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Self-Duality of Green-Schwarz Sigma-Models

Amit Dekel, Yaron Oz

TL;DR

The paper develops a general framework to test self-duality under fermionic T-duality for Green-Schwarz sigma-models on AdS backgrounds, using three algebraic conditions derived from the $\mathbb{Z}$-gradation and $\mathbb{Z}_4$ automorphism of superconformal algebras. By analyzing the flat-connection and its transformation under a $z$-dependent automorphism, the authors classify backgrounds into fully quantum-self-dual, classically self-dual, or non-self-dual cases, and identify new fermionic T-duality directions. They show that only $AdS_n\times S^n$ with $n=2,3,5$ are fully self-dual at the quantum level, while $AdS_n\times S^1$ ($n=2,3,5$) along with $AdS_4\times S^2$ and $AdS_2\times S^4$ are self-dual only at the classical level with dilaton shifts in the quantum theory. The results connect self-duality to Killing-form degeneracy and hint at deeper links to conformal invariance and dual-superconformal structures in holography, offering new directions for abelian subalgebra T-dualities beyond the standard $P$ and $Q$-directions.

Abstract

We study fermionic T-duality symmetries of integrable Green-Schwarz sigma-models on Anti-de-Sitter backgrounds with Ramond-Ramond fluxes, constructed as Z_4 supercosets of superconformal algebras. We find three algebraic conditions that guarantee self-duality of the backgrounds under fermionic T-duality, we classify those that satisfy them and construct the map of the monodromy matrix. We introduce new T-duality directions, where some of them contain no bosonic directions, along which the backgrounds are self-dual. We find that the only self-dual backgrounds are AdS_n x S^n for n=2,3,5. In addition we find that the backgrounds AdS_n x S^1 for n=2,3,5, AdS_4 x S^2 and AdS_2 x S^4 are self-dual at the level of the classical action, but have a non-trivial transformation of the dilaton.

Self-Duality of Green-Schwarz Sigma-Models

TL;DR

The paper develops a general framework to test self-duality under fermionic T-duality for Green-Schwarz sigma-models on AdS backgrounds, using three algebraic conditions derived from the -gradation and automorphism of superconformal algebras. By analyzing the flat-connection and its transformation under a -dependent automorphism, the authors classify backgrounds into fully quantum-self-dual, classically self-dual, or non-self-dual cases, and identify new fermionic T-duality directions. They show that only with are fully self-dual at the quantum level, while () along with and are self-dual only at the classical level with dilaton shifts in the quantum theory. The results connect self-duality to Killing-form degeneracy and hint at deeper links to conformal invariance and dual-superconformal structures in holography, offering new directions for abelian subalgebra T-dualities beyond the standard and -directions.

Abstract

We study fermionic T-duality symmetries of integrable Green-Schwarz sigma-models on Anti-de-Sitter backgrounds with Ramond-Ramond fluxes, constructed as Z_4 supercosets of superconformal algebras. We find three algebraic conditions that guarantee self-duality of the backgrounds under fermionic T-duality, we classify those that satisfy them and construct the map of the monodromy matrix. We introduce new T-duality directions, where some of them contain no bosonic directions, along which the backgrounds are self-dual. We find that the only self-dual backgrounds are AdS_n x S^n for n=2,3,5. In addition we find that the backgrounds AdS_n x S^1 for n=2,3,5, AdS_4 x S^2 and AdS_2 x S^4 are self-dual at the level of the classical action, but have a non-trivial transformation of the dilaton.

Paper Structure

This paper contains 41 sections, 157 equations, 2 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Properties of Superconformal Algebras
  3. The conformal basis and $\mathbb{Z}$-gradation
  4. $\mathbb{Z}_4$ automorphism
  5. More on $\mathbb{Z}$-gradations
  6. $\mathbb{Z}$-gradation of type I SCA's
  7. $\mathbb{Z}$-gradation of type II SCA's
  8. Green-Schwarz Sigma-models on Semi-Symmetric backgrounds
  9. The action
  10. Integrability
  11. T-duality of the GS Sigma-Models
  12. Assumptions
  13. T-duality of the Green-Schwarz sigma-model
  14. Flat-connection transformation under T-self-duality
  15. Quantum consistency of the T-Self-Duality transformation
  16. ...and 26 more sections

Figures (2)

  • Figure 1: The charge of the SCA's generators under $D$.
  • Figure 2: $\mathbb{Z}$-gradation of type-I SCA's with $4N$-odd generators and R-symmetry $\mathrm{SU}(2M)\times \mathrm{U}(1)$. The abelian subalgebras are circled. (a) Decomposition under $B$ and $D$. In this case the relevant $U(1)$'s are $(\pm)(D\pm B)$ (circled with solid blue contours) where the abelian subalgebra contains $d$-bosonic and $N$-fermionic generators, and $\pm 2B$ (circled with dashed red contours) where the abelian subalgebra contains $2N$-fermionic generators. (b) Decomposition under $B$ and $\check R$. In this case the relevant $U(1)$'s are $(\pm)(\check R\pm B)$ (circled with solid blue contours) where the abelian subalgebra contains $M^2$-bosonic and $2M$-fermionic generators, and $\pm 2B$ (circled with dashed red contours) where the abelian subalgebra contains $2N$-fermionic generators. (c) Decomposition under $D$ and $\check R$. In this case the relevant $U(1)$'s are $(\pm)(\check R\pm D)$ where the abelian subalgebra contains $d+M^2$-bosonic and $N$-fermionic generators. (d) $\mathbb{Z}$-gradation of type-II SCA's with $4N$-odd generators and R-symmetry $R_1\oplus\lambda_0\oplus R_{-1}$. The abelian subalgebras are circled. We have the decomposition under $\check \lambda$ and $D$. In this case the relevant $U(1)$'s are $(\pm)(D\pm \check \lambda)$ where the abelian subalgebra contains $d+\mathrm{dim}(R_1)$-bosonic and $N$-fermionic generators.