An Algebraic Classification of Solution Generating Techniques

Abstract

We consider a two-fold problem: on the one hand, the classification of a family of solution-generating techniques in (modified) supergravity and, on the other hand, the classification of a family of canonical transformations of 2-dimensional $\sigma$-models giving rise to integrable-preserving transformations. Assuming a generalised Scherk-Schwarz ansatz, in fact, the two problems admit essentially the same algebraic formulation, emerging from an underlying double Lie algebra $\mathfrak{d}$. After presenting our derivation of the classification, we discuss in detail the relation to modified supergravity and the additional conditions to recover the standard (unmodified) supergravity. Starting from our master equation - that encodes all the possible continuous deformations allowed in the family of solution-generating techniques - we show that these are classified by the Lie algebra cohomologies $H^2(\mathfrak{h},\mathbb{R})$ and $H^3(\mathfrak{h},\mathbb{R})$ of the maximally isotropic subalgebra $\mathfrak{h}$ of the double Lie algebra $\mathfrak{d}$. {We illustrate our results with a non-trivial example, the bi-Yang-Baxter-Wess-Zumino model.

arXiv comment8 pages

References in corpus (21)

• Review of AdS/CFT Integrability: An Overview
• On integrability of the Yang-Baxter $\si$-model
• An integrable deformation of the AdS_5 x S^5 superstring action
• Duality Symmetries from Non--Abelian Isometries in String Theories
• Dual Non-Abelian Duality and the Drinfeld Double
• Double Field Theory: A Pedagogical Review
• Poisson-Lie T-duality and Loop Groups of Drinfeld Doubles
• Scale invariance of the eta-deformed AdS5 x S5 superstring, T-duality and modified type II equations
• On Nonabelian Duality
• On Non-Abelian Duality
• $\eta$ and $\lambda$ deformations as $\mathcal{E}$-models
• Integrable deformations of T-dual $\sigma$ models
• A Problem with Non-Abelian Duality
• Poisson-Lie T-Duality in Double Field Theory
• Remarks on Non-Abelian Duality
• Poisson-Lie T-duals of the bi-Yang-Baxter models
• Quantum correction to generalized T-dualities
• ${\alpha }^{\prime }$-corrected Poisson-Lie T-duality
• Yang-Baxter Deformation as an O(d,d) Transformation
• On current algebras, generalised fluxes and non-geometry
• Strong integrability of the bi-YB-WZ model

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• Non-isometric U-dualities
• Double Field Theory for Generalized $\lambda$-deformation
• Pin(d,d) covariance of pure spinor equations for supersymmetric vacua and Non-Abelian T-duality