Homotopy theory of net representations

TL;DR

This paper develops a homotopy theory framework for representations of nets of algebras, establishing Quillen adjunctions and equivalences, with applications to homotopy algebraic quantum field theory and Maxwell p-forms.

Contribution

It introduces a homotopy-theoretic approach to net representations, including explicit constructions and applications in quantum field theory.

Findings

Establishes Quillen adjunctions and equivalences for net representations.

Provides explicit constructions for Maxwell p-forms on Lorentzian manifolds.

Applies homotopy theory to algebraic quantum field theory contexts.

Abstract

The homotopy theory of representations of nets of algebras over a (small) category with values in a closed symmetric monoidal model category is developed. We illustrate how each morphism of nets of algebras determines a change-of-net Quillen adjunction between the model categories of net representations, which is furthermore a Quillen equivalence when the morphism is a weak equivalence. These techniques are applied in the context of homotopy algebraic quantum field theory with values in cochain complexes. In particular, an explicit construction is presented that produces constant net representations for Maxwell $p$-forms on a fixed oriented and time-oriented globally hyperbolic Lorentzian manifold.