The Chromatic Brauer Category and Its Linear Representations
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The Chromatic Brauer Category and Its Linear Representations L. Felipe Müller1
· Dominik J. Wrazidlo2
Received: 27 September 2019 / Accepted: 8 November 2020 © The Author(s) 2020
Abstract The Brauer category is a symmetric strict monoidal category that arises as a (horizontal) categorification of the Brauer algebras in the context of Banagl’s framework of positive topological field theories (TFTs). We introduce the chromatic Brauer category as an enrichment of the Brauer category in which the morphisms are component-wise labeled. Linear representations of the (chromatic) Brauer category are symmetric strict monoidal functors into the category of real vector spaces and linear maps equipped with the Schauenburg tensor product. We study representation theory of the (chromatic) Brauer category, and classify all its faithful linear representations. As an application, we use indices of fold lines to construct a refinement of Banagl’s concrete positive TFT based on fold maps into the plane. Keywords Monoidal categories · Brauer category · Schauenburg tensor product · Topological quantum field theories · Semirings · Fold maps · Kervaire spheres Mathematics Subject Classification Primary: 18D10 · 18A22 · 05E10 · 15A69; Secondary: 57R56 · 57R45 · 16Y60
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries on Strict Monoidal Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Communicated by Ross Street. An older version of this manuscript can be found at http://export.arxiv.org/pdf/1902.05517.
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L. Felipe Müller [email protected] Dominik J. Wrazidlo [email protected]
1
Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
2
Institute of Mathematics for Industry, Kyushu University, Motooka 744, Nishi-ku, Fukuoka 819-0395, Japan
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L. F. Müller, D. J. Wrazidlo 2.1 Strict Monoidal Categories . . . . . . . . . . . . . . . . . 2.2 The Schauenburg Tensor Product . . . . . . . . . . . . . . 3 The Chromatic Brauer Category and Its Linear Representations 3.1 Compact Closed Categories . . . . . . . . . . . . . . . . . 3.2 The Chromatic Brauer Category . . . . . . . . . . . . . . 3.3 Linear Representations of the Chromatic Brauer Category . 4 Proof of Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . 5 Positive TFTs, Fold Maps, and Exotic Kervaire Spheres . . . . 5.1 General Framework . . . . . . . . . . . . . . . . . . . . . 5.2 Fold Fields and cBr-Valued Actions . . . . . . . . . . . . 5.2.1 Cobordisms . . . . . . . . . . . . . . . . . . . . . . 5.2.2 System of Fold Fields . . . . . . . . . . . . . . . . . 5.2.3 System of cBr-Valued Action Functionals . . . . . . 5.2.4 Linearization . . . . . . . . . . . . . . . . . . . . . 5.3 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Semirings and Semimodules . . . . . .
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