Blobbed topological recursion from extended loop equations

Hock, Alexander; Wulkenhaar, Raimar

Abstract

We consider the N×N Hermitian matrix model with measure dμE,λ(M)=1Zexp(−λN4tr(M4))dμE,0(M), where dμE,0 is the Gaussian measure with covariance ⟨MklMmn⟩=δknδlmN(Ek+El) for given E1,...,EN>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 ω(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 ∞ gives global linear and quadratic loop equations for the ω(g)n. These global equations provide the ω(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 zi+zj=0 and at zi=0. We deduce a recursion kernel representation valid at least for g≤1.