String connections and Chern-Simons theory
HTML articles powered by AMS MathViewer
- by Konrad Waldorf PDF
- Trans. Amer. Math. Soc. 365 (2013), 4393-4432
Abstract:
We present a finite-dimensional and smooth formulation of string structures on spin bundles. It uses trivializations of the Chern-Simons 2-gerbe associated to this bundle. Our formulation is particularly suitable to deal with string connections: it enables us to prove that every string structure admits a string connection and that the possible choices form an affine space. Further we discover a new relation between string connections, 3-forms on the base manifold, and degree three differential cohomology. We also discuss in detail the relation between our formulation of string connections and the original version of Stolz and Teichner.References
- Orlando Alvarez and I. M. Singer, Beyond the elliptic genus, Nuclear Phys. B 633 (2002), no. 3, 309–344. MR 1910266, DOI 10.1016/S0550-3213(02)00233-X
- John C. Baez, Danny Stevenson, Alissa S. Crans, and Urs Schreiber, From loop groups to 2-groups, Homology Homotopy Appl. 9 (2007), no. 2, 101–135. MR 2366945
- 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. MR 1291698, DOI 10.1215/S0012-7094-94-07518-2
- J.-L. Brylinski and D. A. McLaughlin, The geometry of degree-$4$ characteristic classes and of line bundles on loop spaces. II, Duke Math. J. 83 (1996), no. 1, 105–139. MR 1388845, DOI 10.1215/S0012-7094-96-08305-2
- Jean-Luc Brylinski and Dennis McLaughlin, A geometric construction of the first Pontryagin class, Quantum topology, Ser. Knots Everything, vol. 3, World Sci. Publ., River Edge, NJ, 1993, pp. 209–220. MR 1273576, DOI 10.1142/9789812796387_{0}012
- Alan L. Carey, Stuart Johnson, Michael K. Murray, Danny Stevenson, and Bai-Ling Wang, Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories, Comm. Math. Phys. 259 (2005), no. 3, 577–613. MR 2174418, DOI 10.1007/s00220-005-1376-8
- Shiing Shen Chern and James Simons, Characteristic forms and geometric invariants, Ann. of Math. (2) 99 (1974), 48–69. MR 353327, DOI 10.2307/1971013
- Domenico Fiorenza, Urs Schreiber, and Jim Stasheff, Cech cocycles for differential characteristic classes – An infinity-lie theoretic construction, Preprint. [arxiv:1011.4735]
- Daniel S. Freed, Classical Chern-Simons theory. II, Houston J. Math. 28 (2002), no. 2, 293–310. Special issue for S. S. Chern. MR 1898192
- Daniel S. Freed and Gregory W. Moore, Setting the quantum integrand of M-theory, Comm. Math. Phys. 263 (2006), no. 1, 89–132. MR 2207325, DOI 10.1007/s00220-005-1482-7
- K. Gawędzki, Topological actions in two-dimensional quantum field theories, Nonperturbative quantum field theory (Cargèse, 1987) NATO Adv. Sci. Inst. Ser. B: Phys., vol. 185, Plenum, New York, 1988, pp. 101–141. MR 1008277
- Krzysztof Gawędzki and Nuno Reis, WZW branes and gerbes, Rev. Math. Phys. 14 (2002), no. 12, 1281–1334. MR 1945806, DOI 10.1142/S0129055X02001557
- Kiyonori Gomi and Yuji Terashima, Higher-dimensional parallel transports, Math. Res. Lett. 8 (2001), no. 1-2, 25–33. MR 1825257, DOI 10.4310/MRL.2001.v8.n1.a4
- Henning Hohnhold, Stephan Stolz, and Peter Teichner, From minimal geodesics to supersymmetric field theories, A celebration of the mathematical legacy of Raoul Bott, CRM Proc. Lecture Notes, vol. 50, Amer. Math. Soc., Providence, RI, 2010, pp. 207–274. MR 2648897, DOI 10.1090/crmp/050/20
- M. J. Hopkins and I. M. Singer, Quadratic functions in geometry, topology, and M-theory, J. Differential Geom. 70 (2005), no. 3, 329–452. MR 2192936
- Stuart Johnson, Constructions with bundle gerbes, Ph.D. thesis, University of Adelaide, 2002. [arxiv:math/0312175]
- T. P. Killingback, World-sheet anomalies and loop geometry, Nuclear Phys. B 288 (1987), no. 3-4, 578–588. MR 892061, DOI 10.1016/0550-3213(87)90229-X
- Jacob Lurie, On the classification of topological field theories, Current developments in mathematics, 2008, Int. Press, Somerville, MA, 2009, pp. 129–280. MR 2555928
- Dennis A. McLaughlin, Orientation and string structures on loop space, Pacific J. Math. 155 (1992), no. 1, 143–156. MR 1174481
- Eckhard Meinrenken, The basic gerbe over a compact simple Lie group, Enseign. Math. (2) 49 (2003), no. 3-4, 307–333. MR 2026898
- Jouko Mickelsson, Kac-Moody groups, topology of the Dirac determinant bundle, and fermionization, Comm. Math. Phys. 110 (1987), no. 2, 173–183. MR 887993
- Michael K. Murray, Bundle gerbes, J. Lond. Math. Soc. 54 (1996), 403–416.
