nLab homotopical algebraic quantum field theory

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.

Related concepts

• higher structures

• locally covariant perturbative algebraic quantum field theory

References

Review and exposition:

• Alexander Schenkel, Towards Homotopical Algebraic Quantum Field Theory, talk at Foundational and Structural Aspects of Gauge Theories, Mainz Institute for Theoretical Physics, 29 May – 2 June 2017. (pdf)

• Alexander Schenkel, On the problem of gauge theories in locally covariant QFT, talk at Operator and Geometric Analysis on Quantum Theory Trento, 2014 (pdf) (with further emphasis on this point in the companion talk Schreiber 14)

• Alexander Schenkel, From Fredenhagen’s universal algebra to homotopy theory and operads, talk at Quantum Physics meets Mathematics, Hamburg, December 2017 (pdf slides)

• Marco Benini, Alexander Schenkel, Higher Structures in Algebraic Quantum Field Theory, Proceedings of LMS/EPSRC Symposium Higher Structures in M-Theory 2018, Fortschritte der Physik 2019 (arXiv:1903.02878, doi:10.1002/prop.201910015)

• Simen Bruinsma, Coloring Operads for Algebraic Field Theory, in: Proceedings of Higher Structures in M-Theory 2018 Fortschritte der Physik, Special Issue Volume 67, Issue 8-9 (arXiv:1903.02863 doi:10.1002/prop.201910004)

• Marco Benini, Alexander Schenkel, Operads, homotopy theory and higher categories in algebraic quantum field theory, in Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2305.03372]

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

• Marco Benini, Claudio Dappiaggi, Alexander Schenkel, Quantized Abelian principal connections on Lorentzian manifolds, Communications in Mathematical Physics 2013 (arXiv:1303.2515)

• Marco Benini, Alexander Schenkel, Richard Szabo, Homotopy colimits and global observables in Abelian gauge theory (arXiv:1503.08839)

• Marco Benini, Alexander Schenkel, Quantum field theories on categories fibered in groupoids, Communications in Mathematical Physics November 2017, Volume 356, Issue 1, pp 19–64 (arXiv:1610.06071)

The stack of Yang-Mills gauge fields is discussed in

• Marco Benini, Alexander Schenkel, Urs Schreiber, The stack of Yang-Mills fields on Lorentzian manifolds, Commun. Math. Phys. 359, 765–820 (2018) (arXiv:1704.01378, doi:10.1007/s00220-018-3120-1)

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

• Marco Benini, Alexander Schenkel, Lukas Woike, Operads for algebraic quantum field theory (arXiv:1709.08657)

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

• Marco Benini, Alexander Schenkel, Lukas Woike, Homotopy theory of algebraic quantum field theories, Lett. Math. Phys. 109, 1487-1532 (2019) (arXiv:1805.08795, doi:10.1007/s11005-018-01151-x)

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

• Donald Yau, Homotopical Quantum Field Theory, World Scientific 2019 (arXiv:1802.08101, doi:10.1142/11626)

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

• Marco Benini, Simen Bruinsma, Alexander Schenkel, Linear Yang-Mills theory as a homotopy AQFT, Commun. Math. Phys. 378, 185–218 (2020) (arXiv:1906.00999, doi:10.1007/s00220-019-03640-z)

Discussion of orbifolding via categorification:

• Marco Benini, Marco Perin, Alexander Schenkel, Lukas Woike, Categorification of algebraic quantum field theories, Lett. Math. Phys. 2021 (h)arXiv:2003.13713)

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

• Marco Benini, Alexander Schenkel, Benoit Vicedo, Homotopical analysis of 4d Chern-Simons theory and integrable field theories (arXiv:2008.01829)

More on relevant model category-structures:

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

See also:

• Simen Bruinsma, Christopher J. Fewster, Alexander Schenkel, Relative Cauchy evolution for linear homotopy AQFTs (arXiv:2108.10592)

On the time slice axiom in homotopy AQFT:

• Marco Benini, Victor Carmona, Alexander Schenkel, Strictification theorems for the homotopy time-slice axiom [arxiv:2208.04344]

Discussion of the model structure on modules for representations of higher local nets of observables:

• Angelos Anastopoulos, Marco Benini, Homotopy theory of net representations, Rev. Math. Phys. (2023) [arXiv:2201.06464, doi:10.1142/S0129055X23500083]

Relation to factorization algebras:

• Marco Benini, Giorgio Musante, Alexander Schenkel, Quantization of Lorentzian free BV theories: factorization algebra vs algebraic quantum field theory [arXiv:2212.02546]

Review:

• Alexander Schenkel, Quantum field theories on Lorentzian manifolds, talk at QFT and Cobordism, CQTS (Mar 2023) [web, slides:pdf, video:YT]

Application to the CS/WZW correspondence:

• Marco Benini, Alastair Grant-Stuart, Alexander Schenkel, The linear CS/WZW bulk/boundary system in AQFT, Annales Henri Poincaré (2023) [arXiv:2302.06990]