Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

String connections and Chern-Simons theory


Author: Konrad Waldorf
Journal: Trans. Amer. Math. Soc. 365 (2013), 4393-4432
MSC (2010): Primary 53C08; Secondary 57R56, 57R15
Published electronically: March 5, 2013
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. O. Alvarez and I. M. Singer, Beyond the elliptic genus, Nuclear Phys. B 633 (2002), no. 3, 309-344. MR 1910266 (2003h:58047)
  • 2. John C. Baez, Alissa S. Crans, Danny Stevenson, and Urs Schreiber, From loop groups to 2-groups, Homology, Homotopy Appl. 9 (2007), no. 2, 101-135. MR 2366945 (2009c:22022)
  • 3. Jean-Luc 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 (95m:57038)
  • 4. -, The geometry of degree four characteristic classes and of line bundles on loop spaces II, Duke Math. J. 83 (1996), no. 1, 105-139. MR 1388845 (97j:58157)
  • 5. Jean-Luc Brylinski and Dennis McLaughlin, A geometric construction of the first Pontryagin class, Series on knots and everything (H. Kauffman and Randy A. Baadhio, eds.), World Scientific, Singapore, 1992, pp. 209-220. MR 1273576 (95b:55017)
  • 6. Alan L. Carey, Stuart Johnson, Michael K. Murray, Danny Stevenson, and Bai-Ling Wang, Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories, Commun. Math. Phys. 259 (2005), no. 3, 577-613. MR 2174418 (2007a:58023)
  • 7. Shiing-Shen Chern and James Simons, Characteristic forms and geometric invariants, Ann. of Math. 99 (1974), no. 1, 48-69. MR 0353327 (50:5811)
  • 8. Domenico Fiorenza, Urs Schreiber, and Jim Stasheff, Cech cocycles for differential characteristic classes - An infinity-lie theoretic construction, Preprint. [arxiv:1011.4735]
  • 9. Daniel S. Freed, Classical Chern Simons theory, II, Houston J. Math. 28 (2002), no. 1, 293-310. MR 1898192 (2003f:55022)
  • 10. Daniel S. Freed and Gregory W. Moore, Setting the quantum integrand of M-theory, Commun. Math. Phys. 263 (2006), no. 1, 89-132. MR 2207325 (2006k:58055)
  • 11. Krzysztof Gawedzki, Topological actions in two-dimensional quantum field theories, Non-perturbative quantum field theory (G. 't Hooft, A. Jaffe, G. Mack, K. Mitter, and R. Stora, eds.), Plenum Press, 1988, pp. 101-142. MR 1008277 (90i:81122)
  • 12. Krzysztof Gawedzki and Nuno Reis, WZW branes and gerbes, Rev. Math. Phys. 14 (2002), no. 12, 1281-1334. MR 1945806 (2003m:81222)
  • 13. Kiyonori Gomi and Yuji Terashima, Higher-dimensional parallel transports, Math. Res. Lett. 8 (2001), 25-33. MR 1825257 (2002c:53082)
  • 14. Henning Hohnhold, Stephan Stolz, and Peter Teichner, From minimal geodesics to supersymmetric field theories, A celebration of the mathematical legacy of Raoul Bott (P. Robert Kotiuga, ed.), CRM Proceedings and Lecture Notes, vol. 50, 2010, pp. 207-276. MR 2648897 (2011j:19017)
  • 15. 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 (2007b:53052)
  • 16. Stuart Johnson, Constructions with bundle gerbes, Ph.D. thesis, University of Adelaide, 2002. [arxiv:math/0312175]
  • 17. T. Killingback, World sheet anomalies and loop geometry, Nuclear Phys. B 288 (1987), 578. MR 892061 (88f:53120)
  • 18. Jacob Lurie, On the classification of topological field theories, Current developments in mathematics, vol. 2008, International Press of Boston, 2009, pp. 129-280. MR 2555928 (2010k:57064)
  • 19. Dennis A. McLaughlin, Orientation and string structures on loop space, Pacific J. Math. 155 (1992), no. 1, 143-156. MR 1174481 (93j:57015)
  • 20. Eckhard Meinrenken, The basic gerbe over a compact simple Lie group, Enseign. Math., II. Sér. 49 (2002), no. 3-4, 307-333. MR 2026898 (2004j:53064)
