Duality, asymptotic charges and algebraic topology in mixed symmetry tensor gauge theories and applications

TL;DR

This paper extends the duality and topological charge theorems from p-form gauge theories to mixed symmetry tensor gauge theories, developing new mathematical tools and exploring applications in fractons, string theory, and holography.

Contribution

It introduces a mathematical framework for mixed symmetry tensors, proving the existence and uniqueness of duality maps and topological charges in these theories.

Findings

Duality map exists and is unique for mixed symmetry tensor gauge theories under certain conditions.

Developed an analogue of de Rham complex for mixed symmetry tensors.

Applications demonstrated in fractons, higher symmetries, string theory, and holography.

Abstract

Recently the duality map between electric-like asymptotic charges of $p$-form gauge theories is studied. The outcome is an existence and uniqueness theorem and the topological nature of the duality map. The goal of this work is to extend that theorem in the case of mixed symmetry tensor gauge theories in order to have a deeper understanding of exotic gauge theories, of the non-trivial charges associated to them and of the duality of their observables. Unlike the simpler case of $p$-form gauge theories, here we need to develop some mathematical tools. The crucial points are to view a mixed symmetry tensor as a Young projected object of the $N$-multi-form space and to develop an analogue of de Rham complex for mixed symmetry tensors. As a result, if the underlying manifold satisfy appropriate conditions, the duality map can be proven to exist and to be unique ensuring the charge of a description has information on the dual ones. Moreover, we provide some physical applications ranging form fractons and higher symmetries to string theory and holography.