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Chatterjee, Saikat; Lahiri, Amitabha; Sengupta, Ambar N.

Gauge transformations for categorical bundles. (English) Zbl 1402.18007

J. Geom. Phys. 133, 219-241 (2018).
A categorical bundle is a structure, in terms of category theory, that encodes a classical bundle equipped with connections and some additional entities. The authors have developed the theory of categorical principal bundles in [S. Chatterjee et al., Theory Appl. Categ. 31, 388–417 (2016; Zbl 1343.18009); J. Geom. Phys. 98, 128–149 (2015; Zbl 1339.18005); J. Geom. Phys. 112, 147–174 (2017; Zbl 1355.53018); J. Geom. Phys. 75, 129–161 (2014; Zbl 1280.81087); Theory Appl. Categ. 29, 215–255 (2014; Zbl 1305.18014)]. The principal objective in this paper is to develop a counterpart of the classical gauge transformation in the setting of categorical bundles. There is a considerable literature on category-theoretic approaches to gauge theories [J. C. Baez and D. K. Wise, Commun. Math. Phys. 333, No. 1, 153–186 (2015; Zbl 1308.83017)], [J. C. Baez and J. Huerta, Gen. Relativ. Gravitation 43, No. 9, 2335–2392 (2011; Zbl 1225.83001)], [J. C. Baez and U. Schreiber, Contemp. Math. 431, 7–30 (2007; Zbl 1132.55007)], [J. F. Martins and R. Picken, Differ. Geom. Appl. 29, No. 2, 179–206 (2011; Zbl 1217.53049)], [J. F. Martins and R. Picken, Adv. Math. 226, No. 4, 3309–3366 (2011; Zbl 1214.53043)], [U. Schreiber and K. Waldorf, J. Homotopy Relat. Struct. 12, No. 3, 617–658 (2017; Zbl 1404.18015); Theory Appl. Categ. 28, 476–540 (2013; Zbl 1279.53024)], [A. J. Parzygnat, Theory Appl. Categ. 30, 1319–1428 (2015; Zbl 1341.53078)], [D. Bambusi, in: Multiscale methods in quantum mechanics. Theory and experiment. Papers from the meeting, Rome, Italy, December 16–20, 2002. Boston, MA: Birkhäuser. 23–39 (2004; Zbl 1322.81046)], [E. Soncini and R. Zucchini, J. Geom. Phys. 95, 28–73 (2015; Zbl 1322.81064); J. High Energy Phys. 2014, No. 10, Paper No. 079, 64 p. (2014; Zbl 1333.81272)], [W. Wang, J. Math. Phys. 55, No. 4, 043506, 32 p. (2014; Zbl 1300.81065); Theory Appl. Categ. 30, 1999–2047 (2015; Zbl 1353.18006)], [K. Waldorf, Trans. Am. Math. Soc. 365, No. 8, 4393–4432 (2013; Zbl 1277.53024)]. The authors’ motivation is more differential geometric than category theoretic, and the typical examples are decorated bundles (§2.14) and twisted-product bundels (§3) appearing to find their due place only in the authors’ approach. A categorical connection (§2.11) in the authors’ approach involves several distinct \(1\)-forms and \(2\)-forms that are to be used to prescribe parallel transport.
Reviewer: Hirokazu Nishimura (Tsukuba)

Cited in 1 Document

MSC:

18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
20C99 Representation theory of groups

Keywords:

categorical groups; categorical geometry; principal bundles; gauge theory

Citations:

Zbl 1343.18009; Zbl 1339.18005; Zbl 1355.53018; Zbl 1280.81087; Zbl 1305.18014; Zbl 1308.83017; Zbl 1225.83001; Zbl 1132.55007; Zbl 1217.53049; Zbl 1214.53043; Zbl 1279.53024; Zbl 1341.53078; Zbl 1322.81046; Zbl 1322.81064; Zbl 1333.81272; Zbl 1300.81065; Zbl 1353.18006; Zbl 1277.53024; Zbl 1404.18015
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\textit{S. Chatterjee} et al., J. Geom. Phys. 133, 219--241 (2018; Zbl 1402.18007)
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References:

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