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A brief review of E theory

Peter West

Department of Mathematics, King’s College, London WC2R 2LS, UK

https://doi.org/10.1142/S0217751X1630043XCited by:17 (Source: Crossref)

Abstract

I begin with some memories of Abdus Salam who was my PhD supervisor. After reviewing the theory of nonlinear realisations and Kac–Moody algebras, I explain how to construct the nonlinear realisation based on the Kac–Moody algebra $E_{11}$ $E_{11}$ and its vector representation. I explain how this field theory leads to dynamical equations which contain an infinite number of fields defined on a space–time with an infinite number of coordinates. I then show that these unique dynamical equations, when truncated to low level fields and the usual coordinates of space–time, lead to precisely the equations of motion of 11-dimensional supergravity theory. By taking different group decompositions of $E_{11}$ $E_{11}$ we find all the maximal supergravity theories, including the gauged maximal supergravities, and as a result the nonlinear realisation should be thought of as a unified theory that is the low energy effective action for type II strings and branes. These results essentially confirm the $E_{11}$ $E_{11}$ conjecture given many years ago.

Based on an invited talk given at the Memorial Meeting for Nobel Laureate Prof. Abdus Salam’s 90th Birthday, 25–28 January 2016, NTU, Singapore.

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References

1. J. Wess and B. Zumino, Supergauge transformations in four dimensions, Nucl. Phys. B 70, 139 (1974); A Lagrangian model invariant under supergauge transformations, Phys. Lett. B 49, 52 (1974). Google Scholar
2. A. Salam and J. Strathdee, On superfields and Fermi-Bose symmetry, Phys. Rev. D 11, 1521 (1975). Web of Science, ADS, Google Scholar
3. L. O’Raifeartaigh, Spontaneous symmetry breaking for chiral scalar superfields, Nucl. Phys. B 96, 331 (1975). Web of Science, ADS, Google Scholar
4. P. Fayet, Spontaneous supersymmetry breaking without gauge invariance, Phys. Lett. B 58, 67 (1975). Web of Science, ADS, Google Scholar
5. P. West, Supersymmetric effective potential, Nucl. Phys. B 106, 219 (1976). Web of Science, ADS, Google Scholar
6. E. Witten, Dynamical breaking of supersymmetry, Nucl. Phys. B 188, 513 (1981). Web of Science, ADS, Google Scholar
7. S. Weinberg, The Quantum Theory of Fields, Vol. 1 (Cambridge University Press, 1995); M. Veltman, Facts and Mysteries in Elementary Particle Physics (World Scientific, 2003); Contributions in: Shelter Island II, eds. R. Jackiw, N. Khuri, S. Weinberg and E. Witten (MIT Press, 1985); C. N. Yang, Selected Papers 1945-1980 (W. H. Freeman and Company, 1983); Contributions of: S. Coleman, M. Gell-Mann, S. Glashow and B. Zumino, Hadrons and their interactions, in 1967 International School of Physics Ettore Majorana, ed. A. Zichichi (Academic Press, 1968). The reader may like to read the wonderful lectures of Zumino that explain how the results of current algebra can be found in a very simple way from the non-linear realisation of $S U (2) \otimes S U (2)$ $S U (2) \otimes S U (2)$ . Google Scholar
8. V. Alessandrini, A general approach to dual multiloop diagrams, Nuovo Cimento A 2, 321 (1971); V. Allessandrini and D. Amati, Properties of dual multiloop amplitudes, Nuovo Cimento A 4, 793 (1971); C. Lovelace, M-loop generalized Veneziano formula, Phys. Lett. 32, 703 (1970). Google Scholar
9. V. Alessandrini, D. Amati, M. Le Bellac and D. I. Olive, The operator approach to dual multiparticle theory, Phys. Rep. 1C, 170 (1971), as well as the additional supplement by D. Olive which appears immediately after this article. Google Scholar
10. A. Neveu and J. Scherk, Gauge invariance and uniqueness of the renormalisation of dual models with unit intercept, Nucl. Phys. B 36, 155 (1972). Web of Science, ADS, Google Scholar
