Integrable 2d sigma models: Quantum corrections to geometry from RG flow

The so-called homogeneous Yang-Baxter (YB) deformations can be considered a non-abelian generalization of T-duality-shift-T-duality (TsT) transformations. TsT transformations are known to preserve conformal symmetry to all orders in α ′ . Here we argue that (unimodular) YB deformations of a bosonic string also preserve conformal symmetry, at least to two-loop order. We do this by showing that, starting from a background with no NSNS-flux, the deformed background solves the α ′ -corrected supergravity equations to second order in the deformation parameter. At the same time we determine the required α ′ -corrections of the deformed background, which take a relatively simple form. In examples that can be constructed using, possibly non-commuting sequences of, TsT transformations we show how to obtain the first α ′ -correction to all orders in the deformation parameter by making use of the α ′ -corrected T-duality rules. We demonstrate this on the specific example of YB deformations of a Bianchi type II background. sigma model with isometries. This was carried out for the Green-Schwarz superstring in [14] and rules for writing the supergravity background directly in terms of the R-matrix were derived. 2The simplest class of such YB deformations is when R is defined on an abelian subalgebra of the isometry algebra. In this case the deformation is equivalent to a T-duality-shift-T-duality (TsT) transformation [17]. These are also known as O(d, d)-transformations [18,19] and they have been argued to map a consistent string background to another consistent string background, i.e. there should exist corrections to the background fields such that the corrected background solves the α ′ -corrected supergravity equations to all orders in α ′ [20,21,22,23,24,25]. 3 Here we want to ask what happens for YB deformations in general at the quantum level. 4 Unimodular YB deformations are known to give a conformal theory at one loop, i.e. the background solves the (super)gravity equations. Here we will analyze the two-loop equations in the bosonic string case. For simplicity we will restrict to deformations of backgrounds with vanishing NSNS-flux. We will show, to second order in the deformation parameter, that the deformed background can be corrected so that it solves the 2-loop equations. Furthermore the correction to the background fields can be cast in a relatively simple form, giving hope that the result can be extended to all orders in the deformation parameter and perhaps higher orders in α ′ .Since the homogeneous YB deformations can be constructed using NATD, our results indicate that also NATD should preserve conformality at two loops, and possibly all orders in α ′ . Another piece of evidence for this comes from the recent analysis of renormalizability of deformed sigma models with two-dimensional target space in [28], and very recently [29]. Some of the deformations considered have a limit where they reduce to NATD and it was found that the models behave nicely beyond lowest order in α ′ suggesting that things should wor...

Section: Discussionmentioning

confidence: 53%

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confidence: 99%

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confidence: 99%

See 1 more Smart Citation

Two-loop conformal invariance for Yang-Baxter deformed strings

Borsato

Wulff

2020

“…This question of higher loop orders was addressed recently in [3], where we showed that, starting from 2 loops, renormalizability requires a specific deformation of the classical target space geometry, which may be interpreted as the result of adding finite local counterterms.…”

Section: Introductionmentioning

confidence: 95%

Integrable sigma models and 2-loop RG flow

Hoare

Levine

Tseytlin

2019

Self Cite

Following arXiv:1907.04737, we continue our investigation of the relation between the renormalizability (with finitely many couplings) and integrability in 2d σ-models. We focus on the "λ-model," an integrable model associated to a group or symmetric space and containing as special limits a (gauged) WZW model and an "interpolating model" for non-abelian duality. The parameters are the WZ level k and the coupling λ, and the fields are g, valued in a group G, and a 2d vector A ± in the corresponding algebra. We formulate the λ-model as a σ-model on an extended G×G×G configuration space (g, h,h), defining h andh byOur central observation is that the model on this extended configuration space is renormalizable without any deformation, with only λ running. This is in contrast to the standard σ-model found by integrating out A ± , whose 2-loop renormalizability is only obtained after the addition of specific finite local counterterms, resulting in a quantum deformation of the target space geometry. We compute the 2-loop β-function of the λ-model for general group and symmetric spaces, and illustrate our results on the examples of SU (2)/U (1) and SU (2). Similar conclusions apply in the non-abelian dual limit implying that non-abelian duality commutes with the RG flow. We also find the 2-loop β-function of a "squashed" principal chiral model. 1 bhoare@ethz.ch 2 n.levine17@imperial.ac.uk 3 Also at the Institute of Theoretical and Mathematical Physics, MSU and Lebedev Institute, Moscow. tseytlin@imperial.ac.uk 1 Our notation and conventions are summarized in Appendix A. In particular, we use hermitian generators T a of the Lie algebra so that if g = e v ∈ G then v = i T a v a ∈ Lie(G) is anti-hermitian. The action is defined as S = 1 4π d 2 σL so that L has extra factor of 2 compared to the "conventional" normalization.

“…The two-loop RG equation of this model was also recently considered in[44]. The background metric was modified by a quantum correction (determinant) arising from the integration of the gauge fields.…”

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confidence: 99%

An exact symmetry in λ-deformed CFTs

Georgiou

Sagkrioti

Sfetsos

et al. 2020

We consider λ-deformed current algebra CFTs at level k, interpolating between an exact CFT in the UV and a PCM in the IR. By employing gravitational techniques, we derive the two-loop, in the large k expansion, β-function. We find that this is covariant under a remarkable exact symmetry involving the coupling λ, the level k and the adjoint quadratic Casimir of the group. Using this symmetry and CFT techniques, we are able to compute the Zamolodchikov metric, the anomalous dimension of the bilinear operator and the Zamolodchikov C-function at two-loops in the large k expansion, as exact functions of the deformation parameter. Finally, we extend the above results to λ-deformed parafermionic algebra coset CFTs which interpolate between exact coset CFTs in the UV and a symmetric coset space in the IR.