In mathematics, the ELSV formula, named after its four authors Torsten Ekedahl, Sergei Lando, Michael Shapiro, Alek Vainshtein, is an equality between a Hurwitz number (counting ramified coverings of the sphere) and an integral over the moduli space of stable curves.
Contents
- The formula
- The Hurwitz numbers
- The integral over the moduli space
- Example
- History
- Idea of proof
- References
Several fundamental results in the intersection theory of moduli spaces of curves can be deduced from the ELSV formula, including the Witten conjecture, the Virasoro constraints, and the
The formula
Define the Hurwitz number
as the number of ramified coverings of the complex projective line (Riemann sphere, P1(C)) that are connected curves of genus g, with n numbered preimages of the point at infinity having multiplicities k1, ..., kn and m more simple branch points. Here if a covering has a nontrivial automorphism group G it should be counted with weight 1/|G|.
The ELSV formula then reads
Here the notation is as follows:
The numbers
in the left-hand side have a combinatorial definition and satisfy properties that can be proved combinatorially. Each of these properties translates into a statement on the integrals on the right-hand side of the ELSV formula (Kazarian 2009).
The Hurwitz numbers
The Hurwitz numbers
also have a definition in purely algebraic terms. With K = k1 + ... + kn and m = K + n + 2g − 2, let τ1, ..., τm be transpositions in the symmetric group SK and σ a permutation with n numbered cycles of lengths k1, ..., kn. Then
is a transitive factorization of identity of type (k1, ..., kn) if the product
equals the identity permutation and the group generated by
is transitive.
Definition.
Example A. The number
The equivalence between the two definitions of Hurwitz numbers (counting ramified coverings of the sphere, or counting transitive factorizations) is established by describing a ramified covering by its monodromy. More precisely: choose a base point on the sphere, number its preimages from 1 to K (this introduces a factor of K!, which explains the division by it), and consider the monodromies of the covering about the branch point. This leads to a transitive factorization.
The integral over the moduli space
The moduli space
The Hodge bundle E is the rank g vector bundle over the moduli space
We have
The ψ-classes. Introduce line bundles
The integrand. The fraction
The integral as a polynomial. It follows that the integral
is a symmetric polynomial in variables k1, ..., kn, whose monomials have degrees between 3g − 3 + n and 2g − 3 + n. The coefficient of the monomial
where
Remark. The polynomiality of the numbers
was first conjectured by I. P. Goulden and D. M. Jackson. No proof independent from the ELSV formula is known.
Example B. Let g = n = 1. Then
Example
Let n = g = 1. To simplify the notation, denote k1 by k. We have m = K + n + 2g − 2 = k + 1.
According to Example B, the ELSV formula in this case reads
On the other hand, according to Example A, the Hurwitz number h1, k equals 1/k times the number of ways to decompose a k-cycle in the symmetric group Sk into a product of k + 1 transpositions. In particular, h1, 1 = 0 (since there are no transpositions in S1), while h1, 2 = 1/2 (since there is a unique factorization of the transposition (1 2) in S2 into a product of three transpositions).
Plugging these two values into the ELSV formula we find
From which we deduce
History
The ELSV formula was announced by Ekedahl et al. (1999), but with an erroneous sign. Fantechi & Pandharipande (2002) proved it for k1 = ... = kn = 1 (with the corrected sign). Graber & Vakil (2003) proved the formula in full generality using the localization techniques. The proof announced by the four initial authors followed (Ekedahl et al. 2001). Now that the space of stable maps to the projective line relative to a point has been constructed by Li (2001), a proof can be obtained immediately by applying the virtual localization to this space.
Kazarian (2009), building on preceding work of several people, gave a unified way to deduce most known results in the intersection theory of
Idea of proof
Let
The branching morphism br or the Lyashko–Looijenga map assigns to
The branching morphism is of finite degree, but has infinite fibers. Our aim is now to compute its degree in two different ways.
The first way is to count the preimages of a generic point in the image. In other words, we count the ramified coverings of P1(C) with a branch point of type (k1, ..., kn) at ∞ and m more fixed simple branch points. This is precisely the Hurwitz number
The second way to find the degree of br is to look at the preimage of the most degenerate point, namely, to put all m branch points together at 0 in C.
The preimage of this point in
Thus the ELSV formula expresses the equality between two ways to compute the degree of the branching morphism.