A Laplacian to compute intersection numbers on ${\bar{\mathcal{M}}}_{g,n}$ and correlation functions in NCQFT

Abstract

Let $F_g(t)$ be the generating function of intersection numbers on the moduli spaces ${\bar{\mathcal{M}}}_{g,n}$ of complex curves of genus $g$. As by-product of a complete solution of all non-planar correlation functions of the renormalised ${\Phi }^3$-matrical QFT model, we explicitly construct a Laplacian ${\Delta }_t$ on the space of formal parameters $t_i$ satisfying $\exp ({\sum }_{g\ge 2}{N}^{2-2g}{F}_g(t))=\exp ((-{\Delta }_t+F_2(t))/N^2)1$ for any $N>0$. The result is achieved via Dyson-Schwinger equations from noncommutative quantum field theory combined with residue techniques from topological recursion. The genus-$g$ correlation functions of the ${\Phi }^3$-matricial QFT model are obtained by repeated application of another differential operator to $F_g(t)$ and taking for $t_i$ the renormalised moments of a measure constructed from the covariance of the model.

39 pages, LaTeX. v2: references added, appendix suppressed (still contained in *.tex). A Mathematica implementation to compute all intersection numbers up to genus 10 (but easily extended) is provided as ancillary file. v3: relation to kappa classes added, a larger gap in proof of Prop 5.2 filled, minor change of conventions, typos corrected

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Cited by in corpus (7)

• Solution of all quartic matrix models
• Solution of the self-dual ${\Phi }^4$ QFT-model on four-dimensional Moyal space
• Perturbative and Geometric Analysis of the Quartic Kontsevich Model
• $x-y$ duality in Topological Recursion for exponential variables via Quantum Dilogarithm
• KP integrability of triple Hodge integrals. III. Cut-and-join description, KdV reduction, and topological recursions
• Blobbed topological recursion from extended loop equations
• Exact Solutions v.s. Perturbative Calculations of Finite ${\Phi }^3$-${\Phi }^4$ Hybrid-Matrix-Model