String^{𝑐} structures and modular invariants

Huang, Ruizhi; Han, Fei; Duan, Haibao

doi:10.1090/tran/8311

String structures and modular invariants

Authors: Ruizhi Huang, Fei Han and Haibao Duan
Journal: Trans. Amer. Math. Soc. 374 (2021), 3491-3533
MSC (2020): Primary 53C27, 55R35, 57S15; Secondary 57R20, 22E67, 55R40
DOI: https://doi.org/10.1090/tran/8311
Published electronically: January 27, 2021
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we study some algebraic topology aspects of String structures, more precisely, from the perspective of Whitehead tower and the perspective of the loop group of . We also extend the generalized Witten genera constructed for the first time by Chen et al. [J. Differential Geom. 88 (2011), pp. 1-40] to correspond to String structures of various levels and give vanishing results for them.

[1] M. F. Atiyah and R. Bott, The Lefschetz fixed point theorems for elliptic complexes, I, II, in Atiyah, M. F., Collected works, Vol. 3, Oxford Sci. Pub., Oxford. Univ. Press, NY, 1988, 91-170.
[2] M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. {236952}, https://doi.org/10.2307/1970717
[3] Ulrich Bunke, String structures and trivialisations of a Pfaffian line bundle, Comm. Math. Phys. 307 (2011), no. 3, 675–712. {2842963}, https://doi.org/10.1007/s00220-011-1348-0
[4] K. Chandrasekharan, Elliptic functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 281, Springer-Verlag, Berlin, 1985. {808396}
[5] Qingtao Chen, Fei Han, and Weiping Zhang, Witten genus and vanishing results on complete intersections, C. R. Math. Acad. Sci. Paris 348 (2010), no. 5-6, 295–298 (English, with English and French summaries). {2600126}, https://doi.org/10.1016/j.crma.2010.02.005
[6] Qingtao Chen, Fei Han, and Weiping Zhang, Generalized Witten genus and vanishing theorems, J. Differential Geom. 88 (2011), no. 1, 1–40. {2819754}
[7] F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, The double suspension and exponents of the homotopy groups of spheres, Ann. of Math. (2) 110 (1979), no. 3, 549–565. {554384}, https://doi.org/10.2307/1971238
[8] A. Dessai, The Witten genus and -actions on manifolds, preprint, 1994, Preprint-Reihe des Fachbereichs Mathematik, Univ. Mainz, Nr. 6, February 1994.
[9] A. Dessai, 𝑆𝑝𝑖𝑛^{𝑐}-manifolds with 𝑃𝑖𝑛(2)-action, Math. Ann. 315 (1999), no. 4, 511–528. {1731460}, https://doi.org/10.1007/s002080050327
[10] Anand Dessai and Burkhard Wilking, Torus actions on homotopy complex projective spaces, Math. Z. 247 (2004), no. 3, 505–511. {2114425}, https://doi.org/10.1007/s00209-003-0614-z
[11] H. Duan, Characteristic classes and invariants of Spin geometry, preprint, 2018.
[12] F. Han and V. Mathai, Witten Genus and Elliptic genera for proper actions, 1807.06863.
[13] John R. Harper, Secondary cohomology operations, Graduate Studies in Mathematics, vol. 49, American Mathematical Society, Providence, RI, 2002. {1913285}
[14] Akio Hattori, 𝑆𝑝𝑖𝑛^{𝑐}-structures and 𝑆¹-actions, Invent. Math. 48 (1978), no. 1, 7–31. {508087}, https://doi.org/10.1007/BF01390060
[15] Akio Hattori and Tomoyoshi Yoshida, Lifting compact group actions in fiber bundles, Japan. J. Math. (N.S.) 2 (1976), no. 1, 13–25. {461538}, https://doi.org/10.4099/math1924.2.13
[16] 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. {1189136}
[17] M. J. Hopkins, Algebraic topology and modular forms, Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 291–317. {1989190}
[18] T. P. Killingback, World-sheet anomalies and loop geometry, Nuclear Phys. B 288 (1987), no. 3-4, 578–588. {892061}, https://doi.org/10.1016/0550-3213(87)90229-X
[19] Daisuke Kishimoto and Akira Kono, On the cohomology of free and twisted loop spaces, J. Pure Appl. Algebra 214 (2010), no. 5, 646–653. {2577671}, https://doi.org/10.1016/j.jpaa.2009.07.006
[20] C. Kottke and R. Melrose, Equivalence of string and fusion loop-spin structures, 1309.0210, 2013.
[21] Chris Kottke and Richard B. Melrose, Loop-fusion cohomology and transgression, Math. Res. Lett. 22 (2015), no. 4, 1177–1192. {3391882}, https://doi.org/10.4310/MRL.2015.v22.n4.a11
[22] Katsuhiko Kuribayashi, Module derivations and the adjoint action of a finite loop space, J. Math. Kyoto Univ. 39 (1999), no. 1, 67–85. {1684168}, https://doi.org/10.1215/kjm/1250517954
[23] H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. {1031992}
[24] Kefeng Liu, Modular invariance and characteristic numbers, Comm. Math. Phys. 174 (1995), no. 1, 29–42. {1372798}
[25] Kefeng Liu, On modular invariance and rigidity theorems, J. Differential Geom. 41 (1995), no. 2, 343–396. {1331972}
