mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Mathai–Quillen formalism is an approach to

topological quantum field theory In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory that computes topological invariants. While TQFTs were invented by physicists, they are also of mathemati ...

introduced by , based on the Mathai–Quillen form constructed in . In more detail, using the superconnection formalism of Quillen, they obtained a refinement of the Riemann–Roch formula, which links together the Thom classes in

K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...

and

cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...

, as an equality on the level of differential forms. This has an interpretation in physics as the computation of the classical and quantum (super)

partition functions Partition may refer to: Arts and entertainment Film and television * Partition (1987 film), ''Partition'' (1987 film), directed by Ken McMullen * Partition (2007 film), ''Partition'' (2007 film), directed by Vic Sarin * ''Partition: 1947'', or '' ...

for the fermionic analogue of a

harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force ''F'' proportional to the displacement ''x'': \vec F = -k \vec x, where ''k'' is a positive const ...

with source term. In particular, they obtained a nice Gaussian shape representative of the Thom class in cohomology, which has a peak along the zero section.

References

* * * Algebraic topology {{Topology-stub