nLab
homological quantum field theory

Contents

Idea

A homological TQFT is a representation of the cobordism category that depends only on the homology of the hom-spaces.

In d=2d = 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 BordBord for any given cobordism category, regarded as a Top-enriched category. A homological TQFT is a symmetric monoidal Ab-enriched functor

Z:H (Bord)Ab. 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

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

Revised on March 16, 2013 17:43:21 by Thomas Wasserman? (163.1.130.189)