Symplectic 3-Algebras and D=3, N=4,5,6,8 superconformal Chern-Simons-Matter theories

M-theory is the underlying theory of five di fferent string theories and 11D supergravity theory. While strings (1+1D) are fundamental objects in string theory, M2-branes (1+2D) are fundamental objects in M-theory. According to the Gauge/Gravity duality, a gravity theory is equivalent to a gauge theory. Extended (N ? 4) superconformal Chern-Simonsmatter (CSM) theories in 3D are natural candidates of the dual gauge theories of multi M2-branes. In the last two years, the N = 4; 5; 6 CSM theories were constructed by using ordinary Lie 2-algebras, and the N = 8 theory was constructed by using 3-algebra. However, it remains unclear whether these theories can be constructed in a uni ed 3-algebra approach or not. It is also natural to ask whether there are new examples of the extended superconformal CSM theories. In this thesis, we propose to solve these two problems. We de fine a 3-algebra with structure constants being symmetric in the fi rst two indices. We also introduce an invariant antisymmetric tensor into this 3-algebra and call it a symplectic 3-algebra. The D = 3;N = 4; 5; 6; 8 CSM theories are constructed in terms of this unifi ed 3-algebraic structure, and some new examples of the N = 4 quiver gauge theories are derived as well. In particular, in order to realize the 3-algebra used to construct the N = 4 quiver gauge theories, we `fuse' two simple super Lie algebras into a single new super Lie algebra, by requiring that the even parts of these two simple super Lie algebras share one simple factor. We demonstrate how to construct this class of new super Lie algebras by presenting an explicit example. Finally, a quantization scheme for the 3-brackets is proposed.

SYMPLECTIC 3-ALGEBRAS AND D = 3, N = 4; 5; 6; 8 SUPERCONFORMAL CHERN-SIMONS-MATTER THEORIES by Famin Chen A dissertation submitted to the faculty of The University of Utah in partial ful llment of the requirements for the degree of Doctor of Philosophy in Physics Department of Physics and Astronomy The University of Utah December 2010 Copyright c Famin Chen 2010 All Rights Reserved THE UNIVERSITY OF UTAH The Graduate School STATEMENT OF DISSERTATION APPROVAL The dissertation of____________Famin Chen__________________________ has been approved by the following supervisory committee members: Yong-Shi Wu_ , Chair Date Approved 8/20/10 Adam Bolton , Member Date Approved 8/20/10 Carleton DeTar , Member Date Approved 9/2/10 Paolo Gondolo , Member Date Approved 8/26/2010 Aaron Bertram , Member Date Approved 8/20/10 and by David Kieda, Chair of the Department of Physics and Astronomy and Charles A. Wight, Dean of The Graduate School ABSTRACT M-theory is the underlying theory of ve di erent string theories and 11D supergravity theory. While strings (1+1D) are fundamental objects in string theory, M2-branes (1+2D) are fundamental objects in M-theory. According to the Gauge/Gravity duality, a gravity theory is equivalent to a gauge theory. Extended (N 4) superconformal Chern-Simons- matter (CSM) theories in 3D are natural candidates of the dual gauge theories of multi M2-branes. In the last two years, the N = 4; 5; 6 CSM theories were constructed by using ordinary Lie 2-algebras, and the N = 8 theory was constructed by using 3-algebra. However, it remains unclear whether these theories can be constructed in a uni ed 3-algebra approach or not. It is also natural to ask whether there are new examples of the extended superconformal CSM theories. In this thesis, we propose to solve these two problems. We de ne a 3-algebra with structure constants being symmetric in the rst two indices. We also introduce an invariant antisymmetric tensor into this 3-algebra and call it a symplectic 3-algebra. The D = 3;N = 4; 5; 6; 8 CSM theories are constructed in terms of this uni ed 3-algebraic structure, and some new examples of the N = 4 quiver gauge theories are derived as well. In particular, in order to realize the 3-algebra used to construct the N = 4 quiver gauge theories, we `fuse' two simple super Lie algebras into a single new super Lie algebra, by requiring that the even parts of these two simple super Lie algebras share one simple factor. We demonstrate how to construct this class of new super Lie algebras by presenting an explicit example. Finally, a quantization scheme for the 3-brackets is proposed. To Jing Bian. CONTENTS ABSTRACT : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : iii ACKNOWLEDGMENTS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : vii CHAPTERS 1. INTRODUCTION : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 2. SYMPLECTIC THREE-ALGEBRAS : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 3. N = 5 THEORIES AND 3-ALGEBRAS : : : : : : : : : : : : : : : : : : : : : : : : : : 10 3.1 N = 5 Theories in Terms of 3-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Closure of the N = 5 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4. N = 4 THEORIES AND SYMPLECTIC 3-ALGEBRAS : : : : : : : : : : : 22 4.1 N = 4 Theories by Starting from N = 5 Theories . . . . . . . . . . . . . . . . . . . 22 4.2 Closure of the N=4 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5. 3-ALGEBRAS, LIE SUPERALGEBRAS AND EMBEDDING TENSORS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 32 5.1 3-algebras and Lie Superalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.2 Three-algebras and Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.3 Three-algebras and Embedding Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6. N=4, 5 THEORIES IN TERMS OF THE BOSONIC PARTS OF SUPERALGEBRAS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 39 6.1 N = 5 Theories in Terms of the Bosonic Parts of Superalgebras . . . . . . . . 39 6.2 N = 4 Theories in Terms of the Bosonic Parts of Superalgebras . . . . . . . . 40 6.2.1 Sp(2N1) SO(N2) Sp(2N3) Example . . . . . . . . . . . . . . . . . . . . . . . 40 6.2.2 Examples of N = 4 Quiver Gauge Theories . . . . . . . . . . . . . . . . . . . . 43 6.2.3 N = 4 GW Theory in Terms of Lie Algebras . . . . . . . . . . . . . . . . . . . 