## The Genus Expanded Cut-and-join Operator Algebra and Hurwitz Number

Time: 2016-10-25
Published By: Ningbo Lu

**Speaker(s): ** Quan Zheng （SCU）

**Time: ** 15:15-17:15 October 12, 2016

**Venue: ** Room 9, Quan Zhai, BICMR

Abstract. To distinguish the contributions to the generalized Hurwitz number of the source Riemann surface with different genus, by observing carefully the symplectic surgery and the gluing formulas of the relative GW-invariants, we define the genus expanded cut-and-join operators, which can form a differential operator algebra. Moreover, the differential operator algebra is isomorphic to the central subalgebra of the symmetric group algebra for the finite case, and is isomorphic is isomorphic to the central subalgebra of infinite

symmetric group algebra for the shifted case. As an application, we get some differential equations for the generating functions of the Hurwitz numbers for the source Riemann surface with different genus, thus we can express the generating functions in terms of the genus expanded cut-and-join operators.

symmetric group algebra for the shifted case. As an application, we get some differential equations for the generating functions of the Hurwitz numbers for the source Riemann surface with different genus, thus we can express the generating functions in terms of the genus expanded cut-and-join operators.