Blobbed topological recursion from extended loop equations

We consider the N × N Hermitian matrix model with measure d μ E , λ ( M ) = 1 Z exp ⁡ ( − λ N 4 tr ( M 4 ) ) d μ E , 0 ( M ) , where d μ E , 0 is the Gaußian measure with covariance 〈 M k l M m n 〉 = δ k n δ l m 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 ω n ( g ) 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 n ( g ) . These global equations provide the ω n ( g ) 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 ≤ 1 .