nLab homological quantum field theory

Idea

A homological TQFT is a representation of the 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 called TCFT.

Definition

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

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

Applications

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

References

For 2-dimensional cobordisms with closed boundary HCFT has been considered in

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

A detailed treatment in $d = 2$ involving arbitrary sets of branes is in section 2 of

• Sander Kupers, String topology operations MSc thesis (2011)