Kac-Moody and Borcherds Symmetries of Six-Dimensional Chiral Supergravity

TL;DR

This paper explores the conjectured infinite-dimensional symmetries of six-dimensional chiral supergravity, analyzing geodesic equations on a coset space and describing the p-form hierarchy via a Borcherds superalgebra.

Contribution

It provides a detailed analysis of the hidden symmetries and the p-form hierarchy in six-dimensional chiral supergravity, connecting them to Kac-Moody and Borcherds algebras.

Findings

Geodesic equations match supergravity equations up to dual graviton level.

Self-duality condition is automatically satisfied without dual potential.

Constructed the Borcherds superalgebra for the p-form hierarchy.

Abstract

We investigate the conjectured infinite-dimensional hidden symmetries of six-dimensional chiral supergravity coupled to two vector multiplets and two tensor multiplets, which is known to possess the $F_{4,4}$ symmetry upon dimensional reduction to three spacetime dimensions. Two things are done. (i) First, we analyze the geodesic equations on the coset space $F_{4,4}^{++}/K(F_{4,4}^{++})$ using the level decomposition associated with the subalgebra $\mathfrak{gl}(5)\oplus \mathfrak{sl}(2)$ of $F_{4,4}^{++}$ and show their equivalence with the bosonic equations of motion of six-dimensional chiral supergravity up to the level where the dual graviton appears. In particular, the self-duality condition on the chiral $2$-form is automatically implemented in the sense that no dual potential appears for that $2$-form, in contradistinction with what occurs for the non chiral $p$-forms. (ii) Second, we describe the $p$-form hierarchy of the model in terms of its $V$-duality Borcherds superalgebra, of which we compute the Cartan matrix.