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An invitation to higher gauge theory. (English) Zbl 1225.83001

Higher gauge theory is a generalization of the familiar gauge theory, which is concerned with transports of point particles, to higher-dimensional objects. It should not be surprising that the emerging theory should be applicable to string theory and loop quantum gravity, both of which agree that we need higher-dimensional extended objects, though always disputing in almost all other points concerned. This paper is to sketch how to generalize the theory of parallel transport from point particles to \(1\)-dimensional objects with such a bare minimum of such prerequisites as manifolds, differential forms, Lie groups, Lie algebras and the traditional theory of bundles and connections. In place of a connection, which tells us how particles transform as they move along paths, one should speak of a \(2\)-connection, which tells us how strings transform as they sweep out surfaces. Six interesting examples, such as the Poincaré \(2\)-group leading to spin foam model for Minkowski spacetime, are discussed. For more applications, one can visit, e.g., [H. Sati, Proceedings of Symposia in Pure Mathematics 81, 181–236 (2010; Zbl 1210.81089), J. Aust. Math. Soc. 90, No. 1, 93–108 (2011; Zbl 1217.81131)].

MSC:

83-02 Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
83A05 Special relativity
83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
83C22 Einstein-Maxwell equations
83E30 String and superstring theories in gravitational theory
78A25 Electromagnetic theory (general)
17B45 Lie algebras of linear algebraic groups
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
81T20 Quantum field theory on curved space or space-time backgrounds
83E50 Supergravity
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[1] Aschieri P., Cantini L., Jurčo B.: Nonabelian bundle gerbes, their differential geometry and gauge theory. Commun. Math. Phys. 254, 367–400 (2005) Also available as arXiv:hep-th/0312154 · Zbl 1092.53020 · doi:10.1007/s00220-004-1220-6
[2] Aschieri P., Jurčo B.: Gerbes, M5-brane anomalies and E 8 gauge theory. JHEP 10, 068 (2004) Also available as arXiv:hep-th/0409200 · doi:10.1088/1126-6708/2004/10/068
[3] Ashtekar A., Lewandowski J.: Differential geometry on the space of connections via graphs and projective limits. J. Geom. Phys. 17, 191–230 (1995) Also available as arXiv:hep-th/9412073 · Zbl 0851.53014 · doi:10.1016/0393-0440(95)00028-G
[4] Baez J.: Four-dimensional BF theory as a topological quantum field theory. Lett. Math. Phys. 38, 129–143 (1996) · Zbl 0858.57022 · doi:10.1007/BF00398315
[5] Baez J.: Spin foam models. Class. Quantum Grav. 15, 1827–1858 (1998) Also available as arXiv: gr-qc/9709052 · Zbl 0932.83014 · doi:10.1088/0264-9381/15/7/004
[6] Baez J.: An introduction to spin foam models of BF theory and quantum gravity. In: Gausterer, H., Grosse, H. (eds) Geometry and Quantum Physics, pp. 25–93. Springer, Berlin (2000) · Zbl 0978.81043
[7] Baez, J., Baratin, A., Freidel, L., Wise, D.: Infinite-dimensional representations of 2-groups. Available as arXiv:0812.4969 · Zbl 1342.18008
[8] Baez J., Crans A.: Higher dimensional algebra VI: Lie 2-algebras. Theory Appl Categories 12, 492–538 (2004) Also available as arXiv:math.0307263 · Zbl 1057.17011
[9] Baez J., Crans A., Schreiber U., Stevenson D.: From loop groups to 2-groups. HHA 9, 101–135 (2007) Also available as arXiv:math.QA/0504123
