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UC Berkeley UC Berkeley Electronic Theses and Dissertations UC Berkeley UC Berkeley Electronic Theses and Dissertations Title On Freudenthal Duality and Gauge Theories Permalink https://escholarship.org/uc/item/2217p9m4 Author Tagliaferro, Anthony Joseph Publication Date 2012 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California On Freudenthal Duality and Gauge Theories by Anthony Joseph Tagliaferro A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Bruno Zumino, Chair Professor Ori Ganor Professor Michael Hutchings Fall 2012 On Freudenthal Duality and Gauge Theories Copyright 2012 by Anthony Joseph Tagliaferro 1 Abstract On Freudenthal Duality and Gauge Theories by Anthony Joseph Tagliaferro Doctor of Philosophy in Physics University of California, Berkeley Professor Bruno Zumino, Chair In this thesis, I write down a Lagrangian for an exotic gauge theory defined using Freudenthal Triple Systems (FTS). FTSs are algebraic systems that arise in the context of Lie algebras and have have been found useful in D=4 Supergravity. These systems come with a sym- metry known as Freudenthal Duality (or F-duality) which preserves a certain degree four polynomials ∆(x). The Lagrangian I write down is invariant under both the exotic gauge theory defined by the FTS and Freudenthal Duality. In prepration for discussing these top- ics, I review FTS and touch on their relationship to Lie algebras. I then discuss F-duality and present a novel proof that only depends on the axioms of the FTS on not on a direct calculation of any particular realization. I then review N=2 Maxwell-Einstein Supergravity in 4D (and 5D) and go over how F-duality arose in the first place. The final main chapter goes over my main results, which is taken from a paper which will be published soon, with several coathurs. i I dedicate this to my friends and family. I really can’t say enough about the people who have supported me during my time as a graduate student. ii Contents List of Tables iv Acknowledgments v 1 Introduction 1 1.1 Introduction.................................... 1 2 Freudenthal Triple Sytems 3 2.1 Introduction to Freudenthal Triple Systems . ......... 3 2.2 Definition of a Freudenthal Triple System . ....... 3 2.2.1 TheDefiningPropertiesofaFTS . 3 2.2.2 TheLeftMultiplicationOperator . ... 4 2.2.3 UsefulIdentities.............................. 6 2.2.4 Relations Between Various FTS in the Literature . ...... 7 2.3 Lie Algebras and Freudenthal Triple Systems . ........ 9 2.3.1 ConstructingaLiealgebrafromaFTS . .. 9 2.3.2 ConstructingaFTSfromaLiealgebra . .. 9 2.3.3 An Easy Example: SU(3) ........................ 11 2.4 SimpleLieAlgebrasandFTS . 12 2.5 Explicit Construction of FTS using Jordan algebras . .......... 14 3 Freudenthal Duality 17 3.1 IntroductiontoF-duality. .... 17 3.2 ProofofFreudenthalDuality. .... 18 3.3 WhyF-dualityworksthewayitdoes . ... 19 3.4 ThecontinuousF-duality. ... 20 4 Maxwell-Einstein Supergravity Theories and Freudenthal Triple Systems 23 4.1 Introduction.................................... 23 4.2 5 Dimensional =2MESGTs ......................... 23 N 4.2.1 TheActionfor5DMESGT . .. .. 23 4.3 Jordan Algebras and Entropy of 5D Black Holes . ...... 25 Contents iii 4.3.1 Euclidean Jordan Algebras of Degree 3 . ... 25 4.3.2 The Rotation, Lorentz, and Conformal Groups of Jordan Algebras . 27 4.3.3 The Attractor Equations and Black Hole Entropy in 5D . ...... 28 4.4 4D = 2 MESGT and Freudenthal Triple Systems . 30 N 4.4.1 4DMESGTfrom5DMESGT .. .. 30 4.4.2 4DBlackHoleEntropy.......................... 32 4.5 Freudenthal Duality and Black Hole Entropy . ....... 33 5 Freudenthal Gauge Theory 35 5.1 Introduction.................................... 35 5.2 Freudenthal Triple Systems (FTS’s)....................... 38 5.2.1 Rank-3JordanAlgebrasandLieAlgebras . ... 38 5.2.2 GeneralCase ............................... 39 5.2.3 AxiomaticDefinition ........................... 40 5.2.4 FTS Structure Constants and their Invariance . 42 5.2.5 FreudenthalDuality ........................... 43 5.3 FreudenthalGaugeTheory(FGT). ... 45 5.3.1 From Global Symmetry............................ 45 5.3.2 ...to Gauge Symmetry .......................... 46 5.3.3 TheLagrangian.............................. 48 5.3.4 Gauge Algebras of Type e7 ........................ 51 5.3.5 FGTandSupergravity .......................... 52 5.4 Generalization? ................................. 54 5.4.1 CouplingtoaVectorSpace . 54 5.4.2 A No-Go Theorem ............................ 56 5.5 FGTand ( = 3, D =3)SCCSMGaugeTheories . 58 5.6 ConcludingRemarksN ............................... 62 5.7 FreudenthalDuality. .. .. .. 65 5.8 Space-Time Symmetry of Scalar Kinetic Term . ...... 66 5.9 Axioms of V .................................... 66 6 Conclusions 68 6.1 ConcludingRemarks ............................... 68 Bibliography 70 iv List of Tables 2.1 GradingofLieAlgebra-FTSperspective . ..... 9 2.2 ProductRuleforLieBracket . .. 