Blobbed topological recursion from extended loop equations

Abstract

We consider the $N\times N$ Hermitian matrix model with measure $d\mu_{E,\lambda}(M)=\frac{1}{Z} \exp(-\frac{\lambda N}{4} \mathrm{tr}(M^4)) d\mu_{E,0}(M)$, where $d\mu_{E,0}$ is the Gaussian measure with covariance $\langle M_{kl}M_{mn}\rangle=\frac{\delta_{kn}\delta_{lm}}{N(E_k+E_l)}$ for given $E_1,...,E_N>0$. It was previously understood that 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 conjectured to obey blobbed topological recursion due to Borot and Shadrin. We develop a new approach to this problem via a system of six meromorphic functions which satisfy extended loop equations. Two of these functions are symmetric in the preimages of $x$ and can be determined from their consistency relations. An expansion at $\infty$ 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 $z_i=0$. We deduce a recursion kernel representation valid at least for $g\leq 1$.