Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Mod 3 congruence and twisted signature of 24 dimensional string manifolds


Authors: Qingtao Chen and Fei Han
Journal: Trans. Amer. Math. Soc. 367 (2015), 2959-2977
MSC (2010): Primary 58J26, 53C27
DOI: https://doi.org/10.1090/S0002-9947-2014-06241-7
Published electronically: August 8, 2014
MathSciNet review: 3301888
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, by combining modularity of the Witten genus and the modular forms constructed by Liu and Wang, we establish mod 3 congruence properties of certain twisted signatures of 24 dimensional string manifolds.


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

  • [1] M. F. Atiyah, $ K$-theory, Lecture notes by D. W. Anderson, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0224083 (36 #7130)
  • [2] M. F. Atiyah and F. Hirzebruch, Riemann-Roch theorems for differentiable manifolds, Bull. Amer. Math. Soc. 65 (1959), 276-281. MR 0110106 (22 #989)
  • [3] M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546-604. MR 0236952 (38 #5245)
  • [4] K. Chandrasekharan, Elliptic functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 281, Springer-Verlag, Berlin, 1985. MR 808396 (87e:11058)
  • [5] Qingtao Chen and Fei Han, Modular invariance and twisted cancellations of characteristic numbers, Trans. Amer. Math. Soc. 361 (2009), no. 3, 1463-1493. MR 2457406 (2010a:58029), https://doi.org/10.1090/S0002-9947-08-04703-X
  • [6] Peter B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, 2nd ed., Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1396308 (98b:58156)
  • [7] Fei Han, Kefeng Liu, and Weiping Zhang, Modular forms and generalized anomaly cancellation formulas, J. Geom. Phys. 62 (2012), no. 5, 1038-1053. MR 2901844, https://doi.org/10.1016/j.geomphys.2012.01.016
  • [8] F. Hirzebruch, Topological methods in algebraic geometry, Third enlarged edition. New appendix and translation from the second German edition by R. L. E. Schwarzenberger, with an additional section by A. Borel. Die Grundlehren der Mathematischen Wissenschaften, Band 131, Springer-Verlag New York, Inc., New York, 1966. MR 0202713 (34 #2573)
  • [9] F. Hirzebruch, Mannigfaltigkeiten und Modulformen, Jahresberichte der Deutschen Mathematiker Vereinigung, Jber. d. Dt. Math.-Verein, 1992, pp. 20-38.
  • [10] Friedrich Hirzebruch, Thomas Berger, and Rainer Jung, Manifolds and modular forms, Aspects of Mathematics, E20, Friedr. Vieweg & Sohn, Braunschweig, 1992. With appendices by Nils-Peter Skoruppa and by Paul Baum. MR 1189136 (94d:57001)
  • [11] M. J. Hopkins, Algebraic topology and modular forms (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 291-317. MR 1989190 (2004g:11032)
  • [12] Masanobu Kaneko and Don Zagier, A generalized Jacobi theta function and quasimodular forms, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 165-172. MR 1363056 (96m:11030)
  • [13] Peter S. Landweber, Elliptic cohomology and modular forms, Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986), Lecture Notes in Math., vol. 1326, Springer, Berlin, 1988, pp. 55-68. MR 970281, https://doi.org/10.1007/BFb0078038
  • [14] H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992 (91g:53001)
  • [15] Gerd Laures, $ K(1)$-local topological modular forms, Invent. Math. 157 (2004), no. 2, 371-403. MR 2076927 (2005h:55003), https://doi.org/10.1007/s00222-003-0355-y
  • [16] Kefeng Liu, Modular invariance and characteristic numbers, Comm. Math. Phys. 174 (1995), no. 1, 29-42. MR 1372798 (96m:57034)
  • [17] KeFeng Liu and Yong Wang, A note on modular forms and generalized anomaly cancellation formulas, Sci. China Math. 56 (2013), no. 1, 55-65. MR 3016582, https://doi.org/10.1007/s11425-012-4479-7
  • [18] John W. Milnor and Michel A. Kervaire, Bernoulli numbers, homotopy groups, and a theorem of Rohlin, Proc. Internat. Congress Math. 1958, Cambridge Univ. Press, New York, 1960, pp. 454-458. MR 0121801 (22 #12531)
  • [19] S. Ochanine, Signature modulo 16, invariants de Kervaire géneralisés et nombre caractéristiques dans la $ K$-théorie reelle, Mémoire Soc. Math. France, Tom. 109 (1987), 1-141.
  • [20] V. A. Rohlin, New results in the theory of four-dimensional manifolds, Doklady Akad. Nauk SSSR (N.S.) 84 (1952), 221-224 (Russian). MR 0052101 (14,573b)
  • [21] P. Teichner, Elliptic cohomology via Conformal Field Theory, Lecture Notes at UC Berkeley.
  • [22] 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, https://doi.org/10.1007/BFb0078045
  • [23] Edward Witten, Elliptic genera and quantum field theory, Comm. Math. Phys. 109 (1987), no. 4, 525-536. MR 885560 (89i:57017)
  • [24] Don Zagier, Note on the Landweber-Stong elliptic genus, Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986), Lecture Notes in Math., vol. 1326, Springer, Berlin, 1988, pp. 216-224. MR 970290, https://doi.org/10.1007/BFb0078047
  • [25] Weiping Zhang, Lectures on Chern-Weil theory and Witten deformations, Nankai Tracts in Mathematics, vol. 4, World Scientific Publishing Co. Inc., River Edge, NJ, 2001. MR 1864735 (2002m:58032)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 58J26, 53C27

Retrieve articles in all journals with MSC (2010): 58J26, 53C27


Additional Information

Qingtao Chen
Affiliation: Mathematics Section, International Center for Theoretical Physics, Strada Costiera, 11, I - 34151 Trieste, Italy
Email: qchen1@ictp.it

Fei Han
Affiliation: Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore 119076
Email: mathanf@nus.edu.sg

DOI: https://doi.org/10.1090/S0002-9947-2014-06241-7
Received by editor(s): December 3, 2012
Received by editor(s) in revised form: June 26, 2013
Published electronically: August 8, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society