nLab invertible field theory

Stub

Idea

An invertible (extended) functorial field theory is one which is an invertible object under the tensor product on TQFTs [Freed & Moore (2006), Def. 5.7], which essentially means that it is one whose target category is a rigid symmetric monoidal groupoid, a Picard groupoid (DGP 19, p.20).

References

As quantum anomalies of lower-dimensional non-invertible (and possibly non-topological) quantum field theories:

• Daniel S. Freed, Gregory W. Moore, Def. 5.7 in: Setting the quantum integrand of M-theory, Commun. Math. Phys. 263 (2006) 89-132 [arXiv:hep-th/0409135, doi:10.1007/s00220-005-1482-7]

• Daniel Freed, Anomalies and Invertible Field Theories, talk at StringMath2013 [arXiv.1404.7224]

As formalizing short-range entanglement in topological phases of matter:

• Daniel S. Freed, Short-range entanglement and invertible field theories [arXiv:1406.7278]

On classification (via reflection positivity):

• Daniel Freed, Michael Hopkins, Reflection positivity and invertible topological phases, Geometry & Topology 25 (2021) 1165–1330 [arXiv:1604.06527, doi:10.2140/gt.2021.25.1165]

Review:

• Arun Debray, Soren Galatius, Martin Palmer, Lectures on invertible field theories, [arXiv:1912.08706]

Relation to cutting and pasting of manifolds:

• Carmen Rovi, Matthew Schoenbauer, Relating Cut and Paste Invariants and TQFTs, The Quarterly Journal of Mathematics 73 2 (2022) 579–607 [arXiv:1803.02939, doi:10.1093/qmath/haab044]

• Carmen Rovi, Relating cut and paste invariants and TQFTS, talk at CQTS (Apr. 2023) [video: YT]

• Mayuko Yamashita, Invertible QFTs and differential Anderson duals [arXiv:2304.08833]