Solution of all quartic matrix models
Harald Grosse, Alexander Hock, Raimar Wulkenhaar
Abstract
We consider the quartic analogue of the Kontsevich model, which is defined by a measure exp ( − N Tr ( E Φ 2 + ( λ / 4 ) Φ 4 ) ) d Φ 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 ω g , 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 λ Φ 4 4 -model on noncommutative Moyal space does not have a triviality problem.