# Affine sphere equation, Hitchin system and Painleve III

### Prim Plansangkate (Cambridge)

Tuesday 24th February, 2009 15:00-16:00 Mathematics Building, room 204

#### Abstract

We give a gauge invariant characterisation of the symmetry reduction from the anti-self-dual Yang-Mills system on $R^4$ with gauge group $SU(2,1)$ to the affine sphere equation $\psi_{z \bar z} + 1/2 e^{\psi} + |U|^2 e^{-2\psi} = 0,\quad U_{\bar z}=0,$ which arises in the context of Strominger-Yau-Zaslow conjecture in Mirror Symmetry. The radially symmetric solutions of the affine sphere equation are characterised by solutions of Painleve III with special values of parameters.