In mathematics, the ELSV formula, named after its four authors Torsten Ekedahl [sv], Sergei Lando [ru], 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
that are connected curves of genus g, with n numbered preimages of the point at infinity having multiplicities
Here if a covering has a nontrivial automorphism group G it should be counted with weight
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 also have a definition in purely algebraic terms.
is the number of transitive factorizations of identity of type (k1, ..., kn) divided by K!.
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.
is a smooth Deligne–Mumford stack of (complex) dimension 3g − 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.)
whose fiber over a curve (C, x1, ..., xn) with n marked points is the space of abelian differentials on C. Its Chern classes are denoted by We have The ψ-classes.
, where the sum can be cut at degree 3g − 3 + n (the dimension of the moduli space).
We expand this product, extract from it the part of degree 3g − 3 + n and integrate it over the moduli space.
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.
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.
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
be the space of stable maps f from a genus g curve to P1(C) such that f has exactly n poles of orders
the unordered set of its m branch points in C with multiplicities taken into account.
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 ft(x, y) = x + y.
As t → 0, two branch points of ft tend towards the value of f0 at the node of C0.
The branching morphism is of finite degree, but has infinite fibers.
The first way is to count the preimages of a generic point in the image.
is an infinite fiber of br isomorphic to the moduli space
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
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.