*Citation:* Turaev, Vladimir G. Homotopy quantum field theory. Vol. 10. European Mathematical Society, 2010.

*Web:* https://math.univ-lille1.fr/~virelizi/HQFT-eflyer.pdf

*Tags:* Foundational, SPT/SETs, TQFT

In this paper, Turaev gives a textbook treatment of homotopy quantum field theory (HQFT). HQFT is a theory that was introduced and developped by Turaev over the course of six papers:

> Turaev, Vladimir. "Homotopy field theory in dimension 2 and group-algebras." arXiv preprint math/9910010 (1999).

> Turaev, Vladimir. "Homotopy field theory in dimension 3 and crossed group-categories." arXiv preprint math/0005291 (2000).

> Turaev, Vladimir. "Dijkgraaf-Witten invariants of surfaces and projective representations of groups." Journal of geometry and physics 57.11 (2007): 2419-2430.

> Turaev, Vladimir. "On certain enumeration problems in two-dimensional topology." arXiv preprint arXiv:0804.1489 (2008).

> Turaev, Vladimir. "Sections of fiber bundles over surfaces." arXiv preprint arXiv:0904.2692 (2009).

> Turaev, Vladimir. "Crossed group-categories." Arabian Journal for Science and Engineering 33.2 C (2008): 483-504.

Perhaps a better name for HQFT is "G-crossed TQFT", or, "X-crossed TQFT" where X is a topological space. When X=K(G,1), you recover the G-crossed story. Given a topological space X, the X-crossed generalization of a manifold is called an "X-manifold", and is defined to be a continuous map M->X where M is a manifold. There is a nice generalization of bordism called an X-bordism, from which you can thus construct an X-TQFT, or "X-HQFT". TQFTs are X-HQFTs when X={*} is the point. Throughout Turaev's work, we consistently take X=K(G,1) to be the classifying space of some finite group.

For (1+1)-dimensional HQFTs, we have a bijection between isomorphism classes of G-HQFTs and isomorphism classes of G-crossed Frobenius algebras.

For (2+1)-dimensional HQFTs, there is a construction coming from G-crossed modular tensor categories. There is a nice construction of G-crossed MTCs coming from quasitriangular G-crossed Hopf algebras. The author states, however, that the issue of systematically constructing quasitraingular G-crossed Hopf algebras is "largely open".

It is important to note that it is in HQFT that G-crossed categories were introduced. This is very interesting, since G-crossed MTCs are now very famous in the context of twists defects and boundaries. A good reference for G-crossed fusion categories to read after Turaev is

> Etingof, Pavel, Dmitri Nikshych, and Victor Ostrik. "Fusion categories and homotopy theory." Quantum topology 1.3 (2010): 209-273.

Another nice paper in this area is the one which proved a Verlinde formula for crossed modular categories:

> Deshpande, Tanmay, and Swarnava Mukhopadhyay. "Crossed modular categories and the Verlinde formula for twisted conformal blocks." arXiv preprint arXiv:1909.10799 (2019).