Higher holonomy maps for hyperplane arrangements
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Higher holonomy maps for hyperplane arrangements Toshitake Kohno1 Received: 30 April 2019 / Revised: 4 October 2019 / Accepted: 17 October 2019 © Springer Nature Switzerland AG 2019
Abstract We develop a method to construct representations of the homotopy 2-groupoid of a manifold as a 2-category by means of Chen’s formal homology connections. As an application we describe 2-holonomy maps for hyperplane arrangements and discuss representations of the category of braid cobordisms. Keywords Braid group · Iterated integral · Formal homology connection · Hyperplane arrangement · Higher holonomy · 2-Category · Braid cobordism Mathematics Subject Classification 20F36 · 57M25 · 55P62
1 Introduction The 2-categories play an important role in higher gauge theory [1]. In particular, the 2-holonomy maps have been investigated in the framework of 2-connections. The notion of formal homology connections was developed by Kuo-tsai Chen in the theory of iterated integrals of differential forms in order to describe the homology group of the loop space of a manifold M by the chain complex formed by the tensor algebra of the homology group of M, see [3–5]. The purpose of this article is to give a systematic treatment of representations of the homotopy 2-groupoid of a manifold as a 2-category by means of Chen’s formal homology connections. We apply the method of formal homology connections to the complement of complex hyperplane arrangements. In this case because of the formality of the space the formal homology connection can be described by quadratic derivations. In particular, we discuss in detail the 2-flatness condition in the case of the configuration space
The author is partially supported by Grant-in-Aid for Scientific Research, KAKENHI 16H03931, Japan Society of Promotion of Science and by World Premier Research Center Initiative, MEXT, Japan.
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Toshitake Kohno [email protected] Kavli IPMU, Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
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of ordered distinct points in the complex plane. We describe categorified infinitesimal pure braid relations in this setting. It is an important problem to construct a categorification of the Knizhnik–Zamolodchikov (KZ) connections. In this regard, Cirio and Martins [8] provided a categorification of the KZ connections by means of 2-Yang–Baxter operators for sl2 (C). In this paper we give a general expression for 2-holonomy maps based on the formal homology connections. One of our motivations is to apply such methods to braided surfaces in 4-space studied by Carter, Kamada and Saito, see [2,10]. We discuss an application of 2-holonomy maps to construction of representations of 2-category of braid cobordisms. The paper is organized as follows. In Sect. 2 we briefly review Chen’s iterated integrals and their basic properties. In particular, we recall the formula for the composition of plots. In Sect. 3 we recall the notion of formal homology connections. In particular, we explain the notions of 2-conne
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