Topological field theory is the path integral quantization theory of sigma model field in mathematical physics.

Sigma model is a mapping from a real two-dimensional surface to a fixed space, plus the smooth sections of some clusters on this two-dimensional surface. The mapping part is called boson field, and the cross section part is called Fermi field. The main purpose of this theory is to calculate the partition function through path integral.

In some special cases, the original integral of partition function in infinite space can be simplified to the integral in finite space by localization method. For different functions, this process gives several counting theories of algebraic geometry, including:

Gromov Witten Invariant (i.e. Type IIA String Theory)

The number of holomorphic curves in symplectic manifold

Seeberger Witten

Chen Simon number gauge field