Mathematisch-Physikalisches Kolloquium mit Prof. Dr. Ezra Getzler (Northwestern)

Chern-Weil theory was used by Shulman (Berkeley thesis, 1972) to give explicit closed simplicial differential forms on the simplicial manifold B•GL(n), which realize the Chern classes. For example, the only nonzero component of the differential form c1 is
Tr(g−1 dg) ∈ A1(GL(n)),
and the fact that it defines a closed simplicial form on B•GL(n) amounts to the multiplicativity of the determinant. Likewise, there are two nonzero components of c2 , namely
1/6  Tr((g −1dg)3 )  ∈ A3(GL(n))
and
1/2 Tr((g1-1dg1)(dg2 g2-1)) − Tr(g1-1dg)Tr(g2-1dg2 ) ∈ A2(GL(n) × GL(n)),
and the fact that the simplicial differential form that these forms comprise is closed is
known as the Polyakov-Wiegmann formula.

In this talk, I will define an extension of these forms to the classifying stack of perfect
complexes (that is, extend these formulas to the case where g is a quasi-invertible map
between finite-dimensional complexes). These extensions were proved to exist by Toen and Vezzosi, but the explicit formulas are new. In the case of c1 , we obtain a new perspective on the determinant of quasi-invertible maps, defined by Knudsen and Mumford.