45 7. N = 6; 8 CSM THEORIES AND 3-ALGEBRAS : : : : : : : : : : : : : : : : : : : 47 7.1 General N=6 CSM Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 7.2 N=8 CSM Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 7.3 Closure of the N = 6 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7.4 Examples of the N = 6 Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 7.4.1 N = 6, Sp(2N) U(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 7.4.2 N = 6, U(M) U(N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 8. N=6, 8 THEORIES IN TERMS OF THE BOSONIC PARTS OF SUPERALGEBRAS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 58 8.1 N = 6 Theories in Terms of the Bosonic Parts of Superalgebras . . . . . . . . 58 8.2 N = 8 Theory in Terms of the Bosonic Part of PSU(2j2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 9. CONCLUSIONS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 64 APPENDICES A. CONVENTIONS AND USEFUL IDENTITIES : : : : : : : : : : : : : : : : : : : 67 B. VERIFICATION OF SP(4) GLOBAL SYMMETRY OF THE N = 5 BOSONIC POTENTIAL : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 73 C. SOME EXPLICIT EXAMPLES OF N = 5; 6 THEORIES : : : : : : : : : : 75 REFERENCES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 81 vi ACKNOWLEDGMENTS I am grateful to my thesis advisor, Prof. Yong-Shi Wu, for teaching and encouraging me. I am also grateful to Prof. Eric Sharpe and Prof. Katrin Becker for teaching me. I thank my family for their support and encouragement. CHAPTER 1 INTRODUCTION String theory is a plausible candidate for unifying quantum gravity and elementary particle forces. There are ve di erent string theories. All known string theories and 11D supergravity theory arise as di erent limits of a single theory: M-theory. M2-branes (1+2D) are important in that they are fundamental objects in M-theory. According to the Gauge/Gravity duality in string theory, a non-Abelian gauge theory is equivalent to a quantum gravity theory. In the last two years, extended (N 4)1 supersymmetric Chern-Simons-matter (CSM) theories in 3D have attracted a lot of interest in the string/M theory community, because they are natural candidates of the dual gauge theories of multi M2-branes in M-theory. For example, in the next page we will see that M-theory on AdS4 S7=Zk (k > 2) is equivalent to an N = 6 superconformal CSM theory in 3D. These two theories have the same amount of supersymmetries. Less extended supersymmetric (N < 4) CSM theories with arbitrary gauge groups were constructed and investigated long time ago [1]-[6]. (To our knowledge, their dual gravity theories are still under construction.) And generic Chern-Simons gauge theories with or without (massless) matter were demonstrated to be conformally invariant even at the quantum level [7, 8, 9, 10, 11]. However, it was much more di cult until recently to construct N 4 CSM theories, since only some special gauge groups are allowed in these theories. By virtue of the Nambu 3-algebra structure [12, 13], the maximally supersymmetric N = 8 CSM theory with SO(4) gauge group was rst constructed independently by Bagger and Lambert [14] and by Gustavsson [15] (BLG). The BLG theory was conjectured to be the dual gauge theory of two M2-branes [16, 17, 18, 19]. The Nambu 3-algebra, equipped with a symmetric and positive-de nite metric, has the limitation that it can only 1Here `N' stands for N copies of supersymmetries. In 3D, if N = 1, there are two independent fermionic generators. 2 generate an SO(4) gauge theory [20, 21], much too restrictive for a low-energy e ective description of M2-branes. Very soon Aharony, Bergman, Ja eris and Maldacena (ABJM) observed [25] that an N = 2 superconformal CSM theory, with gauge group U(N) U(N), actually has an SU(4) R-symmetry, hence an enhanced supersymmetry N = 6. The same theory was also obtained by taking the infrared limit of a brane construction. In their formulation, the Nambu 3-algebra structure did not play any role, though the ABJM theory with SU(2) SU(2) gauge group is equivalent to the BLG theory. Based on the brane construction, ABJM conjectured that at level k their theory describes the low energy limit of N M2- branes probing a C4=Zk singularity. In the special cases of k = 1; 2, the theory has the maximal supersymmetries (N = 8) [25, 26, 27, 28]. In a large-N limit the ABJM theory is then dual to M-theory on AdS4 S7=Zk [25]. The superspace formulation and a manifest SU(4) R-symmetry formulation of the ABJM theory can be found in Ref. [29] and [30], respectively. In Ref. [31, 32], some extended superconformal gauge theories were constructed by taking a conformal limit of D = 3 gauged supergravity theories. In this approach, the embedding tensors play a crucial role. Gaiotto and Witten (GW) [33] have been able to construct a large class of N = 4 CSM theories by a method that enhances N = 1 supersymmetry to N = 4. They also demonstrated that the gauge groups can be classi ed by super Lie algebras. In Ref. [34], the GW theory was extended to include additional twisted hypermultiplets; in particular, the extended GW theory with SO(4) gauge group was demonstrated to be equivalent to the BLG theory. In Ref. [35], two new theories, N = 5, Sp(2M) O(N) and N = 6, Sp(2M) O(2) CSM theories, were constructed by further enhancing the R-symmetry to Sp(4) and SU(4), respectively, and the N = 6, U(M) U(N) CSM theory was rederived. The gravity duals of N = 5, Sp(2M) O(N) and N = 6, U(M) U(N) theories were studied in Ref. [36]. By using group representation theory and applying GW's super-Lie-algebra method for classifying gauge groups, the N = 1 to N = 8 CSM theories were constructed systematically in a recent paper [37]. The progress mentioned in the last two paragraphs was made using mainly ordinary Lie algebras. On the other hand, Bagger and Lambert (BL) have been able to construct the N = 6;U(M) U(N) theory in terms of a modi ed 3-algebra [38]. Unlike the Nambu 3-algebra with totally antisymmetric structure constants, the structure constants of the 3 modi ed 3-algebra are antisymmetric only in the rst two indices. By introducing an invariant antisymmetric tensor into a 3-algebra, hence called a `symplectic 3-algebra', another class of N = 6 CSM theories, with gauge group Sp(2M) O(2), has been constructed in Ref. [39]. It is also demonstrated that the N = 6;U(M) U(N) theory can be recast into the symplectic 3-algebraic formalism [39]. In Ref. [40], both the general N = 5 and N = 6 CSM theories have been formulated in a uni ed