A Laplacian to compute intersection numbers on M_{g,n} and correlation functions in NCQFT

Grosse, Harald; Hock, Alexander; Wulkenhaar, Raimar

Abstract

Let Fg(t) be the generating function of intersection numbers of ψ-classes on the moduli spaces Mg,n of stable complex curves of genus g. As by-product of a complete solution of all non-planar correlation functions of the renormalised Φ3-matrical QFT model, we explicitly construct a Laplacian ∆t on a space of formal parameters ti which satisfies exp(∑g≥2 N2−2g Fg(t)) =exp((−∆t + F2(t))/N2)1 as formal power series in 1/N2. 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 Φ3-matricial QFT model are obtained by repeated application of another differential operator to Fg (t) and taking for ti the renormalised moments of a measure constructed from the covariance of the model.