The Rényi entropies constitute a family of information measures that generalizes the wellknown Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies or mutual information, and have found many applications in information theory and beyond. Various generalizations of Rényi entropies to the quantum setting have been proposed, most prominently Petz's quasi-entropies and Renner's conditional min-, max-and collision entropy. However, these quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of Rényi entropies that contains the von Neumann entropy, min-entropy, collision entropy and the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities, a duality relation, and an entropic uncertainty relation.

We prove that the K-theoretic Nekrasov instanton partition functions have a positive radius of convergence in the instanton counting parameter and are holomorphic functions of the Coulomb parameters in a suitable domain. We discuss the implications for the AGT correspondence and the analyticity of the norm of Gaiotto states for the deformed Virasoro algebra. The proof is based on random matrix techniques and relies on an integral representation of the partition function, due to Nekrasov, which we also prove.

Abstract. BRST complexes are differential graded Poisson algebras. They are associated to a coisotropic ideal J of a Poisson algebra P and provide a description of the Poisson algebra (P/J) J as their cohomology in degree zero. Using the notion of stable equivalence introduced in [6], we prove that any two BRST complexes associated to the same coisotropic ideal are quasi-isomorphic in the case P = R[V ] where V is a finite-dimensional symplectic vector space and the bracket on P is induced by the symplectic structure on V . As a corollary, the cohomology of the BRST complexes is canonically associated to the coisotropic ideal J in the symplectic case. We do not require any regularity assumptions on the constraints generating the ideal J. We finally quantize the BRST complex rigorously in the presence of infinitely many ghost variables and discuss uniqueness of the quantization procedure.

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