The Pennsylvania State University The Graduate School Eberly College of Science MINIMAL REPRESENTATIONS OF U-DUALITY GROUPS A Thesis in Physics by Oleksandr Pavlyk c 2005 Oleksandr Pavlyk Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2005 The thesis of Oleksandr Pavlyk was reviewed and approved∗ by the following: Murat G¨unaydin Professor of Physics Thesis Adviser Chair of Committee Vincent H. Crespi Professor of Physics Richard W. Robinett Professor of Physics Victor Nistor Professor of Mathematics Jayanth Banavar Professor of Physics Head of the Department of Physics ∗Signatures are on file in the Graduate School. iii Abstract Equations of motion of eleven dimensional supergravity compactified to d dimensions on (11 − d)-torus have a hidden non-compact global symmetry group E11−d(11−d), being examples of U-duality groups. U-duality groups are conjectured to be broken to their discrete subgroups E11−d(11−d) (Z) in the non-perturbatve M-theory. In this thesis we study algebraic structures and unitary realizations of U-duality groups, in particular those arising from Maxwell-Einstein supergravity (MESGT) theories in d = 5, 4, 3 dimensions. These algebraic structures and their related symmetry groups arise also in the study of generalized space-times defined by Jordan algebras and related algebraic con- structs. After reviewing the previous work on the Lorentz groups and conformal groups of spacetimes defined by Jordan algebras we study their quasi-conformal groups. In particular we give the geometric realization of these quasi-conformal groups in a basis covariant with respect to their generalized Lorentz groups. Minimal realization of U-duality groups associated with d = 3 supergravities is con- structed as an extension of conformal quantum mechanics using ideas and methods pio- neered by G¨unaydin, Koepsell and Nicolai. Minimal unitary representations of such U- duality groups are presented ellucidating connection with alternative construction of mini- mal representations by Kazhdan, Pioline and Waldron. iv Table of Contents List of Figures ....................................... vi List of Tables ....................................... vii Acknowledgments ..................................... viii Chapter 1. Introduction ................................. 1 Chapter 2. A glimpse of supergravity theories ..................... 7 2.1 Short review of supersymmetry ......................... 7 2.2 11-dimensional supergravity theory ....................... 10 2.3 Extension of supersymmetry algebra in d = 11 ................ 12 2.4 Dimensional reduction .............................. 13 2.5 Scalar manifold and hidden symmetry ..................... 16 2.6 N = 2 supergravity in d = 5 ........................... 19 2.7 Dimensional reduction to d = 4 ......................... 24 2.8 Geometries of the three dimensional MESGTs defined by Jordan algebras of degree 3 ...................................... 26 2.9 U-duality groups and entropy of BPS black holes in supergravity theories . 26 Chapter 3. Structure of U-duality algebras and Jordan algebras . 28 3.1 Jordan algebras .................................. 28 3.2 Linear fractional group of Jordan algebras as generalised conformal group . 30 3.3 Generalised rotation, Lorentz and conformal groups ............. 31 3.4 Positive energy unitary representations of generalised conformal groups .. 34 3.5 Quasiconformal groups and Freudenthal triple systems ............ 36 A 3.6 Space-times over J3 as dilatonic and spinorial extensions of Minkowskian space-times .................................... 40 3.7 Geometric realizations of SO (d + 2, 4) as quasiconformal groups ...... 42 3.8 Geometric realizations of E8(−24), E7(−5), E6(2) and F4(4) as quasiconformal groups ....................................... 44 3.8.1 Geometric realization of the quasiconformal group E8(−24) ..... 44 3.8.2 Geometric realization of the quasiconformal group E7(−5) ...... 47 3.8.3 Geometric realization of the quasiconformal group E6(2) ....... 49 3.8.4 Geometric realization of the quasiconformal group F4(4) ....... 51 3.9 Summary ..................................... 53 Chapter 4. Minimal representations of U-duality groups . 55 4.1 Conformal quantum mechanics ......................... 56 4.2 Minimal realization of e8(−24) .......................... 57 4.2.1 Exceptional Lie Algebra e8(−24) .................... 59 v ∗ 4.2.2 The Minimal Unitary Realization of e8(−24) in su (8) Basis ..... 62 4.2.3 The Minimal Unitary Realization of e8(−24) in su(6, 2) Basis ..... 64 4.3 Truncations of the minimal unitary realization of e8(−24) ........... 67 4.3.1 Truncation to the minimal unitary realization of e7(−5) as a quasicon- formal subalgebra ............................ 