In algebraic geometry, the formula_1-conjecture gives a particularly simple formula for certain integrals on the Deligne–Mumford compactification formula_2 of the moduli space of curves with marked points. It was first found as a consequence of the Virasoro conjecture by E. Getzler and R. Pandharipande (1998). Later, it was proven by C. Faber and R. Pandharipande (2003) using virtual localization in Gromov–Witten theory. It is named after the factor of formula_1, the "g"th Chern class of the Hodge bundle, appearing in its integrand. The other factor is a monomial in the formula_4, the first Chern classes of the "n" cotangent line bundles, as in Witten's conjecture.
Let formula_5 be positive integers such that:
formula_6
Then the formula_1-formula can be stated as follows:
 formula_8
The formula_1-formula in combination withge
formula_10
where the "B"2"g" are Bernoulli numbers, gives a way to calculate all integrals on formula_2 involving products in formula_12-classes and a factor of formula_1.
References.
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