Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 024, 28 pages      arXiv:1203.4179      https://doi.org/10.3842/SIGMA.2014.024

M-Theory with Framed Corners and Tertiary Index Invariants

Hisham Sati
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA

Received March 19, 2013, in final form March 01, 2014; Published online March 14, 2014

Abstract
The study of the partition function in M-theory involves the use of index theory on a twelve-dimensional bounding manifold. In eleven dimensions, viewed as a boundary, this is given by secondary index invariants such as the Atiyah-Patodi-Singer eta-invariant, the Chern-Simons invariant, or the Adams e-invariant. If the eleven-dimensional manifold itself has a boundary, the resulting ten-dimensional manifold can be viewed as a codimension two corner. The partition function in this context has been studied by the author in relation to index theory for manifolds with corners, essentially on the product of two intervals. In this paper, we focus on the case of framed manifolds (which are automatically Spin) and provide a formulation of the refined partition function using a tertiary index invariant, namely the f-invariant introduced by Laures within elliptic cohomology. We describe the context globally, connecting the various spaces and theories around M-theory, and providing a physical realization and interpretation of some ingredients appearing in the constructions due to Bunke-Naumann and Bodecker. The formulation leads to a natural interpretation of anomalies using corners and uncovers some resulting constraints in the heterotic corner. The analysis for type IIA leads to a physical identification of various components of eta-forms appearing in the formula for the phase of the partition function.

Key words: anomalies; manifolds with corners; tertiary index invariants; M-theory; elliptic genera; partition functions; eta-forms.

