Lambda g conjecture explained

In algebraic geometry, the

λ _{g} -conjecture gives a particularly simple formula for certain integrals on the Deligne–Mumford compactification \overline{ l M} _{g,} 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 Gromov–Witten theory. It is named after the factor of λ _{g} , the gth Chern class of the Hodge bundle, appearing in its integrand. The other factor is a monomial in the \psi _{i} , the first Chern classes of the n cotangent line bundles, as in Witten's conjecture.

Let

a _{1,} \ldots, a _{n} be positive integers such that:a _{1} + … + a _{n} = 2 g -3+ n.

Then the

λ _{g} -formula can be stated as follows:\int _{\overline{lM} _{g,} } \psi_1^ \cdots \psi_n^\lambda_g = \binom \int_ \psi_1^\lambda_g.

The

λ _{g} -formula in combination withge\int _{\overline{lM} _{g,} } \psi_1^\lambda_g = \frac \frac,

where the B_2g are Bernoulli numbers, gives a way to calculate all integrals on

\overline{ l M} _{g,} involving products in \psi -classes and a factor of λ _{g}