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.

arXiv comment39 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

References in corpus (24)

• 2D Gravity and Random Matrices
• Renormalisation of ϕ^4-theory on noncommutative R^4 in the matrix base
• Duality in Scalar Field Theory on Noncommutative Phase Spaces
• Vanishing of Beta Function of Non Commutative ${\Phi }_4^4$ Theory to all orders
• Free energy topological expansion for the 2-matrix model
• Exact Solution of Quantum Field Theory on Noncommutative Phase Spaces
• Combinatorics of the Modular Group II: the Kontsevich integrals
• Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves
• Self-dual noncommutative ϕ^4-theory in four dimensions is a non-perturbatively solvable and non-trivial quantum field theory
• Cut-and-Join operator representation for Kontsevich-Witten tau-function
• admcycles -- a Sage package for calculations in the tautological ring of the moduli space of stable curves
• Renormalization of the noncommutative phi^3 model through the Kontsevich model
• Blobbed topological recursion: properties and applications
• A short overview of the "Topological recursion"
• Blobbed topological recursion of the quartic Kontsevich model I: Loop equations and conjectures
• A nontrivial solvable noncommutative ϕ^3 model in 4 dimensions
• Lambert-W solves the noncommutative ${\Phi }^4$-model
• Solution of the self-dual ${\Phi }^4$ QFT-model on four-dimensional Moyal space
• Exact solution of matricial ${\Phi }_2^3$ quantum field theory
• From scalar fields on quantum spaces to blobbed topological recursion
• Perturbative and Geometric Analysis of the Quartic Kontsevich Model
• Nested Catalan tables and a recurrence relation in noncommutative quantum field theory
• Complete solution of the LSZ Model via Topological Recursion
• Noncommutative 3-colour scalar quantum field theory model in 2D

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