2019/06/11 by Grosse, Harald, Hock, Alexander, Wulkenhaar, Raimar
#14H81 #30E20 #39B32 #81Q80 #Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.1906.04600
We consider the quartic analogue of the Kontsevich model, which is defined by a measure exp(-N Tr(EΦ2+(λ/4)Φ4)) dΦ on Hermitian N×N-matrices, where E is any positive matrix and λ a scalar. It was previously established that the large-N limit of the second moment (the planar two-point function) satisfies a non-linear integral equation. By employing tools from complex analysis, in particular the Lagrange-Bürmann inversion formula, we identify the exact solution of this non-linear problem, both for finite N and for a large-N limit to unbounded operators E of spectral dimension ≤ 4. For finite N, the two-point function is a rational function evaluated at the preimages of another rational function R constructed from the spectrum of E. Subsequent work has constructed from this formula a family ωg,n of meromorphic differentials which obey blobbed topological recursion. For unbounded operators E, the renormalised two-point function is given by an integral formula involving a regularisation of R. This allowed a proof, in subsequent work, that the λΦ44-model on noncommutative Moyal space does not have a triviality problem.