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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. Moreover all normalized the genus expanded cut-and-join operators with same degree form a differential algebra, which is isomorphic to the central subalgebra of the symmetric group algebra. As an application, we get some differential equations for the generating functions of the generalized 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.

Genus expanded cut-and-join operators and generalized Hurwtiz numbers