11. T. Yoneya, Connection of dual models to electrodynamics and gravidynamics, Progr. Theor. Phys. 51, 1907 (1974); J. Scherk and J. Schwarz, Dual models for nonhadrons, Nucl. Phys. B 81, 118 (1974). Google Scholar
12. C. Campbell and P. West, $N = 2$ $N = 2$ $D = 10$ $D = 10$ nonchiral supergravity and its spontaneous compactification, Nucl. Phys. B 243, 112 (1984); M. Huq and M. Namazie, Kaluza–Klein supergravity in ten dimensions, Class. Quantum Grav. 2, 597 (1985); F. Giani and M. Pernici, $N = 2$ $N = 2$ supergravity in ten dimensions, Phys. Rev. D 30, 325 (1984). Google Scholar
13. J. Schwarz and P. West, Symmetries and transformation of chiral $N = 2$ $N = 2$ $D = 10$ $D = 10$ supergravity, Phys. Lett. B 126, 301 (1983). Web of Science, ADS, Google Scholar
14. P. Howe and P. West, The complete $N = 2$ $N = 2$ $D = 10$ $D = 10$ supergravity, Nucl. Phys. B 238, 181 (1984). Web of Science, ADS, Google Scholar
15. J. Schwarz, Covariant field equations of chiral $N = 2$ $N = 2$ $D = 10$ $D = 10$ supergravity, Nucl. Phys. B 226, 269 (1983). Web of Science, ADS, Google Scholar
16. E. Cremmer, B. Julia and J. Scherk, Supergravity theory in eleven-dimensions, Phys. Lett. B 76, 409 (1978). Web of Science, ADS, Google Scholar
17. E. Cremmer and B. Julia, The $N = 8$ $N = 8$ supergravity theory. I. The Lagrangian, Phys. Lett. B 80, 48 (1978). Web of Science, ADS, Google Scholar
18. B. Julia, Group disintegrations, in Superspace Supergravity, eds. S. W. Hawking and M. Roček (Cambridge University Press, 1981), p. 331; E. Cremmer, Dimensional Reduction in Field Theory and Hidden Symmetries in Extended Supergravity (Trieste Supergravity School, 1981), p. 313; Supergravities in 5 dimensions, in Superspace Supergravity, eds. S. W. Hawking and M. Roček (Cambridge University Press, 1981), p. 331. Google Scholar
19. A. Font, L. Ibanez, D. Lust and F. D. Quevedo, Strong-weak coupling duality and nonperturbative effects in string theory, Phys. Lett. B 249, 35 (1990); S. J. Rey, The confining phase of superstrings and axionic strings, Phys. Rev. D 43, 526 (1991). Google Scholar
20. C. M. Hull and P. K. Townsend, Unity of superstring dualities, Nucl. Phys. B 438, 109 (1995), arXiv:hep-th/9410167. Web of Science, ADS, Google Scholar
21. P. Townsend, The eleven dimensional supermembrane revisited, Phys. Lett. B 350, 184 (1995), arXiv:hep-th/9501068; E. Witten, String theory dynamics in various dimensions, Nucl. Phys. B 443, 85 (1995), arXiv:hep-th/9503124. Google Scholar
22. S. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 1, Phys. Rev. 177, 2239 (1969); C. Callan, S. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 2, Phys. Rev. 177, 2247 (1969). Google Scholar
23. P. West, Introduction to Strings and Branes (Cambridge University Press, 2012). Google Scholar
24. A. Salam and J. Strathdee, Nonlinear realizations. 1: The role of Goldstone bosons, Phys. Rev. 184, 1750 (1969), C. Isham, A. Salam and J. Strathdee, Spontaneous, breakdown of conformal symmetry, Phys. Lett. B 31, 300 (1970). Google Scholar
25. A. Borisov and V. Ogievetsky, Theory of dynamical affine and conformal symmetries as gravity theory of the gravitational field, Theor. Math. Phys. 21, 1179 (1975); V. Ogievetsky, Infinite-dimensional algebra of general covariance group as the closure of the finite dimensional algebras of conformal and linear groups, Nuovo Cimento 8, 988 (1973). Google Scholar
26. D. V. Volkov, Phenomological Lagrangians, Sov. J. Part. Nucl. 4, 3 (1973). Google Scholar
27. V. Kac, Graded Lie algebras and symmetric spaces, Funct. Anal. Appl. 2, 183 (1968); V. Kac, Infinite Dimensional Lie Algebras (Birkhauser, 1983). Google Scholar
28. R. Moody, A new class of Lie algebras, J. Algebra 10, 211 (1968). Web of Science, Google Scholar
