Flat Connections and Quantum Groups

Kontsevich deformation quantization and flat connections

In arXiv:math/0105152, the second author used the Kontsevich deformation quantization technique to define a natural connection ω_n on the compactified configuration spaces of n points on the upper half-plane. This connection takes values in the Lie algebra of derivations of the free Lie algebra with n generators. In this paper, we show that ω_n is flat. The configuration space contains a boundary stratum at infinity which coincides with the (compactified) configuration space of n points on the complex plane. When restricted to this stratum, ω_n gives rise to a flat connection ω_n^\infty. We sh…

从霍普夫群代数出发的4维流形平坦连接不变量

Invariants of flat connections on 4-manifolds from Hopf group-algebras

本文提出通过霍普夫G-代数构造4维流形平坦连接的不变量，并在G为有限群时定义了基于所有平坦连接的不变量求和方法。

平坦 $GL(1|1)$-联络与胖图

Flat $GL(1|1)$-connections and fatgraphs

本文利用胖图上的图联络坐标研究带穿孔曲面的平坦 $GL(1|1)$-联络模空间，在三价图情形给出 Whitehead 移动的显式公式，并讨论其不变泊松括号。

平坦联络的周期与拟动机铅笔

Periodic and quasi-motivic pencils of flat connections

本文提出平坦联络的周期铅笔与拟动机铅笔概念，证明其单值性和奇点结构的显著性质，并验证 Knizhnik-Zamolodchikov、Dunkl、Casimir 等联络均为拟动机周期铅笔，为后续计算正特征约化的 $p$-曲率特征值奠定基础。

Poisson structure on moduli of flat connections on Riemann surfaces and $r$-matrix

We consider the space of graph connections (lattice gauge fields) which can be endowed with a Poisson structure in terms of a ciliated fat graph. (A ciliated fat graph is a graph with a fixed linear order of ends of edges at each vertex.) Our aim is however to study the Poisson structure on the moduli space of locally flat vector bundles on a Riemann surface with holes (i.e. with boundary). It is shown that this moduli space can be obtained as a quotient of the space of graph connections by the Poisson action of a lattice gauge group endowed with a Poisson-Lie structure. The present paper cont…

Invariants of tangles with flat connections in their complements

Let G be a simple complex algebraic group. By using a notion of a G-category we define invariants of tangles with flat G-connections in their complements. We also show that quantized universal enveloping algebras at roots of unity provide examples of G-categories.

The Weyl quantization and the quantum group quantization of the moduli space of flat SU(2)-connections on the torus are the same

We prove that, for the moduli space of flat SU(2)-connections on the torus, the Weyl quantization and the quantization using the quantum group of SL(2,C) are unitarily equivalent. This is done by comparing the matrices of the operators associated by the two quantization to cosine functions. We also discuss the *-product of the Weyl quantization and show that it satisfies the product-to-sum formula for noncommutative cosines on the noncommutative torus.

A Note on Quantum Geometric Langlands Duality, Gauge Theory, and Quantization of the Moduli Space of Flat Connections

Montonen-Olive duality implies that the categories of A-branes on the moduli spaces of Higgs bundles on a Riemann surface C for a pair of Langlands-dual groups are equivalent. We reformulate this as a statement about categories of B-branes on the quantized moduli spaces of flat connections for these groups. We show that it implies the statement of the Quantum Geometric Langlands duality with a purely imaginary ``quantum parameter'' which is proportional to the inverse of the Planck constant of the gauge theory. The ramified version of the story is also considered.

Quantum Moduli Spaces of Flat Connections

Using the formalism of discrete quantum group gauge theory, one can construct the quantum algebras of observables for the Hamiltonian Chern-Simons model. The resulting moduli algebras provide quantizations of the algebra of functions on the moduli spaces of flat connections on a punctured 2-dimensional surface. In this note we describe some features of these moduli algebras with special emphasis on the natural action of mapping class groups. This leads, in particular, to a closed formula for representations of the mapping class groups on conformal blocks.

Classification of Connections on Higher-Dimensional Non-Commutative Tori

If X is a full, finitely generated, projective module over a non-commutative torus, the Yang-Mills functional attains its minimum exactly on the flat connections on X. We classify the flat connections on modules admitting integrable connections.

The hyperbolic moduli space of flat connections and the isomorphism of symplectic multiplicity spaces

Let $G$ be a simple complex Lie group, $\alg{g}$ be its Lie algebra, $K$ be a maximal compact form of $G$ and $\alg{k}$ be a Lie algebra of $K$. We denote by $X\rightarrow \overline{X}$ the anti-involution of $\alg{g}$ which singles out the compact form $\alg{k}$. Consider the space of flat $\alg{g}$-valued connections on a Riemann sphere with three holes which satisfy the additional condition $\overline{A(z)}=-A(\overline{z})$. We call the quotient of this space over the action of the gauge group $\overline{g(z)}=g^{-1}(\overline{z})$ a \emph{hyperbolic} moduli space of flat connections. We p…

Holonomy braidings, biquandles and quantum invariants of links with $SL_2(\mathbb C)$ flat connections

R. Kashaev and N. Reshetikhin introduced the notion of holonomy braiding extending V. Turaev's homotopy braiding to describe the behavior of cyclic representations of the unrestricted quantum group $U_qsl_2$ at root of unity. In this paper, using quandles and biquandles we develop a general theory for Reshetikhin-Turaev ribbon type functor for tangles with quandle representations. This theory applies to the unrestricted quantum group $U_qsl_2$ and produces an invariant of links with a gauge class of quandle representations.

Categorifying the $sl(2,C)$ Knizhnik-Zamolodchikov Connection via an Infinitesimal 2-Yang-Baxter Operator in the String Lie-2-Algebra

We construct a flat (and fake-flat) 2-connection in the configuration space of $n$ indistinguishable particles in the complex plane, which categorifies the $sl(2,C)$-Knizhnik-Zamolodchikov connection obtained from the adjoint representation of $sl(2,C)$. This will be done by considering the adjoint categorical representation of the string Lie 2-algebra and the notion of an infinitesimal 2-Yang-Baxter operator in a differential crossed module. Specifically, we find an infinitesimal 2-Yang-Baxter operator in the string Lie 2-algebra, proving that any (strict) categorical representation of the st…