In algebraic geometry, the -conjecture gives a particularly simple formula for certain integrals on the DeligneâÂÂMumford compactification of the moduli space of curves with marked points. It was first found as a consequence of the Virasoro conjecture by . Later, it was proven by using virtual localization in GromovâÂÂWitten theory. It is named after the factor of , the gth Chern class of the Hodge bundle, appearing in its integrand. The other factor is a monomial in the , the first Chern classes of the n cotangent line bundles, as in Witten's conjecture.
Let be positive integers such that:
Then the -formula can be stated as follows:
The -formula in combination with
where the B<sub>2g</sub> are Bernoulli numbers, gives a way to calculate all integrals on involving products in -classes and a factor of .