# Geometry and Mathematical Physics Seminar 2013/2014

**Thursday 17 October 2013. ** *The Frank Adams Room (Room 1.212), the Alan Turing Building. 3.15pm*
##
On Duistermaat-Heckman Localisation Formula

Hovhannes Khudaverdian (University of Manchester)

`khudian@manchester.ac.uk`

The talk is devoted to a review of the powerful Duistermaat-Heckman (DH) localisation formula.

It is well known that the asymptotic expansion of the integral $Z(h)=\int e^{iS(x)/h}\,|F(x)|Dx|$ is localised in a vicinity of the stationary points of the function $S(x)$. If $S(x)=S(x_0)+S_{ij}x^ix^j+\dots$, then
$$
\log Z(h)=i\log S(x_0)+c\left(h^n/\det S_{ij}\right)^{1/2}+\dots \,.
$$
This can be applied to different integrals, from the Stirling formula for
$n!$, where the zero term of the expansion is nothing but $(n/e)^n$ to calculation
of partition function in QFT, where the zero term corresponds to classical
mechanics and the first term to quasiclassical mechanics.

The DH localisation formula covers an interesting class of geometric
constructions for which the exact answer coincides with the quasiclassical
approximation. We consider the basic example and discuss the formulation of
this construction in terms of geometry with even and odd variables
(supergeometry).

For references you may look at the file "Duistermaat-Heckman
localisation formula and...", which you can find in the subdirectory
Études/Geometry of my personal homepage: http://www.maths.manchester.ac.uk/~khudian/.