symplectic 3-algebraic framework. These theories based on 3-algebras are constructed by requiring that the supersymmetries be closed on-shell. The main goal of the thesis is to combine the superspace formalism with the symplectic 3-algebra, then construct all D = 3 extended (N = 4; 5; 6; 8) superconformal CSM theo- ries in a uni ed symplectic 3-algebraic framework. The ordinary Lie algebra counterparts of these superconformal CSM theories are derived by using a super Lie algebra to realize the symplectic 3-algebra. We also derive all known examples of the D = 3, N = 4; 5; 6; 8 superconformal CSM theories, and construct some new example of the N = 4 quiver gauge theories.2 We rst combine the superspace formalism with the 3-algebra, then rederive the general N = 5 theories by using the Giatto-Witten enhancement mechanism. Previously the N = 5 theories were derived from the N = 4 theories by carefully choosing the gauge groups [35, 37]. So the construction of N = 5 theories by enhancing N = 1 supersymmetry is interesting in its own right, especially in a 3-algebraic framework. It provides insight into the relationship between the 3-algebra and conventional Lie-algebra approach. We then construct general N = 4 theories in the (quaternion) 3-algebra framework, in which there are two similar sets of complex 3-algebra generators. These N = 4 theories are 3-algebra version of Chern-Simons quiver gauge theories. We start from N = 5 super-multiplets, decompose them and the symplectic 3-algebra generators properly, and propose a new superpotential which is N=4 superconformally invariant. We demonstrate that the N = 5 supersymmetry can be enhanced to N = 6 by decom- posing the symplectic 3-algebra and the elds properly, and the fundamental identity and the symmetry and reality properties of the structure constants of the hermitian 3-algebra (used to construct N = 6 theories) can be derived from their N = 5 counterparts. 2The gauge group of a quiver gauge theory is a product of gi factors; and the matter elds are in the bifundamental representations. 4 In the special case that the structure constants are totally antisymmetric, the hermi- tian algebra becomes the Nambu 3-algebra. As a result, the N = 6 supersymmetry is promoted to N = 8, and the corresponding theory becomes the BLG theory. Therefore N = 4; 5; 6; 8 superconformal CSM theories are described by a uni ed (sympletic) 3-algebraic framework. We systematically investigate the relations between the 3-algebras, Lie superalgebras, ordinary Lie algebras and embedding tensors that are used to build D = 3 extended supergravity theories in Ref. [32]. The relations between the 3-algebras and Lie super- algebras are explored in Ref. [37, 42, 46], using representation theory. They did not discuss the relations between the embedding tensors in Ref. [32] and 3-algebras or Lie superalgebras. We ll this gap by a more physical approach. We demonstrate that the symplectic 3-algebra can be realized in terms of a super Lie algebra. The generators of the 3-algebra TI can be realized as the fermionic generators of the super Lie algebra QI , and the 3-bracket is realized in terms of a double graded bracket: [TI ; TJ ; TK] := [fQI ;QJg;QK]. In this realization, the fundamental identity (FI) of the symplectic 3-algebra can be converted into the MMQ Jacobi identity of the super Lie algebra (M is a bosonic generator). It will be shown that the structure constants of the symplectic 3-algebra furnish a quaternion representation of the bosonic part of the super Lie algebra, and play the role of Killing-Cartan metric of the bosonic part of the super Lie algebra. Then the FI of the 3-algebra are rewritten as ordinary commutator, whose structure constants are totally antisymmetric. Moreover, we prove that the structure constants of the symplectic 3-algebra are the components of the embedding tensor proposed in [32], if we realize the symplectic 3-algebra in terms of the super Lie algebra. The general N = 5; 6; 8 superconformal Chern-Simons-matter theories in terms of ordinary Lie algebras can be rederived from our super-Lie-algebra realization of the symplectic 3-algebras. Not only all known examples of N = 4; 5; 6; 8 ordinary CSM theories, but also N = 4 CSM quiver gauge theories (including some new examples), can be produced as well. Therefore, our superspace formulation for the super-Lie-algebra realization of symplectic 3-algebras provide a uni ed treatment of all known N = 4; 5; 6; 8 CSM theories, including new examples of N = 4 quiver gauge theories as well. In order to classify the gauge groups of the N = 4 quiver gauge theories, we `fuse' two simple super Lie algebras into a single super Lie algebra, by requiring that the 5 bosonic parts of these two simple super Lie algebras share one simple factor. As a result, the fermionic generators Qa and Qb0 of this pair of super Lie algebras have nontrivial anticommutators, i.e., fQa;Qb0g 6= 0. An explicit example is presented to demonstrate how to construct this kind of new super Lie algebras: we `fuse' the simple super Lie algebras OSp(N2j2N1) and OSp(N2j2N3) (N1 6= N3) into a single super Lie algebra, whose bosonic part is the Lie algebra of the group Sp(2N1) SO(2N2) Sp(2N3) (N1 6= N3). This group can be selected as a gauge group of the N = 4 quiver gauge theory. The (N = 6) hermitian 3-algebras and the Nambu 3-algebras are also realized in terms of super Lie algebras. We propose a quantization scheme for the symplectic, hermitian and Nambu 3-brackets, by promoting the generators and the double graded commutators of the corresponding super Lie algebras as quantum mechanical operators and double graded commutators, respectively. We also derive the same N = 4; 5; 6 theories by requiring that the supersymmetry transformations are closed on-shell, i.e., we also examine the closure of the N = 4; 5; 6 algebras. The closure of N = 4 algebra in the GW theory (without the twisted hyper- multiplets) has been checked in Ref. [33]. However, to our knowledge, the closure of the algebra in theories with the twisted hypermultiplets has not been explicitly checked in the literature. So our calculation will ll this gap. The closure of N = 6 algebra is rst checked in Ref. [38], using a hermitian 3-algebra approach. Our approach to the N = 6 theories is slightly di erent from