Communications in Mathematical Physics

, Volume 367, Issue 2, pp 417–453 | Cite as

Supersymmetric Field Theories from Twisted Vector Bundles

  • Augusto StoffelEmail author
Open Access


We give a description of the delocalized twisted cohomology of an orbifold and the Chern character of a twisted vector bundle in terms of supersymmetric Euclidean field theories. This includes the construction of a twist functor for \({1\vert1}\)-dimensional EFTs from the data of a gerbe with connection.



Open access funding provided by Max Planck Society. This paper is based on a part of my Ph.D. thesis [24], and I would like to thank my advisor, Stephan Stolz, for the guidance. I would also like to thank Matthias Ludewig, Byungdo Park, Peter Teichner, and Peter Ulrickson for valuable discussions, and Karsten Grove for the financial support during my last semester as a graduate student (NSF Grant DMS-1209387).


  1. 1.
    Adem, A., Ruan, Y.: Twisted orbifold K-theory. Commun. Math. Phys. 237(3), 533–556 (2003). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baum, P., Connes, A.: Chern character for discrete groups. In: Matsumoto, Y., Mizutani, T., Morita, S., (eds.) A Fête of Topology, pp. 163–232. Academic Press, Boston (1988).
  3. 3.
    Behrend, K., Xu, P.: Differentiable stacks and gerbes. J. Symplectic Geom. 9(3), 285–341 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bernšteĭn, I.N., Leĭtes, D.A.: Integral forms and the Stokes formula on supermanifolds. Funct. Anal. Appl. 11(1), 55–56 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Berwick-Evans, D.: Twisted equivariant elliptic cohomology with complex coefficients from gauged sigma models. Preprint. arXiv:1410.5500 [math.AT]
  6. 6.
    Berwick-Evans, D., Han, F.: The equivariant Chern character as super holonomy on loop stacks, 10 (2016). Preprint. aXiv:1610.02362 [math.AT]
  7. 7.
    Bouwknegt, P., Carey, A.L., Mathai, V., Murray M.K., Stevenson, D.: Twisted K-theory and K-theory of bundle gerbes. Commun. Math. Phys. 228(1), 17–45 (2002).
  8. 8.
    Brylinski, J.-L.: Loop spaces, characteristic classes and geometric quantization. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston (2008). ISBN 978-0-8176-4730-8. Reprint of the 1993 edition
  9. 9.
    Deligne, P., Morgan, J.W.: Notes on supersymmetry (following Joseph Bernstein). In: Deligne, P., Etingof, P., Freed, D.S., Jeffrey, L.C., Kazhdan, D., Morgan, J.W., Morrison, D.R., Witten, E. (eds.) Quantum Fields and Strings: A Course for Mathematicians, vol. 1, 2 (Princeton, NJ, 1996/1997), pp. 41–97. American Mathematical Society, Providence (1999)Google Scholar
  10. 10.
    Dumitrescu, F.: A geometric view of the Chern character. Preprint. arXiv:1202.2719 [math.AT]
  11. 11.
    Dumitrescu, F.: Superconnections and parallel transport. Pac. J. Math. 236(2), 307–332 (2008).
  12. 12.
    Fiorenza, D., Valentino, A.: Boundary conditions for topological quantum field theories, anomalies and projective modular functors. Commun. Math. Phys. 338(3), 1043–1074 (2015).
  13. 13.
    Freed, D.S.: Anomalies and invertible field theories. In: String-Math 2013, Volume 88 of Proceedings of Symposium Pure Mathematics, pp. 25–45. American Mathematical Society, Providence (2014).
  14. 14.
    Han F.: Supersymmetric QFT, super loop spaces and Bismut-Chern character. Preprint. arXiv:0711.3862 [math.DG]
  15. 15.
    Hohnhold, H., Stolz, S., Teichner, P.: From minimal geodesics to supersymmetric field theories. In: A Celebration of the Mathematical Legacy of Raoul Bott, Volume 50 of CRM Proceedings of Lecture Notes, pp. 207–274. American Mathematical Society, Providence (2010)Google Scholar
  16. 16.
    Hohnhold, H., Kreck, M., Stolz, S., Teichner, P.: Differential forms and 0-dimensional supersymmetric field theories. Quantum Topol. 2(1), 1–41 (2011).
  17. 17.
    Lupercio, E., Uribe, B.: Holonomy for gerbes over orbifolds. J. Geom. Phys. 56(9), 1534–1560 (2006). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Martins-Ferreira, N.: Pseudo-categories. J. Homotopy Relat. Struct. 1(1), 47–78 (2006)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Park, B.: Geometric models of twisted differential K-theory I, 02 (2016). PreprintGoogle Scholar
  20. 20.
    Quillen, D.: Superconnections and the Chern character. Topology. 24(1), 89–95 (1985). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Segal, G.: The definition of conformal field theory. In: Tillmann, U. (ed.) Topology, Geometry and Guantum Field Theory, Volume 308 of London Mathematical Society Lecture Note Series, pp. 421–577. Cambridge University Press, Cambridge (2004)Google Scholar
  22. 22.
    Shulman, M.A.: Constructing symmetric monoidal bicategories. Preprint. arXiv:1004.0993 [math.CT]
  23. 23.
    Stoffel, A.: Dimensional reduction and the equivariant Chern character. Algebraic Geom Topol 19(1), 109–150 (2019).
  24. 24.
    Stoffel, A.: Supersymmetric Field Theories and Orbifold Cohomology. ProQuest LLC, Ann Arbor (2016). Thesis (Ph.D.), University of Notre Dame. ISBN 978-1339-97974-8. Accessed 12 Feb 2018
  25. 25.
    Stolz, S., Teichner, P.: What is an elliptic object? In: Tillmann, U. (ed.) Topology, Geometry and Quantum Field Theory, Volume 308 of London Mathematical Society Lecture Note Series, pp. 247–343. Cambridge University Press, Cambridge (2004).
  26. 26.
    Stolz, S., Teichner, P.: Supersymmetric field theories and generalized cohomology. In: Mathematical foundations of quantum field theory and perturbative string theory, Volume 83 of Proceedings of Symposium Pure Mathematics, pp. 279–340. American Mathematical Society, Providence (2011).
  27. 27.
    Tu, J.-L., Xu, P.: Chern character for twisted K-theory of orbifolds. Adv. Math. 207(2), 455–483 (2006). MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Max Planck Institute for MathematicsBonnGermany

Personalised recommendations