Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

On $SL(2)-GL(n)$ strange duality

There is a generalized theta divisor on a moduli space of vector bundles on an algebraic curve. The dimension of the vector space of generalized theta divisors is given by the Verlinde formula. By this formula, the dimension of the vector space of $n$-th tensor of generalized theta divisors on a moduli space of rank $r$ vector bundles with trivial determinant on a curve of genus $g$ and that of the vector space of $r$-th tensor of generalized theta divisors on a moduli space of rank $n$ bundles with degree $n(g-1)$ are the same. The strange duality conjecture says that these two vector spaces are dual to each other. This conjecture has recently been proved affirmatively by Belkale (math.AG/0602018), and Marian and Oprea (math.AG/0605097). Their excellent method of proof is using the equality of the Verlinde number and a certain Gromov-Witten invariant, the intersection theory on a quot scheme. In the talk, in the case of $r=2$ (for a generic curve), I will give an alternative proof. This proof uses degeneration arguments. More precisely, it is based on the observation of the moduli (stack) of rank $2$ vector bundles with trivial determinant on a nodal curve, and the factorization of generalized theta divisors.