ELSV formula

The Hurwitz numbers

The Hurwitz numbers

${\displaystyle h_{g;k_1,\dots ,k_n}}$

also have a definition in purely algebraic terms. With K = k₁ + ... + k_n and m = K + n + 2g − 2, let τ₁, ..., τ_m be transpositions in the symmetric group S_K and σ a permutation with n numbered cycles of lengths k₁, ..., k_n. Then

${\displaystyle ({\tau }_1,\dots ,{\tau }_m,\sigma )}$

is a transitive factorization of identity of type (k₁, ..., k_n) if the product

${\displaystyle {\tau }_1\cdots {\tau }_m\sigma }$

equals the identity permutation and the group generated by

${\displaystyle {\tau }_1,\dots ,{\tau }_m}$

is transitive.

Definition. ${\displaystyle h_{g;k_1,\dots ,k_n}}$ is the number of transitive factorizations of identity of type (k₁, ..., k_n) divided by K!.

Example A. The number ${\displaystyle h_{g;k}}$ is 1/k! times the number of lists of transpositions ${\displaystyle ({\tau }_1,\dots ,{\tau }_{k+2g-1})}$ whose product is a k-cycle. In other words, ${\displaystyle h_{g;k}}$ is 1/k times the number of factorizations of a given k-cycle into a product of k + 2g − 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 ${\displaystyle {\overline{\mathcal{M}}}_{g,n}}$ 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.)

The Hodge bundle E is the rank g vector bundle over the moduli space ${\displaystyle {\overline{\mathcal{M}}}_{g,n}}$ whose fiber over a curve (C, x₁, ..., x_n) with n marked points is the space of abelian differentials on C. Its Chern classes are denoted by

${\displaystyle {\lambda }_j=c_j(E)\in H^{2j}({\overline{\mathcal{M}}}_{g,n},\mathbf{Q}).}$

We have

${\displaystyle c(E^{\ast })=1-{\lambda }_1+{\lambda }_2-\cdots +(-1{)}^g{\lambda }_g.}$

The ψ-classes. Introduce line bundles ${\displaystyle {\mathcal{L}}_1,\dots ,{\mathcal{L}}_n}$ over ${\displaystyle {\overline{\mathcal{M}}}_{g,n}}$. The fiber of ${\displaystyle {\mathcal{L}}_i}$ over a curve (C, x₁, ..., x_n) is the cotangent line to C at x_i. The first Chern class of ${\displaystyle {\mathcal{L}}_i}$ is denoted by

${\displaystyle {\psi }_i=c_1({\mathcal{L}}_i)\in H^2({\overline{\mathcal{M}}}_{g,n},\mathbf{Q}).}$

The integrand. The fraction ${\displaystyle 1/(1-k_i{\psi }_i)}$ is interpreted as ${\displaystyle 1+k_i{\psi }_i+k_i^2{\psi }_i^2+\cdots }$, where the sum can be cut at degree 3g − 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 3g − 3 + n and integrate it over the moduli space.

The integral as a polynomial. It follows that the integral

${\displaystyle {\int }_{{\overline{\mathcal{M}}}_{g,n}}\frac{c(E^{\ast })}{(1-k_1{\psi }_1)\cdots (1-k_n{\psi }_n)}}$

is a symmetric polynomial in variables k₁, ..., k_n, whose monomials have degrees between 3g − 3 + n and 2g − 3 + n. The coefficient of the monomial ${\displaystyle k_1^{d_1}\cdots k_n^{d_n}}$ equals

${\displaystyle {\int }_{{\overline{\mathcal{M}}}_{g,n}}(-1{)}^j{\lambda }_j{\psi }_1^{d_1}\cdots {\psi }_n^{d_n},}$

where

${\displaystyle j=3g-3+n-\sum d_i.}$

Remark. The polynomiality of the numbers

${\displaystyle \frac{h_{g;k_1,\dots ,k_n}}{m!}\prod _{i=1}^n\frac{k_i!}{k_i^{k_i}}}$

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

${\displaystyle {\int }_{{\overline{\mathcal{M}}}_{g,n}}\frac{c(E^{\ast })}{(1-k_1{\psi }_1)\cdots (1-k_n{\psi }_n)}={\int }_{{\overline{\mathcal{M}}}_{1,1}}\frac{1-{\lambda }_1}{1-k_1{\psi }_1}=\left[{\int }_{{\overline{\mathcal{M}}}_{1,1}}{\psi }_1\right]k_1-\left[{\int }_{{\overline{\mathcal{M}}}_{1,1}}{\lambda }_1\right].}$

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

${\displaystyle h_{1;k}=(k+1)!\frac{k^k}{k!}{\int }_{{\overline{\mathcal{M}}}_{1,1}}\frac{1-{\lambda }_1}{1-k{\psi }_1}=(k+1)k^k\left\{\left[{\int }_{{\overline{\mathcal{M}}}_{1,1}}{\psi }_1\right]k-\left[{\int }_{{\overline{\mathcal{M}}}_{1,1}}{\lambda }_1\right]\right\}.}$

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 S₁), while h_{1, 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

${\displaystyle {\int }_{{\overline{\mathcal{M}}}_{1,1}}{\psi }_1={\int }_{{\overline{\mathcal{M}}}_{1,1}}{\lambda }_1=\frac{1}{24}.}$

From which we deduce

${\displaystyle h_{1;k}=\frac{(k^2-1)k^k}{24}.}$

History

The ELSV formula was announced by Ekedahl et al. (1999), but with an erroneous sign. Fantechi & Pandharipande (2002) proved it for k₁ = ... = k_n = 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 ${\displaystyle {\overline{\mathcal{M}}}_{g,n}}$ from the ELSV formula.

Idea of proof

Let ${\displaystyle {\mathcal{M}}_{g;k_1,\dots ,k_n}}$ be the space of stable maps f from a genus g curve to P¹(C) such that f has exactly n poles of orders ${\displaystyle k_1,\dots ,k_n}$.

The branching morphism br or the Lyashko–Looijenga map assigns to ${\displaystyle f\in {\mathcal{M}}_{g;k_1,\dots ,k_n}}$ 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 C_t 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 f₀ at the node of C₀.

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, ..., k_n) at ∞ and m more fixed simple branch points. This is precisely the Hurwitz number ${\displaystyle h_{g;k_1,\dots ,k_n}}$.

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 ${\displaystyle {\mathcal{M}}_{g;k_1,\dots ,k_n}}$ is an infinite fiber of br isomorphic to the moduli space ${\displaystyle {\overline{\mathcal{M}}}_{g,n}}$. 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 ${\displaystyle z\mapsto z^{k_1},\dots ,z\mapsto z^{k_n}}$. Thus we obtain all stable maps in ${\displaystyle {\mathcal{M}}_{g;k_1,\dots ,k_n}}$ 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.