Pontrjagin duality on multiplicative gerbes. (English) Zbl 07761274

Blanco, Jaider; Uribe, Bernardo; Waldorf, Konrad

J. Noncommut. Geom. 17, No. 4, 1469-1520 (2023).

Summary: We use Segal-Mitchison’s cohomology of topological groups to define a convenient model for topological gerbes. We introduce multiplicative gerbes over topological groups in this setup and define their representations. For a specific choice of representation, we construct its category of endomorphisms, and we show that it induces a new multiplicative gerbe over another topological group. This new induced group is fiberwise Pontrjagin dual of the original one, and therefore we called the pair of multiplicative gerbes “Pontrjagin dual”. We show that Pontrjagin dual multiplicative gerbes have equivalent categories of representations. In addition, we show that their monoidal centers are equivalent. Examples of Pontrjagin dual multiplicative gerbes over finite and discrete, as well as compact and non-compact, Lie groups are provided.

MSC:

55U30	Duality in applied homological algebra and category theory (aspects of algebraic topology)
53C08	Differential geometric aspects of gerbes and differential characters
55N30	Sheaf cohomology in algebraic topology
18M20	Fusion categories, modular tensor categories, modular functors
18N10	2-categories, bicategories, double categories
22A22	Topological groupoids (including differentiable and Lie groupoids)

Keywords:

multiplicative gerbe; topological group; Pontrjagin duality; Morita equivalence

References:

