1 year ago

M-Theory from the Superpoint


3. Under

3. Under the above identifications the symmetic bilinear Spin-equivariant spinor-to-vector pairing is given by N ± ⊗ N ± −→ Mat herm 2×2 (K) ( ψ, φ ↦→ 1 2 ψφ † + φψ †) . 4. There is in addition a bilinear symmetric, non-degenerate and Spin-invariant spinor-to-scalar pairing give by N ± ⊗ N ∓ −→ R ψ, φ ↦→ Re(ψ † · φ) Acknowledgements. We thank David Corfield for comments on an earlier version of this article. We thank the Max Planck Institute for Mathematics in Bonn for kind hospitality while the result reported here was conceived. U.S. thanks Roger Picken for an invitation to Instituto Superior Técnico, Lisbon, where parts of this article was written. U.S. was supported by RVO:67985840. J.H. was supported by the Portuguese science foundation grant SFRH/BPD/92915/2013. References [1] A. Achúcarro, J. Evans, P. Townsend, D. Wiltshire, Super p-Branes, Phys. Lett. B 198 (1987) [2] R. D’Auria and P. Fré, Geometric supergravity in D = 11 and its hidden supergroup, Nucl. Phys. B201 (1982) 101–140 [3] J. de Azcrraga, J. M. Izquierdo, Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics, Cambridge Monographs of Mathematical Physics (1995) [4] J. de Azcárraga, P. Townsend, Superspace geometry and the classification of supersymmetric extended objects, Phys. Rev. Lett. Volume 62, Number 22 (1989) [5] J. Baez, The Octonions, Bull. Amer. Math. Soc. 39 (2002), 145-205, Errata in Bull. Amer. Math. Soc. 42 (2005), 213. [6] J. Baez, J. Huerta, Division algebras and supersymmetry I, in R. Doran, G. Friedman, J. Rosenberg (eds.), Superstrings, Geometry, Topology, and C ∗ -algebras, Proc. Symp. Pure Math. 81, AMS, Providenc (2010) arXiv:0909.0551 [7] J. Baez, J. Huerta, Division algebras and supersymmetry II, Adv. Math. Theor. Phys. 15 (2011) arXiv:1003.3436 [8] I. Bandos, K. Lechner, A. Nurmagambetov, P. Pasti, D. Sorokin and M. Tonin, Covariant action for the super-fivebrane of M-theory, Phys. Rev. Lett. 78 (1997), 4332 arXiv:hep-th/9701149 [9] E. Bergshoeff, E. Sezgin, P. Townsend, Supermembranes and eleven dimensional supergravity, Phys.Lett. B189 (1987) 75-78, [10] M. Blencowe, M. Duff, Supersingletons, Physics Letters B, Volume 203, Issue 3 (1988) [11] F. Brandt, Supersymmetry algebra cohomology II: Primitive elements in 2 and 3 dimensions, J. Math. Phys. 51 (2010) 112303, arXiv:1004.2978 [12] F. Brandt, Supersymmetry algebra cohomology III: Primitive elements in four and five dimensions, J. Math. Phys. 52 (2011), 052301, arXiv:1005.2102 22

[13] F. Brandt, Supersymmetry algebra cohomology IV: Primitive elements in all dimensions from D = 4 to D = 11, J. Math. Phys. 54 (2013), 052302, arXiv:1303.6211 [14] A. Candiello, K. Lechner, Duality in Supergravity Theories, Nucl.Phys. B412 (1994) 479-501 arXiv:hep-th/9309143 [15] C. Chryssomalakos, J. de Azcárraga, J. Izquierdo, and C. Pérez Bueno, The geometry of branes and extended superspaces, Nucl. Phys. B 567 (2000), 293-330, [arXiv:hep-th/9904137]. [16] P. Claus, R. Kallosh, A. Van Proeyen, M5-brane and superconformal (0, 2) tensor multiplet in 6 dimensions, Nucl.Phys. B518 (1998) 117-150 arXiv:hep-th/9711161 [17] P. Claus, R. Kallosh, J. Kumar, P. Townsend, A. Van Proeyen, Conformal Theory of M2, D3, M5 and D1+D5 Branes‘, JHEP 9806 (1998) 004 arXiv:hep-th/9801206 [18] M. Duff, Near-horizon brane-scan revived, Nucl.Phys.B810:193-209 (2009) [19] M. Duff, The World in Eleven Dimensions: Supergravity, Supermembranes and M-theory, IoP (1999) [20] M. Duff, P. Howe, T. Inami, K. Stelle, Superstrings in D = 10 from Supermembranes in D = 11, Phys. Lett. B191 (1987) 70 [21] M. Duff, C. Sutton, The Membrane at the End of the Universe, New Sci. 118 (1988) [22] J. M. Evans, Supersymmetric Yang–Mills theories and division algebras, Nucl. Phys. B298 (1988), 92–108. [23] D. Fiorenza, C. Rogers, and U. Schreiber, L ∞ -algebras of local observables, Homology, Homotopy and Applications, vol. 16(2) (2014) arXiv:1304.6292 [24] D. Fiorenza, H. Sati, and U. Schreiber, Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory, Adv. Theor. Math. Phys. Volume 18, Number 2 (2014) p. 229321 (2014) arXiv:1201.5277 [25] D. Fiorenza, H. Sati, and U. Schreiber, Super Lie n-algebra extensions, higher WZW models, and super p-branes with tensor multiplet fields, Intern. J. Geom. Meth. Mod. Phys. 12 (2015) 1550018 , [arXiv:1308.5264]. [26] D. Fiorenza, H. Sati, and U. Schreiber, Rational sphere valued supercocycles in M-theory and type IIA string theory, Journal of Geometry and Physics, Volume 114, April 2017, arXiv:1606.03206 [27] D. Fiorenza, H. Sati, and U. Schreiber, T-Duality from super Lie n-algebra cocycles for super p-branes arXiv:1611.06536 [28] D. Freed, Five lectures on supersymmetry, American Mathematical Society (1999) [29] M. Green, J. Schwarz, Covariant description of superstrings, Phys. Lett. B136 (1984), 367370 [30] M. Henneaux, L. Mezincescu, A Sigma Model Interpretation of Green–Schwarz Covariant Superstring Action, Phys.Lett. B152 (1985) 340 [31] G. Horowitz, L. Susskind, Bosonic M Theory, Journal of Mathematical Physics, Volume 42, Issue 7 arXiv:hep-th/0012037 23

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