nLab homological quantum field theory

Idea

A homological TQFT is a representation of (meaning: functor on, cf. functorial QFT)) a cobordism category that depends only on the homology of the hom-spaces.

In $d = 2$ this is also called a homological conformal field theory. The passage to homology forgets the conformal structure. The refinement of this concept from homology groups to chain complexes is calledTCFT.

Definition

Write $Bord$ for any given cobordism category, regarded as a Top-enriched category. A homological TQFT is a symmetric monoidal Ab-enriched functor

$$Z \colon H_\bullet(Bord) \to Ab \,.$$

Examples

• The string topology operations of a manifold are part of an HTQFT. See there for details.

References

For 2-dimensional cobordisms without boundary:

• Veronique Godin: Higher string topology operations (2007) [arXiv:0711.4859]

and generalized to arbitrary sets of boundary labels/branes:

• Sander Kupers: HCFTs with and without branes, chapter 2 of: String topology operations, MS thesis, Utrecht (2011) [hdl:20.500.12932/7283, pdf]