Mathai–Quillen formalism

In mathematics, the Mathai–Quillen formalism is an approach to topological quantum field theory introduced by Atiyah and Jeffrey (1990), based on the Mathai–Quillen form constructed in Mathai and Quillen (1986). 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 and cohomology, 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 for the fermionic analogue of a harmonic oscillator 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

• Atiyah, Michael Francis; Jeffrey, L. (1990), "Topological Lagrangians and cohomology", Journal of Geometry and Physics, 7 (1): 119–136, Bibcode:1990JGP.....7..119A, doi:10.1016/0393-0440(90)90023-V, ISSN 0393-0440, MR 1094734
• Blau, Matthias (1993), "The Mathai-Quillen formalism and topological field theory", Journal of Geometry and Physics, 11 (1): 95–127, arXiv:hep-th/9203026, Bibcode:1993JGP....11...95B, doi:10.1016/0393-0440(93)90049-K, ISSN 0393-0440, MR 1230422
• Mathai, Varghese; Quillen, Daniel (1986), "Superconnections, Thom classes, and equivariant differential forms", Topology, 25 (1): 85–110, doi:10.1016/0040-9383(86)90007-8, ISSN 0040-9383, MR 0836726