nLab unitary functorial field theory

Idea

A cobordism category $Cob$ typically is canonically a dagger-category with dagger-involution given by reversal of the orientation of cobordisms.

Given a symmetric monoidal dagger category $\mathcal{C}$ as coefficients, a 1-functorial field theory

$$Cob \longrightarrow \mathcal{C}$$

is called “unitary” if it is a dagger functor.

(This is the “hermitian axiom” of Atiyah 1989.)

References

The original notion:

• Michael Atiyah, p. 8 of: Topological quantum field theories, Publications Mathématiques de l’IHÉS 86 (1989) 175-186 [numdam:PMIHES_1988__68__175_0]

• Vladimir Turaev, Hermitian and unitary TQFTs, §III.5 in: Quantum invariants of knots and 3-manifolds, Studies in Mathematics 18, de Gruyter (1994) [doi:10.1515/9783110435221]

Exposition and review:

• John Baez, p. 11 of Quantum Quandaries: a Category-Theoretic Perspective, in D. Rickles et al. (ed.) The structural foundations of quantum gravity, Clarendon Press (2006) 240-265 [arXiv:quant-ph/0404040, ISBN:9780199269693]

• Luuk Stehouwer, Unitary fermionic topological field theory (PhD thesis)

Further discussion:

• Honglin Zhu, The Hermitian axiom on two-dimensional topological quantum field theories, J. Math. Phys. 64 (2023) 022301 [arXiv:2206.07193, doi:10.1063/5.0121440]

On generalization to extended functorial field theory via higher dagger categories:

• Theo Johnson-Freyd, Higher Dagger Categories, talk at CQTS (08 Nov 2023) [slides:pdf, pdf]

See also:

• Lukas Müller, Luuk Stehouwer, around Def 4.1 in: Reflection Structures and Spin Statistics in Low Dimensions [arXiv:2301.06664]