Lambda g conjecture

In algebraic geometry, the $$\lambda_g$$-conjecture gives a particularly simple formula for certain integrals on the Deligne–Mumford compactification $$\overline{\mathcal M}_{g, n}$$ 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 $$\lambda_g$$, 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:


 * $$a_1 + \cdots + a_n = 2g-3+n.$$

Then the $$\lambda_g$$-formula can be stated as follows:


 * $$\int_{\overline{\mathcal M}_{g, n}} \psi_1^{a_1} \cdots \psi_n^{a_n}\lambda_g = \binom{2g + n - 3}{a_1, \ldots, a_n} \int_{\overline{\mathcal M}_{g, 1}} \psi_1^{2g - 2}\lambda_g.$$

The $$\lambda_g$$-formula in combination withge


 * $$\int_{\overline{\mathcal M}_{g, 1}} \psi_1^{2g - 2}\lambda_g = \frac{2^{2g-1} - 1}{2^{2g - 1}} \frac{|B_{2g}|}{(2g)!},$$

where the B2g are Bernoulli numbers, gives a way to calculate all integrals on $$\overline{\mathcal M}_{g, n}$$ involving products in $$\psi$$-classes and a factor of $$\lambda_g$$.