that of [38], in that we use the symplectic 3-algebra to construct the N = 6 theories. As a result, ours is more suited to the case with gauge group Sp(2N) U(1). This thesis is organized as follows. In Chapter 2, we introduce the symplectic three- algebras and de ne the notations. Section 3.1 is devoted to the construction of the N = 5 theories by enhancing the supersymmetry from N = 1 to N = 5 in a 3-algebraic framework. The (on-shell) closure of the N = 5 algebra is explicitly veri ed in section 3.2. In section 4.1, we derive the N=4 theories by decomposing the N = 5 super-multiplets and the symplectic 3-algebra properly and proposing a new superpotential. The closure of the N = 4 super algebra is explicitly veri ed in section 4.2. In Chapter 5, we discuss the relations between 3-algebras, super Lie algebras, ordinary Lie algebras and the embedding tensors proposed in Ref. [32]. In section 6.1, we present how to reproduce the general Lie algebra version of N = 5 theory from the 3-algebra approach. In Chapter 6, we present all known examples of N = 4; 5 theories, and derive some new N = 4 quiver 6 gauge theories. In Chapter 7, we derive the N=6 theories by decomposing the N = 5 elds and the symplectic 3-algebra properly. The N = 8 BLG theory is derived as a special case of the N=6 theory. We also derive all known examples of the N = 6; 8 theories by specifying the structure constants of the 3-brackets. In section 7.3, the N = 6 theories are also constructed by requiring that the supersymmetry transformations are closed on-shell. In Chapter 8, we rederive a super Lie algebra which can be used to realize the hermitian 3-algebra and the Nambu 3-algebra. By using the super-Lie-algebra realization of 3-algebras, the ordinary Lie algebra constructions of the N = 6; 8 theories are rederived. In section 8.2, we propose a quantization scheme for the 3-brackets. The last chapter is devoted to conclusions. Our convention and useful identities are given in Appendix A. We verify the Sp(4) global symmetry of the N = 5 bosonic potential in Appendix B. In Appendix C, we present some explicit examples of the N = 5; 6 theories. CHAPTER 2 SYMPLECTIC THREE-ALGEBRAS A 3-algebra is a complex vector space equipped with a 3-bracket, mapping three vectors to one vector [40]: [TI ; TJ ; TK] = fIJK LTL; (2.1) where TI (I = 1; 2; ;M) is a set of generators. The set of complex numbers fIJK L are called the structure constants. We de ne the global transformation of a eld X valued in this 3-algebra (X = XKTK) as [14]: ~ X = IJ [TI ; TJ ;X]; (2.2) where the parameter IJ is independent of spacetime coordinate. (The symmetry trans- formation (2.2) will be gauged later). The above equation is the natural generalization of X = a[Ta;X] in an ordinary Lie 2-algebra. For an ordinary Lie 2-algebra, the Jacobi identity is equivalent to ([X; Y ]) = [ X; Y ] + [X; Y ]: (2.3) That is, X = a[Ta;X] must act as a derivative. Analogously, for (2.2) to be a symmetry, one has to require that it acts as a derivative [14]: ~ ([X; Y ;Z]) = [ ~ X; Y ;Z] + [X; ~ Y ;Z] + [X; Y ; ~ Z]; (2.4) where Y = Y NTN and Z = ZKTK. Canceling IJ ;XM; Y N and ZK from both sides, we obtain the following fundamental identity (FI) satis ed by the generators: [TI ; TJ ; [TM; TN; TK]] = [[TI ; TJ ; TM]; TN; TK]+[TM; [TI ; TJ ; TN]; TK]+[TM; TN; [TI ; TJ ; TK]]: (2.5) The FI is a generalization of the Jacobi identity of an ordinary Lie algebra. Combining the three-bracket (2.1) and the FI (2.5), we nd that the FI satis ed by the structure constants is fMNK OfIJO L = fIJM OfONK L + fIJN OfMOK L + fIJK OfMNO L: (2.6) 8 To de ne a symplectic 3-algebra, we introduce a symplectic bilinear form into the 3-algebra: !(X; Y ) = !IJXIY J : (2.7) We denote the inverse of the antisymmetric tensor !IJ as !IJ . 1 The existence of the inverse implies that a 3-algebra index I must run from 1 to M = 2L. We will use !IJ and !IJ to lower or raise 3-algebra indices; for instance, fIJKL !LMfIJK M. The symplectic bilinear form must be invariant under an arbitrary global transformation: ~ (!IJXIY J ) = LM(fLMI K!KJ + fLMJ K!IK)XIY J = 0: (2.8) It turns out that the structure constants must be symmetric in the last two indices: fLMIJ = fLMJI : (2.9) From point of view of ordinary Lie group, the in nitesimal matrices ~ K I LMfLM K I (2.10) must form the Lie algebra Sp(2L;C). We call the 3-algebra de ned by the above equations a symplectic 3-algebra. Since the 3-algebra is also a complex vector space, one can de ne a hermitian bilinear form h(X; Y ) = X IY I (2.11) (with X I the complex conjugate of XI ), which is naturally positive-de nite and will be used to construct the Lagrangians. The hermitian bilinear form is also required to be invariant under the global transformation: ~ (X IY I ) = ( LMf LMI K + LMfLMK I )X IY K = 0: (2.12) To solve the above equation, we assume that the parameter LM is hermitian: LM = ML. Since it also carries two symplectic 3-algebra indices, it obeys the natural real- 1In order to close the N 4 super Pioncare algebras, one must introduce the antisymmetric tensor into the theories (see sections 3.2 and 4.2). 9 ity condition LM = !LI!MJ IJ . These two equations imply that the parameter is symmetric, i.e., ML = LM. In summary, we have LM = ML = LM: (2.13) Now since the parameter IJ is symmetric, re-examining the global transformation (2.2) leads us to require that the structure constants are symmetric in the rst two indices: fIJKL = fJIKL: (2.14) With Eq. (2.13) and (2.14), we nd that Eq. (2.12) can be satis ed if we impose the following reality condition on the structure constants: f LMIK = fMLKI or f L M I K = fM L K I : (2.15) Now both the symplectic bilinear form (2.7) and the hermitian bilinear (2.11) form are invariant under the global transformation (2.2). So from point of view of ordinary Lie group, the symmetry group generated by the 3-algebra transformations (2.2) is nothing but Sp(2L), which is the intersection of U(2L) and Sp(2L;C). By using the FI (2.6), one can prove that the structure constants fIJK L are also preserved under the global symmetry transformations [38]: ~ fMNK L = 􀀀~ O MfONK L 􀀀 ~ O NfMOK L 􀀀 ~ O KfMNO L + ~ L OfMNK O = IJ (􀀀fIJM OfONKL 􀀀 fIJN OfMOK L 􀀀 fIJK OfMNO L + fIJO LfMNK O) = 0; (2.16) where we have used the FI (2.6) in the second line. In other words, Eq. (2.16) is equivalent to the FI (2.6). Thus we can use !IJ and fIJK L to construct invariant Lagrangians, when the symmetry is gauged. Later we will see, to enhance the super-symmetry from N = 1 to N = 5, we will require the 3-bracket to satisfy an additional constraint condition: !