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 GL(∞) group. Then, from a W 1+∞ constraint of the partition function of r-spin intersection numbers, we get a W 1+∞ constraint for the Hodge partition function.The W 1+∞ constraint completely determines the Schur polynomials expansion of the Hodge partition function.

Punctures and p-spin curves from matrix models III. Dl type and logarithmic potential

“…Thus we confirm that the three point function of (5.23) gives a correct generating function of the intersection numbers for three marked point for p = 3. In general p, it also gives the intersection numbers, which are consistent with the results of p = 4, 5 derived through recursion relations [11,13].…”

“…This reformulation enables us to obtain easily the intersection numbers for integer p (Neveu-Schwarz punctures) for n point functions, which should be consistent with the results obtained by the recursive method [10,11,13] due to Gelfand-Dikii equation. The evaluation of several marked points in general p and for genus g was obtained in the recursive calculations [11], and we show in this article that our method of the Laurent expansion agrees with them for the lower orders, especially for three point function.…”

WDVV equations and invariant bi-Hamiltonian formalism

“…The Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations are an overdetermined system of Partial Differential Equations (PDEs) that originated in two-dimensional topological field theory [17,68]. Since then, they have been a central subject in Theoretical Physics, with applications ranging from supersymmetric quantum mechanics [5,37,46], topological quantum field theory [39], string theory [10,18] and supersymmetric gauge theory [40].…”

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

“…It is an interesting project for mathematic physics, geometry, representation theory, integrable systems, Hodge integral, etc. Hurwitz numbers have many different expressions in the different fields, for example, [1], [2], [3], [4], [5], [6], [7], [13], [14], [15]. One of the important geometric tools to deal with Hurwitz numbers is the so-called symplectic surgery: cutting and gluing [8], [9], [10], in the views of algebra and differential equations, which is equivalent to the so-called cut-and-join operators [4], [5], [6], [11], [16].…”