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Yang-Baxter deformations of the OSP(1|2) WZW model

Ali Eghbali, Yaghoub Samadi, Adel Rezaei-Aghdam

TL;DR

This work classifies inequivalent real r-matrices for $osp(1|2)$ by solving the graded (m)CYBE and, up to automorphisms, identifies five distinct CrM families. It then implements Yang-Baxter deformations of the $OSP(1|2)$ WZW model using these r-matrices, deriving explicit deformed backgrounds whose metric and $B$-field are modified by the deformation parameters. All CrMs are shown to be non-Abelian and non-unimodular, and the deformed backgrounds do not satisfy the graded generalized supergravity equations, consistent with the model not being a Green-Schwarz string. The undeformed background, when augmented by a Killing supervector $I$, does satisfy the graded generalized supergravity equations, illustrating a clear dichotomy between the undeformed and deformed theories and highlighting the rarity and technical richness of the osp(1|2) deformations.

Abstract

We obtain inequivalent classical r-matrices of the $osp(1|2)$ Lie superalgebra as real solutions of the graded (modified) classical Yang-Baxter equation, in such a way that the corresponding automorphism transformation is employed. Then, Yang-Baxter deformations of the Wess-Zumino-Witten model based on the OSP$(1|2)$ Lie supergroup are specified by super skew-symmetric classical r-matrices. In this regard, the effect coming from the deformation is reflected as the coefficient of both metric and $B$-field. Furthermore, it is shown that all resulting classical r-matrices are non-Abelian and also non-unimodular, which leads us to graded generalized supergravity equations. We show that the background of undeformed model is a solution of the graded generalized supergravity equations when supplemented by an appropriate supervector field obtaining from the linear combination of the Killing supervectors corresponding to the background, while the deformed models do not satisfy these equations. This is consistent with our expectations, since the deformed models under consideration do not describe a Green-Schwarz superstring. However, the deformed backgrounds are interesting, in particular in the OSP$(1|2)$ case they are rare, much hard technical work was needed to obtain them.

Yang-Baxter deformations of the OSP(1|2) WZW model

TL;DR

This work classifies inequivalent real r-matrices for by solving the graded (m)CYBE and, up to automorphisms, identifies five distinct CrM families. It then implements Yang-Baxter deformations of the WZW model using these r-matrices, deriving explicit deformed backgrounds whose metric and -field are modified by the deformation parameters. All CrMs are shown to be non-Abelian and non-unimodular, and the deformed backgrounds do not satisfy the graded generalized supergravity equations, consistent with the model not being a Green-Schwarz string. The undeformed background, when augmented by a Killing supervector , does satisfy the graded generalized supergravity equations, illustrating a clear dichotomy between the undeformed and deformed theories and highlighting the rarity and technical richness of the osp(1|2) deformations.

Abstract

We obtain inequivalent classical r-matrices of the Lie superalgebra as real solutions of the graded (modified) classical Yang-Baxter equation, in such a way that the corresponding automorphism transformation is employed. Then, Yang-Baxter deformations of the Wess-Zumino-Witten model based on the OSP Lie supergroup are specified by super skew-symmetric classical r-matrices. In this regard, the effect coming from the deformation is reflected as the coefficient of both metric and -field. Furthermore, it is shown that all resulting classical r-matrices are non-Abelian and also non-unimodular, which leads us to graded generalized supergravity equations. We show that the background of undeformed model is a solution of the graded generalized supergravity equations when supplemented by an appropriate supervector field obtaining from the linear combination of the Killing supervectors corresponding to the background, while the deformed models do not satisfy these equations. This is consistent with our expectations, since the deformed models under consideration do not describe a Green-Schwarz superstring. However, the deformed backgrounds are interesting, in particular in the OSP case they are rare, much hard technical work was needed to obtain them.

Paper Structure

This paper contains 13 sections, 1 theorem, 50 equations.

Table of Contents

  1. Introduction
  2. Brief review of YB deformation of WZW model on a Lie supergroup
  3. YB deformations of the OSP$(1|2)$ WZW model
  4. The OSP$(1|2)$ WZW model
  5. Inequivalent solutions of the graded (m)CYBE
  6. YB deformed backgrounds of the OSP$(1|2)$ WZW model
  7. Deformation by the ${r_{_{i, \epsilon}}}$: YB deformed backgrounds OSP$(1|2)_{{i, \epsilon=\pm 1}}^{(\eta,\tilde{A},\kappa)}$
  8. Deformation by the ${r_{_{ii, \epsilon}}}$: YB deformed backgrounds OSP$(1|2)_{{ii, \epsilon=\pm 1}}^{(\eta,\tilde{A},\kappa)}$
  9. Deformation by the ${r_{_{\omega}}}$: YB deformed background OSP$(1|2)_{{\omega}}^{(\eta,\tilde{A},\kappa)}$
  10. Non-unimodularity and solutions of graded generalized supergravity equations
  11. The OSP$(1|2)$ WZW model and the graded generalized supergravity equations
  12. The YB deformed backgrounds OSP$(1|2)_{{i, \epsilon=\pm 1}}^{(\eta,\tilde{A},\kappa)}$ and the graded generalized supergravity equations
  13. Conclusions

Key Result

Theorem 1

Any real CrM of the $osp(1|2)$ Lie superalgebra as a solution of the graded (m)CYBE belongs just to one of the following five inequivalent classes

Theorems & Definitions (1)

  • Theorem 1