# Geometry and Mathematical Physics Seminar 2013/2014

**Thursday 24 October 2013. ** *The Frank Adams Room (Room 1.212), the Alan Turing Building. 3.15pm*
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On Duistermaat-Heckman Localisation Formula. II

Hovhannes Khudaverdian (University of Manchester)

`khudian@manchester.ac.uk`

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.