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Three Approaches to M-Theory

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Three Approaches to M-Theory Three Approaches to M-theory Luca Carlevaro PhD. thesis submitted to obtain the degree of \docteur `es sciences de l'Universit´e de Neuch^atel" on monday, the 25th of september 2006 PhD. advisors: Prof. Jean-Pierre Derendinger Prof. Adel Bilal Thesis comittee: Prof. Matthias Blau Prof. Matthias R. Gaberdiel Universit´e de Neuch^atel Facult´e des Sciences Institut de Physique Rue A.-L. Breguet 1 CH-2001 Neuch^atel Suisse This thesis is based on researches supported the Swiss National Science Foundation and the Commission of the European Communities under contract MRTN-CT-2004-005104. i Mots-cl´es: Th´eorie des cordes, Th´eorie de M(atrice) , condensation de gaugino, brisure de supersym´etrie en th´eorie M effective, effets d'instantons de membrane, sym´etries cach´ees de la supergravit´e, conjecture sur E10 Keywords: String theory, M(atrix) theory, gaugino condensation, supersymmetry breaking in effective M-theory, membrane instanton effects, hidden symmetries in supergravity, E10 con- jecture ii Summary: Since the early days of its discovery, when the term was used to characterise the quantum com- pletion of eleven-dimensional supergravity and to determine the strong-coupling limit of type IIA superstring theory, the idea of M-theory has developed with time into a unifying framework for all known superstring the- ories. This evolution in conception has followed the progressive discovery of a large web of dualities relating seemingly dissimilar string theories, such as theories of closed and oriented strings (type II theories), of closed and open unoriented strings (type I theory), and theories with built in gauge groups (heterotic theories). In particular the eleven-dimensional Lorentz invariance of M-theory has been further highlighted by the discovery of how it descends by an orbifold compactification to the weakly-coupled heterotic E8 × E8 string theory. Furthermore the derivation from M-theory of one-loop corrections to type II and heterotic low-energy effective supergravity has shed new light on our understanding of anomaly cancellation mechanisms which are so crucial for the consistency of superstring theories and are also of phenomenological importance, since the fermion content of the Standard Model is chiral. Despite these successes, M-theory remains somewhat elusive and is still in want of a complete and closed formulation. In particular, its fundamental degrees of freedom are still unknown, which in principle prevents us from establishing the full quantum theory. Proposals have been made, advocating D0-branes or M2-branes as its elementary states. However, even the most fruitful of these proposals, based on a partonic description of M-theory and known as the BFSS (Bank-Fischler-Seiberg-Susskind) conjecture or M(atrix) theory, has stumbled on a number of issues, one of the most nagging being its failure to correctly reproduce quantum corrections to two- and three-graviton tree-level scattering amplitudes known from supergravity one-loop calculations. In the absence of a closed and definite description, we are bound to adopt a fragmentary approach to M- theory. This work naturally follows this route. Three facets of M-theory are thus uncovered or further investigated in the course of the present thesis. The first part deals with the matrix model approach to M-theory and studies the relevancy of an alternative to the BFSS proposal, defined in purely algebraic terms, which is desirable from the point of view of a background independent formulation of M-theory. It implements overt eleven-dimensional Lorentz invariance by resorting to a osp(1j32) realisation of its group of symmetry, thereby incorporating M2- and M5-branes as fundamental degrees of freedom on the same footing as the original D0-branes of the BFSS model. This seems more in phase with the concept of M-theory which does not appear to give preference to strings over more extended objects. Finally, this construction partially solves the puzzle about the absence of transverse five-brane degrees of freedom in M(atrix) theory and opens the perspective of a description of M-theory in a curved AdS11 background. The second part of this thesis focuses on the determination of non-perturbative effects in low-energy effective heterotic supergravity, investigating some phenomenology related consequences of the Hoˇrava-Witten scenario. The latter provides the M-theory origin of the strongly-coupled heterotic E8 × E8 superstring theory in terms of an orbifold compactification on an interval, with twisted sectors given by two copies of E8 super Yang-Mills multiplets propagating on the ten-dimensional boundary hyperplanes sitting at the orbifold fixed points. This setups can accomodate, under certain