Quantum Gravity and Random Geometry

Blobbed topological recursion of the λϕ4 matrix model

18 janv. 2023, 09:30

50m

Amphithéâtre Hermite (Institut Henri Poincaré)

Orateur

Raimar Wulkenhaar (University of Münster)

Description

We consider an $N\times N$ Hermitian matrix model with measure $d{\mu }_{E,\lambda }(\mathrm{\Phi })=\frac{1}{Z}\exp (-\frac{\lambda N}{4}\mathrm{tr}({\mathrm{\Phi }}^4))d{\mu }_{E,0}(\mathrm{\Phi })$ where $d{\mu }_{E,0}$ is the Gau\ss{}ian measure with covariance $⟨{\mathrm{\Phi }}_{kl}{\mathrm{\Phi }}_{mn}⟩=\frac{{\delta }_{kn}{\delta }_{lm}}{N(E_k+E_l)}$ for given $E_1,...,E_N>0$. We explain how this setting gives rise to two ramified coverings $x,y$ of the Riemann sphere strongly tied by $y(z)=-x(-z)$ and a family ${\omega }_{g,n}$ of meromorphic differentials. We provide strong evidence that the ${\omega }_{g,n}$ obey blobbed topological recursion due to Borot and Shadrin. A key step is to extract from the matrix model a system of six meromorphic functions which satisfy interwoven Dyson-Schwinger equations. Two of these functions are symmetric in the preimages of $x$ and can be determined from their consistency relations. Their expansion at $\mathrm{\infty }$ gives global linear and quadratic loop equations for the ${\omega }_{g,n}$. These global equations provide the ${\omega }_{g,n}$ not only in the vicinity of the ramification points of $x$ but also in the vicinity of all other poles located at opposite diagonals $z_i+z_j=0$ and at $z_i=0$.