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

General

The proposal to understand quantum anomalies of lower-dimensional non-invertible (and possibly non-topological) quantum field theories as invertible topological 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 S. Freed, Michael J. Hopkins, Constantin Teleman, very last sentence of: Consistent Orientation of Moduli Spaces [arXiv:0711.1909]

• Daniel Freed, Anomalies and Invertible Field Theories, talk at StringMath2013, Proc. Symp. Pure Math, Proc. Sympos. Pure Math. 88, Amer. Math. Soc. (2014) 25–45 [arXiv.1404.7224]

• Daniel S. Freed: What is an anomaly?, talk at Mathematical Picture Language Seminar, Feb 2023 [arXiv:2307.08147, video:YT, slides:pdf]

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:

• 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]

  (via reflection positivity)

• Yosuke Kubota: Stable homotopy theory of invertible gapped quantum spin systems I: Kitaev’s $\Omega$-spectrum [arXiv:2503.12618]

• Yosuke Kubota: Stable homotopy theory of invertible gapped quantum spin systems, talk at CQTS@NYUAD (Nov 2025) [slides:pdf]

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]

• Renee Hoekzema, Luuk Stehouwer, Simona Veselá: SKK groups of manifolds and non-unitary invertible TQFTs [arXiv:2504.07917]

Application to anomalies of gapped systems:

• Clay Córdova, Daniel S. Freed, Constantin Teleman: Gapped theories have torsion anomalies [arXiv:2408.15148]

Application to SPTs

Further on the application to the description of short-range entangled symmetry protected topological phases of matter:

• Omar Antolín Camarena, Arun Debray, Cameron Krulewski, Natalia Pacheco-Tallaj, Daniel Sheinbaum, Luuk Stehouwer: Weak topological phases in the presence of interactions [arXiv:2410.10031]

• Cameron Krulewski: Invertible Field Theory Models for SPTs in the Bott Spiral, talk at CQTS @ NYUAD (Apr 2026) [pdf]