String^{𝑐} structures and modular invariants

Huang, Ruizhi; Han, Fei; Duan, Haibao

doi:10.1090/tran/8311

String$^c$ 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
MathSciNet review: 4237954
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Abstract: In this paper, we study some algebraic topology aspects of String$^c$ structures, more precisely, from the perspective of Whitehead tower and the perspective of the loop group of $Spin^c(n)$. 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$^c$ structures of various levels and give vanishing results for them.

References

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.

M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. MR 236952, DOI https://doi.org/10.2307/1970717

Ulrich Bunke, String structures and trivialisations of a Pfaffian line bundle, Comm. Math. Phys. 307 (2011), no. 3, 675–712. MR 2842963, DOI https://doi.org/10.1007/s00220-011-1348-0

K. Chandrasekharan, Elliptic functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 281, Springer-Verlag, Berlin, 1985. MR 808396

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). MR 2600126, DOI https://doi.org/10.1016/j.crma.2010.02.005

Qingtao Chen, Fei Han, and Weiping Zhang, Generalized Witten genus and vanishing theorems, J. Differential Geom. 88 (2011), no. 1, 1–40. MR 2819754

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. MR 554384, DOI https://doi.org/10.2307/1971238

A. Dessai, The Witten genus and $S^3$-actions on manifolds, preprint, 1994, Preprint-Reihe des Fachbereichs Mathematik, Univ. Mainz, Nr. 6, February 1994.

A. Dessai, ${\rm Spin}^c$-manifolds with ${\rm Pin}(2)$-action, Math. Ann. 315 (1999), no. 4, 511–528. MR 1731460, DOI https://doi.org/10.1007/s002080050327

Anand Dessai and Burkhard Wilking, Torus actions on homotopy complex projective spaces, Math. Z. 247 (2004), no. 3, 505–511. MR 2114425, DOI https://doi.org/10.1007/s00209-003-0614-z

H. Duan, Characteristic classes and invariants of Spin geometry, preprint, 2018.

F. Han and V. Mathai, Witten Genus and Elliptic genera for proper actions, arXiv:1807.06863.

John R. Harper, Secondary cohomology operations, Graduate Studies in Mathematics, vol. 49, American Mathematical Society, Providence, RI, 2002. MR 1913285

Akio Hattori, ${\rm Spin}^{c}$-structures and $S^{1}$-actions, Invent. Math. 48 (1978), no. 1, 7–31. MR 508087, DOI https://doi.org/10.1007/BF01390060

Akio Hattori and Tomoyoshi Yoshida, Lifting compact group actions in fiber bundles, Japan. J. Math. (N.S.) 2 (1976), no. 1, 13–25. MR 461538, DOI https://doi.org/10.4099/math1924.2.13

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

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. MR 1989190

T. P. Killingback, World-sheet anomalies and loop geometry, Nuclear Phys. B 288 (1987), no. 3-4, 578–588. MR 892061, DOI https://doi.org/10.1016/0550-3213%2887%2990229-X

Daisuke Kishimoto and Akira Kono, On the cohomology of free and twisted loop spaces, J. Pure Appl. Algebra 214 (2010), no. 5, 646–653. MR 2577671, DOI https://doi.org/10.1016/j.jpaa.2009.07.006

C. Kottke and R. Melrose, Equivalence of string and fusion loop-spin structures, arXiv:1309.0210, 2013.

Chris Kottke and Richard B. Melrose, Loop-fusion cohomology and transgression, Math. Res. Lett. 22 (2015), no. 4, 1177–1192. MR 3391882, DOI https://doi.org/10.4310/MRL.2015.v22.n4.a11

Katsuhiko Kuribayashi, Module derivations and the adjoint action of a finite loop space, J. Math. Kyoto Univ. 39 (1999), no. 1, 67–85. MR 1684168, DOI https://doi.org/10.1215/kjm/1250517954

H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992

Kefeng Liu, Modular invariance and characteristic numbers, Comm. Math. Phys. 174 (1995), no. 1, 29–42. MR 1372798

Kefeng Liu, On modular invariance and rigidity theorems, J. Differential Geom. 41 (1995), no. 2, 343–396. MR 1331972

Kefeng Liu, On elliptic genera and theta-functions, Topology 35 (1996), no. 3, 617–640. MR 1396769, DOI https://doi.org/10.1016/0040-9383%2895%2900042-9

Kefeng Liu and Xiaonan Ma, On family rigidity theorems. I, Duke Math. J. 102 (2000), no. 3, 451–474. MR 1756105, DOI https://doi.org/10.1215/S0012-7094-00-10234-7