- Michael K. Murray, An introduction to bundle gerbes, The many facets of geometry, Oxford Univ. Press, Oxford, 2010, pp. 237–260. MR 2681698, DOI 10.1093/acprof:oso/9780199534920.003.0012
- Michael K. Murray and Daniel Stevenson, Bundle gerbes: stable isomorphism and local theory, J. London Math. Soc. (2) 62 (2000), no. 3, 925–937. MR 1794295, DOI 10.1112/S0024610700001551
- Thomas Nikolaus, Äquivariante Gerben und Abstieg, Diplomarbeit, Universität Hamburg, 2009.
- Corbett Redden and Konrad Waldorf, Private discussion.
- D. Corbett Redden, Canonical metric connections associated to string structures, ProQuest LLC, Ann Arbor, MI, 2006. Thesis (Ph.D.)–University of Notre Dame. MR 2709440
- Christopher J. Schommer-Pries, Central extensions of smooth 2-groups and a finite-dimensional string 2-group, Geom. Topol. 15 (2011), no. 2, 609–676. MR 2800361, DOI 10.2140/gt.2011.15.609
- Urs Schreiber and Konrad Waldorf, Parallel transport and functors, J. Homotopy Relat. Struct. 4 (2009), no. 1, 187–244. MR 2520993
- Christoph Schweigert and Konrad Waldorf, Gerbes and Lie groups, Developments and trends in infinite-dimensional Lie theory (Karl-Hermann Neeb and Arturo Pianzola, eds.), Progr. Math., vol. 600, Birkhäuser, 2010. [arxiv:0710.5467]
- James Simons and Dennis Sullivan, Axiomatic characterization of ordinary differential cohomology, J. Topol. 1 (2008), no. 1, 45–56. MR 2365651, DOI 10.1112/jtopol/jtm006
- Mauro Spera and Tilmann Wurzbacher, Twistor spaces and spinors over loop spaces, Math. Ann. 338 (2007), no. 4, 801–843. MR 2317752, DOI 10.1007/s00208-007-0085-3
- Daniel Stevenson, The geometry of bundle gerbes, Ph.D. thesis, University of Adelaide, 2000. [arxiv:math.DG/1004117]
- Daniel Stevenson, Bundle 2-gerbes, Proc. London Math. Soc. (3) 88 (2004), no. 2, 405–435. MR 2032513, DOI 10.1112/S0024611503014357
- Stephan Stolz and Peter Teichner, Supersymmetric Euclidean field theories and generalized cohomology, Preprint.
- Stephan Stolz and Peter Teichner, What is an elliptic object?, Topology, geometry and quantum field theory, London Math. Soc. Lecture Note Ser., vol. 308, Cambridge Univ. Press, Cambridge, 2004, pp. 247–343. MR 2079378, DOI 10.1017/CBO9780511526398.013
- Konrad Waldorf, A construction of string $2$-group models using a transgression-regression technique, Preprint. [arxiv:1201.5052].
- —, Transgression to loop spaces and its inverse, I: Diffeological bundles and fusion maps, Cah. Topol. Géom. Différ. Catég. (to appear) [arxiv:0911.3212].
- —, Transgression to loop spaces and its inverse, II: Gerbes and fusion bundles with connection, Preprint. [arxiv:1004:0031]
- —, Algebraic structures for bundle gerbes and the Wess-Zumino term in conformal field theory, Ph.D. thesis, Universität Hamburg, 2007.
- Konrad Waldorf, More morphisms between bundle gerbes, Theory Appl. Categ. 18 (2007), No. 9, 240–273. MR 2318389
- Konrad Waldorf, Multiplicative bundle gerbes with connection, Differential Geom. Appl. 28 (2010), no. 3, 313–340. MR 2610397, DOI 10.1016/j.difgeo.2009.10.006
- Konrad Waldorf, A loop space formulation for geometric lifting problems, J. Aust. Math. Soc. 90 (2011), no. 1, 129–144. MR 2810948, DOI 10.1017/S1446788711001182
- Edward Witten, The index of the Dirac operator in loop space, Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986) Lecture Notes in Math., vol. 1326, Springer, Berlin, 1988, pp. 161–181. MR 970288, DOI 10.1007/BFb0078045
Additional Information
- Konrad Waldorf
- Affiliation: Department of Mathematics, 970 Evans Hall #3840, University of California, Berkeley, Berkeley, California 94720
- Address at time of publication: Fakultät für Mathematik, Universität Regensburg, Universitätsstrasse 31, 93053 Regensburg, Germany
- Email: waldorf@math.berkeley.edu
- Received by editor(s): June 30, 2011
- Received by editor(s) in revised form: January 29, 2012
- Published electronically: March 5, 2013
- Additional Notes: The author gratefully acknowledges support by a Feodor-Lynen scholarship, granted by the Alexander von Humboldt Foundation. The author thanks Martin Olbermann, Arturo Prat-Waldron, Urs Schreiber and Peter Teichner for exciting discussions, and two referees for their helpful comments and suggestions.
- © Copyright 2013 by the author
- Journal: Trans. Amer. Math. Soc. 365 (2013), 4393-4432
- MSC (2010): Primary 53C08; Secondary 57R56, 57R15
- DOI: https://doi.org/10.1090/S0002-9947-2013-05816-3
- MathSciNet review: 3055700