10 2.3 Grading of Lie Algebra - root space perspective . ........ 10 2.4 Grading of SU(3)................................. 11 2.5 SimpleJordanAlgebras ............................ 15 3.1 Multiplication table for the subalgebra generated by a single element . 19 3.2 Multiplication table for the subalgebra generated by φ, φ ........... 19 4.1 Scalar Manifolds Associated with Magical Supergravitiese ........... 27 4.2 Rotation, Lorentz, and Conformal groups of various Jordanalgebras. 27 4.3 ScalarManifoldsfor4DMESGT . 32 5.1 Conformal conf(J) and quasi-conformal qconf(J) Lie algebras associated to rank-3 Euclidean Jordan algebras. The relevant symplectic irrep. R of conf(J) is also reported.b In particular, 14′ denotes theb rank-3 antisymmetric irrep. of sp(6, R), whereas 32 and 32′ are the two chiral spinor irreps. of so∗ (12)b. As As Note that conf(J3 ) and qconf(J3 ) are the maximally non-compact (split) real forms of the corresponding compact Lie algebra. M1,2 (O) is the JTS generated by 2 1 vectors over O [7; 8]. Note the Jordan algebraic isomor- phisms Γ R× R, and Γ R. The number of spinor supercharges 1,1 ∼ ⊕ 1,0 ∼ N of the corresponding supergravity theory in D =4(cfr. Subsec. 5.3.5) is also listed. ....................................... 52 v Acknowledgments I would like to thank my advisor, Bruno Zumino, for his continual support and encour- agement during my time as his advisee. I would like to thank Darren Shih, for getting me out of my doldrums and making me productive again. His work ethic and encouragement greatly helped me in finishing my program. I also thank Alessio Marrani for all of the time and effort he has put in to help me find a project and bring it to completion. His knowledge and support made our work possible. 1 Chapter 1 Introduction 1.1 Introduction Freudenthal Triple Systems (FTS) have long been in the mathematical literature, and more recently have started appearing in the physics literature. Hans Freudenthal intro- duced them when studying them in connection to the exceptional Lie algebras [28]. These algebras have some very nice propterties and enjoy a close relationship with Lie algebras, particularly the exceptional Lie algebras (G2, F4, E6, E7, E8) as well as the symplectic Lie algebras (Sp(2n)). Many of their properties had been worked out decades ago, but more re- cently, they’ve been appearing in physical applications, especially in connection to Maxwell- Einstein Supergravity Theories, U-duality, and black hole entropy. The properties that define a Freudenthal triple system not only make them useful in studying gauge theories and U-dualities in supergravity, they also are useful in defining new types of gauge theories [10]. To begin, a Freudenthal Triple System consists of a triple product, xyz, and a antisym- metric bilinear form, x, y , satisfies the following axioms h i xyz = yxz (1.1) xyz = xzy +2λ y, z x λ z, x y λ x, y z (1.2) h i − h i − h i uv(xyz)=(uvx)yz + x(uvy)z + xy(uvz) (1.3) xyz,w = z,xyw (1.4) h i −h i where λ is a real number; different FTS in the literature have different values of λ, but they all can be put in a form which satisfies these axioms. From this system it is possible to construct a degree 4 polynomial, of the form 1 ∆(x)= xxx, x . (1.5) 2h i This object is used to calculate the entropy of a black hole, which is the first case in which Freudenthal duality arose. As explained in [9] (and references contained therein), black hole Section 1.1. Introduction 2 entropy takes the form (in some 4D models) I4 = π ∆(x) (1.6) where x is a FTS element representing the electricp and magnetic charges of the black hole. The reason it takes this form is that they come from 5D models whose structure is determined by Jordan algebras. By dimensional reduction the Jordan algebras become Freudenthal Triple Systems using a well-known construction. It turns out that the object ∆(x) (and hence the black hole entropy) was invariant under a duality operation dubbed Freudenthal Duality [14]. This duality relation showed that two black holes with very different charge configurations had the same entropy. Freudenthal duality (F-duality) is a nonlinear operation on the FTS which leaves the object ∆(x) invariant. It acts as follows: sgn(∆(x)) (x)= xxx. (1.7) F 6 λ∆(x) | | Using these ingredients, it is possible top construct a gauge theory which is based on the Freudenthal Triple System, which has the axioms listed above, and which is invariant under the F-Duality. The gauge symmetry, which is constructed similarly to the Lie 3-algebra gauge theories of the BLG model [3], acts on scalar fields as follows, ( φ(x))d = f dΛab(x)φc(x) (1.8) LΛ abc d where fabc are the structure constants for the FTS (which satisfies the axioms listed above), Λab(x) are the local gauge parameters, and φ(x) is the scalar field. From this, we construct a covariant derivative, which transforms in the usual way under a gauge transformation, (D φ)d = ∂ φd f dAabφc (1.9) µ µ − abc µ for gauge fields Aab.
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