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Duality of generalized Maxwell theories as an equivalence in derived geometry

Chris Elliott, Owen Gwilliam, Ingmar Saberi, Brian R. Williams

Abstract

We propose a non-perturbative description of the moduli spaces encoding p-form generalized Maxwell theories in any dimension, using derived differential geometry. Our approach synthesizes the Batalin--Vilkovisky formalism with differential cohomology. Within this framework we formulate Dirac charge quantization and show how such charge-quantized moduli spaces exhibit abelian duality between generalized Maxwell theories of different types. We also describe the compactification of generalized Maxwell theories along closed Riemannian manifolds by computing the pushforward of the underlying sheaves of cochain complexes that model differential cohomology.

Duality of generalized Maxwell theories as an equivalence in derived geometry

Abstract

We propose a non-perturbative description of the moduli spaces encoding p-form generalized Maxwell theories in any dimension, using derived differential geometry. Our approach synthesizes the Batalin--Vilkovisky formalism with differential cohomology. Within this framework we formulate Dirac charge quantization and show how such charge-quantized moduli spaces exhibit abelian duality between generalized Maxwell theories of different types. We also describe the compactification of generalized Maxwell theories along closed Riemannian manifolds by computing the pushforward of the underlying sheaves of cochain complexes that model differential cohomology.
Paper Structure (56 sections, 9 theorems, 268 equations)

This paper contains 56 sections, 9 theorems, 268 equations.

Table of Contents

  1. Introduction
  2. Abelian duality as an equivalence of derived stacks: an overview in three dimensions
  3. 3d electromagnetism and cochain complexes
  4. 3d compact boson
  5. Duality as an isomorphism of cochain complexes
  6. An overview of our main results
  7. On prior work and the novelty here
  8. Acknowledgements
  9. Our mathematical setting for higher form gauge theory
  10. Our working context, for physicists or the computationally minded, and a puzzle
  11. Highbrow commentary on this context
  12. Notations and conventions
  13. On categories
  14. On sheaves
  15. On higher bundles
  16. ...and 41 more sections

Key Result

Theorem 5.3

There is an isomorphism as sheaves on $\mathop{\mathrm{\mathsf{Riem}}}\nolimits_n^{\mathrm{or}}$ (or $\mathop{\mathrm{\mathsf{Lrtz}}}\nolimits_n^{\mathrm{or}}$) valued in derived stacks for all natural numbers $n$ and $0\leq p \leq n-2$ and for all coupling constants $e, \kappa \in \mathbb R^\times$.

Theorems & Definitions (80)

  • Example 2.1
  • Example 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • ...and 70 more