The Eynard-Orantin topological recursion relies on the geometry of a Riemann surface S and two meromorphic functions x and y on S. To formulate the recursion, one must assume that x has only simple ramification points. In this paper we propose a generalized topological recursion that is valid for x with arbitrary ramification. We justify our proposal by studying degenerations of Riemann surfaces. We check in various examples that our generalized recursion is compatible with invariance of the free energies under the transformation (x, y) → (y, x), where either x or y (or both) have higher order ramification, and that it satisfies some of the most important properties of the original recursion. Along the way, we show that invariance under (x, y) → (y, x) is in fact more subtle than expected; we show that there exists a number of counter examples, already in the case of the original Eynard-Orantin recursion, that deserve further study.1. the ramification points of x and y must not coincide; 2. the ramification points of x must all be simple. The Eynard-Orantin topological recursionIn this section we review the original topological recursion formulated by Eynard and Orantin in [16,18]. We will use the notation put forward by Prats Ferrer in [27].

The B p -algebras are a family of vertex operator algebras parameterized by p ∈ Z ≥2 . They are important examples of logarithmic CFTs and appear as chiral algebras of type (A 1 , A 2p−3 ) Argyres-Douglas theories. The first member of this series, the B 2 -algebra, are the well-known symplectic bosons also often called the βγ vertex operator algebra.We study categories related to the B p vertex operator algebras using their conjectural relation to unrolled restricted quantum groups of sl 2 . These categories are braided, rigid and non semisimple tensor categories. We list their simple and projective objects, their tensor products and their Hopf links. The latter are succesfully compared to modular data of characters thus confirming a proposed Verlinde formula of David Ridout and the second author.

Motivated by connections to the singlet vertex operator algebra in the g = sl 2 case, we study the unrolled restricted quantum groups U H q (g) at arbitrary roots of unity with a focus on its category of weight modules. We show that the braid group action on the Drinfeld-Jimbo algebra U q (g) naturally extends to the unrolled quantum groups and that the category of weight modules is a generically semi-simple ribbon category (previously known only for odd roots) with trivial Müger center. Projective covers of simple modules are shown to be self-dual, and some preliminary connections to the higher rank singlet vertex operator algebras are motivated.

We construct families of commutative (super) algebra objects in the category of weight modules for the unrolled restricted quantum group [Formula: see text] of a simple Lie algebra [Formula: see text] at roots of unity, and study their categories of local modules. We determine their simple modules and derive conditions for these categories being finite, non-degenerate, and ribbon. Motivated by numerous examples in the [Formula: see text] case, we expect some of these categories to compare nicely to categories of modules for vertex operator algebras. We focus in particular on examples expected to correspond to the higher rank triplet vertex algebra [Formula: see text] of Feigin and Tipunin and the [Formula: see text] algebras.

Let U be a braided tensor category, typically unknown, complicated and in particular non-semisimple. We characterize U under the assumption that there exists a commutative algebra A in U with certain properties: Let C be the category of local A-modules in U and B the category of A-modules in U, which are in our set-up usually much simpler categories than U. Then we can characterize U as a relative Drinfeld center Z C (B) and B as representations of a certain Hopf algebra inside C.In particular this allows us to reduce braided tensor equivalences to the knowledge of abelian equivalences, e.g. if we already know that U is abelian equivalent to the category of modules of some quantum group U q or some generalization thereof, and if C is braided equivalent to a category of graded vector spaces, and if A has a certain form, then we already obtain a braided tensor equivalence between U and Rep(U q ).A main application of our theory is to prove logarithmic Kazhdan-Lusztig correspondences, that is, equivalences of braided tensor categories of representations of vertex algebras and of quantum groups. Here, the algebra A and the corresponding category C are a-priori given by a free-field realization of the vertex algebra and by a Nichols algebra. We illustrate this in those examples where the representation theory of the vertex algebra is well enough understood. In particular we prove the conjectured correspondences between the singlet vertex algebra M(p) and the unrolled small quantum groups of sl 2 at 2p-th root of unity. Another new example is the Kazhdan-Lusztig correspondence between the affine vertex algebra of gl 1|1 and an unrolled quantum group of gl 1|1 . * this should be q = exp πi r ∨ (k+h ∨ ) with h ∨ the dual Coxeter number of g and r ∨ the lacing number of the even subalgebra.