2T J2+ ;
1 J2 + D2T 2J1
B+ = i I
= i I
1 2T J2+ + D2 1T J1+ ;
2 J1 + D1T 1J2
the -model action written in matrix notation as
Sk; 1; 2(g1; g2) = Sk(g1) + Sk(g2)
J1+J2+
where we have de ned the matrices
12 = I
21 = I
proportional to the identity it has the following global symmetry
For other choices this symmetry is broken partially or all together.
For small elements of the matrices i's the action (2.11) becomes
Sk; 1; 2 (g1; g2) = Sk(g1) + Sk(g2) +
Due to the similarity with the well known
-deformations and taking into account
be named as a double -deformation.
Integrability
choices of the matrices
2. In particular, we will show that it is integrable for
the choices of the matrices
2 and only for those, which give rise to integrable
-deformed models corresponding to (1.6) (for
= 1 and
= 2, separately).
2Combining with the model interchanging symmetry this is equivalent to
2 ! 1 1 ; g1 ! g1 1 g2 ! g2 1 :
3It is also distinct from the case of (1.6) for the -deformation of the direct product Gk1
Gk2 WZW
model. For instance in that case the analog of (2.17) is
Sk1;k2; (g1; g2) = Sk1 (g1) + Sk2 (g2) +
ab(s1J1a+ + s2J2a+)(s1J1b + s2J2b ) +
respectively. Varying with respect to g1 and g2 we obtain that
D (D+g1g1 1) = F +(A) ;
D (D+g2g2 1) = F +(B) ;
F +(A) = @+A
[A+; A ] ;
F +(B) = @+B
[B+; B ] :
Equivalently, these can be written as
D+(g1 1D g1) = F +(B) ;
D+(g2 1 (...truncated)