29. P. West, Hidden superconformal symmetry in $M$ $M$ theory, J. High Energy Phys. 0008, 007 (2000), arXiv:hep-th/0005270. Web of Science, ADS, Google Scholar
30. P. West, $E_{11}$ $E_{11}$ and $M$ $M$ theory, Class. Quantum Grav. 18, 4443 (2001), arXiv:hep-th/0104081. Web of Science, ADS, Google Scholar
31. P. West, $E_{11}$ $E_{11}$ , SL(32) and central charges, Phys. Lett. B 575, 333 (2003), arXiv:hep-th/0307098. Web of Science, ADS, Google Scholar
32. A. Kleinschmidt and P. West, Representations of $G^{+ + +}$ $G^{+ + +}$ and the role of space-time, J. High Energy Phys. 0402, 033 (2004), arXiv:hep-th/0312247. ADS, Google Scholar
33. P. Cook and P. West, Charge multiplets and masses for $E_{11}$ $E_{11}$ , J. High Energy Phys. 11, 091 (2008), arXiv:0805.4451. Web of Science, ADS, Google Scholar
34. P. West, $E_{11}$ $E_{11}$ origin of brane charges and U-duality multiplets, J. High Energy Phys. 0408, 052 (2004), arXiv:hep-th/0406150. ADS, Google Scholar
35. T. Nutma, SimpLie, a simple program for Lie algebras, https://code.google.com/p/simplie/. Google Scholar
36. F. Englert and L. Houart, $G^{+ + +}$ $G^{+ + +}$ invariant formulation of gravity and M-theories: Exact BPS solutions, J. High Energy Phys. 0401, 002 (2004), arXiv:hep-th/0311255. Web of Science, ADS, Google Scholar
37. A. Salam, Gauge unification of fundamental forces, Nobel lecture on December 8, 1979, Rev. Mod. Phys. 52, 306 (1980). Web of Science, Google Scholar
38. P. West, Generalised geometry, eleven dimensions and $E_{11}$ $E_{11}$ , J. High Energy Phys. 1202, 018 (2012), arXiv:1111.1642. Web of Science, ADS, Google Scholar
39. A. Tumanov and P. West, Generalised vielbeins and non-linear realisations, J. High Energy Phys. 1410, 009 (2014), arXiv:1405.7894. Web of Science, ADS, Google Scholar
40. A. Tumanov and P. West, $E_{11}$ $E_{11}$ must be a symmetry of strings and branes, Phys. Lett. B 759, 663 (2016), arXiv:1512.01644. Web of Science, ADS, Google Scholar
41. A. Tumanov and P. West, $E_{11}$ $E_{11}$ in 11D, Phys. Lett. B 758, 278 (2016), arXiv:1601.03974. Web of Science, ADS, Google Scholar
42. I. Schnakenburg and P. West, Kac–Moody symmetries of IIB supergravity, Phys. Lett. B 517, 421 (2001), arXiv:hep-th/0107181. Web of Science, ADS, Google Scholar
43. P. West, The IIA, IIB and eleven dimensional theories and their common $E_{11}$ $E_{11}$ origin, Nucl. Phys. B 693, 76 (2004), arXiv:hep-th/0402140. Web of Science, ADS, Google Scholar
44. F. Riccioni and P. West, The $E_{11}$ $E_{11}$ origin of all maximal supergravities, J. High Energy Phys. 0707, 063 (2007), arXiv:0705.0752. Web of Science, ADS, Google Scholar
45. F. Riccioni and P. West, $E_{11}$ $E_{11}$ -extended spacetime and gauged supergravities, J. High Energy Phys. 0802, 039 (2008), arXiv:0712.1795. Web of Science, ADS, Google Scholar
46. A. Kleinschmidt, I. Schnakenburg and P. West, Very-extended Kac–Moody algebras and their interpretation at low levels, Class. Quantum Grav. 21, 2493 (2004), arXiv:hep-th/0309198; P. West, $E_{11},$ $E_{11},$ ten forms and supergravity, J. High Energy Phys. 0603, 072 (2006), arXiv:hep-th/0511153. Google Scholar
47. P. West, $E_{11}$ $E_{11}$ , ten forms and supergravity, J. High Energy Phys. 0603, 072 (2006), arXiv:hep-th/0511153. ADS, Google Scholar
48. P. Messen and T. Ortin, An $S L (2, Z)$ $S L (2, Z)$ multiplet of nine-dimensional type II supergravity theories, Nucl. Phys. B 541, 195 (1999), arXiv:hep-th/9806120; G. Dall’Agata, K. Lechner and M. Tonin, $D = 10$ $D = 10$ , $N = IIB$ $N = IIB$ supergravity: Lorentz-invariant actions and duality, J. High Energy Phys. 9807, 017 (1998), arXiv:hep-th/9806140; E. Bergshoeff, U. Gran and D. Roest, Type IIB seven-brane solutions from nine-dimensional domain walls, Class. Quantum Grav. 19, 4207 (2002), arXiv:hep-th/0203202. Google Scholar
