From $r$-Spin Intersection Numbers to Hodge Integrals

Abstract

Generalized Kontsevich Matrix Model (GKMM) with a certain given potential is the partition function of $r$-spin intersection numbers. We represent this GKMM in terms of fermions and expand it in terms of the Schur polynomials by boson-fermion correspondence, and link it with a Hurwitz partition function and a Hodge partition by operators in a $\widehat{GL}(\infty )$ group. Then, from a $W_{1+\infty }$ constraint of the partition function of $r$-spin intersection numbers, we get a $W_{1+\infty }$ constraint for the Hodge partition function. The $W_{1+\infty }$ constraint completely determines the Schur polynomials expansion of the Hodge partition function.

51 pages, 1 figure

References in corpus (5)

• Generation of Matrix Models by W-operators
• Virasoro constraints for Kontsevich-Hurwitz partition function
• Enumerative geometry, tau-functions and Heisenberg-Virasoro algebra
• Intersection numbers of Riemann surfaces from Gaussian matrix models
• Uniqueness Theorem of W-Constraints for Simple Singularities

Cited by in corpus (3)

• WDVV equations and invariant bi-Hamiltonian formalism
• From Kontsevich-Witten to linear Hodge integrals via Virasoro operators
• 2D Toda $\tau$ Functions, Weighted Hurwitz Numbers and the Cayley Graph: Determinant Representation and Recursion Formula