  • 21. Jouko Mickelsson, Kac-Moody groups, topology of the Dirac determinant bundle and fermionization, Commun. Math. Phys. 110 (1987), 173-183. MR 887993 (89a:22036)
  • 22. Michael K. Murray, Bundle gerbes, J. Lond. Math. Soc. 54 (1996), 403-416.
  • 23. -, An introduction to bundle gerbes, The many facets of geometry. A tribute to Nigel Hitchin (Oscar Garcia-Prada, Jean Pierre Bourguignon, and Simon Salamon, eds.), Oxford Univ. Press, 2010. MR 2681698 (2011h:53026)
  • 24. Michael K. Murray and Daniel Stevenson, Bundle gerbes: stable isomorphism and local theory, J. Lond. Math. Soc. 62 (2000), 925-937. MR 1794295 (2001j:55019)
  • 25. Thomas Nikolaus, Äquivariante Gerben und Abstieg, Diplomarbeit, Universität Hamburg, 2009.
  • 26. Corbett Redden and Konrad Waldorf, Private discussion.
  • 27. D. Corbett Redden, Canonical metric connections associated to string structures, Ph.D. thesis, University of Notre Dame, 2006. MR 2709440
  • 28. Chris Schommer-Pries, Central extensions of smooth $ 2$-groups and a finite-dimensional string $ 2$-group, Geom. Topol. 15 (2011), 609-676. MR 2800361
  • 29. Urs Schreiber and Konrad Waldorf, Parallel transport and functors, J. Homotopy Relat. Struct. 4 (2009), 187-244. MR 2520993 (2010k:53039)
  • 30. 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]
  • 31. James Simons and Dennis Sullivan, Axiomatic characterization of ordinary differential cohomology, J. Topology 1 (2008), no. 1, 45-56. MR 2365651 (2009e:58035)
  • 32. Mauro Spera and Tilmann Wurzbacher, Twistor spaces and spinors over loop spaces, Math. Ann. 338 (2007), 801-843. MR 2317752 (2008i:53063)
  • 33. Daniel Stevenson, The geometry of bundle gerbes, Ph.D. thesis, University of Adelaide, 2000. [arxiv:math.DG/1004117]
  • 34. -, Bundle 2-gerbes, Proc. Lond. Math. Soc. 88 (2004), 405-435. MR 2032513 (2005b:18010)
  • 35. Stephan Stolz and Peter Teichner, Supersymmetric Euclidean field theories and generalized cohomology, Preprint.
  • 36. -, What is an elliptic object?, Topology, geometry and quantum field theory, London Math. Soc. Lecture Note Ser., vol. 308, Cambridge Univ. Press, 2004, pp. 247-343. MR 2079378 (2005m:58048)
  • 37. Konrad Waldorf, A construction of string $ 2$-group models using a transgression-regression technique, Preprint. [arxiv:1201.5052].
  • 38. -, Transgression to loop spaces and its inverse, I: Diffeological bundles and fusion maps, Cah. Topol. Géom. Différ. Catég. (to appear) [arxiv:0911.3212].
  • 39. -, Transgression to loop spaces and its inverse, II: Gerbes and fusion bundles with connection, Preprint. [arxiv:1004:0031]
  • 40. -, Algebraic structures for bundle gerbes and the Wess-Zumino term in conformal field theory, Ph.D. thesis, Universität Hamburg, 2007.
  • 41. -, More morphisms between bundle gerbes, Theory Appl. Categ. 18 (2007), no. 9, 240-273. MR 2318389 (2008k:53101)
  • 42. -, Multiplicative bundle gerbes with connection, Differential Geom. Appl. 28 (2010), no. 3, 313-340. MR 2610397 (2011c:53044)
  • 43. -, A loop space formulation for geometric lifting problems, J. Aust. Math. Soc. 90 (2011), 129-144. MR 2810948
  • 44. Edward Witten, The index of the Dirac operator on loop space, Elliptic curves and modular forms in algebraic topology, Lecture Notes in Math., no. 1326, Springer, 1986, pp. 161-181. MR 970288

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53C08, 57R56, 57R15

Retrieve articles in all journals with MSC (2010): 53C08, 57R56, 57R15


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

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05816-3
PII: S 0002-9947(2013)05816-3
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.
Article copyright: © Copyright 2013 by the author