49. E. A. Bergshoeff, M. de Roo, S. F. Kerstan and F. Riccioni, IIB supergravity revisited, J. High Energy Phys. 0508, 098 (2005), arXiv:hep-th/0506013; E. A. Bergshoeff, M. de Roo, S. F. Kerstan, T. Ortin and F. Riccioni, IIA ten-forms and the gauge algebras of maximal supergravity theories, J. High Energy Phys. 0607, 018 (2006), arXiv:hep-th/0602280. Google Scholar
50. P. West, $E_{11}$ $E_{11}$ , generalised space-time and IIA string theory, Phys. Lett. B 696, 403 (2011), arXiv:1009.2624. Web of Science, ADS, Google Scholar
51. W. Siegel, Two vielbein formalism for string inspired axionic gravity, Phys. Rev. D 47, 5453 (1993), arXiv:hep-th/9302036. Web of Science, ADS, Google Scholar
52. W. Siegel, Superspace duality in low-energy superstrings, Phys. Rev. D 48, 2826 (1993), arXiv:hep-th/9305073; Manifest duality in low-energy superstrings, in Proceedings, Strings ’93, Berkeley, 1993, p. 353, arXiv:hep-th/9308133. Google Scholar
53. O. Hohm and S. Kwak, Frame-like geometry of double field theory, J. Phys. A 44, 085404 (2011), arXiv:1011.4101. Google Scholar
54. A. Rocen and P. West, $E_{11}$ $E_{11}$ , generalised space-time and IIA string theory; the $R \otimes R$ $R \otimes R$ sector, in Strings, Gauge Fields and the Geometry Behind: The Legacy of Maximilian Kreuzer, eds. A. Rebhan, L. Katzarkov, J. Knapp, R. Rashkov and E. Scheid (World Scientific, 2013), arXiv:1012.2744. Google Scholar
55. P. West, $E_{11}$ $E_{11}$ , generalised space-time and equations of motion in four dimensions, J. High Energy Phys. 1212, 068 (2012), arXiv:1206.7045. Web of Science, ADS, Google Scholar
56. E. Bergshoeff, I. De Baetselier and T. Nutma, $E_{11}$ $E_{11}$ and the embedding tensor, J. High Energy Phys. 0709, 047 (2007), arXiv:0705.1304. Web of Science, ADS, Google Scholar
57. F. Riccioni, D. Steele and P. West, The $E_{11}$ $E_{11}$ origin of all maximal supergravities — the hierarchy of field-strengths, J. High Energy Phys. 0909, 095 (2009), arXiv:0906.1177. Web of Science, ADS, Google Scholar
58. F. Riccioni and P. West, Dual fields and $E_{11}$ $E_{11}$ , Phys. Lett. B 645, 286 (2007), arXiv:hep-th/0612001; F. Riccioni, D. Steele and P. West, Duality symmetries and $G^{+ + +}$ $G^{+ + +}$ theories, Class. Quantum Grav. 25, 045012 (2008), arXiv:0706.3659. Google Scholar
59. L. J. Romans, Massive $N = 2 A$ $N = 2 A$ supergravity in ten dimensions, Phys. Lett. B 169, 374 (1986). Web of Science, ADS, Google Scholar
60. B. de Wit, H. Samtleben and M. Trigiante, The maximal $D = 5$ $D = 5$ supergravities, Nucl. Phys. B 716, 215 (2005), arXiv:hep-th/0412173; B. de Wit and H. Samtleben, Gauged maximal supergravities and hierarchies of non-Abelian vector-tensor systems, Fortschr. Phys. 53, 442 (2005), arXiv:hep-th/0501243, and references therein. Google Scholar
61. P. West, Brane dynamics, central charges and $E_{11}$ $E_{11}$ , J. High Energy Phys. 0503, 077 (2005), arXiv:hep-th/0412336. ADS, Google Scholar
62. P. West, Very extended $E_{8}$ $E_{8}$ and $A_{8}$ $A_{8}$ at low levels, gravity and supergravity, Class. Quantum Grav. 20, 2393 (2003), arXiv:hep-th/0212291. Web of Science, ADS, Google Scholar
63. P. West, $E_{11}$ $E_{11}$ and higher spin theories, Phys. Lett. B 650, 197 (2007), arXiv:hep-th/0701026. Web of Science, ADS, Google Scholar
64. D. Steele and P. West, $E_{11}$ $E_{11}$ and supersymmetry, J. High Energy Phys. 1102, 101 (2011), arXiv:1011.5820. Web of Science, ADS, Google Scholar
65. S. de Buyl, M. Henneaux and L. Paulot, Extended $E_{8}$ $E_{8}$ invariance of 11-dimensional supergravity, J. High Energy Phys. 0602, 056 (2006), arXiv:hep-th/0512292; T. Damour, A. Kleinschmidt and H. Nicolai, Hidden symmetries and the fermionic sector of eleven-dimensional supergravity, Phys. Lett. B 634, 319 (2006), arXiv:hep-th/0512163; S. de Buyl, M. Henneaux and L. Paulot Hidden symmetries and Dirac fermions, Class. Quantum Grav. 22, 3595 (2005), arXiv:hep-th/0506009. Google Scholar