[10] Baez J., Crans A., Wise D.: Exotic statistics for strings in 4d BF theory. Adv. Theor. Math. Phys. 11, 707–749 (2007) Also available as arXiv:gr-qc/0603085 · Zbl 1134.81039 · doi:10.4310/ATMP.2007.v11.n5.a1
[11] Baez J., Hoffnung A., Rogers C.: Categorified symplectic geometry and the classical string. Commun. Math. Phys. 293, 701–715 (2010) Also available as arXiv:0808.0246 · Zbl 1192.81208 · doi:10.1007/s00220-009-0951-9
[12] Baez, J., Huerta, J.: Division algebras and supersymmetry II. Available as arXiv:1003.3436
[13] Baez, J., Lauda, A.: A prehistory of n-categorical physics. Available as arXiv:0908.2469
[14] Baez J., Lauda A.: Higher dimensional algebra V: 2-groups. Theory Appl. Categories 12, 423–491 (2004) Also available as arXiv:math.0307200 · Zbl 1056.18002
[15] Baez J., Perez A.: Quantization of strings and branes coupled to BF theory. Adv. Theor. Math. Phys. 11, 1–19 (2007) Also available as arXiv:gr-qc/0605087 · Zbl 1200.17014 · doi:10.4310/ATMP.2007.v11.n1.a1
[16] Baez, J., Rogers, C.: Categorified symplectic geometry and the string Lie 2-algebra. Homol. Homotopy Appl. (to appear). Available as arXiv:0901.4721
[17] Baez J., Schreiber U. et al.: Higher gauge theory. In: Davydov, A. et al. (eds) Categories in Algebra, Geometry and Mathematical Physics. Contemp. Math., vol. 431, pp. 7–30. AMS, Providence (2007) Also available arXiv:math.0511710 · Zbl 1132.55007
[18] Baez J., Sawin S.: Functional integration on spaces of connections. J. Funct. Analysis 150, 1–26 (1997) Also available as arXiv:q-alg/9507023 · Zbl 0891.46040 · doi:10.1006/jfan.1997.3108
[19] Baez J., Stevenson D.: The classifying space of a topological 2-group. In: Baas, N., Friedlander, E., Jahren, B., Østvær, P.A. (eds) Algebraic Topology: The Abel Symposium 2007, Springer, Berlin (2009)
[20] Bakovic, I., Jurčo, B.: The classifying topos of a topological bicategory. Available as arXiv:0902.1750
[21] Balachandran A.P., Lizzi F., Sparano G.: A new approach to strings and superstrings. Nucl. Phys. B 277, 359–387 (1986) · doi:10.1016/0550-3213(86)90447-5
[22] Baratin A., Freidel L.: Hidden quantum gravity in 3d Feynman diagrams. Class. Quant. Grav. 24, 1993–2026 (2007) Also available as arXiv:gr-qc/0604016 · Zbl 1113.83302 · doi:10.1088/0264-9381/24/8/006
[23] Baratin A., Freidel L.: Hidden quantum gravity in 4d Feynman diagrams: emergence of spin foams. Class. Quant. Grav. 24, 2027–2060 (2007) Also available as arXiv:hep-th/0611042 · Zbl 1113.83303 · doi:10.1088/0264-9381/24/8/007
[24] Baratin, A., Wise, D.: 2-Group representations for spin foams. Available as arXiv:0910.1542
[25] Barrett J.W., Mackaay M.: Categorical representations of categorical groups. Theory Appl. Categories 16, 529–557 (2006) Also available as arXiv:math.0407463
[26] Bartels, T.: Higher gauge theory: 2-bundles. Available as arXiv:math.CT/0410328
[27] Breen L.: Notes on 1- and 2-gerbes. In: Baez, J., May, P. (eds) Towards Higher Categories, pp. 193–235. Springer, Berlin (2009) Also available as arXiv:math.0611317
[28] Breen, L.: Differential geometry of gerbes and differential forms. Available as arXiv:0802.1833
[29] Breen L., Messing W.: Differential geometry of gerbes. Adv. Math. 198, 732–846 (2005) Available as arXiv:math.AG/0106083 · Zbl 1102.14013 · doi:10.1016/j.aim.2005.06.014
[30] Brylinski J.-L.: Loop Spaces, Characteristic Classes and Geometric Quantization. Birkhäuser, Boston (1993) · Zbl 0823.55002