67 4.3.2 Truncation to the minimal unitary realization of e6(2) as a quasicon- formal subalgebra ............................ 70 4.3.3 Truncation to the minimal unitary realization of e6(−14) as a quasi- conformal subalgebra ........................... 71 4.3.4 Truncation to the minimal unitary realization of f4(4) as a quasicon- formal subalgebra ............................ 72 4.3.5 Truncation to the minimal unitary realization of so (4, 4) as a quasiconformal subalgebra ............... 74 4.4 Truncation to the minimal unitary realization of e7(−25) as a quasiconformal subalgebra ..................................... 76 4.5 Minimal unitary realizations of the quasiconformal groups SO(d + 2, 4) ... 77 4.6 Minimal realizations as non-compact groups and conformal quantum mechanics 80 4.7 Representations of non-compact supergroups and conformal quantum me- chanics ....................................... 86 Chapter 5. Minimal unitary representations of U-duality Lie algebras . 89 5.1 K-types ...................................... 89 5.2 Constructing spherical vector for split algebras ................ 91 5.3 Spherical vector of D4(4) ............................. 92 5.4 Recasting into conformal QM basis ....................... 95 Appendix A. Reference material ............................. 98 A.1 Clifford algebras and spinors .......................... 98 A.2 Lie algebras .................................... 101 A.3 Real forms of Lie algebras ............................ 104 A.4 Going from su∗(8) to su(6, 2) basis ....................... 107 ∗ A.5 Minimal realization of e8(8) in su (8) basis ................... 108 Bibliography ........................................ 111 vi List of Figures 1.1 The web of dualities between string theories. ................. 3 2.1 Dynkin diagram corresponding to the Lie algebra E8. Labels denote gener- ~ ators of the root lattice formed by ~aijk and bij. ................ 18 5.1 Root system of d4. ................................ 93 vii List of Tables 1.1 Duality symmetries of toroidally compactified type II string. ......... 4 1.2 Duality symmetries of supergravities obtained by compactifications of N = 2 d = 5 MESGTs with scalar manifolds being symmetric spaces. ........ 5 2.1 Number of scalars and k-form field strengths in theories obtained from 11D supergravity by toroidal compactification ................... 15 2.2 Hidden symmetries of dimensionally reduced supergravity theories d > 5 .. 17 2.3 Hidden symmetries of dimensionally reduced supergravity theories d 6 5 .. 17 2.4 Maximal compact subgroups of maximally split maximal supergravity sym- metry groups G = Ed(d) ............................. 18 3.1 Rotation, Lorentz and conformal groups for generalised space-times defined by simple Euclidean Jordan algebras ...................... 33 3.2 U-duality groups for exceptional MESGTs in d = 3, 4, 5. ........... 53 4.1 Minimal number of degrees of freedom necessary to realize given simple Lie algebra as a symmetry of quantum mechanical system. ............ 55 4.2 Quasi-conformal algebras based on simple (complex) g and irreducible ρ .. 84 4.3 Dimensions and dual Coxeter numbers of simple complex Lie algebras. ... 85 A.1 Admissible reality conditions in various space-times. ............. 99 A.2 Dynkin diagrams of finite dimensional simple Lie algebras. .......... 103 A.3 Real forms of simple Lie algebras. ........................ 105 A.4 Real forms of classical Lie algebras and reality conditions. .......... 106 viii Acknowledgments I would like thank my adviser Murat G¨unaydin for the wealth of knowledge I have learned under his guidence, for his encouragement and enthusiasm, and for his readily available advice. I am grateful to my thesis committee members Prof. Richard Robinett, Prof. Vincent Crespi and Prof. Victor Nistor for serving on my committee, their comments, and their flexibility regarding the date of the thesis defense. I would like to thank fellow students Sudarshan Fernando, Orcan Ogetbil and Sean McRey- nolds for numerous discussions, during our weekly seminars, in graduate offices, and beyond. It has been by pleasure working with Sudarshan at early stages during my graduate years at Penn State. I enjoyed working with Prof. Vincent Crespi on the departmental website, the project which proved invaluable on my resume and was a lot of fun to work on. My special thanks go to Randi Neshteruk. Her knowledge of intricacies of formal require- ments and readiness to assist have been of great help to me. And last, but not least, I thank my son and wife, Dima and Iana, for their infinite patience, love and support through my years at graduate school. ix To my parents 1 Chapter 1 Introduction In the 1970’s Veneziano [110] proposed the dual amplitude Γ(−α(s))Γ(−α(t)) A (s, t) = α(s) = α +