There are many ways for an article to be exceptional. A paper by Goulden, Jackson, and Vakil, introduces an idea (double Hurwitz numbers), establishes some properties, conjectures some more, and connects the idea with several areas of mathematics. A great review of this paper would tell you all these things. And that is precisely what Hsian-Hua Tseng does.
MR2183250
Goulden, I. P.(3-WTRL-B); Jackson, D. M.(3-WTRL-B); Vakil, R.(1-STF)
Towards the geometry of double Hurwitz numbers. (English summary)
Adv. Math. 198 (2005), no. 1, 43–92.
Hurwitz numbers count the number of branched covers of $\Bbb{C}{\rm P}^1$ with prescribed branching over a point and a certain number of other simple branchings (the number is determined by the Riemann-Hurwitz formula). Hurwitz numbers are known to have strong connections to other aspects of mathematics. For example, the famous formula of Ekedahl-Lando-Shapiro-Vainshtein (ELSV for short) [T. Ekedahl et al., Invent. Math. 146 (2001), no. 2, 297–327; MR1864018 (2002j:14034)] expresses Hurwitz numbers as Hodge integrals, providing a connection to the intersection theory of the moduli spaces of stable curves. Representation theoretic methods lead to a proof [A. Okounkov, Math. Res. Lett. 7 (2000), no. 4, 447–453; MR1783622 (2001i:14047)] that the generating function of Hurwitz numbers satisfies the KP hierarchy, providing a connection to integrable systems. (In fact, the work [A. Okounkov, op. cit.] showed that the generating function of double Hurwitz numbers satisfies the Toda hierarchy, and the statement about Hurwitz numbers was obtained as a special case.) It is quite amazing that such a simple-looking object has such a rich structure.
The paper under review studies a generalization of Hurwitz numbers called the double Hurwitz numbers. By definition, these are the numbers of branched covers of ${\Bbb C}{\rm P}^1$ with prescribed branching over two points and a certain number of other simple branchings. The main theme of this paper is that double Hurwitz numbers should also have a strong connection to geometry. More precisely, the authors conjecture an ELSV-type formula for double Hurwitz numbers, asserting that double Hurwitz numbers should be certain top intersection numbers on some (yet-to-be-found) moduli space of curves with additional structures (likely some universal Picard variety over the moduli space of curves). Since it is known that the generating function of double Hurwitz numbers satisfies the Toda hierarchy, this conjecture will likely lead to constraints on the intersection theory of those moduli spaces, in a way similar to the proof of Witten’s conjecture given in [A. Okounkov and R. Pandharipande, “Gromov-Witten theory, Hurwitz numbers, and matrix models. I”, preprint, arxiv.org/abs/math/0101147]. Work along this line has been pursued in [S. Shadrin and D. Zvonkine, “Changes of variables in ELSV-type formulas”, preprint, arxiv.org/abs/math/0602457].
Throughout the paper, the authors prove many properties of the double Hurwitz numbers, giving supporting evidence for their conjectural ELSV-type formula. Let $H_{\alpha,\beta}^g$ denote the double Hurwitz number enumerating degree-$d$ branched covers from a genus-$g$ curve with two prescribed branchings determined by partitions $\alpha={(\alpha_1\geq\alpha_2\geq\dots \geq\alpha_m)}, \beta={(\beta_1\geq\beta_2\geq\dots \geq\beta_n)}$ of $d$. The authors prove that for fixed $m,n$, the number $H_{\alpha,\beta}^g$ is piecewise polynomial in $\alpha_1,\dots,\alpha_m,\beta_1,\dots,\beta_n$, by translating the problem into an enumeration of lattice points in a polytope. The authors also show that the highest degree of this piecewise polynomial is $4g-3+m+n$, and conjecture that the lowest degree is $2g-3+m+n$. In case $g=0$ a geometric proof of this conjecture about the lowest degree is also given. Such a piecewise polynomiality suggests the existence of an ELSV-type formula (the analogy here is that polynomiality of Hurwitz numbers is a consequence of the ELSV formula).
The authors also study extensively the case when $\alpha=(d)$ has only one part (in this case $H_{(d), \beta}^g$ are called one-part double Hurwitz numbers). Using character theory the authors find explicit formulas for the generating function of $H_{(d),\beta}^g$. Character theory is also used to prove (less explicit) formulas for generating functions of other double Hurwitz numbers $H_{\alpha,\beta}^g$. Formulas of the generating function of $H_{(d),\beta}^g$ lead to a precise conjecture of an ELSV-type formula for $H_{(d),\beta}^g$. The authors verify the conjecture in genus $0$ and $1$. Next, the authors proceed to introduce a symbol $\langle\langle\cdots\rangle\rangle_g$ defined via $H_{(d),\beta}^g$. Various properties of this symbol are proven (not by geometric means), including explicit formulas for $\langle\langle\cdots\rangle\rangle_g$ and an Itzykson-Zuber style genus expansion ansatz. Interestingly, the authors prove string and dilaton equations for $\langle\langle\cdots \rangle\rangle_g$ and they attempt to find Virasoro constraints. At this point, it is very tempting to speculate that the symbols$\langle\langle\cdots\rangle\rangle_g$ give rise to an axiomatic Gromov-Witten theory. For this to be true, what remains to be done is to search for certain forms of topological recursion relations. The authors prove results along this line, but for multi-part double Hurwitz numbers.
It is clear that a lot remains to be understood about the geometry of double Hurwitz numbers. This paper presents a very important part of the whole picture, and the mixed use of algebraic, combinatorial, and geometric techniques is remarkable. One hopes that the study of moduli spaces of curves with additional structures will soon lead to a proof of the conjectural ELSV-type formula. Once such a formula is established, we may be able to derive many properties of double Hurwitz numbers from the geometry of these moduli spaces. The structure of double Hurwitz numbers should also shed light on other related problems. For instance, the authors announce a proof of Faber’s intersection number conjecture in some cases [I. P. Goulden, D. M. Jackson, and R. Vakil, “The moduli space of curves, double Hurwitz numbers, and Faber’s intersection number conjecture”, in preparation]. On the other hand, it is worth pointing out that double Hurwitz numbers may be expressed as relative Gromov-Witten invariants. One may wonder how much we can learn about double Hurwitz numbers via the geometry of the relevant moduli space of relative stable maps.
Reviewed by Hsian-Hua Tseng