Kefeng Liu and Xiaonan Ma, On family rigidity theorems for ${\rm Spin}^c$ manifolds, Mirror symmetry, IV (Montreal, QC, 2000) AMS/IP Stud. Adv. Math., vol. 33, Amer. Math. Soc., Providence, RI, 2002, pp. 343–360. MR 1969037

Kefeng Liu, Xiaonan Ma, and Weiping Zhang, ${\rm Spin}^c$ 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. MR 1870666, DOI https://doi.org/10.4310/AJM.2000.v4.n4.a12

Kefeng Liu, Xiaonan Ma, and Weiping Zhang, Rigidity and vanishing theorems in $K$-theory, Comm. Anal. Geom. 11 (2003), no. 1, 121–180. MR 2016198, DOI https://doi.org/10.4310/CAG.2003.v11.n1.a6

Kefeng Liu, Xiaonan Ma, and Weiping Zhang, On elliptic genera and foliations, Math. Res. Lett. 8 (2001), no. 3, 361–376. MR 1839484, DOI https://doi.org/10.4310/MRL.2001.v8.n3.a11

B. Liu and J. Yu, On the Witten Rigidity Theorem for String$^c$ Manifolds, Pacific J. Math., 2013, 266(2): 477-508.

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. MR 815458

John McCleary, A user’s guide to spectral sequences, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 58, Cambridge University Press, Cambridge, 2001. MR 1793722

Dennis A. McLaughlin, Orientation and string structures on loop space, Pacific J. Math. 155 (1992), no. 1, 143–156. MR 1174481

Paul Melvin and Jeffrey Parker, $4$-manifolds with large symmetry groups, Topology 25 (1986), no. 1, 71–83. MR 836725, DOI https://doi.org/10.1016/0040-9383%2886%2990006-6

Brian A. Munson and Ismar Volić, Cubical homotopy theory, New Mathematical Monographs, vol. 25, Cambridge University Press, Cambridge, 2015. MR 3559153

Thomas Nikolaus, Christoph Sachse, and Christoph Wockel, A smooth model for the string group, Int. Math. Res. Not. IMRN 16 (2013), 3678–3721. MR 3090706, DOI https://doi.org/10.1093/imrn/rns154

Ted Petrie, Smooth $S^{1}$ actions on homotopy complex projective spaces and related topics, Bull. Amer. Math. Soc. 78 (1972), 105–153. MR 296970, DOI https://doi.org/10.1090/S0002-9904-1972-12898-2

Corbett Redden, String structures and canonical 3-forms, Pacific J. Math. 249 (2011), no. 2, 447–484. MR 2782680, DOI https://doi.org/10.2140/pjm.2011.249.447

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. MR 2810946, DOI https://doi.org/10.1017/S1446788711001261

Hisham Sati, Urs Schreiber, and Jim Stasheff, Twisted differential string and fivebrane structures, Comm. Math. Phys. 315 (2012), no. 1, 169–213. MR 2966944, DOI https://doi.org/10.1007/s00220-012-1510-3

Stephan Stolz, A conjecture concerning positive Ricci curvature and the Witten genus, Math. Ann. 304 (1996), no. 4, 785–800. MR 1380455, DOI https://doi.org/10.1007/BF01446319

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. MR 2079378, DOI https://doi.org/10.1017/CBO9780511526398.013

S. Stolz and P. Teichner, The Spinor bundle on loop space, MPIM preprint, 2005.

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)]. MR 1886843

Emery Thomas, On the cohomology groups of the classifying space for the stable spinor groups, Bol. Soc. Mat. Mexicana (2) 7 (1962), 57–69. MR 153027

Konrad Waldorf, Transgression to loop spaces and its inverse, III: Gerbes and thin fusion bundles, Adv. Math. 231 (2012), no. 6, 3445–3472. MR 2980505, DOI https://doi.org/10.1016/j.aim.2012.08.016

Konrad Waldorf, String geometry vs. spin geometry on loop spaces, J. Geom. Phys. 97 (2015), 190–226. MR 3385126, DOI https://doi.org/10.1016/j.geomphys.2015.07.003

Konrad Waldorf, Spin structures on loop spaces that characterize string manifolds, Algebr. Geom. Topol. 16 (2016), no. 2, 675–709. MR 3493404, DOI https://doi.org/10.2140/agt.2016.16.675

George W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR 516508

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, DOI https://doi.org/10.1007/BFb0078045

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, DOI https://doi.org/10.1007/BFb0078047