[Paper Review] Supercoset construction of Yang-Baxter deformed AdS$_5 imes$S$^5$ backgrounds

Hideki Kyono, Kentaroh Yoshida|arXiv (Cornell University)|May 9, 2016

Black Holes and Theoretical PhysicsPhysics and Astronomy58 references22 citations

TL;DR

This paper develops a supercoset construction framework to derive Yang-Baxter deformed AdS$_5\times$S$^5$ backgrounds in type IIB supergravity, explicitly computing the full metric, Ramond-Ramond fluxes, and dilaton from classical $r$-matrices. It establishes a master formula for the dilaton in terms of $r$-matrices and confirms perfect agreement with known backgrounds—such as $\gamma$-deformations and Schr"odinger spacetimes—even in non-supersymmetric cases, demonstrating the robustness of the supercoset method beyond maximally supersymmetric settings.

ABSTRACT

We proceed to study Yang-Baxter deformations of the AdS$_5 imes$S$^5$ superstring with the classical Yang-Baxter equation. We make a general argument on the supercoset construction and present the master formula to describe the dilaton in terms of classical $r$-matrices. The supercoset construction is explicitly performed for some classical $r$-matrices and the full backgrounds including the Ramond-Ramond (R-R) sector and dilaton are derived. Within the class of abelian $r$-matrices, the perfect agreement is shown for well-known examples including gravity duals of non-commutative gauge theories, $γ$-deformations of S$^5$ and Schrödinger spacetimes. It would be remarkable that the supercoset construction works well, even if the resulting backgrounds are not maximally supersymmetric. In particular, three-parameter $γ$-deformations of S$^5$ and Schrödinger spacetimes do not preserve any supersymmetries. As for non-abelian $r$-matrices, we will focus upon a specific example. The resulting background does not satisfy the equation of motion of the Neveu-Schwarz-Neveu-Schwarz (NS-NS) two-form because the R-R three-form is not closed.

Motivation & Objective

• To extend the supercoset construction to Yang-Baxter deformed AdS$_5\times$S$^5$ backgrounds beyond maximally supersymmetric cases.
• To derive the complete type IIB supergravity background, including the Ramond-Ramond (R-R) sector and dilaton, from classical $r$-matrices.
• To establish a general formula for the dilaton in terms of classical $r$-matrices for arbitrary Yang-Baxter deformations.
• To verify the consistency of the supercoset method with known integrable deformations, including non-abelian $r$-matrices and non-supersymmetric backgrounds.
• To investigate whether the resulting backgrounds satisfy the equations of motion of type IIB supergravity, particularly the Bianchi identities for R-R fluxes.

Proposed method

• The supercoset construction is applied to the $\mathfrak{su}(2,2|4)$ superalgebra, using a left-invariant current and a $\mathbb{Z}_4$-graded decomposition to derive the deformed Lagrangian.
• A canonical form of the Lagrangian is obtained via field redefinitions involving shifts of $X$ and rotations of $\theta$, enabling systematic extraction of background fields.
• The dilaton is derived from the $r$-matrix through a master formula involving the supertrace of products of $r$-matrix components and gamma matrices.
• The method is applied to abelian $r$-matrices, reproducing known backgrounds such as $\gamma$-deformations of S$^5$, non-commutative gauge theory duals, and Schr"odinger spacetimes.
• For non-abelian $r$-matrices, a specific example is analyzed, showing that the resulting background fails to satisfy the NS-NS two-form equation of motion due to a non-closed R-R three-form.
• The $16\times16$ and $32\times32$ matrix representations of $\mathfrak{su}(2,2|4)$ are used to lift spinor bilinears and extract component fields of type IIB supergravity.

Experimental results

Research questions

• RQ1 Can the supercoset construction be systematically applied to Yang-Baxter deformed AdS$_5\times$S$^5$ backgrounds with general classical $r$-matrices?
• RQ2 What is the general formula for the dilaton in terms of the classical $r$-matrix in such deformations?
• RQ3 Does the supercoset method consistently reproduce known integrable backgrounds like $\gamma$-deformations and Schr"odinger spacetimes, even when they break all supersymmetries?
• RQ4 Why do some deformed backgrounds fail to satisfy the equations of motion of type IIB supergravity, particularly the Bianchi identity for the R-R three-form?
• RQ5 Can the supercoset method be extended to non-abelian $r$-matrices, and what are the implications for the consistency of the resulting backgrounds?

Key findings

• The supercoset construction successfully reproduces the full background, including the dilaton, for all known abelian $r$-matrix deformations such as $\gamma$-deformations of S$^5$, non-commutative gauge theory duals, and Schr"odinger spacetimes.
• A master formula for the dilaton is derived in terms of the classical $r$-matrix, valid for any $r$-matrix satisfying the classical Yang-Baxter equation.
• The method works even for backgrounds that break all supersymmetries, such as three-parameter $\gamma$-deformations of S$^5$ and Schr"odinger spacetimes, indicating the robustness of the supercoset approach.
• For a specific non-abelian $r$-matrix, the resulting background does not satisfy the equation of motion for the NS-NS two-form because the R-R three-form is not closed, signaling a breakdown of standard supergravity equations.
• The analysis confirms that the supercoset method is consistent with the interpretation of these deformations as TsT transformations, particularly in the abelian case.
• The $32\times32$ matrix representation is systematically constructed from the $16\times16$ one via insertion of $\Gamma_{01234}$, enabling the extraction of all component fields in type IIB supergravity.