A Laplacian to Compute Intersection Numbers on $$\overline{{{\mathcal {M}}}}_{g,n}$$ and Correlation Functions in NCQFT

Abstract

AbstractLet$$F_g(t)$$Fg(t)be the generating function of intersection numbers of$$\psi $$ψ-classes on the moduli spaces$$\overline{{{\mathcal {M}}}}_{g,n}$$M¯g,nof stable complex curves of genusg. As by-product of a complete solution of all non-planar correlation functions of the renormalised$$\Phi ^3$$Φ3-matrical QFT model, we explicitly construct a Laplacian$$\Delta _t$$Δton a space of formal parameters$$t_i$$tiwhich satisfies$$\exp (\sum _{g\ge 2} N^{2-2g}F_g(t))=\exp ((-\Delta _t+F_2(t))/N^2)1$$exp(∑g≥2N2-2gFg(t))=exp((-Δt+F2(t))/N2)1as formal power series in$$1/N^2$$1/N2. The result is achieved via Dyson-Schwinger equations from noncommutative quantum field theory combined with residue techniques from topological recursion. The genus-gcorrelation functions of the$$\Phi ^3$$Φ3-matricial QFT model are obtained by repeated application of another differential operator to$$F_g(t)$$Fg(t)and taking for$$t_i$$tithe renormalised moments of a measure constructed from the covariance of the model.