Related papers: Quantum mechanics and non-abelian theta functions …
Abelian Chern-Simons theory relates classical theta functions to the topological quantum field theory of the linking number of knots. In this paper we explain how to derive the constructs of abelian Chern-Simons theory directly from the…
In this paper we construct the quantum group, at roots of unity, of abelian Chern-Simons theory. We then use it to model classical theta functions and the actions of the Heisenberg and modular groups on them.
In this paper we describe progress made toward the construction of the Witten-Reshetikhin-Turaev theory of knot invariants from the geometric point of view. This is done in the perspective of a joint result of the author with A. Uribe which…
We construct the functional integral of Abelian Chern-Simons theory with toral gauge group $\mathbb T=\mathfrak t/\Lambda \cong U(1)^n$ at level $K$, where $K:\Lambda\times\Lambda\to\mathbb Z$ is an even, integral, nondegenerate symmetric…
Classical and quantum Chern-Simons with gauge group $\text{U}(1)^N$ were classified by Belov and Moore in \cite{belov_moore}. They studied both ordinary topological quantum field theories as well as spin theories. On the other hand a…
In this paper, the well-known relationship between theta functions and Heisenberg group actions thereon is resumed by combining complex algebraic and noncommutative geometric techniques in that we describe Hermitian-Einstein vector bundles…
In this paper, we introduce and study two new types of non-abelian zeta functions for curves over finite fields, which are defined by using (moduli spaces of) semi-stable vector bundles and non-stable bundles. A Riemann-Weil type hypothesis…
We study the infra-red limit of non-abelian Chern-Simons gauge theory perturbed by a non-topological, albeit gauge invariant, mass term. It is shown that, in this limit, we may construct an infinite class of integrable quantum mechanical…
We address two linked problems at the interface of quantum topology and number theory: deriving asymptotic expansions of the Witten--Reshetikhin--Turaev invariants for 3-manifolds and establishing quantum modularity of false theta…
We consider the quantum mechanics of a particle on a noncommutative two-sphere with the coordinates obeying an SU(2)-algebra. The momentum operator can be constructed in terms of an $SU(2)\times SU(2)$-extension and the Heisenberg algebra…
We study theta functions of a Riemann surface of genus g from the view point of tau function of a hierarchy of soliton equations. We study two kinds of series expansions. One is the Taylor expansion at any point of the theta divisor. We…
In the previous paper arXiv:1711.07122, we show that a holomorphic zero-mode wave function in abelian Chern-Simons theory on the torus can be considered as a quantum version of a modular form of weight 2. Motivated by this result, in this…
We study Chern-Simons Gauge Theory in axial gauge on ${\mathbb R}^3.$ This theory has a quadratic Lagrangian and therefore expectations can be computed nonperturbatively by explicit formulas, giving an (unbounded) linear functional on a…
We consider the link invariants defined by the quantum Chern-Simons field theory with compact gauge group U(1) in a closed oriented 3-manifold M. The relation of the abelian link invariants with the homology group of the complement of the…
We construct from first principles the operator 'A-hat' that annihilates the partition functions (or wavefunctions) of three-dimensional Chern-Simons theory with gauge groups SU(2), SL(2,R), or SL(2,C) on a knot complement M. The operator…
An approximate procedure for performing nonperturbative calculations in quantum field theories is presented. The focus will be quantum non-Abelian gauge theories with the goal of understanding some of the open questions of these theories…
We investigate the Reshetikhin--Turaev invariants associated to SU(2) for the 3-manifolds M obtained by doing any rational surgery along the figure 8 knot. In particular, we express these invariants in terms of certain complex double…
The classical counterpart of noncommutative quantum mechanics is a constrained system containing only second class constraints. The embedding procedure formulated by Batalin, Fradkin and Tyutin (BFT) enables one to transform this system…
Tensor-network methods enable probing dynamics of strongly interacting quantum many-body systems, including gauge theories, via Hamiltonian simulation, hence bypassing sign problems. They also have the potential to inform efficient…
Non-abelian gauge theories play an important role in the standard model of particle physics, and unfold a partially unexplored world of exciting physical phenomena. In this letter, we suggest a realization of a non-abelian lattice gauge…