Combinatorial Quantisation of GL(1|1) Chern-Simons Theory I: The Torus
Combinatorial Quantisation of GL(1|1) Chern-Simons Theory I: The Torus
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GL(1|1) Chern-Simons 理论的组合量化 I：环面
DOI：
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发表时间：
2018
期刊：
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通讯作者：
V. Schomerus
中科院分区：
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作者：
N. Aghaei;A. Gainutdinov;M. Pawelkiewicz;V. Schomerus
具有规范超群的陈 - 西蒙斯理论自然地出现在弦理论中，并且它们在数学中具有有趣的应用，例如用于构建纽结和链环不变量。本文是系列文章中的第一篇，在其中我们针对形如\(\Sigma\times\mathbb{R}\)的三维流形上的这类超群陈 - 西蒙斯理论提出了一种新的量子化方案。它基于亏格为\(g\)的\(n\) - 穿孔黎曼曲面\(\Sigma = \Sigma_{g,n}\)的单纯分解，并允许从基本构建块（最重要的是所谓的单值化代数）为任意的\(g\)和\(n\)构建量子理论的可观测量。在本文中，我们局限于环面情形，即我们假设\(\Sigma = T^2\)，并且规范超群\(G = GL(1|1)\)。我们构建了整数级\(k\)的陈 - 西蒙斯理论的相应量子态空间，以及模群\(SL(2,\mathbb{Z})\)在这些态上的一个显式表示。后者被证明等价于在本原\(k\)次单位根处的李超代数\(gl(1|1)\)的量子化泛包络代数的受限版本的中心上的柳巴琴科 - 马吉德作用。
Chern-Simons Theories with gauge super-groups appear naturally in string theory and they possess interesting applications in mathematics, e.g. for the construction of knot and link invariants. This paper is the first in a series where we propose a new quantisation scheme for such super-group Chern-Simons theories on 3-manifolds of the form $Sigma imes mathbb{R}$. It is based on a simplicial decomposition of an n-punctured Riemann surface $Sigma=Sigma_{g,n}$ of genus g and allows to construct observables of the quantum theory for any g and n from basic building blocks, most importantly the so-called monodromy algebra. In this paper we restrict to the torus case, i.e. we assume that $Sigma = T^2$, and to the gauge super-group G=GL(1|1). We construct the corresponding space of quantum states for the integer level k Chern-Simons theory along with an explicit representation of the modular group SL(2,Z) on these states. The latter is shown to be equivalent to the Lyubachenko-Majid action on the centre of a restricted version of the quantised universal enveloping algebra of the Lie super-algebra gl(1|1) at the primitive k-th root of unity.