Geometric and topological structures related to M-branes



physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics

This entry provides hyperlinked keywords and further pointers to the literature for the articles

  • Hisham Sati, Geometric and topological structures related to M-branes ,

    part I, Proc. Symp. Pure Math. 81 (2010), 181-236 (arXiv:1001.5020),

    part II: Twisted StringString and String cString^c structures, J. Australian Math. Soc. 90 (2011), 93-108 (arXiv:1007.5419);

    part III: Twisted higher structures, Int. J. Geom. Meth. Mod. Phys. 8 (2011), 1097-1116 (arXiv:1008.1755)

on phenomena of higher geometry and generalized cohomology encountered in string theory and specifically when going towards M-theory.

Apart from original work this provides an exhaustive bibliography of the relevant existing literature, which we reproduce hyperlinked below.



We consider the topological and geometric structures associated with cohomological and homological objects in M-theory. For the latter, we have M2-branes and M5-branes, the analysis of which requires the underlying spacetime to admit a String structure and a Fivebrane structure, respectively. For the former, we study how the fields in M-theory are associated with the above structures, with homotopy algebras, with twisted cohomology, and with generalized cohomology. We also explain how the corresponding charges should take values in topological modular forms. We survey background material and related results in the process.

Higher cohomological charges

Discussion of elliptic cohomology and Morava K-theory as the home for higher analogs of D-brane charges (M5-brane charge, M9-brane charge…) and the corresponding orientation in generalized cohomology as higher quantum anomaly conditions (such as the Diaconescu-Moore-Witten anomaly):

Bibliography of part I

{AKMW} Orlando Alvarez, T. P. Killingback, M. L. Mangano, and P. Windey, String theory and loop space index theorems, Commun. Math. Phys. 111 (1987) 1-10, euclid, MR0896755

{ASi} Orlando Alvarez and I. M. Singer, Beyond the elliptic genus, Nuclear Phys. B 633 (2002) 309-344, arXiv:hep-th/0104199.

{And} Matthew Ando, Power operations in elliptic cohomology and representations of loop groups, Trans. AMS 352 (2000) 5619–5666, doi.

{Atwist} Matthew Ando, Andrew Blumberg, and David Gepner, Twists of K-theory and TMF, arxiv/1002.3004, to appear in Proc. Symp. Pure Math.

{AHR} Matthew Ando, Mike Hopkins, and Charles Rezk, Multiplicative orientations of KO-theory and of the spectrum of topological modular forms, preprint 2009,

{AHS} Matthew Ando, Mike Hopkins, and N. P. Strickland, Elliptic spectra, the Witten genus and the theorem of the cube, Invent. Math. 146 (2001), no. 3, 595–687.

{As} A. Asada, Four lectures on the geometry of loop group and non abelian de Rham theory, Chalmers University of Technology/The University of Göteborg, 1990.

{AJ} P. Aschieri and Branislav Jurčo, Gerbes, M5-brane anomalies and E 8E_8 gauge theory, J. High Energy Phys. 0410 (2004) 068, arXiv:hep-th/0409200.

{AS1} M. Atiyah and G. Segal, Twisted KK-theory, Ukr. Math. Bull. 1 (2004), no. 3, 291–334, arXiv:math.KT/0407054.

{AS2} Michael Atiyah and Graeme Segal, Twisted K-theory and cohomology, Inspired by S. S. Chern, 5-43, Nankai Tracts Math. 11, World Sci. Publ., Hackensack, NJ, 2006, arXiv:math/0510674.

{BBK} Nils Baas, M. B{"o}kstedt, and T. A. Kro, Two-categorical bundles and their classifying spaces, [arXiv:math.AT/0612549][].

{BCSS} John Baez, Alissa Crans, Urs Schreiber, and Danny Stevenson, From loop groups to 2-groups, Homology, Homotopy Appl. 9 (2007), no. 2, 101–135, arXiv:math.QA/0504123.

{BS} John Baez, and Urs Schreiber, Higher gauge theory, Categories in Algebra, Geometry and Mathematical Physics, 7–30, Contemp. Math., 431, Amer. Math. Soc., Providence, RI, 2007, arXiv:math/0511710v2.

{BSt} John Baez and Danny Stevenson, The classifying space of a topological 2-group, arXiv:math.AT/0801.3843.

{BST} E. Bergshoeff, E. Sezgin, and P.K. Townsend, Supermembranes and eleven-dimensional supergravity, Phys. Lett. B189 (1987) 75-78.

{BCMMS} P. Bouwknegt, A. L. Carey, V. Mathai, M. K. Murray, and D. Stevenson, Twisted K-theory and K-theory of bundle gerbes, Commun. Math. Phys. 228 (2002) 17-49, hep-th/0106194.

{BEJMS} P. Bouwknegt, J. Evslin, B. Jurčo, V. Mathai, and H. Sati, Flux compactification on projective spaces and the S-duality puzzle, Adv. Theor. Math. Phys. 10 (2006) 345-94, hep-th/0501110.

{BM} P. Bouwknegt and V. Mathai, D-branes, B-fields and twisted K-theory, J. High Energy Phys. 0003 (2000) 007, hep-th/0002023.

{BKTV} P. Bressler, M. Kapranov, B. Tsygan, and E. Vasserot, Riemann-Roch for real varieties, math/0612410.

{BMRS} J. Brodzki, V. Mathai, J. Rosenberg, and R. J. Szabo, D-branes, RR-fields and duality on noncommutative manifolds, Commun. Math. Phys. 277, no. 3 (2008) 643-706, hep-th/0607020v3.

{CJM} A. L. Carey, S. Johnson and M. K. Murray, Holonomy on D-branes, J. Geom. Phys. 52 (2004) 186-216, hep-th/0204199.

{CJMSW} A. L. Carey, S. Johnson, M. K. Murray, D. Stevenson, and B.-L. Wang, Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories, Commun. Math. Phys. 259 (2005) 577-613, arXiv:math.DG/0410013.

{CW} A. L. Carey and B.-L. Wang, Thom isomorphism and push-forward map in twisted K-theory, math.KT/0507414v4.

{CW2} A. L. Carey and B.-L. Wang, Riemann-Roch and index formulae in twisted K-theory, arXiv:0909.4848.

{CN} A. H. Chamseddine and H. Nicolai, Coupling the SO(2)SO(2) supergravity through dimensional reduction, Phys. Lett. B96 (1980) 89-93.

{CS} J. Cheeger and J. Simons, Differential characters and geometric invariants, In Geometry and Topology, 50-80, LNM 1167, Springer, 1985.

{CP} R. Coquereaux and K. Pilch, String structures on loop bundles, Commun. Math. Phys. 120 (1989) 353-378.

{C} E. Cremmer, Supergravities in 5 dimensions, in Superspace and supergravity, eds. S.W. Hawking and M. Rocek, Cambridge Univ. Press, Cambridge 1981.

{CJ} E. Cremmer and B. Julia, The SO(8)SO(8) supergravity, Nuclear Phys. B159 (1979) 141–212.

{CJLP} E. Cremmer, B. Julia, H. Lu, and C.N. Pope, Dualization of dualities II, Nucl. Phys. B535 (1998) 242–292, hep-th/9806106.

{CJS} E. Cremmer, B. Julia, and J. Scherk, Supergravity theory in eleven-dimensions, Phys. Lett. B76 (1978) 409-412.

{Dai} X. Dai, Adiabatic limits, nonmultiplicativity of signature, and Leray spectral sequence, J. Amer. Math. Soc. 4 (1991), no. 2, 265–321.

{dWN} B. de Wit and H. Nicolai, Hidden symmetries, central charges and all that, Class. Quant. Grav. 18 (2001) 3095-3112, hep-th/0011239.

{DFM} E. Diaconescu, D. S. Freed and G. Moore, The M-theory 3-form and E 8E_8 gauge theory, Elliptic cohomology, 44–88, London Math. Soc. Lecture Note Ser., 342, Cambridge Univ. Press, Cambridge, 2007, hep-th/0312069.

{DMW} E. Diaconescu, G. Moore and E. Witten, E 8E_8 gauge theory, and a derivation of K-Theory from M-Theory, Adv. Theor. Math. Phys. 6 (2003) 1031–1134, hep-th/0005090.

{dk} P. Donovan and M. Karoubi, Graded Brauer groups and KK-theory with local coefficients, Inst. Hautes 'Etudes Sci. Publ. Math. 38 (1970) 5–25.

{D} C. L. Douglas, On the twisted KK-homology of simple Lie groups, Topology 45 (2006), no. 6, 955–988.

{Du} M. J. Duff, E 8×SO(16)E_8 \times SO(16) symmetry of d=11d = 11 supergravity?, in Quantum Field Theory and Quantum Statistics, eds. I. A. Batalin, C. Isham and G. A. Vilkovisky, Adam Hilger, Bristol, UK 1986.

{DGT} M. J. Duff, G. W. Gibbons and Paul Townsend, Phys. Lett. 332B (1994) 321.

{DHIS} Mike Duff, Paul Howe, T. Inami and K. S. Stelle, Superstrings in D=10D=10 from supermembranes in D=11D=11, Phys. Lett. 191B (1987) 70–74.

{DLM} Mike Duff, J. T. Liu, and R. Minasian, Eleven-dimensional origin of string-string duality: a one loop test, Nucl. Phys. B452 (1995) 261-282, hep-th/9506126.

{DNP} M. J. Duff, B. E. W. Nilsson, and C. N. Pope, Kaluza-Klein supergravity, Phys. Rept. 130 (1986) 1-142.

{Englert} F. Englert, Spontaneous compactification of eleven-dimensional supergravity, Phys. Lett. B119 (1982) 339–342.

{ES} J. Evslin and H. Sati, SUSY vs E 8E_8 gauge theory in 11 dimensions, J. High Energy Phys. 0305 (2003) 048, hep-th/0210090.

{ES2} J. Evslin and Hisham Sati, Can D-branes wrap nonrepresentable cycles?, J. High Energy Phys. 0610 (2006) 050, hep-th/0607045.

{FS} J. Figueroa-O’Farrill, J. Simon, Supersymmetric Kaluza-Klein reductions of M2 and M5-branes, Adv. Theor. Math. Phys. 6 (2003) 703-793, hep-th/0208107.

{Fr} Dan Freed, Dirac charge quantization and generalized differential cohomology, in Surv. Differ. Geom. VII, 129–194, Int. Press, Somerville, MA, 2000, hep-th/0011220.

{CSII} Dan Freed, Classical Chern-Simons theory II, Houston J. Math. 28 (2002), no. 2, 293–310.

{FH} Dan Freed and Mike Hopkins, On Ramond-Ramond fields and K-theory, J. High Energy Phys. 05 (2000) 044, hep-th/0002027.

{FMS} Dan Freed, Greg Moore, and G. Segal, The uncertainty of fluxes, Commun. Math. Phys. 271 (2007) 247-274, hep-th/0605198.

{FW} Dan Freed and Edward Witten, Anomalies in string theory with D-branes, Asian J. Math. 3 (1999), no. 4, 819–851, hep-th/9907189.

{For} R. Forman, Spectral sequences and adiabatic limits, Comm. Math. Phys. 168 (1995), no. 1, 57–116, euclid.

{Go} Paul Goerss, Topological modular forms (after Hopkins, Miller, and Lurie), arXiv:math.AT/0910.5130.

{GHK} J. M.Gomez, P. Hu and I. Kriz, Stringy bundles and infinite loop space theory, preprint 2009.

{GS} M. B. Green and J. H. Schwarz, Anomaly cancellation in supersymmetric D=10D=10 gauge theory and superstring theory, Phys. Lett. B 149 (1984) 117-122.

{Gu} R. G"ven, Black pp-brane solutions of D=11D=11 supergravity theory, Phys. Lett. B276 (1992) 49–55.

{G1} A. Gustavsson, The non-Abelian tensor multiplet in loop space, J. High Energy Phys. 0601 (2006) 165, arXiv:hep-th/0512341.

{G2} A. Gustavsson, Loop space, (2,0)(2,0) theory, and solitonic strings, J. High Energy Phys. 0612 (2006) 066, hep-th/0608141.

{G3} A. Gustavsson, Selfdual strings and loop space Nahm equations, J. High Energy Phys. 0804 (2008) 083, arxiv:0802.3456.

{HM} J. A. Harvey and G. Moore, Superpotentials and membrane instantons, hep-th/9907026.

{H} A. Henriques, Integrating L L_\infty-algebras, Compos. Math. 144 (2008), no. 4, 1017–1045, [math.AT/0603563][].

{Ho} Mike Hopkins, Algebraic topology and modular forms, Proceedings of the ICM, Beijing 2002, vol. 1, 283–309, math.AT/0212397.

{HS} Mike Hopkins and I. M. Singer, Quadratic functions in geometry, topology, and M-theory, J. Differential Geom. 70 (2005), no. 3, 329–452, arXiv:math.AT/0211216.

{HK1} P. Hu and I. Kriz, Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence, Topology 40 (2001) 317–399.

{HK2} P. Hu and I. Kriz, Conformal field theory and elliptic cohomology, Adv. Math. 189 (2004), no. 2, 325–412.

{J} B. Julia, in Lectures in Applied Mathematics, AMS-SIAM, vol. 21, (1985), p. 335.

{Kap} A. Kapustin, D-branes in a topologically nontrivial B-field, Adv. Theor. Math. Phys. 4 (2000) 127-154, hep-th/9909089.

{Ki} T. P. Killingback, Global anomalies, string theory and space-time topology, Class. Quant. Grav. 5 (1988) 1169–1186.

{KS1} I. Kriz and H. Sati, M Theory, type IIA superstrings, and elliptic cohomology, Adv. Theor. Math. Phys. 8 (2004) 345–395, hep-th/0404013.

{KS2} Igor Kriz and Hisham Sati, Type IIB string theory, S-duality and generalized cohomology, Nucl. Phys. B715 (2005) 639–664, hep-th/0410293.

{KS3} I. Kriz and H. Sati, Type II string theory and modularity, J. High Energy Phys. 08 (2005) 038, hep-th/0501060.

{Ku} K. Kuribayashi, On the vanishing problem of string classes, J. Austral. Math. Soc. (series A) 61 (1996), 258-266.

{LNSW} W. Lerche, B. E. W. Nilsson1, A. N. Schellekens and N. P. Warner, Anomaly cancelling terms from the elliptic genus, Nucl. Phys. B 299 (1988) 91–116.

{Li} B. Li and H. Duan, Spin characteristic classes and reduced KSpinKSpin group of a low-dimensional complex, Proc. Amer. Math. Soc. 113 (1991) 479–491.

{LLW} W. Li, X. Liu, and H. Wang, On a spectral sequence for twisted cohomologies, arXiv:0911.1417.

{Liu} K. Liu, On modular invariance and rigidity theorems, Harvard thesis 1993.

{Lo} J. Lott, Higher degree analogs of the determinant line bundles, Comm. Math. Phys. 230 (2002) 41–69, [math.DG/0106177][].

{Lu} Jacob Lurie, A survey of elliptic cohomology, preprint, 2007,

{TDMW} V. Mathai and H. Sati, Some relations between twisted K-theory and Esb8E\sb8 gauge theory, J. High Energy Phys. 03 (2004) 016, hep-th/0312033.

{MSt} V. Mathai and D. Stevenson, Chern character in twisted K-theory: equivariant and holomorphic cases, Commun. Math. Phys. 236 (2003) 161-186, hep-th/0201010.

{MW} V. Mathai and S. Wu, Analytic torsion for twisted de Rham complexes, arXiv:math.DG/0810.4204.

{MM} R. Mazzeo and R. Melrose, The adiabatic limit, Hodge cohomology and Leray’s spectral sequence for a fibration, J. Differential Geom. 31 (1990), no. 1, 185–213.

{Mc} D. A. McLaughlin, Orientation and string structures on loop space, Pacific J. Math. 155 (1992) 143-156.

{Mic} J. Mickelsson, Current Algebras and Groups, Plenum Press, London and New York, 1989.

{MP} J. Mickelsson and R. Percacci, Global aspects of pp-branes, J. Geom. and Phys. 15 (1995) 369-380, hep-th/9304054.

{MM} R. Minasian and G. Moore, K-theory and Ramond-Ramond charge, J. High Energy Phys. 11 (1997) 002, hep-th/9710230.

{GaussĐ G. W. Moore, Anomalies, Gauss laws, and Page charges in M-theory, Comptes Rendus Physique 6 (2005) 251-259, hep-th/0409158.

{MW} G. Moore and E. Witten, Self duality, Ramond-Ramond fields, and K-theory, J. High Energy Phys. 05 (2000) 032, hep-th/9912279.

{Mo} Jack Morava, Forms of KK-theory, Math. Zeitschrift 201 (1989) 401–428, (scan at goettingen).

{Mu} Michael Murray, Bundle gerbes, J. London Math. Soc. (2) 54 (1996), no. 2, 403Đ416.

{MuS} Michael Murray and D. Stevenson, Bundle gerbes: Stable isomorphsim and local theory, J. London Math. Soc. (2). 62 (2002), no. 3, 925-937, [math.DG/9908135][}]).

{PSW} K. Pilch, A. N. Schellekens, and N. P. Warner, Path integral calculation of string anomalies, Nucl. Phys. B287 (1987) 362–380.

{PW} K. Pilch and N. P. Warner, String structures and the index of the Dirac-Ramond operator on orbifolds, Commun. Math. Phys. 115 (1988) 191–212, euclid.

{Pol} J. Polchinski, Dirichlet-branes and Ramond-Ramond charges, Phys. Rev. Lett. 75 (1995) 4724–4727, arXiv:hep-th/9510017.

{Red0} C. Redden, Canonical metric connections associated to String structures, PhD dissertation, University of Notre Dame, Indiana, 2006.

{Red} C. Redden, String structures and canonical three-forms, arxiv:math.DG/0912.2086.

{Red2} C. Redden, Harmonic forms on principal bundles, arXiv:0810.4578.

{RS} C. Redden and H. Sati, work in progress.

{Ros} J. Rosenberg, Continuous trace C *C^*-algebras from the bundle theoretic point of view, J. Aust. Math. Soc. A47 (1989) 368–381.

{S1} H. Sati, M-theory and characteristic classes, J. High Energy Phys. 0508 (2005) 020, hep-th/0501245.

{S2} H. Sati, Flux quantization and the M-theoretic characters, Nucl. Phys. B727 (2005) 461–470, hep-th/0507106.

{S3} H. Sati, Duality symmetry and the form-fields in M-theory, J. High Energy Phys. 0606 (2006) 062, arXiv:hep-th/0509046.

{S4} H. Sati, The Elliptic curves in gauge theory, string theory, and cohomology, J. High Energy Phys. 0603 (2006) 096, arXiv:hep-th/0511087.

{S5} Hisham Sati, E 8E_8 gauge theory and gerbes in string theory, hep-th/0608190.

{7twist} Hisham Sati, A higher twist in string theory, J. Geom. Phys. 59 (2009) 369-373, hep-th/0701232.

{KSpin} H. Sati, An approach to anomalies in M-theory via KSpinKSpin, J. Geom. Phys. 58 (2008) 387-401, arXiv:hep-th/0705.3484.

{MO8} H. Sati, On anomalies of E 8E_8 gauge theory on String manifolds, arXiv:0807.4940.

{SSS1} H. Sati, U. Schreiber and J. Stasheff, L L_\infty-connections and applications to String- and Chern-Simons nn-transport, in Recent Developments in QFT, eds. B. Fauser et al., Birkh{"a}user, Basel (2008), arXiv:0801.3480.

{SSS2} Hisham Sati, Urs Schreiber, and Jim Stasheff, Fivebrane structures, to appear in Rev. Math. Phys. arXiv:math.AT/0805.0564.

{SSS3} Hisham Sati, Urs Schreiber, and Jim Stasheff, Twisted differential String- and Fivebrane structures, arXiv:math.AT/0910.4001.

{SW} N. Seiberg and Edward Witten, Spin structures in string theory, Nucl. Phys. B276 (1986) 272-290.

{SWa} A. N. Schellekens and N.P. Warner, Anomalies and modular invariance in string theory, Phys. Lett. B 177 (1986) 317–323.