[31] Brylinski, J.-L.: Differentiable cohomology of gauge groups. Available as arXiv:math.DG/0011069
[32] Brylinski J.-L., McLaughlin D.A.: The geometry of degree-four characteristic classes and of line bundles on loop spaces I. Duke Math. J. 75, 603–638 (1994) · Zbl 0844.57025 · doi:10.1215/S0012-7094-94-07518-2
[33] Brylinski J.-L., McLaughlin D.A.: The geometry of degree-four characteristic classes and of line bundles on loop spaces II. Duke Math. J. 83, 105–139 (1996) · Zbl 0864.57026 · doi:10.1215/S0012-7094-96-08305-2
[34] Carey A.L., Johnson S., Murray M.K.: Holonomy on D-branes. J. Geom. Phys. 52, 186–216 (2004) Also available as arXiv:hep-th/0204199 · Zbl 1092.81055 · doi:10.1016/j.geomphys.2004.02.008
[35] Castellani L., D’Auria R., Fré P.: Supergravity and Superstrings: A Geometric Perspective. World Scientific, Singapore (1991)
[36] Crane, L., Kauffman, L., Yetter, D.: State-sum invariants of 4-manifolds I. Available as arXiv: hep-th/9409167
[37] Crane, L., Sheppeard, M.D.: 2-Categorical Poincaré representations and state sum applications. Available as arXiv:math.0306440
[38] Crane L., Yetter D.: A categorical construction of 4d TQFTs. In: Kauffman, L., Baadhio, R. (eds) Quantum Topology, pp. 120–130. World Scientific, Singapore (1993) Also available as arXiv: hep-th/9301062 · Zbl 0841.57030
[39] Crane L., Yetter D.N.: Measurable categories and 2-groups. Appl. Categorical Struct 13, 501–516 (2005) Also available as arXiv:math.0305176 · Zbl 1107.18004 · doi:10.1007/s10485-005-9004-5
[40] DeDonder T.: Theorie Invariantive du Calcul des Variations. Gauthier–Villars, Paris (1935)
[41] DeWitt-Morette C., Maheshwari A., Nelson B.: Path integration in phase space. Gen. Relativ. Gravit. 8, 581–593 (1977) · Zbl 0418.58009 · doi:10.1007/BF00756309
[42] Duff, M.J.: Supermembranes: the first fifteen weeks. Class. Quantum Grav. 5, 189–205 (1988). Also available at http://ccdb4fs.kek.jp/cgi-bin/img_index?8708425
[43] Eckmann B., Hilton P.: Group-like structures in categories. Math. Ann. 145, 227–255 (1962) · Zbl 0099.02101 · doi:10.1007/BF01451367
[44] Elgueta J.: Representation theory of 2-groups on Kapranov and Voevodsky 2-vector spaces. Adv. Math. 213, 53–92 (2007) Also available as arXiv:math.0408120 · Zbl 1118.18003 · doi:10.1016/j.aim.2006.11.010
[45] Fairbairn W., Perez A.: Extended matter coupled to BF theory. Adv. Theor. Math. Phys. D 78, 024013 (2008) Also available as arXiv:0709.4235
[46] Fairbairn, W., Noui, K., Sardelli, F.: Canonical analysis of algebraic string actions. Available as arXiv:0908.0953 · Zbl 1269.83031
[47] Forrester-Barker, M.: Group objects and internal categories. Available as arXiv:math.CT/0212065
[48] Freed D.S., Witten E.: Anomalies in string theory with D-Branes, §6: Additional remarks. Asian J. Math. 3, 819–852 (1999) Also available as arXiv:hep-th/9907189 · Zbl 1028.81052
[49] Freidel L., Louapre D.: Ponzano–Regge model revisited. I: Gauge fixing, observables and interacting spinning particles. Class. Quant. Grav. 21, 5685–5726 (2004) Also available as arXiv:hep-th/0401076 · Zbl 1060.83013 · doi:10.1088/0264-9381/21/24/002
[50] Freidel, L., Louapre, D.: Ponzano–Regge model revisited. II: Equivalence with Chern–Simons. Also available as arXiv:gr-qc/0410141
[51] Freidel, L., Livine, E.: Ponzano–Regge model revisited. III: Feynman diagrams and effective field theory. Also available as arXiv:hep-th/0502106
