Geometry and Mathematical Physics Seminar 2013/2014

Thursday 24 October 2013. The Frank Adams Room (Room 1.212), the Alan Turing Building. 3.15pm

On Duistermaat-Heckman Localisation Formula. II

Hovhannes Khudaverdian (University of Manchester)

In the previous talk we considered the asymptotic expansion of the integral $Z(h)=\int e^{iS(x)/h}\,|F(x)|Dx|$, which is localised in a vicinity of the stationary points of the function $S(x)$. We discussed Duistermaat-Heckman localisation formula when the quasiclassical expansion of the integral above is the exact answer. In particular, we considered the following basic example: let $K$ be a vector field that defines a $U(1)$ action on a manifold $M$ and let $\omega$ be $1$-form which is invariant with respect to this vector field. Then we considered the integral $I=\int e^{i(d \omega+i_K \omega)/h}$. This integral in fact does not depend on the parameter $h$ and its quasiclassical approximation coincides with the exact answer.

In this talk, basing on the constructions above, we will prove the following version of Duistermaat-Heckman localisation formula:

Let $K$ be a vector field that defines a $U(1)$ action on a manifold $M$ and let $H(x,dx)$ be an arbitrary form which is invariant with respect to the odd vector field $d+i_K: dH+i_KH=0$. Then the integral of the form $H$ over the manifold $M$ is localised at the zero locus of $K$. We present the exact formulae for the integral if the zero locus is a finite set of points and $K$ is non-degenerate at the zero locus.