On Paragrassmann Differential Calculus

Explicit general constructions of paragrassmann calculus with one and many variables are given. Relations of the paragrassmann calculus to quantum groups are outlined and possible physics applications are briefly discussed. This paper is the same as the original 9210075 except added Appendix and minor changes in Acknowledgements and References. IMPORTANT NOTE: This paper bears the same title as the Dubna preprint E5-92-392 but is NOT identical to it, containing new results, extended discussions, and references.Comment: 19p

Generalized Twisted Quantum Doubles and the McKay Correspondence

We consider a class of quasi-Hopf algebras which we call \emph{generalized twisted quantum doubles}. They are abelian extensions H = \mb{C}[\bar{G}] \bowtie \mb{C}[G] ($G$ is a finite group and $\bar{G}$ a homomorphic image), possibly twisted by a 3-cocycle, and are a natural generalization of the twisted quantum double construction of Dijkgraaf, Pasquier and Roche. We show that if $G$ is a subgroup of SU_2(\mb{C}) then $H$ exhibits an orbifold McKay Correspondence: certain fusion rules of $H$ define a graph with connected components indexed by conjugacy classes of $\bar{G}$, each connected component being an extended affine Diagram of type ADE whose McKay correspondent is the subgroup of $G$ stabilizing an element in the conjugacy class. This reduces to the original McKay Correspondence when $\bar{G} = 1$.Comment: 5 figure

Asymptotics of classical spin networks

A spin network is a cubic ribbon graph labeled by representations of $\mathrm{SU}(2)$. Spin networks are important in various areas of Mathematics (3-dimensional Quantum Topology), Physics (Angular Momentum, Classical and Quantum Gravity) and Chemistry (Atomic Spectroscopy). The evaluation of a spin network is an integer number. The main results of our paper are: (a) an existence theorem for the asymptotics of evaluations of arbitrary spin networks (using the theory of $G$-functions), (b) a rationality property of the generating series of all evaluations with a fixed underlying graph (using the combinatorics of the chromatic evaluation of a spin network), (c) rigorous effective computations of our results for some $6j$-symbols using the Wilf-Zeilberger theory, and (d) a complete analysis of the regular Cube $12j$ spin network (including a non-rigorous guess of its Stokes constants), in the appendix.Comment: 24 pages, 32 figure

3d Modularity

We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d $\mathcal{N}=2$ theories where such structures a priori are not manifest. These modular structures include: mock modular forms, $SL(2,\mathbb{Z})$ Weil representations, quantum modular forms, non-semisimple modular tensor categories, and chiral algebras of logarithmic CFTs.Comment: 119 pages, 10 figures and 20 table