ELSV formula

In mathematics, the ELSV formula, named after its four authors , , 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.
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 formula_1-conjecture.
It is generalized by the Gopakumar–Mariño–Vafa formula.
The formula.
Define the "Hurwitz number"
as the number of ramified coverings of the complex projective line (Riemann sphere, formula_3 that are connected curves of genus "g", with "n" numbered preimages of the point at infinity having multiplicities formula_4 and "m" more simple branch points. Here if a covering has a nontrivial automorphism group "G" it should be counted with weight formula_5.
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 .
The Hurwitz numbers.
The Hurwitz numbers
also have a definition in purely algebraic terms. With "K" = "k"1 + ... + "k""n" and "m" = "K" + "n" + 2"g" − 2, let τ1, ..., τ"m" be transpositions in the symmetric group "S""K" and σ a permutation with "n" numbered cycles of lengths "k"1, ..., "k""n". Then
is a transitive factorization of identity of type ("k"1, ..., "k""n") if the product
equals the identity permutation and the group generated by
is transitive.
Definition. formula_14 is the number of transitive factorizations of identity of type ("k"1, ..., "k""n") divided by "K"!.
Example A. The number formula_20 is 1/"k"! times the number of lists of transpositions formula_21 whose product is a "k"-cycle. In other words, formula_22 is 1/"k" times the number of factorizations of a given "k"-cycle into a product of "k" + 2"g" − 1 transpositions.
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 formula_23 is a smooth Deligne–Mumford stack of (complex) dimension 3"g" − 3 + "n". (Heuristically this behaves much like complex manifold, except that integrals of characteristic classes that are integers for manifolds are rational numbers for Deligne–Mumford stacks.)
The Hodge bundle "E" is the rank "g" vector bundle over the moduli space formula_23 whose fiber over a curve ("C", "x"1, ..., "x""n") with "n" marked points is the space of abelian differentials on "C". Its Chern classes are denoted by
We have
The ψ-classes. Introduce line bundles formula_27 over formula_23. The fiber of formula_29 over a curve ("C", "x"1, ..., "x""n") is the cotangent line to "C" at "x""i". The first Chern class of formula_29 is denoted by
The integrand. The fraction formula_32 is interpreted as formula_33, where the sum can be cut at degree 3"g" − 3 + "n" (the dimension of the moduli space). Thus the integrand is a product of "n" + 1 factors. We expand this product, extract from it the part of degree 3"g" − 3 + "n" and integrate it over the moduli space.
The integral as a polynomial. It follows that the integral
is a symmetric polynomial in variables "k"1, ..., "k""n", whose monomials have degrees between 3"g" − 3 + "n" and 2"g" − 3 + "n". The coefficient of the monomial formula_35 equals
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 "k"1 by "k". We have "m" = "K" + "n" + 2"g" − 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 "h"1, "k" equals 1/"k" times the number of ways to decompose a "k"-cycle in the symmetric group "S""k" into a product of "k" + 1 transpositions. In particular, "h"1, 1 = 0 (since there are no transpositions in "S"1), while "h"1, 2 = 1/2 (since there is a unique factorization of the transposition (1 2) in "S"2 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 , but with an erroneous sign. proved it for "k"1 = ... = "k""n" = 1 (with the corrected sign). proved the formula in full generality using the localization techniques. The proof announced by the four initial authors followed . Now that the space of stable maps to the projective line relative to a point has been constructed by , a proof can be obtained immediately by applying the virtual localization to this space.
, building on preceding work of several people, gave a unified way to deduce most known results in the intersection theory of formula_43 from the ELSV formula.
Idea of proof.
Let formula_44 be the space of stable maps "f" from a genus "g" curve to P1(C) such that "f" has exactly "n" poles of orders formula_45.
The "branching morphism" "br" or the "Lyashko–Looijenga map" assigns to formula_46 the unordered set of its "m" branch points in C with multiplicities taken into account. Actually, this definition only works if "f" is a smooth map. But it has a natural extension to the space of stable maps. For instance, the value of "f" on a node is considered a double branch point, as can be seen by looking at the family of curves "Ct" given by the equation "xy" = "t" and the family of maps "f""t"("x", "y") = "x" + "y". As "t" → 0, two branch points of "f""t" tend towards the value of "f"0 at the node of "C"0.
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 ("k"1, ..., "k""n") at ∞ and "m" more fixed simple branch points. This is precisely the Hurwitz number formula_14.
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 formula_48 is an infinite fiber of "br" isomorphic to the moduli space formula_49. Indeed, given a stable curve with "n" marked points we send this curve to 0 in P1(C) and attach to its marked points "n" rational components on which the stable map has the form formula_50. Thus we obtain all stable maps in formula_44 unramified outside 0 and ∞. Standard methods of algebraic geometry allow one to find the degree of a map by looking at an infinite fiber and its normal bundle. The result is expressed as an integral of certain characteristic classes over the infinite fiber. In our case this integral happens to be equal to the right-hand side of the ELSV formula.
Thus the ELSV formula expresses the equality between two ways to compute the degree of the branching morphism.