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)

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: