Abstract
A study of the intersection theory on the moduli space of Riemann surfaces with boundary was recently initiated in a work of R. Pandharipande, J.P. Solomon and the third author, where they introduced open intersection numbers in genus 0. Their construction was later generalized to all genera by J.P. Solomon and the third author. In this paper we consider a refinement of the open intersection numbers by distinguishing contributions from surfaces with different numbers of boundary components, and we calculate all these numbers. We then construct a matrix model for the generating series of the refined open intersection numbers and conjecture that it is equivalent to the Kontsevich-Penner matrix model. An evidence for the conjecture is presented. Another refinement of the open intersection numbers, which describes the distribution of the boundary marked points on the boundary components, is also discussed.
Highlights
Is satisfied, all the ai are non-negative integers and the dimension constraint 3g−3+l = ai holds
We construct a matrix model for the generating series of the refined open intersection numbers and conjecture that it is equivalent to the Kontsevich-Penner matrix model
Kontsevich ([13]), says that the closed partition function τ c becomes a tau-function of the KdV hierarchy after the change of variables tn = (2n + 1)!!T2n+1
Summary
We prove that the extended refined open partition function τNo,ext is related to the very refined open partition function by a simple transformation. We prove the string and the dilaton equations for τNo,ext.
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