([TI ; T(J ; TK]; TL)) = 0; (2.17) or simply fI(JKL) = 0. Combining Eq. (2.17) with (2.9) and (2.14), we have that f(IJK)L = 0 and fIJKL = fKLIJ . In summary, the structure constants fIJKL enjoy the symmetry properties fIJKL = fJIKL = fJILK = fKLIJ : (2.18) CHAPTER 3 N = 5 THEORIES AND 3-ALGEBRAS In this chapter, we will generalize Giaotto and Witten's idea and method [33] to enhance the super-symmetry from N = 1 to N = 5 [44]. We will work in a three-algebraic framework. The closure of the N = 5 algebra will be checked explicitly [40]. 3.1 N = 5 Theories in Terms of 3-Algebras Let us rst explain the mechanism for supersymmery enhancement. We assume that the N = 1 super elds for the matter elds are 3-algebra valued (our notation and convention are summarized in Appendix A): I A = ZIA + i A B IB 􀀀 i 2 2FIA ; (3.1) where I is a 3-algebra index, A;B are Sp(4) = SO(5) indices (A;B = 1; :::; 4); and A B is a Hermitian SO(5) Sp(4) gamma matrix, satisfying A B B C = A C. 1 The super eld satis es the reality condition: AI = yI A = !AB!IJ J B: (3.2) The purpose for introducing the gamma matrix into the second term of (3.1) is the following: after we promote the supersymmetry from N = 1 to N = 5, we want the supercharges and the matter elds to transform as the 5 and 4 of Sp(4), respectively, with the gamma matrix being the couplings. Despite the fact that I A carries an Sp(4) index, it is still an N = 1 super elds in that it just depends on one copy of fermionic coordinates . Generally speaking, if we use (3:1) to construct an N = 1 CSM theory, the Yukawa couplings will contain the gamma 1Generally A B cm m A B (m = 1; :::; 5), where m A B are the SO(5) gamma matrices (see Appendix A.4), and cm real coe cients. We normalize the parameters cm so that mncmcn = 1. The nonuniqueness of this gamma matrix is exactly what is allowed by the R-symmetry SO(5). 11 matrix A B, which is not Sp(4) invariant. 2 As a result, the CSM theory is generally not Sp(4) invariant. However, we are be able to remove the gamma matrix A B from the theory by adjusting the superspace couplings. The resulting theory then has an Sp(4) global symmetry, which does not commute with the N = 1 supersymmetry. Namely the supercharge transforms nontrivially under the Sp(4) global symmetry group. More precisely, the supercharges transform in the vector representation of SO(5) or 5 of Sp(4). As a result, the supersymmetry gets enhanced from N = 1 to N = 5. We will explain this point in detail when we examine the supersymmetry transformations. To construct the N = 1 CSM theory, we rst gauge the global symmetry transforma- tion (2.2). We de ne the gauge transformation of the super eld I as ~ I A = KLfKL I J J A = ~ I J J A; (3.3) where the parameter KL is a super eld, depending on the coordinates of the superspace. We then de ne the covariant derivatives as (D )I J = D I J + ~􀀀 I J and (D )I J = @ I J + ~􀀀 I J ; (3.4) where D is the supercovariant derivative, de ned by Eq. (A.9). In accordance with our basic de nition (2.2), it is natural to assume that the superconnections take the following forms ~􀀀 I J 􀀀KL fKL I J and ~􀀀 I J 􀀀KL fKL I J ; (3.5) transforming as 3 ~ ~􀀀 I J = 􀀀D ~ I J and ~ ~􀀀 I J = 􀀀D ~ I J; (3.6) respectively. In the Wess-Zumino gauge, the superconnection ~􀀀 takes the form ~􀀀 I J = i A~ I J + 2 ~ I J = (i AKL + 2 KL )fKL I J ; (3.7) where ~ I J is superpartner of the gauge eld A~ I J . In accordance with (2:13), we assume that AKL and KL are hermitian and symmetric in KL. The two superconnections (3.5) 2With the standard de nition A B 1 2!mn mnB A , where mn = 1 4 [ m; n], we note that A B A C C B 􀀀 C B A C = !mncn mB A : Thus, A B is not Sp(4) invariant. 3In this section, we de ne a general tilde eld ~ as ~ I J KLfKL I J , where KL can be a super eld or an ordinary eld. 12 should not be independent, since there is only one gauge symmetry. Actually, imposing the conventional constraint [41] fD ;D g = 2iD (3.8) determines the vector superconnection: ~􀀀 I J = A~ I J 􀀀 i ~ I J 􀀀 i ~ I J + i 2 2 ~ F I J ; (3.9) where the eld strength is de ned as ~ F I J = 1 2 (@ A~ I J + @ A~ I J ) + 1 2 [A~ ;A~ ]I J ; ~ F I J = 1 2 ( ) ~ F I J : (3.10) The super eld 􀀀KL = 􀀀1 2 􀀀KL in Eq. (3.5) can be read o from Eq. (3.9) by rewriting the eld strength as a product of a eld and the structure constants: ~ F I J = 1 2 [@ AKL + @ AKL + (A~ )L MAMK + (A~ )K MAML ]fKL I J FKL fKL I J : (3.11) In the rst line we have used the FI (2.6). To be self-consistent, the covariant derivative D must satisfy the Jacobi identity: [D ; fD ;D g] + [D ; fD ;D g] + [D ; fD ;D g] = 0: (3.12) The Jacobi identity can be solved by introducing a super eld strength ~ W [41]: [D ;D ] = i ~W + i ~ W : (3.13) By direct calculation, we obtain ~W I J = ~ I J + ~ F I J 􀀀 i 2 2(D ~ )I J = [ KL + FKL 􀀀 i 2 2(D )KL]fKL I J WKL fKL I J ; (3.14) with (D )KLfKL I J [@ KL + (A~ )L M MK + (A~ )K M MJ ]fKL I J : (3.15) In deriving the above equation, we have used the FI (2.6) again. Here we would like to make one comment on the relation between the FI (2.6) and the anticommutator (3.8) 13 and the Jacobi identity (3.12). Without consulting the FI, one would not be able to derive Eq. (3.11) and write ~􀀀 I J as 􀀀KL fKL I J . This would be inconsistent with our assumption (3.5) or the basic de nition (2.2). Similarly, the super eld strength would not take the form ~W I J = WKL fKL I J without the FI (see Eq. (3.14)). Recall that the vector superconnection and the super eld strength are de ned through (3.8) and (3.12), respectively. So, had we not introduced the FI in section 2, we would have to introduce the FI in this subsection for making the 3-bracket (2.2) consistent with (3.8) and (3.12). After gauging the symmetry (2.2) in the superspace, we are ready to construct an N = 1 CSM theory. A general N = 1 CSM theory consists of three parts: L = Lkin+LCS+LW, where Lkin is the Lagrangian of the kinetic terms of the matter elds, LCS the Chern- Simons term and LW the superpotential. The rst part Lkin is standard: Lkin = 1 8 Z d2 D AI D I A (3.16) = 1 2 (􀀀D ZA I D ZIA + i A I D IA + 2ifIJKL B A BK IJZL A + FA I FIA ): The covariant derivatives are given by D ZA I = @ ZA I 􀀀 A~ J IZA J ; (3.17) D ZIA = @ ZIA + A~ I JZJA : (3.18) We propose the Chern-Simons term as LCS = 1 8 Z d2 [􀀀ifIJKL􀀀 IJWKL + 1 3 fIJK OfOLMN􀀀 IJ􀀀 KL􀀀MN ] (3.19) = 1 2 (fIJKLAIJ @ AKL + 2 3 fIJK OfOLMNAIJ AKL AMN ) + i 2 fIJKL IJ KL : The rst part of the second line is precisely the `twisted' Chern-Simons term in Ref. [40], while the gaugino is just an auxiliary eld, whose equation of motion is IJ = 􀀀 B A B(IZJ) A : (3.20) Substituting it into (3.16) and (3.19) gives a Yukawa coupling: 􀀀 i 2 ZIA ZJB K C LD fIKJL AC BD: (3.21) Note that this term is not Sp(4) invariant, because the gamma matrix is not Sp(4) invariant (see footnote 2). Let us now consider the superpotential W( ). It must satisfy two conditions. First, for conformal invariance, the superpotential must be homogeneous and quartic in ; 14 schematically, W( ) . Secondly, after combining (3.21) with the Yukawa terms arising from W( ), the nal expression must be Sp(4) invariant. Before proposing W( ), it is useful to look at the structure of (3.21): it contains AC BD. The essential obser- vation is that [AC BD] has to be proportional to the totally antisymmetric (invariant) tensor "ABCD, since this tensor is unique in Sp(4). The precise expression is 􀀀"ABCD = AC BD 􀀀 BC AD + BA CD (3.22) = !AB!CD 􀀀 !AC!BD + !AD!BC: Namely, our problem may be solved if the nal expression for (3.21) plus the Yukawa terms arising from W( ) are somehow related to (3.22). So we are inspired to propose the following superpotential W( ) = 1 4 (gIJKL!AB!CD I A J B KC L D + ~gIJKL AB CD I A J B KC L D); (3.23) where the 3-algebra tensor g satis es gIJKL = 􀀀gJIKL = 􀀀gIJLK = gKLIJ , and ~g has the same symmetry properties. We require that the tensors g and ~g are gauge invariant. This implies that g and ~g can be expressed in terms of !IJ and fIJKL, the only two gauge invariant quantities. After carrying out the Berezin integration i 2 R d2 W( ), we obtain LW = 􀀀 i 2 ZIA ZJB K C LD (gIJKL!AB!CD + 2gIKJL AC BD + ~gIJKL AB CD (3.24) +2~gIKJL!AC!BD) 􀀀 (gIJKL!AB!CD + ~gIJKL AB CD)ZJB ZK C ZLD FIA : The rst and last term of the rst line are already Sp(4) invariant. Combining the middle two terms of the rst line with (3.21) gives 􀀀 i 2 ZIA ZJB K C LD [(2gIKJL + fIKJL) AC BD + ~gIJKL AB CD]: (3.25) Since we wish to use Eq. (3.22), we rst have to antisymmetrize AB in the expression AC BD. Equivalently, we have to set the part proportional to Z(I (AZJ) B) to be zero: gIKJL + gJKIL + 1 2 fIKJL + 1 2 fJKIL = 0: (3.26) Now the remaining part of (3.25) is antisymmetric in AB: i 2 ZIA ZJB K C LD [(2gIKJL + fIKJL) C[A B]D 􀀀 ~gIJKL AB CD]: (3.27) It can be seen that if we set ~gIJKL = 􀀀 1 2 (gIKJL 􀀀 gJKIL + 1 2 fIKJL 􀀀 1 2 fJKIL) (3.28) 15 and apply the key identity (3.22), then Eq. (3.27) becomes i 2 ZIA ZJB K C LD ~gIJKL(!AB!CD 􀀀 !AC!BD + !AD!BC): (3.29) Now Eq. (3.29) is manifestly Sp(4) invariant. However we still need to solve (3.26) and (3.28) in terms of fIJKL and !IJ . An equation similar to (3.26) is rst derived by GW [33]: gIKJL + gJKIL + 3 4 kmn m IK n JL + 3 4 kmn m JK n IL = 0; (3.30) where the set of matrices m IK is in the fundamental representation of Sp(2L) or its subalgebra, and kmn is the Killing-Cartan metric. Although the (N = 4) GW theory is not an N = 5 theory, the similarity between (3.26) and (3.30) strongly suggests that fIJKL can be speci ed as kmn m IJ nK L (up to an unimportant constant). This is indeed the case: the FI (2.6) does admit an explicit solution in terms of the tensor product fIJKL = kmn m IJ nK L. It is straightforward to verify that fIJKL = kmn m IJ nK L satisfy the FI (2.6). This solution is rst found by Gustavsson by converting the FI into two independent commutators of ordinary Lie algebra [15]. Later we will discuss the relations between the 3-algebra and the ordinary Lie algebra in detail. Eq. (3.26) can be easily solved by adopting a method in Ref. [33]. Summing (3.26) over cyclic permutations of IKJ gives f(IKJ)L = 0; or fI(KJL) = 0: (3.31) This is precisely (2.17), as we stated earlier. The above equation is also derived by requiring that the N = 5 supersymmetry transformations are closed on-shell [40]. Eq. (3.26) is solved by setting gIKJL = 1 6 (fIJKL 􀀀 fILKJ ): (3.32) Substituting (3.32) into (3.28), we obtain ~gIJKL = 1 3 (fILJK 􀀀 fIKJL): (3.33) Substituting (3.33) into (3.29), then combining (3.29) with the rst and the last term of the rst line of (3.24), we reach the nal expression for all Yukawa terms: 􀀀 i 2 !AB!CDfIJKL(ZIA ZK B JC LD 􀀀 2ZIA ZK D JC LB ): (3.34) 16 Finally we integrate out the auxiliary eld FIA appearing in (3.16) and (3.24): FA I = 1 3 fIKLJ!BC!ADZK B ZL CZJD 􀀀 2 3 fIKLJ BC ADZK B ZL CZJD : (3.35) Now it is straightforward to calculate the bosonic potential: 􀀀 1 2 FA I FIA = 1 18 fIJKOfO LMN(􀀀!AC!BE!DF + 2!AC BE DF +2!DF AC BE 􀀀 4!BE AC DF )ZIA ZJB ZK C ZLD ZM E ZN F : (3.36) Note that V = 1 2 FA I FIA is positive de nite, though it is not manifestly Sp(4) invariant due to the presence of the gamma matrix. However, by taking advantage of the key identity (3.22) and the constraint condition f(IJK)L = 0, we are able to prove that (3.36) is indeed Sp(4) invariant (see Appendix B). The nal expression for the bosonic potential is V = 􀀀 1 60 (2fIJK OfOLMN 􀀀 9fKLI OfONMJ + 2fIJL OfOKMN)ZN A ZAIZJB ZBKZL CZCM: (3.37) In summary, the full Lagrangian in terms of the symplectic 3-algebra is given by L = 1 2 (􀀀D ZA I D ZIA + i A I D IA ) 􀀀 i 2 !AB!CDfIJKL(ZIA ZK B JC LD 􀀀 2ZIA ZK D JC LB ) + 1 2 (fIJKLAIJ @ AKL + 2 3 fIJK OfOLMNAIJ AKL AMN ) (3.38) + 1 60 (2fIJK OfOLMN 􀀀 9fKLI OfONMJ + 2fIJL OfOKMN)ZN A ZAIZJB ZBKZL CZCM: This Lagrangian is exactly the same as the N = 5 Lagrangian derived by requiring that the supersymmetry transformations are closed on-shell [40]. Using the reality condition (2.15), one can recast the potential term into the following form: V = 2 15 ( L ABC) L ABC; (3.39) where L ABC fIJK L(ZIA ZJB ZK C + 1 4 !BCZIA ZJD ZDK): (3.40) Now the potential term is manifestly positive de nite. 17 Let us consider the supersymmetry transformations. The N = 1 supersymmetry transformation of the scalar eld is QZIA = i A B I B: (3.41) On the other hand, the action (3.38) is invariant under the Sp(4) global symmetry