Abstract

AbstractA brief overview of the recent developments of operadic and higher categorical techniques in algebraic quantum field theory is given. The relevance of such mathematical structures for the description of gauge theories is discussed.

Highlights

  • Background on AQFTAlgebraic quantum field theory (AQFT) is a mathematical framework to formalize and investigate quantum field theories (QFTs) on Lorentzian manifolds, i.e. on spacetimes in the sense of general relativity

  • The original framework of Haag and Kastler [44] was restricted to QFTs on Minkowski spacetime, but a more flexible version of AQFT that works on all Lorentzian manifolds was later developed by Brunetti, Fredenhagen and Verch [23]

  • We call a pair of Loc-morphisms ( f1 : M1 → N, f2 : M2 → N ) to a common target causally disjoint if their images f1(M1) ⊆ N and f2(M2) ⊆ N are causally disjoint subsets of N, i.e. there exists no causal curve connecting f1(M1) and f2(M2)

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Summary

Background on AQFT

Algebraic quantum field theory (AQFT) is a mathematical framework to formalize and investigate quantum field theories (QFTs) on Lorentzian manifolds, i.e. on spacetimes in the sense of general relativity. In order to avoid pathologies, one typically considers only globally hyperbolic Lorentzian manifolds These are Lorentzian manifolds M for which there exists a Cauchy surface Σ ⊂ M , i.e. a codimension 1 hypersurface that is intersected precisely once by every inextensible causal curve. We collect all oriented and time-oriented globally hyperbolic Lorentzian manifolds (of a fixed dimension m ≥ 2) in a category that we denote by Loc. The morphisms f : M → N in Loc are orientation and timeorientation preserving isometric embeddings of M into N such that the image f (M ) ⊆ N is open and causally convex, i.e. every causal curve in N that starts and ends in f (M ) is entirely contained in f (M ).

Einstein causality
Orthogonal categories and AQFTs
AQFT operads
Universal constructions
Higher structures in gauge theory
Groupoids of gauge fields
The role of stacks
Smooth cochain algebras on stacks
Derived geometry of linear gauge fields
Homotopy theory of AQFTs
Homotopy theory of algebras over dg-operads
AQFT model categories and Quillen adjunctions
Homotopy-coherent AQFTs
Derived local-to-global constructions
Examples from homotopy invariants
A On the cosheaf condition in AQFT
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