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Supergravity background of \(\lambda\)-deformed model for \(\mathrm{AdS}_{2} \times S^{2}\) supercoset. (English) Zbl 1332.81170
Summary: Starting with the \(\hat{F} / G\) supercoset model corresponding to the \(\mathrm{AdS}_n \times S^n\) superstring one can define the \(\lambda\)-model of [T. Hollowood et al., “An integrable deformation of the \(\mathrm{AdS}_5\times\mathrm S_5\) superstring”, Preprint, arxiv:1409.1538] either as a deformation of the \(\hat{F} / \hat{F}\) gauged WZW model or as an integrable one-parameter generalisation of the non-abelian T-dual of the \(AdS_n \times S^n\) superstring sigma model with respect to the whole supergroup \(\hat{F}\). Here we consider the case of \(n = 2\) and find the explicit form of the 4d target space background for the \(\lambda\)-model for the \(\operatorname{PSU}(1, 1 | 2) / \operatorname{SO}(1, 1) \times \operatorname{SO}(2)\) supercoset. We show that this background represents a solution of type IIB 10d supergravity compactified on a 6-torus with only metric, dilaton {\(\Phi\)} and the RR 5-form (represented by a 2-form \(F\) in 4d) being non-trivial. This implies that the \(\lambda\)-model is Weyl invariant at the quantum level and thus defines a consistent superstring sigma model. The supergravity solution we find is different from the one in [K. Sfetsos et al., arxiv:1410.1886, 35 p. (2014)] which should correspond to a version of the \(\lambda\)-model where only the bosonic subgroup of \(\hat{F}\) is gauged. Still, the two solutions have equivalent scaling limit of [B. Hoare and the second author, arxiv:1504.07213, 28 p. (2015)] leading to the isometric background for the metric and \(e^{\Phi} F\) which is related to the \(\eta\)-deformed \(\mathrm{AdS}_2 \times S^2\) sigma model of [F. Delduc et al., “Integrable deformation of the \(\mathrm{AdS}_5\times\mathrm S_5\) superstring action”, Phys. Rev. Lett. 112, No. 5, Article ID 051601, 5 p. (2014; doi:10.1103/PhysRevLett.112.051601)]. Similar results are expected in the \(\mathrm{AdS}_3 \times S^3\) and \(\mathrm{AdS}_5 \times S^5\) cases.

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T60 Supersymmetric field theories in quantum mechanics
14D15 Formal methods and deformations in algebraic geometry
83E50 Supergravity
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
Full Text: DOI arXiv
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