Blobbed topological recursion of the \lambda \Phi^4 matrix model

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Abstract

We consider an 𝑁×𝑁 Hermitian matrix model with measure 𝑑𝜇𝐸,𝜆(Φ)=1/𝑍 exp(−𝜆𝑁/4tr(Φ4))𝑑𝜇𝐸,0(Φ) where 𝑑𝜇𝐸,0 is the Gaußian measure with covariance ⟨Φ𝑘𝑙Φ𝑚𝑛⟩=𝛿𝑘𝑛𝛿𝑙𝑚/(𝑁(𝐸𝑘+𝐸𝑙)) for given 𝐸1,...,𝐸𝑁0. 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 𝜔𝑔,𝑛 of meromorphic differentials. We provide strong evidence that the 𝜔𝑔,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 ∞ gives global linear and quadratic loop equations for the 𝜔𝑔,𝑛. These global equations provide the 𝜔𝑔,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.

Speakers from the University of Münster

Wulkenhaar, Raimar

Professur für Reine Mathematik (Prof. Wulkenhaar)