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References

1. Aganagic M., Bouchard V., Klemm A., Topological strings and (almost) modular forms, Comm. Math. Phys. 277 (2008), 771-819, hep-th/0607100.
2. Ando M., Hopkins M.J., Strickland N.P., Elliptic spectra, the Witten genus and the theorem of the cube, Invent. Math. 146 (2001), 595-687.
3. Astey L., Gitler S., Micha E., Pastor G., Parallelizability of complex projective Stiefel manifolds, Proc. Amer. Math. Soc. 128 (2000), 1527-1530.
4. Astey L., Guest M.A., Pastor G., Lie groups as framed boundaries, Osaka J. Math. 25 (1988), 891-907.
5. Atiyah M.F., Patodi V.K., Singer I.M., Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43-69.
6. Atiyah M.F., Singer I.M., Index theory for skew-adjoint Fredholm operators, Inst. Hautes Études Sci. Publ. Math. (1969), 5-26.
7. Atiyah M.F., Singer I.M., The index of elliptic operators. IV, Ann. of Math. 93 (1971), 119-138.
8. Atiyah M.F., Singer I.M., The index of elliptic operators. V, Ann. of Math. 93 (1971), 139-149.
9. Atiyah M.F., Smith L., Compact Lie groups and the stable homotopy of spheres, Topology 13 (1974), 135-142.
10. Becker J.C., Schultz R.E., Fixed-point indices and left invariant framings, in Geometric Applications of Homotopy Theory, Vol. I (Proc. Conf., Evanston, Ill., 1977), Lecture Notes in Math., Vol. 657, Springer, Berlin, 1978, 1-31.
11. Becker K., Becker M., Dasgupta K., Green P.S., Sharpe E., Compactifications of heterotic strings of non-Kähler complex manifolds. II, Nuclear Phys. B 678 (2004), 19-100, hep-th/0310058.
12. Belov D., Moore G.W., Holographic action for the self-dual field, hep-th/0605038.
13. Bismut J.M., Cheeger J., Families index for manifolds with boundary, superconnections, and cones. I. Families of manifolds with boundary and Dirac operators, J. Funct. Anal. 89 (1990), 313-363.
14. Borel A., Hirzebruch F., Characteristic classes and homogeneous spaces. III, Amer. J. Math. 82 (1960), 491-504.
15. Bouwknegt P., Evslin J., Jurčo B., Mathai V., Sati H., Flux compactifications on projective spaces and the S-duality puzzle, Adv. Theor. Math. Phys. 10 (2006), 345-394, hep-th/0501110.
16. Brasselet J.P., Seade J., Suwa T., Vector fields on singular varieties, Lecture Notes in Math., Vol. 1987, Springer-Verlag, Berlin, 2009.
17. Bredon G.E., Kosiński A., Vector fields on π-manifolds, Ann. of Math. 84 (1966), 85-90.
18. Bunke U., Naumann N., Secondary invariants for String bordism and tmf, arXiv:0912.4875.
19. Bunke U., Naumann N., The f-invariant and index theory, Manuscripta Math. 132 (2010), 365-397, arXiv:0808.0257.
20. Cadek M., Crabb M., G-structures on spheres, Proc. London Math. Soc. 93 (2006), 791-816, math.KT/0510149.
21. Diaconescu E., Moore G., Freed D.S., The M-theory 3-form and E8 gauge theory, in Elliptic cohomology, London Math. Soc. Lecture Note Ser., Vol. 342, Cambridge Univ. Press, Cambridge, 2007, 44-88, hep-th/0312069.
22. Diaconescu E., Moore G., Witten E., E8 gauge theory, and a derivation of K-Theory from M-Theory, Adv. Theor. Math. Phys. 6 (2002), 1031-1134, hep-th/0005090.
23. Distler J., Sharpe E., Heterotic compactifications with principal bundles for general groups and general levels, Adv. Theor. Math. Phys. 14 (2010), 335-397, hep-th/0701244.
24. Duff M.J., Liu J.T., Minasian R., Eleven-dimensional origin of string/string duality: a one-loop test, Nuclear Phys. B 452 (1995), 261-282, hep-th/9506126.
25. Evslin J., Sati H., Can D-branes wrap nonrepresentable cycles?, J. High Energy Phys. 2006 (2006), no. 10, 050, 10 pages, hep-th/0607045.
26. Figueroa-O'Farrill J., Kawano T., Yamaguchi S., Parallelisable heterotic backgrounds, J. High Energy Phys. 2003 (2003), no. 10, 012, 22 pages, hep-th/0308141.
27. Freed D.S., Dirac charge quantization and generalized differential cohomology, in Surveys in Differential Geometry, Surv. Differ. Geom., Vol. VII, Int. Press, Somerville, MA, 2000, 129-194, hep-th/0011220.
28. Galvez Carrillo M.I., Modular invariants for manifolds with boundary, Ph.D. Thesis, Universitat Autonoma de Barcelona, 2001.
29. Gran U., Papadopoulos G., Roest D., Supersymmetric heterotic string backgrounds, Phys. Lett. B 656 (2007), 119-126, arXiv:0706.4407.
30. Gran U., Papadopoulos G., Sloane P., Roest D., Geometry of all supersymmetric type I backgrounds, J. High Energy Phys. 2007 (2007), no. 8, 074, 75 pages, hep-th/0703143.
31. Hirzebruch F., Berger T., Jung R., Manifolds and modular forms, Aspects of Mathematics, Vol. E20, Friedr. Vieweg & Sohn, Braunschweig, 1992.
32. Hopkins M.J., Singer I.M., Quadratic functions in geometry, topology, and M-theory, J. Differential Geom. 70 (2005), 329-452, math.AT/0211216.
33. Kamata M., Minami H., The special orthogonal groups SO(2n) as framed boundaries, Kyushu J. Math. 54 (2000), 147-153.
34. Kawano T., Yamaguchi S., Dilatonic parallelizable NS-NS backgrounds, Phys. Lett. B 568 (2003), 78-82, hep-th/0306038.
35. Kervaire M., Courbure intégrale généralisée et homotopie, Math. Ann. 131 (1956), 219-252.
36. Kervaire M.A., Milnor J.W., Groups of homotopy spheres. I, Ann. of Math. 77 (1963), 504-537.
37. Killingback T.P., Global anomalies, string theory and spacetime topology, Classical Quantum Gravity 5 (1988), 1169-1185.
38. Knapp K., Rank and Adams filtration of a Lie group, Topology 17 (1978), 41-52.
39. Kriz I., Sati H., M-theory, type IIA superstrings, and elliptic cohomology, Adv. Theor. Math. Phys. 8 (2004), 345-394, hep-th/0404013.
40. Kriz I., Sati H., Type II string theory and modularity, J. High Energy Phys. 2005 (2005), no. 8, 038, 30 pages, hep-th/0501060.