[1]	E. Aldrovandi and B. Noohi, Butterflies. I. Morphisms of 2-group stacks. Adv. Math. 221 (2009), no. 3, 687-773 Zbl 1179.18007 MR 2511036 · Zbl 1179.18007 · doi:10.1016/j.aim.2008.12.014
[2]	Pontrjagin duality on multiplicative gerbes 1519
[3]	J. C. Baez and A. D. Lauda, Higher-dimensional algebra. V. 2-groups. Theory Appl. Categ. 12 (2004), 423-491 Zbl 1056.18002 MR 2068521 · Zbl 1056.18002
[4]	L. Breen, Théorie de Schreier supérieure. Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 5, 465-514 Zbl 0795.18009 MR 1191733 · Zbl 0795.18009 · doi:10.24033/asens.1656
[5]	J.-L. Brylinski, Loop spaces, characteristic classes and geometric quantization. Progr. Math. 107, Birkhäuser, Boston, MA, 1993 Zbl 0823.55002 MR 1197353 · Zbl 0823.55002 · doi:10.1007/978-0-8176-4731-5
[6]	J.-L. Brylinski and D. A. McLaughlin, The geometry of degree-four characteristic classes and of line bundles on loop spaces. I. Duke Math. J. 75 (1994), no. 3, 603-638 Zbl 0844.57025 MR 1291698 · Zbl 0844.57025 · doi:10.1215/S0012-7094-94-07518-2
[7]	A. L. Carey, S. Johnson, M. K. Murray, D. Stevenson, and B.-L. Wang, Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories. Comm. Math. Phys. 259 (2005), no. 3, 577-613 Zbl 1088.58018 MR 2174418 · Zbl 1088.58018 · doi:10.1007/s00220-005-1376-8
[8]	P. Etingof, D. Nikshych, and V. Ostrik, On fusion categories. Ann. of Math. (2) 162 (2005), no. 2, 581-642 Zbl 1125.16025 MR 2183279 · Zbl 1125.16025 · doi:10.4007/annals.2005.162.581
[9]	K. Gawȩedzki and K. Waldorf, Polyakov-Wiegmann formula and multiplicative gerbes. J. High Energy Phys. (2009), no. 9, article no. 073 MR 2580721 · doi:10.1088/1126-6708/2009/09/073
[10]	K. Gawȩdzki and N. Reis, Basic gerbe over non-simply connected compact groups. J. Geom. Phys. 50 (2004), no. 1-4, 28-55 Zbl 1067.22009 MR 2078218 · Zbl 1067.22009 · doi:10.1016/j.geomphys.2003.11.004
[11]	A. Haefliger, Groupoïdes d’holonomie et classifiants. Astérisque 116 (1984), 70-97 Zbl 0562.57012 MR 755163 · Zbl 0562.57012
[12]	C. Laurent-Gengoux, M. Stiénon, and P. Xu, Non-abelian differentiable gerbes. Adv. Math. 220 (2009), no. 5, 1357-1427 Zbl 1177.22001 MR 2493616 · Zbl 1177.22001 · doi:10.1016/j.aim.2008.10.018
[13]	K. Maya, A. Mejía Castaño, and B. Uribe, Classification of pointed fusion categories of dimen-sion p 3 up to weak Morita equivalence. J. Algebra Appl. 20 (2021), no. 1, article no. 2140008 Zbl 1460.18017 MR 4209962 · Zbl 1460.18017 · doi:10.1142/S0219498821400089
[14]	E. Meinrenken, The basic gerbe over a compact simple Lie group. Enseign. Math. (2) 49 (2003), no. 3-4, 307-333 Zbl 1061.53034 MR 2026898 · Zbl 1061.53034
[15]	M. Mignard and P. Schauenburg, Morita equivalence of pointed fusion categories of small rank. 2017, arXiv:1708.06538
[16]	I. Moerdijk, Classifying toposes and foliations. Ann. Inst. Fourier (Grenoble) 41 (1991), no. 1, 189-209 Zbl 0727.57029 MR 1112197 · Zbl 0727.57029 · doi:10.5802/aif.1254
[17]	S. A. Morris, Duality and structure of locally compact abelian groups: : :for the layman. Math. Chronicle 8 (1979), 39-56 Zbl 0452.22004 MR 564688 · Zbl 0452.22004
[18]	M. Müger, From subfactors to categories and topology. I. Frobenius algebras in and Morita equivalence of tensor categories. J. Pure Appl. Algebra 180 (2003), no. 1-2, 81-157 Zbl 1033.18002 MR 1966524 · Zbl 1033.18002 · doi:10.1016/S0022-4049(02)00247-5
[19]	M. Müger, From subfactors to categories and topology. II. The quantum double of tensor categories and subfactors. J. Pure Appl. Algebra 180 (2003), no. 1-2, 159-219 Zbl 1033.18003 MR 1966525 · Zbl 1033.18003 · doi:10.1016/S0022-4049(02)00248-7
[20]	Á. Muñoz and B. Uribe, Classification of pointed fusion categories of dimension 8 up to weak Morita equivalence. Comm. Algebra 46 (2018), no. 9, 3873-3888 Zbl 1393.18005 MR 3820602 · Zbl 1393.18005 · doi:10.1080/00927872.2018.1427243
[21]	M. K. Murray, Bundle gerbes. J. London Math. Soc. (2) 54 (1996), no. 2, 403-416 Zbl 0867.55019 MR 1405064 · Zbl 0867.55019 · doi:10.1093/acprof:oso/9780199534920.003.0012
[22]	D. Naidu, Categorical Morita equivalence for group-theoretical categories. Comm. Algebra 35 (2007), no. 11, 3544-3565 Zbl 1143.18009 MR 2362670 · Zbl 1143.18009 · doi:10.1080/00927870701511996
[23]	T. Nikolaus and K. Waldorf, Four equivalent versions of nonabelian gerbes. Pacific J. Math. 264 (2013), no. 2, 355-419 Zbl 1286.55006 MR 3089401 · Zbl 1286.55006 · doi:10.2140/pjm.2013.264.355
[24]	V. Ostrik, Module categories over the Drinfeld double of a finite group. Int. Math. Res. Not. (2003), no. 27, 1507-1520 Zbl 1044.18005 MR 1976233 · Zbl 1044.18005 · doi:10.1155/S1073792803205079
[25]	V. Ostrik, Module categories, weak Hopf algebras and modular invariants. Transform. Groups 8 (2003), no. 2, 177-206 Zbl 1044.18004 MR 1976459 · Zbl 1044.18004 · doi:10.1007/s00031-003-0515-6
[26]	L. Pontrjagin, The theory of topological commutative groups. Ann. of Math. (2) 35 (1934), no. 2, 361-388 Zbl 60.0362.02 MR 1503168 · JFM 60.0362.02 · doi:10.2307/1968438
[27]	D. A. Pronk, Etendues and stacks as bicategories of fractions. Compositio Math. 102 (1996), no. 3, 243-303 Zbl 0871.18003 MR 1401424 · Zbl 0871.18003
[28]	A. Rousseau, Bicatégories monoïdales et extensions de gr-catégories. Homology Homotopy Appl. 5 (2003), no. 1, 437-547 Zbl 1068.18007 MR 2072344 · Zbl 1068.18007 · doi:10.4310/HHA.2003.v5.n1.a19
[29]	L. H. Rowen, Graduate algebra: noncommutative view. Grad. Stud. Math. 91, American Math-ematical Society, Providence, RI, 2008 Zbl 1182.16001 MR 2462400 · Zbl 1182.16001 · doi:10.1090/gsm/091
[30]	C. J. Schommer-Pries, Central extensions of smooth 2-groups and a finite-dimensional string 2-group. Geom. Topol. 15 (2011), no. 2, 609-676 Zbl 1216.22005 MR 2800361 · Zbl 1216.22005 · doi:10.2140/gt.2011.15.609
[31]	G. Segal, Cohomology of topological groups. In Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), pp. 377-387, Academic Press, London, 1970 Zbl 0223.57034 MR 280572 · Zbl 0223.57034
[32]	B. Uribe, On the classification of pointed fusion categories up to weak Morita equivalence. Pacific J. Math. 290 (2017), no. 2, 437-466 Zbl 1387.18015 MR 3681114 · Zbl 1387.18015 · doi:10.2140/pjm.2017.290.437
[33]	K. Waldorf, Multiplicative bundle gerbes with connection. Differential Geom. Appl. 28 (2010), no. 3, 313-340 Zbl 1191.53022 MR 2610397 · Zbl 1191.53022 · doi:10.1016/j.difgeo.2009.10.006
[34]	Jaider Blanco Departamento de Matemáticas y Estadística, Universidad del Norte, Km. 5 Vía Antigua a Puerto Colombia, 080020 Barranquilla, Colombia; [e-mail] Bernardo Uribe Departamento de Matemáticas y Estadística, Universidad del Norte, Km. 5 Vía Antigua a Puerto Colombia, 080020 Barranquilla, Colombia;
[35]	Bonn, Germany; [e-mail], [e-mail] Konrad Waldorf Institut für Mathematik und Informatik, Universität Greifswald, Walther-Rathenau Straße 47, 17487
[36]	Greifswald, Germany; [e-mail]