# nLab homotopical algebraic quantum field theory

Contents

### Context

#### Quantum field theory

functorial quantum field theory

## Contents

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

The axioms of traditional algebraic quantum field theory or of locally covariant perturbative quantum field theory – where one assigns a plain algebra (a C*-algebra for non-perturbative QFT and a formal power series algebra for perturbative QFT) of observables to a spacetime region (a “local net of observables”) – fail when one considers gauge theory on spacetimes more general than Minkowski spacetime (AQFT on curved spacetimes):

The nature of gauge fields implies that to re-construct their global configuration (including non-trivial topology as in instanton sectors) one needs to locally remember the groupoid of gauge transformations (see Eggertsson 14, Schenkel 14, Schreiber 14 for exposition), and hence, after quantization, some higher algebra of observables.

The program of homotopical algebraic quantum field theory (Benini-Schenkel 16, see Schenkel 17 for survey) is to lift the axioms of AQFT on curved spacetimes to certain local nets of homotopical algebras such as to properly capture gauge theory with topologically non-trivial gauge field configurations.

The axioms as considered in Schenkel 17 have some similarity with that of factorization homology, but also do take into account the causality condition for quantum fields on Lorentzian manifolds.

## References

For review and exposition see

Construction and axiomatization of gauge field AQFT via homotopy theory and homotopical algebra is being developed in

The stack of Yang-Mills gauge fields is discussed in

An operad for local nets of observables in AQFT is considered in

and its model structure on algebras over an operad (with respect to the model structure on chain complexes) is discussed in

The Boardman-Vogt resolution of the operad for local nets of observables (Benini-Schenkel-Woike 17), lifting it to homotopy AQFT, is considered in

Assignment of BV-BRST complexes as a homotopy AQFT is discussed in

Discussion of orbifolding via categorification:

Application to semi-topological 4d Chern-Simons theory:

More on relevant model category-structures:

• Victor Carmona, Algebraic Quantum Field Theories: a homotopical view (arXiv:2107.14176)

• Angelos Anastopoulos, Marco Benini, Sec. 2.4 of Homotopy theory of net representations $[$arXiv:2201.06464$]$