[26] Kefeng Liu, On elliptic genera and theta-functions, Topology 35 (1996), no. 3, 617–640. {1396769}, https://doi.org/10.1016/0040-9383(95)00042-9
[27] Kefeng Liu and Xiaonan Ma, On family rigidity theorems. I, Duke Math. J. 102 (2000), no. 3, 451–474. {1756105}, https://doi.org/10.1215/S0012-7094-00-10234-7
[28] Kefeng Liu and Xiaonan Ma, On family rigidity theorems for 𝑆𝑝𝑖𝑛^{𝑐} manifolds, Mirror symmetry, IV (Montreal, QC, 2000) AMS/IP Stud. Adv. Math., vol. 33, Amer. Math. Soc., Providence, RI, 2002, pp. 343–360. {1969037}
[29] Kefeng Liu, Xiaonan Ma, and Weiping Zhang, 𝑆𝑝𝑖𝑛^{𝑐} manifolds and rigidity theorems in 𝐾-theory, Asian J. Math. 4 (2000), no. 4, 933–959. Loo-Keng Hua: a great mathematician of the twentieth century. {1870666}, https://doi.org/10.4310/AJM.2000.v4.n4.a12
[30] Kefeng Liu, Xiaonan Ma, and Weiping Zhang, Rigidity and vanishing theorems in 𝐾-theory, Comm. Anal. Geom. 11 (2003), no. 1, 121–180. {2016198}, https://doi.org/10.4310/CAG.2003.v11.n1.a6
[31] Kefeng Liu, Xiaonan Ma, and Weiping Zhang, On elliptic genera and foliations, Math. Res. Lett. 8 (2001), no. 3, 361–376. {1839484}, https://doi.org/10.4310/MRL.2001.v8.n3.a11
[32] B. Liu and J. Yu, On the Witten Rigidity Theorem for String Manifolds, Pacific J. Math., 2013, 266(2): 477-508.
[33] Mikiya Masuda and Yuh-Dong Tsai, Tangential representations of cyclic group actions on homotopy complex projective spaces, Osaka J. Math. 22 (1985), no. 4, 907–919. {815458}
[34] John McCleary, A user’s guide to spectral sequences, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 58, Cambridge University Press, Cambridge, 2001. {1793722}
[35] Dennis A. McLaughlin, Orientation and string structures on loop space, Pacific J. Math. 155 (1992), no. 1, 143–156. {1174481}
[36] Paul Melvin and Jeffrey Parker, 4-manifolds with large symmetry groups, Topology 25 (1986), no. 1, 71–83. {836725}, https://doi.org/10.1016/0040-9383(86)90006-6
[37] Brian A. Munson and Ismar Volić, Cubical homotopy theory, New Mathematical Monographs, vol. 25, Cambridge University Press, Cambridge, 2015. {3559153}
[38] Thomas Nikolaus, Christoph Sachse, and Christoph Wockel, A smooth model for the string group, Int. Math. Res. Not. IMRN 16 (2013), 3678–3721. {3090706}, https://doi.org/10.1093/imrn/rns154
[39] Ted Petrie, Smooth 𝑆¹ actions on homotopy complex projective spaces and related topics, Bull. Amer. Math. Soc. 78 (1972), 105–153. {296970}, https://doi.org/10.1090/S0002-9904-1972-12898-2
[40] Corbett Redden, String structures and canonical 3-forms, Pacific J. Math. 249 (2011), no. 2, 447–484. {2782680}, https://doi.org/10.2140/pjm.2011.249.447
[41] Hisham Sati, Geometric and topological structures related to M-branes II: Twisted string and string^{𝑐} structures, J. Aust. Math. Soc. 90 (2011), no. 1, 93–108. {2810946}, https://doi.org/10.1017/S1446788711001261
[42] Hisham Sati, Urs Schreiber, and Jim Stasheff, Twisted differential string and fivebrane structures, Comm. Math. Phys. 315 (2012), no. 1, 169–213. {2966944}, https://doi.org/10.1007/s00220-012-1510-3
[43] Stephan Stolz, A conjecture concerning positive Ricci curvature and the Witten genus, Math. Ann. 304 (1996), no. 4, 785–800. {1380455}, https://doi.org/10.1007/BF01446319
[44] 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. {2079378}, https://doi.org/10.1017/CBO9780511526398.013
[45] S. Stolz and P. Teichner, The Spinor bundle on loop space, MPIM preprint, 2005.
[46] Robert M. Switzer, Algebraic topology—homotopy and homology, Classics in Mathematics, Springer-Verlag, Berlin, 2002. Reprint of the 1975 original [Springer, New York; MR0385836 (52 #6695)]. {1886843}
[47] Emery Thomas, On the cohomology groups of the classifying space for the stable spinor groups, Bol. Soc. Mat. Mexicana (2) 7 (1962), 57–69. {153027}
[48] Konrad Waldorf, Transgression to loop spaces and its inverse, III: Gerbes and thin fusion bundles, Adv. Math. 231 (2012), no. 6, 3445–3472. {2980505}, https://doi.org/10.1016/j.aim.2012.08.016
[49] Konrad Waldorf, String geometry vs. spin geometry on loop spaces, J. Geom. Phys. 97 (2015), 190–226. {3385126}, https://doi.org/10.1016/j.geomphys.2015.07.003
[50] Konrad Waldorf, Spin structures on loop spaces that characterize string manifolds, Algebr. Geom. Topol. 16 (2016), no. 2, 675–709. {3493404}, https://doi.org/10.2140/agt.2016.16.675
[51] George W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. {516508}
[52] 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. {970288}, https://doi.org/10.1007/BFb0078045
[53] 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. {970290}, https://doi.org/10.1007/BFb0078047