transformation RZIA = A BZIB ; R IA = A B IB : (3.42) Therefore one can consider the commutator of R and Q: [ R; Q]ZIA = i ( A B B C 􀀀 A B B C) I C: (3.43) So the N = 1 supersymmetry does not commute with the Sp(4) global symmetry. Since the matrix A B contains four independent real parameters, equation (3.43) suggests that there are other 4 independent N = 1 supersymmetries. Therefore one may promote the N = 1 supersymmetry (3.41) to N = 5: ZIA = i A B IB ; (3.44) where the parameter A B = m m A B. One may apply the same argument to the super- symmetry transformations of the fermionic and gauge elds. In summary, we have the following supersymmetry transformations: ZIA = i A B IB ; I A = ( ) D ZIB B A + 1 3 fI JKL!BCZJBZK C ZLD D A 􀀀 2 3 fI JKL!BDZJC ZK D ZL A C B ; A~ K L = i AB ( ) JB ZIA fIJ K L; (3.45) where the parameter AB is antisymmetric in AB, satisfying !AB AB = 0; AB = !AC!BD CD: (3.46) The supersymetry transformations are precisely the ones proposed in Ref. [40]. To verify the mechanism for enhancing the N = 1 to N = 5, it is best to check the closure of (3.45); this will be done in the next section. Later we will see that they are indeed closed on-shell, and the corresponding equations of motion can be derived from the Lagrangian (3.38). So the R-symmetry of the theories is Sp(4). 18 3.2 Closure of the N = 5 Algebra Following BL's strategy [38], we will derive the equations of motion by requiring that the supersymmetry transformations are closed on-shell. Let us rst examine scalar supersymmetry transformation. By virtue of the identities in Appendix A.4, we nd [ 1; 2]ZIA = v D ZIA 􀀀 2 3 fJLK I KLZAJ + 2 3 fKLJ I KLZJA ; (3.47) where v 􀀀 i 2 BD 2 1BD; (3.48) KL 􀀀 i 2 ZK D ZL C( CE 1 2E D 􀀀 CE 2 1E D) = LK; (3.49) and the bilinear is symmetric in CD. While the rst term of Eq. (3.47) is the gauge covariant translation, we have to impose some conditions on the structure constants so that the remaining terms add up to be a gauge transformation. (We will read o the parameter of the gauge transformation by looking the closure of the algebra on the gauge elds.) We tentatively assume that the third term of Eq. (3.47) is proportional to the gauge transformation. So the second term of Eq. (3.47) should be also proportional to the gauge transformation. This leads us to impose an additional constraint condition on the structure constants: 1 2 (fJLK I + fJKL I ) = 2 fKLJ I ; (3.50) where is a constant, to be determined later. Now the second and third term of Eq. (3.47) can be combined as 1 3 (􀀀 + 2)fKLJ I KLZJA ; (3.51) which should be the gauge transformation. Let us now look at the gauge elds: [ 1; 2]A~ I J = v ~ F I J 􀀀 (D KL)fKL I J +v [ ~ F I J 􀀀 " (ZK A D ZAL 􀀀 i 2 BK LB )fKL I J ] +O(Z4); (3.52) where the last term O(Z4) is fourth order in the scalar elds Z. We recognize the second term of the rst line as a gauge transformation 􀀀(D KL)fKL I J = 􀀀D ( KLfKL I J ) (3.53) 19 by a parameter ~ I J = KLfKL I J , since the FI (2.6) or (2.16) implies that D fKL I J = 0 [38]. In accordance with the parameter, now (3.51) must satisfy the following equation4: 1 3 (􀀀 + 2)fKLJ I KLZJA = KLfKL I JZJA : (3.54) This equation can be solved by setting = 􀀀1. In other words, Eq. (3.54) is solved if Eq. (3.50) can be written as f(JKL) I = 0; (3.55) since one can prove that fKLJ I = fKL I J : (3.56) Now Eq. (3.47) becomes [ 1; 2]ZIA = v D ZIA + ~ I JZJA ; (3.57) as expected. Following Gustavsson's approach [15], one can demonstrate that the FI (2.6) admits an explicit solution in terms of a tensor product: fIJKL = kmn m IJ nK L, where kmn is the Killing-Cartan metric of Sp(2L), and m IJ = !IK mK J . The matrix mK J is in the fundamental representation of Sp(2L), and !IK is the Sp(2L)-invariant antisymmetric tensor. Now Eq. (3.55) implies that kmn m (IJ nK )L = 0, which is rst derived by GW [33]. In the GW theories, it is the key requirement for enhancing the N = 1 supersymmetry to the N = 4 supersymmetry. By using the FI (2.6) and the symmetry conditions (2.18), one can prove that the last term of Eq. (3.52) vanishes: O(Z4) = 0: (3.58) So the second line of Eq. (3.52) must be the equations of motion for the gauge elds: ~ F I J 􀀀 " (ZK A D ZAL 􀀀 i 2 BK LB )fKL I J = 0: (3.59) Now only the rst line of Eq. (3.52) remains: [ 1; 2]A~ I J = v ~ F I J 􀀀 D ~ I J ; (3.60) which is the desired result. 4According to our convention, if ~ ZIA = ~ I JZJA , we must set ~ A~ I J = 􀀀D ~ I J so that ~ (D ZIA ) = ~ I J (D ZJA ). 20 Finally we turn to the fermion supersymmetry transformation: [ 1; 2] IA = v D IA + ~ I J JA + i 2 ( BC 1 2BA 􀀀 BC 2 1BA)EIC 􀀀 1 2 v EIA ; (3.61) where EIA = D IA 􀀀 fKLJ IZJB ZBK LA + 2fKLJ IZJB ZK A BL: (3.62) Hence the equations of motion for fermionic elds are EIA = 0. The scalar equations of motion can be derived by taking the super-variation of the fermionic equations of motion: EIA = 0: (3.63) After Fierz transformation, we obtain two independent parts, containing BC and BC, respectively. The part containing BC merely implies the equations of motion for the gauge elds, so we will not write it down here. The part containing BC reads [C A FB]I + GA BCI BC = 0; (3.64) where FBI 􀀀D2ZBI + ifKLJ IZCJ BK L C + 1 3 fMNL OfOKJ IZJC ZCKZLD ZDMZBN; (3.65) and GA BCI BC ifKLJ I ( 3 2 ZBL CK JA + ZK A CJ BL) + ifL[KJ] IZBJ CK LA + 2 3 (fMNL OfOKJ I + fKMJ OfONL I + 2fMJL OfONK I ) ZJD ZDKZCLZBMZN A BC: (3.66) Since the parameters BC are traceless, in the sense that !BC BC = B B = 0, Eq. (3.64) must be equivalent to the following traceless equation: [C A FB]I 􀀀 1 4 !BCFIA + GA BCI 􀀀 1 4 !BC!DEGA EDI = 0: (3.67) Contracting on AC gives the scalar equations of motion: FBI + 4 5 GA BAI 􀀀 1 5 GB A AI = 0: (3.68) 21 After some simpli cation we obtain 0 = 􀀀D2ZB I 􀀀 ifIJK L(ZB L CK JC 􀀀 2ZCK JC B L ) (3.69) 􀀀 1 5 (fIJK OfOLM N + fIJL OfOKM N + 3fIJM OfKLO N 􀀀 3fIJM OfOLK N) ZJA ZAKZL CZCMZB N: All the equations of motion can be derived as the Euler-Lagrangian equations from the action (3.38). CHAPTER 4 N = 4 THEORIES AND SYMPLECTIC 3-ALGEBRAS 4.1 N = 4 Theories by Starting from N = 5 Theories In this section, we will construct the N=4 theories by decomposing the N = 5 super- multiplets and the symplectic 3-algebra properly and proposing a new superpotential term that preserves only N = 4 [44]. Let us rst decompose the N = 5 super elds for matter elds into