Gromov-Witten theory, Hurwitz theory, and completed cycles.

*(English)*Zbl 1105.14076The main goal of the article is to relate the Hurwitz theory to the Gromov-Witten theory for curves.

The Hurwitz theory of a non-singular curve \(X\) concerns the enumeration of the branched covers of \(X\) with specified branching points and ramification pattern. This problem has been initiated in A. Hurwitz’s paper [Math. Ann. 55, 53–66 (1902; JFM 32.0404.04)]. The Hurwitz numbers are defined as the weighted number of non-isomorphic ramified covers with prescribed ramification data. Translating the ramification data in terms of the monodromies around the ramification points on \(X\), one obtains combinatorial expressions for the Hurwitz numbers. These expressions involve the number of conjugacy classes of tuples of permutations in the symmetric group. Following this path, a complete combinatorial solution has been obtained in A. D. Mednykh [Sib. Math. J. 25, 606–625 (1984; Zbl 0598.30058)]. The shortcoming of this approach is that it does not allow to extract the information about the behavior of the numbers, and the underlying geometry.

An important step has been taken in T. Ekedahl, S. Lando, M. Shapiro and A. Vainshtein [Invent. Math. 146, 297–327 (2001; Zbl 1073.14041)], where the authors express the Hurwitz numbers as intersection numbers on the Deligne-Mumford moduli space of stable curves.

The main result of the article under review is the GW/H-correspondence, which states that the Gromov-Witten theory of a non-singular target curve \(X\) is equivalent to the theory of completed cycles: \[ {\left\langle\prod_{i=1}^n\tau_{k_i}(\omega)\right\rangle}^{\bullet X}_d=\frac{1}{\prod_i k_i!}H_d^X\left(\overline{(k_1+1)},\ldots,\overline{(k_n+1)}\right). \] On the left-hand-side, the expression \(\tau_k(\omega)\) is obtained by integrating descendant classes on the virtual fundamental class of the space of stable maps \(\overline M_{g,n}(X,d)\). The right-hand-side of the formula represents the Hurwitz number corresponding to the partitions given by the completed cycles \(\overline{(k_i+1)}\). These latter are elements in the center of the group algebra of the symmetric group \({\mathbb Q}{\mathcal S}(d)\).

In the second part of the paper is proved that the Gromov-Witten potential of \(\mathbb P^1\) relative to the marked points \(\{0,\infty\}\) is governed by the \(2\)nd Toda hierarchy. The corresponding differential equations impose strong conditions on the Gromov-Witten invariants of \(\mathbb P^1\), and hence on the Hurwitz numbers of \(\mathbb P^1\). The proof partially relies on the results obtained in [A. Okounkov and R. Pandharipande, Ann. Math. 163, 561–605 (2006; Zbl 1105.14077)].

The reader is invited to consult also the article [Invent. Math. 163, 47–108 (2006; Zbl 1140.14047)] in which the authors continue the study of the structure of the higher genus Gromov-Witten invariants. Namely, they prove the generalized Virasoro constraints for the relative Gromov-Witten theories of all nonsingular target curves.

The Hurwitz theory of a non-singular curve \(X\) concerns the enumeration of the branched covers of \(X\) with specified branching points and ramification pattern. This problem has been initiated in A. Hurwitz’s paper [Math. Ann. 55, 53–66 (1902; JFM 32.0404.04)]. The Hurwitz numbers are defined as the weighted number of non-isomorphic ramified covers with prescribed ramification data. Translating the ramification data in terms of the monodromies around the ramification points on \(X\), one obtains combinatorial expressions for the Hurwitz numbers. These expressions involve the number of conjugacy classes of tuples of permutations in the symmetric group. Following this path, a complete combinatorial solution has been obtained in A. D. Mednykh [Sib. Math. J. 25, 606–625 (1984; Zbl 0598.30058)]. The shortcoming of this approach is that it does not allow to extract the information about the behavior of the numbers, and the underlying geometry.

An important step has been taken in T. Ekedahl, S. Lando, M. Shapiro and A. Vainshtein [Invent. Math. 146, 297–327 (2001; Zbl 1073.14041)], where the authors express the Hurwitz numbers as intersection numbers on the Deligne-Mumford moduli space of stable curves.

The main result of the article under review is the GW/H-correspondence, which states that the Gromov-Witten theory of a non-singular target curve \(X\) is equivalent to the theory of completed cycles: \[ {\left\langle\prod_{i=1}^n\tau_{k_i}(\omega)\right\rangle}^{\bullet X}_d=\frac{1}{\prod_i k_i!}H_d^X\left(\overline{(k_1+1)},\ldots,\overline{(k_n+1)}\right). \] On the left-hand-side, the expression \(\tau_k(\omega)\) is obtained by integrating descendant classes on the virtual fundamental class of the space of stable maps \(\overline M_{g,n}(X,d)\). The right-hand-side of the formula represents the Hurwitz number corresponding to the partitions given by the completed cycles \(\overline{(k_i+1)}\). These latter are elements in the center of the group algebra of the symmetric group \({\mathbb Q}{\mathcal S}(d)\).

In the second part of the paper is proved that the Gromov-Witten potential of \(\mathbb P^1\) relative to the marked points \(\{0,\infty\}\) is governed by the \(2\)nd Toda hierarchy. The corresponding differential equations impose strong conditions on the Gromov-Witten invariants of \(\mathbb P^1\), and hence on the Hurwitz numbers of \(\mathbb P^1\). The proof partially relies on the results obtained in [A. Okounkov and R. Pandharipande, Ann. Math. 163, 561–605 (2006; Zbl 1105.14077)].

The reader is invited to consult also the article [Invent. Math. 163, 47–108 (2006; Zbl 1140.14047)] in which the authors continue the study of the structure of the higher genus Gromov-Witten invariants. Namely, they prove the generalized Virasoro constraints for the relative Gromov-Witten theories of all nonsingular target curves.

Reviewer: Mihai Halic (Zürich)

##### MSC:

14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |

14H30 | Coverings of curves, fundamental group |

14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |

37K20 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions |