Rational sphere valued supercocycles in \(M\)-theory and type IIA string theory. (English) Zbl 1360.83075
Summary: We show that supercocycles on super \(L_\infty\)-algebras capture, at the rational level, the twisted cohomological charge structure of the fields of \(M\)-theory and of type IIA string theory. We show that rational 4-sphere-valued supercocycles for M-branes in \(M\)-theory descend to supercocycles in type IIA string theory with coefficients in the free loop space of the 4-sphere, to yield the Ramond-Ramond fields in the rational image of twisted \(K\)-theory, with the twist given by the \(B\)-field. In particular, we derive the M2/M5 \(\leftrightarrow\) F1/Dp/NS5 correspondence via dimensional reduction of sphere-valued super-\(L_\infty\)-cocycles.
MSC:
| 83E30 | String and superstring theories in gravitational theory |
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
| 81T60 | Supersymmetric field theories in quantum mechanics |
| 83E50 | Supergravity |
Keywords:
homotopy Lie algebras; supersymmetry; branes; rational homotopy theory; \(M\)-theory; string theory; supercocyclesReferences:
| [1] | de Azcárraga, José A.; Izquierdo, José M., Lie groups, Lie algebras, cohomology and some applications in physics, (Cambridge Monographs on Mathematical Physics (1995), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0836.22027 |
| [2] | Fiorenza, Domenico; Sati, Hisham; Schreiber, Urs, Super-Lie \(n\)-algebra extensions, higher WZW models and super-\(p\)-branes with tensor multiplet fields, Int. J. Geom. Methods Mod. Phys., 12, 2, 1550018 (2015), 35 · Zbl 1309.81216 |
| [3] | Fiorenza, D.; Sati, H.; Schreiber, U., The Wess-Zumino-Witten term of the M5-brane and differential cohomotopy, J. Math. Phys., 56, 10, 102301 (2015), 10 · Zbl 1325.81139 |
| [4] | Sati, Hisham, Geometric and topological structures related to M-branes, (Superstrings, Geometry, Topology, and \(C^\ast \)-algebras. Superstrings, Geometry, Topology, and \(C^\ast \)-algebras, Proc. Sympos. Pure Math., vol. 81 (2010), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 181-236 · Zbl 1210.81089 |
| [7] | Schreiber, Urs, Structure theory for higher WZW terms, (Lectures at Higher Structures in String Theory and Quantum Field Theory (2015), ESI: ESI Vienna) |
| [8] | Sati, Hisham; Schreiber, Urs; Stasheff, Jim, \(L_\infty \)-algebra connections and applications to String- and Chern-Simons \(n\)-transport, (Quantum Field Theory (2009), Birkhäuser: Birkhäuser Basel), 303-424 · Zbl 1183.83099 |
| [9] | D’Auria, R.; Fré, P., Geometric supergravity in \(D = 11\) and its hidden supergroup, Nuclear Phys. B, 201, 1, 101-140 (1982) |
| [10] | Pridham, J. P., Unifying derived deformation theories, Adv. Math., 224, 3, 772-826 (2010) · Zbl 1195.14012 |
| [12] | Shvarts, A. S., On the definition of superspace, Teoret. Mat. Fiz., 60, 1, 37-42 (1984) · Zbl 0575.58005 |
| [13] | Konechny, A.; Schwarz, A., On \((k \oplus l | q)\)-dimensional supermanifolds, (Supersymmetry and Quantum Field Theory (Kharkov, 1997). Supersymmetry and Quantum Field Theory (Kharkov, 1997), Lecture Notes in Phys., vol. 509 (1998), Springer: Springer Berlin), 201-206 · Zbl 0937.58001 |
| [15] | Hirschhorn, Philip S., Model categories and their localizations, (Mathematical Surveys and Monographs, vol. 99 (2003), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 1017.55001 |
| [16] | Bandos, Igor; Lechner, Kurt; Nurmagambetov, Aleksei; Pasti, Paolo; Sorokin, Dmitri; Tonin, Mario, Covariant action for the super-five-brane of \(M\) theory, Phys. Rev. Lett., 78, 23, 4332-4334 (1997) · Zbl 0944.81039 |
| [17] | Chryssomalakos, C.; de Azcárraga, J. A.; Izquierdo, J. M.; Pérez Bueno, J. C., The geometry of branes and extended superspaces, Nuclear Phys. B, 567, 1-2, 293-330 (2000) · Zbl 0951.81041 |
| [18] | Nikolaus, Thomas; Schreiber, Urs; Stevenson, Danny, Principal \(\infty \)-bundles: general theory, J. Homotopy Relat. Struct., 10, 4, 749-801 (2015) · Zbl 1349.18032 |
| [20] | Vigué-Poirrier, Micheline; Sullivan, Dennis, The homology theory of the closed geodesic problem, J. Differential Geom., 11, 4, 633-644 (1976) · Zbl 0361.53058 |
| [21] | Vigué-Poirrier, Micheline; Burghelea, Dan, A model for cyclic homology and algebraic \(K\)-theory of \(1\)-connected topological spaces, J. Differential Geom., 22, 2, 243-253 (1985) · Zbl 0595.55009 |
| [22] | Adams, Allan; Evslin, Jarah, The loop group of \(E_8\) and \(K\)-theory from 11d, J. High Energy Phys., 2, 029 (2003), 14 |
| [23] | Bergman, Aaron; Varadarajan, Uday, Loop groups, Kaluza-Klein reduction and M-theory, J. High Energy Phys., 6, 043 (2005), 28 pp. (electronic) |
| [24] | Mathai, Varghese; Sati, Hisham, Some relations between twisted \(K\)-theory and \(E_8\) gauge theory, J. High Energy Phys., 3, 016 (2004), 22 pp. (electronic) |
| [25] | Sati, Hisham, \(E_8\) gauge theory and gerbes in string theory, Adv. Theor. Math. Phys., 14, 2, 399-437 (2010) · Zbl 1225.81120 |
| [26] | Sati, Hisham, The loop group of \(E_8\) and targets for spacetime, Modern Phys. Lett. A, 24, 1, 25-40 (2009) · Zbl 1165.81344 |
| [28] | Figueroa-O’Farrill, José; Simón, Joan, Supersymmetric Kaluza-Klein reductions of M2 and M5-branes, Adv. Theor. Math. Phys., 6, 4, 703-793 (2002) · Zbl 1020.83027 |
| [30] | Ando, Matthew; Blumberg, Andrew J.; Gepner, David, Twists of \(K\)-theory and TMF, (Superstrings, Geometry, Topology, and \(C^\ast \)-algebras. Superstrings, Geometry, Topology, and \(C^\ast \)-algebras, Proc. Sympos. Pure Math., vol. 81 (2010), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 27-63 · Zbl 1210.81054 |
| [31] | Quillen, Daniel, Rational homotopy theory, Ann. of Math. (2), 90, 205-295 (1969) · Zbl 0191.53702 |
| [32] | Sullivan, Dennis, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math., 47, 269-331 (1977), (1978) · Zbl 0374.57002 |
| [33] | Bousfield, A. K.; Gugenheim, V. K.A. M., On \(PL\) de Rham theory and rational homotopy type, Mem. Amer. Math. Soc., 8, 179 (1976), ix+94 · Zbl 0338.55008 |
| [34] | Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude, Rational homotopy theory, (Graduate Texts in Mathematics, vol. 205 (2001), Springer-Verlag: Springer-Verlag New York) · Zbl 0961.55002 |
| [35] | Félix, Yves; Halperin, Steve; Thomas, Jean-Claude, Rational Homotopy Theory. II (2015), World Scientific Publishing Co. Pte. Ltd.: World Scientific Publishing Co. Pte. Ltd. Hackensack, NJ · Zbl 1325.55001 |
| [36] | Félix, Yves; Oprea, John; Tanré, Daniel, Algebraic models in geometry, (Oxford Graduate Texts in Mathematics, vol. 17 (2008), Oxford University Press: Oxford University Press Oxford) · Zbl 1149.53002 |
| [37] | Griffiths, Phillip A.; Morgan, John W., Rational homotopy theory and differential forms, (Progress in Mathematics, vol. 16 (1981), Birkhäuser: Birkhäuser Boston, Mass) · Zbl 0474.55001 |
| [38] | Hess, Kathryn, Rational homotopy theory: a brief introduction, (Interactions Between Homotopy Theory and Algebra. Interactions Between Homotopy Theory and Algebra, Contemp. Math., vol. 436 (2007), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 175-202 · Zbl 1128.55010 |
| [39] | Hilton, Peter; Mislin, Guido; Roitberg, Joe, Localization of Nilpotent Groups and Spaces (1975), North-Holland Publishing Co.: North-Holland Publishing Co. Amsterdam-Oxford · Zbl 0323.55016 |
| [40] | Castellani, Leonardo; D’Auria, Riccardo; Fré, Pietro, Supergravity and Superstrings. A Geometric Perspective (1991), World Scientific Publishing Co., Inc.: World Scientific Publishing Co., Inc. Teaneck, NJ, Mathematical Foundations · Zbl 0753.53047 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