[52] Gawedzki K.: Topological actions in two-dimensional quantum field theories. In: t’Hooft, G., Jaffe, A., Mack, G., Mitter, P.K., Stora, R. (eds) Nonperturbative Quantum Field Theory, pp. 101–141. Plenum, New York (1988)
[53] Gawedzki K., Reis N.: WZW branes and gerbes. Rev. Math. Phys. 14, 1281–1334 (2002) Also available as arXiv:hep-th/0205233 · Zbl 1033.81067 · doi:10.1142/S0129055X02001557
[54] Getzler, E.: Lie theory for nilpotent L algebras. Available as arXiv:math.0404003
[55] Gotay, M., Isenberg, J., Marsden, J., Montgomery, R.: Momentum maps and classical relativistic fields. Part I: covariant field theory. Available as arXiv:physics/9801019
[56] Girelli F., Pfeiffer H.: Higher gauge theory–differential versus integral formulation. J. Math. Phys. 45, 3949–3971 (2004) Also available as arXiv:hep-th/0309173 · Zbl 1071.53011 · doi:10.1063/1.1790048
[57] Girelli F., Pfeiffer H., Popescu E.M.: Topological higher gauge theory–from BF to BFCG theory. J. Math. Phys. 49, 032503 (2008) Also available as arXiv:0708.3051 · Zbl 1153.81366 · doi:10.1063/1.2888764
[58] Guillemin V., Sternberg S.: Symplectic Techniques in Physics. Cambridge University Press, Cambridge (1984) · Zbl 0576.58012
[59] Henriques, A.: Integrating L algebras. Available as arXiv:math.0603563
[60] Johnson M.: The combinatorics of n-categorical pasting. J. Pure Appl. Algebra 62, 211–225 (1989) · Zbl 0694.18007 · doi:10.1016/0022-4049(89)90136-9
[61] Lack, S.: A 2-categories companion. In: Baez J., May P. (eds) Towards Higher Categories pp. 105–191. . Springer (2009). Also available as arXiv:math.0702535
[62] Lerman, E.: Orbifolds as stacks? Available as arXiv:0806.4160
[63] Lewandowski J., Thiemann T.: Diffeomorphism invariant quantum field theories of connections in terms of webs. Class. Quant. Grav. 16, 2299–2322 (1999) Also available as arXiv:gr-qc/9901015 · Zbl 0939.58014 · doi:10.1088/0264-9381/16/7/311
[64] Mac Lane S., Whitehead J.H.C.: On the 3-type of a complex. Proc. Nat. Acad. Sci. 36, 41–48 (1950) · Zbl 0035.39001 · doi:10.1073/pnas.36.1.41
[65] Mackaay M., Picken R.: Holonomy and parallel transport for Abelian gerbes. Adv. Math. 170, 287–339 (2002) Also available as arXiv:math.DG/0007053 · Zbl 1034.53051
[66] Markl, M., Schnider, S., Stasheff, J.: Operads in Algebra, Topology and Physics. AMS, Providence, Rhode Island (2002) · Zbl 1017.18001
[67] Martins, J.F., Picken, R.: On two-dimensional holonomy. Available as arXiv:0710.4310
[68] Martins, J.F., Picken, R.: A cubical set approach to 2-bundles with connection and Wilson surfaces. Available as arXiv:0808.3964
[69] Martins, J.F., Picken, R.: The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module. Available as arXiv:0907.2566
[70] Milnor, J.: Remarks on infinite dimensional Lie groups. In: Relativity, Groups and Topology II, (Les Houches, 1983), North-Holland, Amsterdam (1984) · Zbl 0557.10031
[71] Moerdijk, I.: Introduction to the language of stacks and gerbes. Available as arXiv:math.AT/0212266
[72] Montesinos M., Perez A.: Two-dimensional topological field theories coupled to four-dimensional BF theory. Phys. Rev. D 77, 104020 (2008) Also available as arXiv:0709.4235 · doi:10.1103/PhysRevD.77.104020
[73] Murray M.K.: Bundle gerbes. J. London Math. Soc. 54, 403–416 (1996) Also available as arXiv: dg-ga/9407015 · Zbl 0867.55019 · doi:10.1112/jlms/54.2.403
