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Spacetimes for λ-deformations

Konstantinos Sfetsos, Daniel C. Thompson

TL;DR

Spacetimes for $\lambda$-deformations systematically constructs and embeds integrable, one-parameter deformations of WZW and coset CFTs into full type II string backgrounds. The approach combines gauging of PCM/WZW systems to produce NS backgrounds, then augments them with a dilaton and RR fluxes via a group-theoretic RR ansatz, yielding consistent supergravity solutions (often requiring type II$^\star$ in simple group cases). Explicit, worked examples include $AdS_3\times S^3\times T^4$ and $AdS_2\times S^2$ deformations, as well as coset deformations like $SO(4)/SO(3)$, with detailed limits: $\lambda\to 0$ recovering the undeformed CFTs and $\lambda\to 1$ yielding non-Abelian T-duals. The results support the interpretation of λ-deformations as worldsheet realizations of quantum group $q$-deformations at roots of unity and illuminate their global structure, singularities, and holographic implications, while highlighting challenges for global real embeddings and potential supersymmetry.

Abstract

We examine a recently proposed class of integrable deformations to two-dimensional conformal field theories. These λ-deformations interpolate between a WZW model and the non-Abelian T-dual of a Principal Chiral Model on a group G or, between a G/H gauged WZW model and the non-Abelian T-dual of the geometric coset G/H. λ-deformations have been conjectured to represent quantum group q-deformations for the case where the deformation parameter is a root of unity. In this work we show how such deformations can be given an embedding as full string backgrounds whose target spaces satisfy the equations of type-II supergravity. One illustrative example is a deformation of the Sl(2,R)/U(1) black-hole CFT. A further example interpolates between the $\frac{SU(2)\times SU(2)}{SU(2)}\times\frac{SL(2,R)\times SL(2,R)}{SL(2,R)} \times U(1)^4$ gauged WZW model and the non-Abelian T-dual of $AdS_3\times S^3\times T^4$ supported with Ramond flux.

Spacetimes for λ-deformations

TL;DR

Spacetimes for -deformations systematically constructs and embeds integrable, one-parameter deformations of WZW and coset CFTs into full type II string backgrounds. The approach combines gauging of PCM/WZW systems to produce NS backgrounds, then augments them with a dilaton and RR fluxes via a group-theoretic RR ansatz, yielding consistent supergravity solutions (often requiring type II in simple group cases). Explicit, worked examples include and deformations, as well as coset deformations like , with detailed limits: recovering the undeformed CFTs and yielding non-Abelian T-duals. The results support the interpretation of λ-deformations as worldsheet realizations of quantum group -deformations at roots of unity and illuminate their global structure, singularities, and holographic implications, while highlighting challenges for global real embeddings and potential supersymmetry.

Abstract

We examine a recently proposed class of integrable deformations to two-dimensional conformal field theories. These λ-deformations interpolate between a WZW model and the non-Abelian T-dual of a Principal Chiral Model on a group G or, between a G/H gauged WZW model and the non-Abelian T-dual of the geometric coset G/H. λ-deformations have been conjectured to represent quantum group q-deformations for the case where the deformation parameter is a root of unity. In this work we show how such deformations can be given an embedding as full string backgrounds whose target spaces satisfy the equations of type-II supergravity. One illustrative example is a deformation of the Sl(2,R)/U(1) black-hole CFT. A further example interpolates between the gauged WZW model and the non-Abelian T-dual of supported with Ramond flux.

Paper Structure

This paper contains 21 sections, 100 equations, 1 figure.

Table of Contents

  1. Introduction
  2. $\lambda$-deformations for groups
  3. Limit properties
  4. Towards a supergravity embedding
  5. Integrable deformation based on $SU(2)$ and $SL(2,{\rm I R})$
  6. The $SU(2)$ integrable deformation
  7. The $SL(2,{\rm I R})$ integrable deformation
  8. Embedding to supergravity
  9. $\lambda$-deformations for cosets
  10. $AdS_2\times S^2$ deformations
  11. Deforming the $SU(2)/U(1)$ exact CFT
  12. Deforming the $SL(2,\mathbb{R})/SO(1,1)$ and $SL(2,\mathbb{R})/U(1)$ exact CFTs
  13. Embedding to supergravity
  14. The non-Abelian T-dual limit and the near singularity region
  15. Global structure
  16. ...and 6 more sections

Figures (1)

  • Figure 1: Kruskal (left) and Penrose (right) diagram of deformed space time showing how the location of singularity (red lines) migrates. Shown in blue is the undeformed $\lambda = 0$ singularity at $uv =1$ and its corresponding horizon is displayed on the Penrose diagram in green. In red are the singular curves corresponding to $f(u,v)=0$ for $\lambda=(0.05,0.2,0.5)$ which forming increasingly sharper tear drop regions in the Penrose diagram as $\lambda \to 1$.