Noncommutative Differentials and Yang-Mills on Permutation Groups S_N

A noncommutative de Finetti theorem: Invariance under quantum permutations is equivalent to freeness with amalgamation

We show that the classical de Finetti theorem has a canonical noncommutative counterpart if we strengthen `exchangeability' (i.e., invariance of the joint distribution of the random variables under the action of the permutation group) to invariance under the action of the quantum permutation group. More precisely, for an infinite sequence of noncommutative random variables, we prove that invariance of their joint distribution under quantum permutations is equivalent to the fact that the random variables are identically distributed and free with respect to the conditional expectation onto their…

非交换背景上的 Yang-Mills 模型

Yang-Mills models on Noncommutative background

本文延续 $A_m$ 型包络代数上的非交换微积分与量子化工作，提出一种在非交换构形空间意义下量子化 Yang-Mills 模型的方法。

Lattice Gauge Fields and Discrete Noncommutative Yang-Mills Theory

We present a lattice formulation of noncommutative Yang-Mills theory in arbitrary even dimensionality. The UV/IR mixing characteristic of noncommutative field theories is demonstrated at a completely nonperturbative level. We prove a discrete Morita equivalence between ordinary Yang-Mills theory with multi-valued gauge fields and noncommutative Yang-Mills theory with periodic gauge fields. Using this equivalence, we show that generic noncommutative gauge theories in the continuum can be regularized nonperturbatively by means of {\it ordinary} lattice gauge theory with 't~Hooft flux. In the cas…

Electromagnetism and Gauge Theory on the Permutation Group $S_3$

Using noncommutative geometry we do U(1) gauge theory on the permutation group $S_3$. Unlike usual lattice gauge theories the use of a nonAbelian group here as spacetime corresponds to a background Riemannian curvature. In this background we solve spin 0, 1/2 and spin 1 equations of motion, including the spin 1 or `photon' case in the presence of sources, i.e. a theory of classical electromagnetism. Moreover, we solve the U(1) Yang-Mills theory (this differs from the U(1) Maxwell theory in noncommutative geometry), including the moduli spaces of flat connections. We show that the Yang-Mills ac…

Lévy Processes on Quantum Permutation Groups

We describe basic motivations behind quantum or noncommutative probability, introduce quantum Lévy processes on compact quantum groups, and discuss several aspects of the study of the latter in the example of quantum permutation groups. The first half of this paper is a survey on quantum probability, compact quantum groups, and Lévy processes on compact quantum groups. In the second half the theory is applied to quantum permutations groups. Explicit examples are constructed and certain classes of such Lévy processes are classified.

Open Wilson Lines and Group Theory of Noncommutative Yang-Mills Theory in Two Dimensions

The correlation functions of open Wilson line operators in two-dimensional Yang-Mills theory on the noncommutative torus are computed exactly. The correlators are expressed in two equivalent forms. An instanton expansion involves only topological numbers of Heisenberg modules and enables extraction of the weak-coupling limit of the gauge theory. A dual algebraic expansion involves only group theoretic quantities, winding numbers and translational zero modes, and enables analysis of the strong-coupling limit of the gauge theory and the high-momentum behaviour of open Wilson lines. The dual expr…