Lambda g conjecture

In algebraic geometry, the ${\lambda }_g$-conjecture gives a particularly simple formula for certain integrals on the Deligne–Mumford compactification ${\overline{ℳ}}_{g,n}$ of the moduli space of curves with marked points. It was first found as a consequence of the Virasoro conjecture by Template:Harvs. Later, it was proven by Template:Harvs 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 $a_1,\dots ,a_n$ be positive integers such that:

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

Then the ${\lambda }_g$-formula can be stated as follows:

$\underset{{\overline{ℳ}}_{g,n}}{\int }{\psi }_1^{a_1}\cdots {\psi }_n^{a_n}{\lambda }_g=(\genfrac{}{}{0ex}{}{2g+n-3}{a_1,\dots ,a_n})\underset{{\overline{ℳ}}_{g,1}}{\int }{\psi }_1^{2g-2}{\lambda }_g.$

The ${\lambda }_g$-formula in combination withge

$\underset{{\overline{ℳ}}_{g,1}}{\int }{\psi }_1^{2g-2}{\lambda }_g=\frac{2^{2g-1}-1}{2^{2g-1}}\frac{|B_{2g}|}{(2g)!},$

where the B_2g are Bernoulli numbers, gives a way to calculate all integrals on ${\overline{ℳ}}_{g,n}$ involving products in $\psi$-classes and a factor of ${\lambda }_g$.

References

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