Virasoro conjecture

The Virasoro conjecture is a series of relations among the full Gromov-Witten invariants of a smooth projective variety $ X$ for all genera. If one encodes all Gromov-Witten invariants (including descendents) in a generating function $ Z_{X}$, these relations are expressed as a sequence of differential operators $ L_{k}$ (for $ k \geq -1$) for which

$\displaystyle L_{k}Z_{X} = 0$


$\displaystyle [L_{k},L_{l}] = (k-l)L_{k+l}.$

That is, these operators give a representation of half of the Virasoro algebra of differential operators $ \{x^{k+1}\frac{d}{dx}, k \in \mathbb{Z}\}$ on $ S^1$. The conjecture is currently known for toric Fano varieties and for curves (as well as some other trivial cases). One strange feature of the Virasoro relations is that their definition relies upon the Hodge decomposition in a weak but nontrivial way. In particular, they are not defined for arbitrary symplectic manifolds, even though Gromov-Witten invariants are defined in this context.

Jeffrey Herschel Giansiracusa 2005-06-27