Super topological recursion and Gaiotto vectors for superconformal blocks

Abstract

For any (possibly singular) hyperelliptic curve, we give the definition of a hyperelliptic refined spectral curve and the hyperelliptic refined topological recursion, generalising the formulation for a special class of genus-zero curves by Kidwai and the author, and also improving the proposal by Chekhov and Eynard. Along the way, we uncover a fundamental geometric structure underlying the hyperelliptic refined topological recursion and investigate its properties -parts of which remain conjectural due to computational difficulties. Moreover, we establish a new recursion valid in the so-called Nekrasov-Shatashivili limit and prove existence of the corresponding quantum curve. Contents1. Introduction 2. Definitions 3. Properties 4. Q-top recursion and the Nekrasov-Shatashivili limit Appendix A.

Quantum curves from refined topological recursion: the genus 0 case

“…It is shown that the fundamental structure underlying the topological recursion is a set of equations relating different ω g,n 's called abstract loop equations [32]. This approach has made it possible to generalize topological recursion in several directions, e.g., blobbed topological recursion [33], topological recursion with higher-order ramifications [34,35], and supersymmetric topological recursion [36,37,38]. Kontsevich and Soibelman proposed an algebraic approach to solve abstract loop equations in terms of so-called Airy structures [39,40] which established a new research direction between topological recursion and vertex operator algebras [35,36,37,38,41].…”

“…We will show that the contribution of (A.22) at each ramification point is symmetric. As explained in [39,35,37,38], in a neighbourhood of r ∈ R, ω 0,2 is expanded in a local coordinate z…”

“…where dξ −2k+1 is a meromorphic differential locally defined around r ∈ R, and whose only pole is of order 2k of coefficient 1. See [39,35,37,38] for the details. Since ω 0,1 has at most a double zero at R by Assumption 2.2, the residue computation in (A.22) at each r ∈ R gives a contribution of c r dξ −1 (p 0 )dξ −1 (p 1 ), (A.…”