Mathai-quillen Formalism

Characteristic classes play an essential role in the study of global properties of vector bundles. Particularly important is the Euler class of real orientable vector bundles. A de Rham representative of the Euler class (for tangent bundles) first appeared in Chern’s generalisation of the Gauss-Bonnet theorem to higher dimensions. The representative is the Pfaffian of the curvature, whose cohomology class does not depend on the choice of connections. The Euler class of a vector bundle is also the obstruction to the existence of a nowhere vanishing section. In fact, it is the Poincaré dual of the zero-set of any section which intersects the zero section transversely. In the case of tangent bundles, it counts (algebraically) the zeros of a vector field on the manifold. That this is equal to the Euler characteristic number is known as the Hopf theorem. Also significant is the Thom class of a vector bundle: it is the Poincaré dual of the zero section in the total space. It induces, by a cup product, the Thom isomorphism between the cohomology of the base space and that of the total space with compact vertical support. Thom isomorphism also exists and plays an important role in K-theory. Mathai and Quillen (1986) obtained a representative of the Thom class by a differential form on the total space of a vector bundle. Instead of having a compact support, the form has a nice Gaussian peak near the zero section and exponentially decays along the fiber directions. The pull-back of Mathai-Quillen’s Thom form by any section is a representative of the Euler class. By scaling the section, one obtains an interpolation between the Pfaffian of the curvature, which distributes smoothly on the manifold, and the Poincaré dual of the zero-set, which localises on the latter. This elegant construction prove to be extremely useful in many situations, from the study of Morse theory, analytic torsion in mathematics to the understanding of topological (cohomological) field theories in physics. In this article, we begin with the construction of Mathai-Quillen’s Thom form. We also consider the case with group actions, with a review of equivariant cohomology and then Mathai-Quillen’s construction in this setting. Next, we show that much of the above can be formulated as a “field theory” on a superspace of one fermionic dimension. Finally, we present the interpretation of topological field theories using the MathaiQuillen formalism. 2. Mathai-Quillen’s Construction