Lambda G Conjecture

In algebraic geometry, the $\lambda_g$-conjecture gives a particularly simple formula for certain integrals on the Deligne–Mumford compactification $\overline_$ 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 ''g''th 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:

: $\int_ \psi_1^ \cdots \psi_n^\lambda_g = \binom \int_ \psi_1^\lambda_g.$

The $\lambda_g$-formula in combination withge

:$\int_ \psi_1^\lambda_g = \frac \frac,$

where the ''B''_2''g'' are Bernoulli numbers, gives a way to calculate all integrals on $\overline_$ involving products in $\psi$-classes and a factor of $\lambda_g$.

References

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*{{Citation , first=C. , last=Faber , first2=R. , last2=Pandharipande , author2-link=Rahul Pandharipande , title=Hodge integrals, partition matrices, and the $\lambda_g$ conjecture, journal=Ann. of Math. , series= 2, volume=157 , issue=1, pages=97–124 , year=2003 , arxiv=math.AG/9908052 , doi=10.4007/annals.2003.157.97

Algebraic curves
Moduli theory