41. Kriz I., Sati H., Type IIB string theory, S-duality, and generalized cohomology, Nuclear Phys. B 715 (2005), 639-664, hep-th/0410293.
42. Laures G., On cobordism of manifolds with corners, Trans. Amer. Math. Soc. 352 (2000), 5667-5688.
43. Laures G., K(1)-local topological modular forms, Invent. Math. 157 (2004), 371-403.
44. Lerche W., Nilsson B.E.W., Schellekens A.N., Warner N.P., Anomaly cancelling terms from the elliptic genus, Nuclear Phys. B 299 (1988), 91-116.
45. Löffler P., Smith L., Line bundles over framed manifolds, Math. Z. 138 (1974), 35-52.
46. Mahowald M., Rezk C., Topological modular forms of level 3, Pure Appl. Math. Q. 5 (2009), 853-872, arXiv:0812.2009.
47. Maldacena J., Moore G., Seiberg N., D-brane instantons and K-theory charges, J. High Energy Phys. 2001 (2001), no. 11, 062, 42 pages, hep-th/0108100.
48. Mathai V., Sati H., Some relations between twisted K-theory and E8 gauge theory, J. High Energy Phys. 2004 (2004), no. 3, 016, 22 pages, hep-th/0312033.
49. Melrose R.B., Piazza P., An index theorem for families of Dirac operators on odd-dimensional manifolds with boundary, J. Differential Geom. 46 (1997), 287-334.
50. Minami H., Stiefel manifolds as framed boundaries, Osaka J. Math. 27 (1990), 185-189.
51. Minami H., Remarks on framed bordism classes of classical Lie groups, Publ. Res. Inst. Math. Sci. 43 (2007), 461-470.
52. Moore G., Saulina N., T-duality, and the K-theoretic partition function of type IIA superstring theory, Nuclear Phys. B 670 (2003), 27-89, hep-th/0206092.
53. Pittie H.V., Smith L., Generalized flag manifolds as framed boundaries, Math. Z. 142 (1975), 191-193.
54. Sadri D., Sheikh-Jabbari M.M., String theory on parallelizable pp-waves, J. High Energy Phys. 2003 (2003), no. 6, 005, 35 pages, hep-th/0304169.
55. Sati H., The elliptic curves in gauge theory, string theory, and cohomology, J. High Energy Phys. 2006 (2006), no. 3, 096, 20 pages, hep-th/0511087.
56. Sati H., An approach to anomalies in M-theory via KSpin, J. Geom. Phys. 58 (2008), 387-401, arXiv:0705.3484.
57. Sati H., OP2 bundles in M-theory, Commun. Number Theory Phys. 3 (2009), 495-530, arXiv:0807.4899.
58. Sati H., The loop group of E8 and targets for spacetime, Modern Phys. Lett. A 24 (2009), 25-40, hep-th/0701231.
59. Sati H., E8 gauge theory and gerbes in string theory, Adv. Theor. Math. Phys. 14 (2010), 399-437, hep-th/0608190.
60. Sati H., Geometric and topological structures related to M-branes, in Superstrings, Geometry, Topology, and C*-Algebras, Proc. Sympos. Pure Math., Vol. 81, Amer. Math. Soc., Providence, RI, 2010, 181-236, arXiv:1001.5020.
61. Sati H., Anomalies of E8 gauge theory on string manifolds, Internat. J. Modern Phys. A 26 (2011), 2177-2197, arXiv:0807.4940.
62. Sati H., Constraints on heterotic M-theory from s-cobordism, Nuclear Phys. B 853 (2011), 739-759, arXiv:1102.1171.
63. Sati H., Corners in M-theory, J. Phys. A: Math. Theor. 44 (2011), 255402, 21 pages, arXiv:1101.2793.
64. Sati H., M-theory, the signature theorem, and geometric invariants, Phys. Rev. D 83 (2011), 126010, 10 pages, arXiv:1012.1300.
65. Sati H., On global anomalies in type IIB string theory, arXiv:1109.4385.
66. Sati H., Topological aspects of the partition function of the NS5-brane, arXiv:1109.4834.
67. Sati H., Duality and cohomology in M-theory with boundary, J. Geom. Phys. 62 (2012), 1284-1297, arXiv:1012.4495.
68. Sati H., Geometry of Spin and Spinc structures in the M-theory partition function, Rev. Math. Phys. 24 (2012), 1250005, 112 pages, arXiv:1005.1700.
69. Sati H., Framed M-branes, corners, and topological invariants, arXiv:1310.1060.
70. Sati H., String theory as local pre-quantum field theory, in preparation.
71. Sati H., Schreiber U., Stasheff J., Fivebrane structures, Rev. Math. Phys. 21 (2009), 1197-1240, arXiv:0805.0564.
72. Sati H., Schreiber U., Stasheff J., Twisted differential string and fivebrane structures, Comm. Math. Phys. 315 (2012), 169-213, arXiv:0910.4001.
73. Sati H., Westerland C., Twisted Morava K-theory and E-theory, arXiv:1109.3867.
74. Schellekens A.N., Warner N.P., Anomalies, characters and strings, Nuclear Phys. B 287 (1987), 317-361.
75. Singhof W., Parallelizability of homogeneous spaces. I, Math. Ann. 260 (1982), 101-116.
76. Singhof W., The d-invariant of compact nilmanifolds, Invent. Math. 78 (1984), 113-115.
77. Singhof W., Wemmer D., Parallelizability of homogeneous spaces. II, Math. Ann. 274 (1986), 157-176.
78. Smith L., Framings of sphere bundles over spheres, the plumbing pairing, and the framed bordism classes of rank two simple Lie groups, Topology 13 (1974), 401-415.
79. Spindel P., Sevrin A., Troost W., Van Proeyen A., Complex structures on parallelised group manifolds and supersymmetric σ-models, Phys. Lett. B 206 (1988), 71-74.
80. Steer B., Orbits and the homotopy class of a compactification of a classical map, Topology 15 (1976), 383-393.
81. Stong R.E., Notes on cobordism theory, Mathematical Notes, Princeton University Press, Princeton, N.J., 1968.
82. Thomas E., Cross-sections of stably equivalent vector bundles, Quart. J. Math. Oxford Ser. (2) 17 (1966), 53-57.
83. von Bodecker H., On the geometry of the f-invariant, arXiv:0808.0428.
84. von Bodecker H., On the f-invariant of products, arXiv:0909.3968.
85. Witten E., The index of the Dirac operator in loop space, in Elliptic Curves and Modular Forms in Algebraic Topology (Princeton, NJ, 1986), Lecture Notes in Math., Vol. 1326, Springer, Berlin, 1988, 161-181.
86. Witten E., On flux quantization in M-theory and the effective action, J. Geom. Phys. 22 (1997), 1-13, hep-th/9609122.
87. Wood R.M.W., Framing the exceptional Lie group G2, Topology 15 (1976), 303-320.