Blobbed topological recursion from extended loop equations

Abstract

We consider the $N\times N$ Hermitian matrix model with measure $d{\mu }_{E,\lambda }\left(M\right)=\frac{1}{Z}\exp (-\frac{\lambda N}{4}\mathrm{tr}(M^4\left)\right)d{\mu }_{E,0}\left(M\right)$, where $d{\mu }_{E,0}$ is the Gaußian measure with covariance $〈M_{kl}{M}_{mn}〉=\frac{{\delta }_{kn}{\delta }_{lm}}{N(E_k+E_l)}$ 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 $y\left(z\right)=-x(-z)$ and a family ${\omega }_n^{\left(g\right)}$ 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 ∞ gives global linear and quadratic loop equations for the ${\omega }_n^{\left(g\right)}$. These global equations provide the ${\omega }_n^{\left(g\right)}$ 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 $g\le 1$.