[74] Murray, M.K.: An introduction to bundle gerbes. Available as arXiv:0712.1651
[75] Murray M.K., Stevenson D.: Higgs fields, bundle gerbes and string structures. Commun. Math. Phys. 243, 541–555 (2003) Also available as arXiv:math.DG/0106179 · Zbl 1085.53019 · doi:10.1007/s00220-003-0984-4
[76] Pfeiffer H.: Higher gauge theory and a non-Abelian generalization of 2-form electromagnetism. Ann. Phys. 308, 447–477 (2003) Also available as arXiv:hep-th/0304074 · Zbl 1056.70013 · doi:10.1016/S0003-4916(03)00147-7
[77] Power A.J.: A 2-categorical pasting theorem. J. Algebra 129, 439–445 (1990) · Zbl 0698.18005 · doi:10.1016/0021-8693(90)90229-H
[78] Pressley A., Segal G.: Loop Groups. Oxford University Press, Oxford (1986) · Zbl 0618.22011
[79] Roberts, D., Schreiber, U.: The inner automorphism 3-group of a strict 2-group. Also available as arXiv:0708.1741
[80] Rovelli, C.: Quantum Gravity. Cambridge University Press, Cambridge (2004). Also available at http://www.cpt.univ-mrs.fr/\(\sim\)rovelli/book.pdf · Zbl 1091.83001
[81] Roytenberg, D.: On weak Lie 2-algebras. Available as arXiv:0712.3461
[82] Sati, H.: Geometric and topological structures related to M-branes. Available as arXiv:1001.5020
[83] Sati, H., Schreiber, U., Stasheff, J.: L algebras and applications to string- and Chern–Simons n-transport. Available as arXiv:0801.3480 · Zbl 1183.83099
[84] Schlessinger M., Stasheff J.: The Lie algebra structure of tangent cohomology and deformation theory. J. Pure Appl. Algebra 38, 313–322 (1985) · Zbl 0576.17008 · doi:10.1016/0022-4049(85)90019-2
[85] Schommer-Pries, C.: A finite-dimensional string 2-group. Available as arXiv:0911.2483
[86] Schreiber, U.: Comments at the n-Category Café. Available at http://golem.ph.utexas.edu/category/2009/09/questions_on_ncurvature.html
[87] Schreiber U., Waldorf K.: Parallel transport and functors. J. Homotopy Relat. Struct. 4, 187–244 (2009) Also available as arXiv:0705.0452 · Zbl 1189.53026
[88] Schreiber, U., Waldorf, K.: Smooth functors vs. differential forms. Available as arXiv:0802.0663 · Zbl 1230.53025
[89] Schreiber, U., Waldorf, K.: Connections on non-abelian gerbes and their holonomy. Available as arXiv:0808.1923
[90] Stevenson, D.: The Geometry of Bundle Gerbes. Ph.D. thesis, University of Adelaide (2000). Also available as arXiv:math.DG/0004117
[91] Stolz S., Teichner P.: What is an elliptic object?. In: Tillmann, U. (ed.) Topology, Geometry and Quantum Field Theory: Proceedings of the 2002 Oxford Symposium in Honour of the 60th Birthday of Graeme Segal, Cambridge Uuniversity Press, Cambridge (2004) · Zbl 1107.55004
[92] Waldorf, K.: String connections and Chern–Simons theory. Available as arXiv:0906.0117 · Zbl 1277.53024
[93] Weyl H.: Geodesic fields in the calculus of variation for multiple integrals. Ann. Math. 36, 607–629 (1935) · Zbl 0013.12002 · doi:10.2307/1968645
[94] Whitehead J.H.C.: Note on a previous paper entitled ’On adding relations to homotopy groups’. Ann. Math. 47, 806–810 (1946) · Zbl 0060.41104 · doi:10.2307/1969237
[95] Whitehead J.H.C.: Combinatorial homotopy II. Bull. Am. Math. Soc. 55, 453–496 (1949) · Zbl 0040.38801 · doi:10.1090/S0002-9904-1949-09213-3
[96] Witten E.: The index of the Dirac operator in loop space. In: Landweber, P.S. (ed.) Elliptic Curves and Modular Forms in Algebraic Topology. Lecture Notes in Mathematics, vol 1326, pp. 161–181. Springer, Berlin (1988)
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