{SP} C. J. Schommer-Pries, A finite-dimensional String 2-group, preprint 2009.

{Sch} J. H. Schwarz, The power of M-theory, Phys. Lett. B367 (1996) 97-103, hep-th/9510086.

{Lec} J. H. Schwarz, Lectures on superstring and M-theory dualities, Nucl. Phys. Proc. Suppl. 55B (1997) 1-32, hep-th/9607201.

{SVW} S. Sethi, C. Vafa, and E. Witten, Constraints on low-dimensional string compactifications, Nucl. Phys. B480 (1996) 213-224, hep-th/9606122.

{ST1} S. Stolz and P. Teichner, What is an elliptic object?, in Topology, geometry and quantum field theory, 247–343, Cambridge Univ. Press 2004.

{ST2} Stefan Stolz and Peter Teichner, Super symmetric field theories and integral modular functions,

{Stg} R. E. Stong, Appendix: calculation of Ω 11 Spin(K(Z,4))\Omega^{Spin}_{11}(K(Z,4)), in Workshop on unified string theories (Santa Barbara, Calif., 1985), 430–437, World Sci. Publishing, Singapore, 1986.

{Str} A. Strominger, Superstrings with torsion, Nucl. Phys. B274 (1986) 253–284, (doi).

{Te} C. Teleman, K-theory of the moduli space of bundles on a surface and deformations of the Verlinde algebra, in Topology, Geometry and Quantum Field Theory, U. Tillmann (ed.), Cambridge University Press, 2004.

{Th} E. Thomas, On the cohomology groups of the classifying space for the stable spinor groups, Bol. Soc. Mat. Mexicana (2) 7 (1962) 57–69.

{Town3} P. K. Townsend, String-membrane duality in seven dimensions, Phys. Lett. B354 (1995) 247-255, hep-th/9504095.

{Town2} P. K. Townsend, D-branes from M-branes, Phys. Lett. B373 (1996) 68-75, arXiv:hep-th/9512062.

{Town} P. K. Townsend, Four lectures on M-theory, Summer School in High Energy Physics and Cosmology Proceedings, E. Gava et. al (eds.), Singapore, World Scientific, 1997, arXiv:hep-th/9612121.

{V} C. Vafa, Evidence for F-theory, Nucl. Phys. B469 (1996) 403–418, arXiv:hep-th/9602022.

{VW} C. Vafa and E. Witten, A one-loop test of string duality, Nucl. Phys. B447 (1995) 261-270, arXiv:hep-th/9505053.

{Wa} B.-L. Wang, Geometric cycles, index theory and twisted K-homology, J . Noncommut. Geom. 2 (2008), 497–552, arXiv:math.KT/0710.1625.

{We} P. C. West, E 11E_{11} and M-theory, Class. Quantum Grav. 18 (2001) 4443-4460.

{CMP} Edward Witten, Global gravitational anomalies, Commun. Math. Phys. 100 (1985) 197–229.

{W} Edward Witten, Elliptic genera and quantum field theory, Commun. Math. Phys. 109 (1987) 525–536, project euclid.

{Wit} Edward Witten, The index of the Dirac operator in loop space, in Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986), 161–181, Lecture Notes in Math. 1326, Springer, Berlin, 1988.

{Jones} Edward Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351–399, project euclid.

{Dynamics} Edward Witten, String theory dynamics in various dimensions, Nucl. Phys. B443 (1995) 85-126, hep-th/9503124.

{Wi1} Edward Witten, D-Branes and K-Theory, J. High Energy Phys. 12 (1998) 019, hep-th/9810188.

{Flux} Edward Witten, On flux quantization in M-theory and the effective action, J. Geom. Phys. 22 (1997) 1-13, hep-th/9609122.

{Effective} Edward Witten, Five-brane effective action in M-theory, J. Geom. Phys. 22 (1997) 103-133, hep-th/9610234.

{Among} Edward Witten, Duality relations among topological effects in string theory, J. High Energy Phys. 0005 (2000) 031, hep-th/9912086.

category: reference

Revised on April 11, 2014 05:56:38 by Urs Schreiber (