Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-23T11:03:49.206Z Has data issue: false hasContentIssue false
  • English
  • Français

GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES II: TWISTED STRING AND STRINGC STRUCTURES

Part of: Fiber spaces and bundles Quantum field theory; related classical field theories Homology and cohomology theories

Published online by Cambridge University Press:  09 June 2011

HISHAM SATI
HISHAM SATI*
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The actions, anomalies and quantization conditions allow the M2-brane and the M5-brane to support, in a natural way, structures beyond spin on their world-volumes. The main examples are twisted string structures. This also extends to twisted stringc structures which we introduce and relate to twisted string structures. The relation of the C-field to Chern–Simons theory suggests the use of the string cobordism category to describe the M2-brane.

Keywords

string structurestringc structureanomaliesM-theory

MSC classification

Secondary: 81T50: Anomalies 55R65: Generalizations of fiber spaces and bundles 55N20: Generalized (extraordinary) homology and cohomology theories
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 90 , Issue 1 , February 2011 , pp. 93 - 108
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Atiyah, M. F., ‘On framings of 3-manifolds’, Topology 29 (1990), 1–7.CrossRefGoogle Scholar
[2]Atiyah, M. F. and Singer, I. M., ‘The index of elliptic operators I’, Ann. of Math. (2) 87 (1968), 484–530.CrossRefGoogle Scholar
[3]Bouwknegt, P., Evslin, J. and Mathai, V., ‘T-duality: topology change from H-flux’, Comm. Math. Phys. 249 (2004), 383–415.CrossRefGoogle Scholar
[4]Bunke, U., ‘String structures and trivialisations of a Pfaffian line bundle’, Preprint, arXiv:0909.0846 [math.KT].Google Scholar
[5]Bunke, U. and Naumann, N., ‘Secondary invariants for string bordism and tmf’, Preprint, arXiv:0912.4875 [math.KT].Google Scholar
[6]Chen, Q., Han, F. and Zhang, W., ‘Generalized Witten genus and vanishing theorems’, Preprint, arXiv:1003.2325 [math.DG].Google Scholar
[7]Coquereaux, R. and Pilch, K., ‘String structures on loop bundles’, Comm. Math. Phys. 120 (1989), 353–378.CrossRefGoogle Scholar
[8]Diaconescu, E., Freed, D. S. and Moore, G., ‘The M-theory 3-form and E 8 gauge theory’, in: Elliptic Cohomology, London Mathematical Society Lecture Note Series, 342 (Cambridge University Press, Cambridge, 2007), pp. 44–88.CrossRefGoogle Scholar
[9]Duff, M. J., Lü, H. and Pope, C. N., ‘AdS 5×S 5 untwisted’, Nuclear Phys. B 532 (1998), 181–209.CrossRefGoogle Scholar
[10]Hopkins, M. J. and Singer, I. M., ‘Quadratic functions in geometry, topology and M-theory’, J. Differential Geom. 70 (2005), 329–452.CrossRefGoogle Scholar
[11]Hovey, M. A., ‘v n-elements in ring spectra and applications to bordism theory’, Duke Math. J. 88 (1997), 327–356.CrossRefGoogle Scholar
[12]Killingback, T. P., ‘Global anomalies, string theory and space-time topology’, Classical Quantum Gravity 5 (1988), 1169–1186.CrossRefGoogle Scholar
[13]Lawson, H. B. and Michelson, M.-L., Spin Geometry (Princeton University Press, Princeton, NJ, 1989).Google Scholar
[14]Milnor, J. W. and Stasheff, J. D., Characteristic Classes (Princeton University Press, Princeton, NJ, 1974).CrossRefGoogle Scholar
[15]Redden, C., ‘String structures and canonical 3-forms’, Pacific J. Math. 249 (2011), 447–484.CrossRefGoogle Scholar
[16]Sati, H., ‘Geometry of spin and spin c structures in the M-theory partition function’, Preprint, arXiv:1005.1700 [hep-th].Google Scholar
[17]Sati, H., ‘Geometric and topological structures related to M-branes’, Proc. Sympos. Pure Math. 81 (2010), 181–236.CrossRefGoogle Scholar
[18]Sati, H., Schreiber, U. and Stasheff, J., ‘Twisted differential string- and fivebrane structures’, Preprint, arXiv:0910.4001 [math.AT].Google Scholar
[19]Sati, H., Schreiber, U. and Stasheff, J., ‘L ∞-connections and applications to string- and Chern–Simons n-transport’, in: Recent Developments in QFT (eds. Fauser, B.et al.) (Birkhäuser, Basel, 2008), pp. 303–424.Google Scholar
[20]Thomas, E., ‘On the cohomology groups of the classifying space for the stable spinor groups’, Bol. Soc. Mat. Mexicana (2) 7 (1962), 57–69.Google Scholar
[21]Waldorf, K., ‘String connections and Chern–Simons theory’, Preprint, arXiv:0906.0117 [math.DG].Google Scholar
[22]Wang, B.-L., ‘Geometric cycles, index theory and twisted K-homology’, J . Noncommut. Geom. 2 (2008), 497–552.CrossRefGoogle Scholar
[23]Witten, E., ‘On flux quantization in M-theory and the effective action’, J. Geom. Phys. 22 (1997), 1–13.CrossRefGoogle Scholar
[24]Witten, E., ‘Five-brane effective action in M-theory’, J. Geom. Phys. 22 (1997), 103–133.CrossRefGoogle Scholar
[25]Witten, E., ‘Duality relations among topological effects in string theory’, J. High Energy Phys. 0005 (2000), 031.CrossRefGoogle Scholar