conditions, the insertion of parallel fivebranes, allowing for Euclidean membranes to stretch between them and the boundary hyperplanes. In this framework, we investigate a Calabi- Yau compactification of this setup to four dimensions, establishing the effective supergravity and computing the resulting instanton corrections to the interaction Lagrangian. In the condensed phase, this allows us to derive the corresponding gauge threshold corrections and determine the resulting non-perturbative superpotential with the correct dependence in the fivebrane modulus, from a purely supergravity calculation. Confirming results determined earlier by instanton calculations this works also renders them more accurate and sets them on a sounder basis, whence the vacuum structure of the effective supergravity can be addressed. The third and last part of this thesis tackles the issue of how to recover information about M-theory by studying hidden symmetries of supergravity actions. In particular, we explore the conjecture stating that M- theory possesses an underlying hyperbolic infinite-dimensional Kac-Moody symmetry encapsulated in the split form e10 10, which occurs as a natural very-extension of the exceptional U-duality algebra of eleven-dimensional j supergravity compactified on a seven-torus. However, this conjecture is supposed to carry on beyond the toroidal case and to give clues about uncompactified M-theory. In the present work, we make use of this conjecture to determine the algebraic structure of Zm orbifold compactifications of M-theory, showing how their untwisted sectors are encoded in a certain class of Borcherds algebras. Furthermore, concentrating on a certain family of Z2 orbifolds of M-theory descending to non-supersymmetric but nonetheless stable orientifolds of type 0A string theory, we establish a dictionary between tilted Dp-/Dp0-brane configurations required in these models by tadpole cancellation and a certain class of threshold-one roots of e10. This latter result is of some phenomenological interest insomuch as it gives a deeper algebraic understanding of the magnetised D9-/D90-brane setup of the T-dual type 00 string theories, which have been recently shown to play a r^ole in moduli stabilisation. iii Remerciements Je d´esire remercier tout particuli`erement Jean-Pierre Derendinger, qui m'a aid´e et soutenu dans mon travail, dont la collaboration a ´et´e tr`es pr´ecieuse et qui m'a donn´e la possibilit´e et la chance de r´ealiser cette th`ese sous sa direction. Au m^eme titre, je remercie Adel Bilal qui fut mon premier directeur de th`ese, me conseilla, m'aida et guida mes pas au d´ebut de cette entreprise. Je remercie aussi Matthias Blau et Matthias Gaberdiel qui ont bien voulu accepter de faire partie du jury de th`ese et qui ont pris le temps de faire une lecture critique de mon travail. Je d´esire en outre remercier Maxime Bagnoud de ce que sa collaboration m'a appris et fait d´ecouvrir. Enfin, je remercie mes parents dont le soutien et l'amour ind´efectible m'ont soutenu tout au long de ce travail de th`ese et m'ont permis de le mener a` bien. iv Contents Introduction 1 I M-theory, Matrix models and the osp(1 32; ) superalgebra 5 j Introduction 6 1 M(atrix)-theory, branes and the BFSS conjecture 9 1.1 A matrix model for membranes . 9 1.1.1 The bosonic membrane in the light-cone frame . 9 1.1.2 Regularising the membrane Matrix theory . 13 1.1.3 The supermembrane on a curved background . 15 1.1.4 Supersymmmetrisation in eleven dimensions . 15 1.1.5 Instabilities and continuous spectrum of states in the supermembrane model 19 1.2 D-branes and Dirac-Born-Infeld theory . 20 1.2.1 Ten-dimensional super Yang-Mills theory and its toroidal reduction . 22 1.2.2 D-branes and T-duality . 23 1.3 The BFSS Conjecture . 25 1.3.1 M-theory and IIA string theory . 25 1.3.2 The BFSS matrix model . 27 1.3.3 The Infinite Momentum Frame . 29 1.3.4 M(atrix) theory: a proposal for DLCQ M-theory . 30 1.3.5 Continuous energy spectrum and second-quantisation in M(atrix) theory . 33 1.3.6 The BFSS matrix model and its supersymmetry algebra . 35 1.4 Brane charges and the BFSS supersymetry algebra . 37 1.4.1 The BFSS supercharge density algebra . 39 1.5 M-theory branes from M(atrix) theory . 42 1.5.1 Transverse membranes . 42 1.5.2 Spherical membranes . 44 1.5.3 Longitudinal fivebranes . 44 1.5.4 Transverse fivebranes . 46 1.6 Outlook . 48 2 Supermatrix models for M-theory based on osp(1 32; R) 50 j 2.1 Introduction . 50 2.2 The osp(1 32; R) superalgebra . 51 j 2.3 The twelve-dimensional case . 52 2.3.1 Embedding of SO(10; 2) in OSp(1 32; R) . 54 j 2.3.2 Supersymmetry transformations of 12D matrix fields . 55 v 2.3.3 sp(32; R) transformations of the fields and their commutation relation with supersymmetries . 56 2.3.4 A note on translational invariance and kinematical supersymmetries .
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