Star product formula of theta functions

False Theta Functions and the Verlinde formula

We discover new analytic properties of classical partial and false theta functions and their potential applications to representation theory of W-algebras and vertex algebras in general. More precisely, motivated by clues from conformal field theory, first, we are able to determine modular-like transformation properties of regularized partial and false theta functions. Then, after suitable identification of regularized partial/false theta functions with the characters of atypical modules for the singlet vertex algebra W(2,2p-1), we formulate and prove a Verlinde-type formula for the fusion rul…

Classical Theta Functions and Quantum Tori

The Schwartz kernel of the multiplication operation on a quantum torus is shown to be the distributional boundary value of a classical multivariate theta function. The kernel satisfies a Schrödinger equation in which the role of time is played by the deformation parameter $\hbar$ and the role of the hamiltonian by a Poisson structure. At least in some special cases, the kernel can be written as a sum of products of single-variable theta functions.

Integral and Theta Formulae for Solutions of Sl_N Knizhnik-Zamolodchikov Equations at Level Zero

The solutions of the sl_N Knizhnik-Zamolodchikov(KZ) equations at level 0 are studied. We present the integral formula which is obtained as a quasi-classical limit of the integral formula for form factors of the SU(N) invariant Thirring model due to F. Smirnov. A proof is given that those integrals satisfy sl_N KZ equation of level 0. The relation of the integral formulae with the chiral Szego kernel is clarified. As a consequence the integral formula with the special choice of cycles is rewritten in terms of the Riemann theta functions associated with the Z_N curve. This formula gives a gener…

Noncommutativity and theta-locality

In this paper, we introduce the condition of theta-locality which can be used as a substitute for microcausality in quantum field theory on noncommutative spacetime. This condition is closely related to the asymptotic commutativity which was previously used in nonlocal QFT. Heuristically, it means that the commutator of observables behaves at large spacelike separation like $\exp(-|x-y|^2/θ)$, where $θ$ is the noncommutativity parameter. The rigorous formulation given in the paper implies averaging fields with suitable test functions. We define a test function space which most closely correspo…

Partial theta functions. I. Beyond the lost notebook

It is shown how many of the partial theta function identities in Ramanujan's lost notebook can be generalized to infinite families of such identities. Key in our construction is the Bailey lemma and a new generalization of the Jacobi triple product identity. By computing residues around the poles of our identities we find a surprising connection between partial theta functions identities and Garret-Ismail-Stanton-type extensions of multisum Rogers-Ramanujan identities.

量子仿射代数和Yangian的幺三角R矩阵：基于Theta级数

Unitriangular R-matrices of quantum affine algebras and Yangians via Theta series

本文通过Theta级数给出量子仿射代数幺三角R矩阵的共轭公式，并推广至Yangian情形，简化了计算。