Geometry and Integrability in Mathematical Physics GIMP'06

May 15 - 19, 2006, Moscow, Russia

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Dimitri Zvonkine

Paris 6, France

Moduli of r-spin structures and ramified coverings of the sphere

In 1991 E. Witten formulated two conjectures relating the intersection theory of moduli spaces of Riemann surfaces and that of r-spin structures (= a Riemann surface + an r-th root of its cotangent line bundle) to integrable hierarchies. At present there exist 5 proofs of the first conjecture, but the second one it still wide open.

We will discuss various methods of dealing with the intersection numbers on the space of r-spin structures. A conjectural ELSV-type formula indicates a relation between these numbers and integrable hierarchies. On the other hand, using spaces of admissible coverings as suggested by E.-N. Ionel, we found, in a joint work with S. Shadrin, an algorithm to compute all the intersection numbers involved in Witten's second conjecture.