Two-loop conformal invariance for Yang-Baxter deformed strings

Abstract

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...

‐Corrected Poisson‐Lie T‐Duality

“…On the contrary, recently computed ′corrections of integrable deformations point very clearly in this direction. [8][9][10] Hence, the objective of this letter is to construct leading order ′ -corrections to the PL T-duality transformation rules in a bosonic -model and to argue that they preserve conformal invariance. Key to this endeavour are three techniques: The formulation of PL symmetric target space geometries in the framework of Double Field Theory (DFT), [11] the ′ -corrected DFT flux formulation introduced by Marqués and Nuñez, [12] and finite generalised Green-Schwarz (gGS) transformations recently presented by Borsato, López, and Wulff.…”

“…Key to this endeavour are three techniques: The formulation of PL symmetric target space geometries in the framework of Double Field Theory (DFT), [11] the ′ -corrected DFT flux formulation introduced by Marqués and Nuñez, [12] and finite generalised Green-Schwarz (gGS) transformations recently presented by Borsato, López, and Wulff. [10] PL T-duality and DFT: Directly at the level of the metric, Bfield, and dilaton, PL symmetric target spaces might look very complicated. But fortunately, their underlying structure becomes much simpler in the framework of DFT, [13] where they are expressed in the language of generalised geometry.…”

“…At leading order, the latter leave by definition the generalised metric invariant. This is the reason why we could safely ignore Λ̄āb in (10). Beyond that order, it has to be included and results in ′ corrected transformation rules.…”

“…While it should be possible to formally integrate , we find it more convenient to make an educated guess of how a finite counterpart might look like and then show that it is compatible with the infinitesimal transformations in [12]. A similar approach allowed [10] to present finite gGS transformations for the metric and B-field. However, at this level, the elegant structure of PL T-duality is not manifest.…”

“…Note added: During the completion of this letter, we learned of the closely related independent work. [26]…”