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

Federico Manzoni

TL;DR

This work addresses the problem of extending the duality of electric-like asymptotic charges to mixed symmetry tensor gauge theories. It develops an analog of the de Rham complex for $N$-multi-forms and introduces $(k_1,...,k_{N-1})$-augmented $N$-de Rham-like complexes, proving a Poincaré-like lemma and an existence/uniqueness theorem for duality maps under vanishing cohomology. The authors formulate a general duality framework, establish cohomological isomorphisms between gauge fields and their field strengths, and demonstrate the duality's consequences across memory effects, higher symmetries, holography, fractons, and string theory fluxes. The results provide a topological foundation linking original and dual descriptions of mixed-symmetry gauge theories, with wide implications for both high-energy and condensed-matter contexts. Overall, the paper unifies topology, mixed-symmetry fields, and dual observables into a cohesive framework that informs how charges and memories translate across dual formulations.

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.

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

TL;DR

This work addresses the problem of extending the duality of electric-like asymptotic charges to mixed symmetry tensor gauge theories. It develops an analog of the de Rham complex for -multi-forms and introduces -augmented -de Rham-like complexes, proving a Poincaré-like lemma and an existence/uniqueness theorem for duality maps under vanishing cohomology. The authors formulate a general duality framework, establish cohomological isomorphisms between gauge fields and their field strengths, and demonstrate the duality's consequences across memory effects, higher symmetries, holography, fractons, and string theory fluxes. The results provide a topological foundation linking original and dual descriptions of mixed-symmetry gauge theories, with wide implications for both high-energy and condensed-matter contexts. Overall, the paper unifies topology, mixed-symmetry fields, and dual observables into a cohesive framework that informs how charges and memories translate across dual formulations.

Abstract

Recently the duality map between electric-like asymptotic charges of -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 -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 -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.
Paper Structure (18 sections, 10 theorems, 89 equations, 1 figure)

This paper contains 18 sections, 10 theorems, 89 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The duality map for $p$-form gauge theories
  3. The $N$-multi-form space
  4. Duality for well defined charges in mixed symmetry gauge theories
  5. The de Rham-like complexes for differential mixed symmetry tensors
  6. The extension of Theorem \ref{['THM3.1']} to mixed symmetry tensors
  7. Physical applications
  8. Mixed symmetry memory effects
  9. Higher symmetries and defects
  10. Holography
  11. Condensed matter and fractons
  12. String theory
  13. Conclusions
  14. Basic elements of shaves theory
  15. Sheaves
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let $(M_D,\boldsymbol{\eta})$ be the $D$-dimensional Minkowski spacetime. Then a duality map $f \in \mathrm{GL}(n_p,\mathbb{C})$, such that the following diagram \begin{tikzcd} \Omega_{\mathrm{AS}}^{p+1}(M_D) \arrow[rr, "\star"] & & \Omega_{\mathrm{AS}}^{q+1}(M_D)\\ \Omega_{\mathrm{AS}}^{p}(M_D) \ar

Figures (1)

  • Figure 1: Hasse diagram of Young's lattice

Theorems & Definitions (48)

  • Theorem 2.1: Existence and uniqueness of the duality map for well defined charges
  • Definition 3.1: Bi-form space
  • Definition 3.2: Left and right differential
  • Proposition 3.1: Left and right differential basic properties
  • proof
  • Definition 3.3: Left and right field strength
  • Definition 3.4: Field strength
  • Definition 3.5: Left and right Hodge morphism
  • Definition 3.6: $N$-multi-form space
  • Definition 3.7: $i$-th differential
  • ...and 38 more