N = 4 super elds: ( I A)N=5 = a A a0 _A = ZaA Za0 _A + i 0 A _A y_A A 0 ! a0 A a _A 􀀀 i 2 2 FaA Fa0 _A : (4.1) The index A of the LHS runs from 1 to 4, while A and _A of the RHS run from 1 to 2. (For the dotted and undotted representation, see Appendix A.3.) The indices a and a0 run from 1 to 2M and 1 to 2N, respectively. The super elds a _A and a0 _A are called untwisted and twisted hypermultiplets, respectively, in the literature [35] (from the N = 4 point of view). The two antisymmetric matrices !IJ and !AB are decomposed as !IJ = !ab 0 0 !a0b0 and !AB = AB 0 0 _A _B (4.2) respectively. Now the reality condition ( AI )N=5 = !AB!IJ J B becomes Aa = AB!ab b B and _A a0 = _A _B !a0b0 b0 _B : (4.3) To be compatible with the decomposition of the N = 5 hypermultiplets (4.1), one may decompose the N = 5 superconnections as 􀀀IJfIJ K L = 􀀀abfab c d + 􀀀a0b0fa0b0 c d 0 0 􀀀a0b0fa0b0 c0 d0 + 􀀀abfab c0 d0 ; (4.4) where 􀀀abfab c d = (i Aab + 2 ab )fab c d; (4.5) and the other 3 super elds of the RHS of (4.4) have similar expressions. In proposing (4.4), we have decomposed the set of 3-algebra generators TI into two sets of generators 23 Ta and Ta0 , and decomposed the 3-bracket (2.1) into 4 sets, with the structure constants fabc d; fabc0 d0 ; fa0b0c d and fa0b0c0 d0 . We have also decomposed the parameter super eld 􀀀IJ into two super elds 􀀀ab and 􀀀a0b0 . If we introduce a `spin up' spinor 1 and a `spin down' spinor 2 , i.e., 1 1 = 1 0 = 1 and 2 = 0 1 = 2 ; (4.6) then in component formalism, we now have fIJKL = fabcd 1 1 1 1 +fabc0d0 1 1 2 2 +fa0b0cd 2 2 1 1 +fa0b0c0d0 2 2 2 2 ; (4.7) (Here we assume that (fabcd 􀀀 fabc0d0) does not vanish identically.) and 􀀀IJ = 􀀀ab 1 1 + 􀀀a0b0 2 2 : (4.8) Substituting (4.7) and (4.8) into 􀀀IJfIJ K L indeed gives (4.4). With the decomposition (4.7), the FI (2.6) are decomposed into 4 sets: fabe gfgfcd + fabf gfegcd 􀀀 fefd gfabcg 􀀀 fefc gfabdg = 0; fabe gfgfc0d0 + fabf gfegc0d0 􀀀 fefd0 g0fabc0g0 􀀀 fefc0 g0fabd0g0 = 0; (4.9) fa0b0e gfgfc0d0 + fa0b0f gfegc0d0 􀀀 fefd0 g0fa0b0c0g0 􀀀 fefc0 g0fa0b0d0g0 = 0; fa0b0e0 g0fg0f0c0d0 + fa0b0f0 g0fe0g0c0d0 􀀀 fe0f0d0 g0fa0b0c0g0 􀀀 fe0f0c0 g0fa0b0d0g0 = 0: In accordance with Eq. (2.18), these structure constants enjoy the symmetry properties fabcd = fbacd = fbadc = fcdab; fabc0d0 = fbac0d0 = fbad0c0 = fc0d0ab; (4.10) fa0b0c0d0 = fb0a0c0d0 = fb0a0d0c0 = fc0d0a0b0 : The reality condition (2.15) is decomposed into f a b c d = fb a d c; f a0 b0 c d = fb0 a0 d c; f a0 b0 c0 d0 = fb0 a0 d0 c0 : (4.11) Under the condition that (fabcd 􀀀 fabc0d0) does not vanish identically, decomposing the constraint condition f(IJK)L = 0 results in f(abc)d = 0, f(a0b0c0)d0 = 0 and fabc0d0 = 0. However, the condition fabc0d0 = 0 turns out to be too restrictive to allow any interaction 1Here the index is not an index of a spacetime spinor. We hope this will not cause any confusion. 24 between the primed elds and the unprimed elds. So we have to give up the constraint fabc0d0 = 0. Namely, we have to give up the constraint condition f(IJK)L = 0 as we decompose fIJKL by Eq. (4.7). Later we will see, to construct an interesting N = 4 quiver gauge theory, we need only to impose constraints on fabcd and fa0b0c0d0 : f(abc)d = 0 and f(a0b0c0)d0 = 0; (4.12) while fabc0d0 are unconstrained. With these decompositions, the Lagrangian for the kinetic terms of the matter elds (3.16) becomes Lkin = 1 2 (􀀀D ZA a D ZaA + i _A a D a _A 􀀀 2i y_B A _B a ~ a bZbA + FA a FaA ) + 1 2 (􀀀D Z _A a0D Za0 _A + i A a0 D a0 A 􀀀 2i B _A B a0 ~ a0 b0Zb0 _A + F _A a0Fa0 _A ); (4.13) where D ZA d = @ ZA d 􀀀 A~ c dZA c ; A~ c d = Aab fab c d + Aa0b0 fa0b0 c d; ~ a0 b0 = cdfcd a0 b0 + c0d0fc0d0 a0 b0 ; (4.14) and similar de nitions for A~ c0 d0 and ~ a b; and the Chern-Simons term (3.19) becomes LCS = 1 2 (fabcdAab @ Acd + 2 3 fabc gfgdefAab Acd Aef ) + 1 2 (fa0b0c0d0Aa0b0 @ Ac0d0 + 2 3 fa0b0c0 g0fg0d0e0f0Aa0b0 Ac0d0 Ae0f0 ) (4.15) + (fabc0d0Aab @ Ac0d0 + fabc gfgde0f0Aab Acd Ae0f0 + fabc0 g0fg0d0e0f0Aab Ac0d0 Ae0f0 ) + i 2 (fabcd ab cd + 2fabc0d0 ab c0d0 + fa0b0c0d0 a0b0 c0d0): The equations of motion for the auxiliary eld (3.20) is decomposed into two sets ab = 􀀀 y_A B _A (aZb) B ; a0b0 = 􀀀 A _B A(a0Zb0) _B : (4.16) Plugging (4:16) into (4.13) and (4.15) gives three Yukawa terms 􀀀 i 2 (facbd A _C B _D ZaA ZbB c _C d _D + fa0c0b0d0 y A_C y _B DZa0 _A Zb0 _B c0 C d0D +2fabc0d0 A _B y _C DZaA Zc0 _C b _B d0D ): (4.17) Alternatively, we can also obtain (4.17) by directly decomposing the N = 5 Yukawa term (3.21). It can be seen that the last term of (4.17) is a mixed term, in which the primed 25 elds couple the unprimed elds through fabc0d0 . So we cannot obtain a nontrivial N = 4 superpotential by decomposing the N = 5 superpotential (3.24), because the N = 5 superpotential (3.24) is desired only if f(IJK)L = 0, which implies that fabc0d0 = 0 as we decompose fIJKL by Eq. (4.7) under the condition that (fabcd 􀀀 fabc0d0) does not vanish identically. So we have to propose a new superpotential for the N = 4 theory, allowing fabc0d0 6= 0. However, unlike the last term of (4.17), the rst two terms of (4.17) are un-mixed terms. This inspires us to decompose the rst term of the N = 5 superpotential (3.23) with fa0c0bd and facb0d0 deleted from fIKJL (hence we denote the `modi ed' structure constants as f0I KJL): W1( ) = 1 12 (f0I KJL!AB!CD I A J B KC L D)N=5 = 1 12 (facbd AB CD a A b B c C d D + fa0c0b0d0 _A _B _C _D a0 _A b0 _B c0 _C d0 _D ): (4.18) where f0I JKL = fabcd 1 1 1 1 + fa0b0c0d0 2 2 2 2 : (4.19) Of course, the `modi ed' structure constants f0I KJL still satisfy the constraint condition f0 (IKJ)L=0, which is equivalent to Eq. (4.12): f(acb)d = 0 and f(a0c0b0)d0 = 0. We will prove that the rst two terms of (4.17) combining the Yukawa terms arising from the superpotential W1 (see (4.20)) are SU(2) SU(2) invariant. Carrying out the Berezin integral i 2 R d 2W1( ) gives: LW1 = 􀀀 i 6 (facbd AB _C _D ZaA ZbB c _C d _D + fa0c0b0d0 _A _B CDZa0 _A Zb0 _B c0 C d0D ) 􀀀 i 6 [(fabcd 􀀀 fadcb) A _C B _D ZaA ZbB c _C d _D +(fa0b0c0d0 􀀀 fa0d0c0b0) yA_C y _B DZa0 _A Zb0 _B c0 C d0D ] 􀀀 1 3 (fabcdZbB ZBcZAdFaA + fa0b0c0d0Zb0 _B Z _B c0Z A_d0Fa0 _A ): (4.20) Let us now combine the rst term of (4.17) and the rst term of the second line of (4.20): 􀀀 i 6 [3facbd + (fabcd 􀀀 fadcb)] A _C B _D ZaA ZbB c _C d _D = 􀀀 i 6 (facbd 􀀀 fbcad)( A _C B _D 􀀀 B _C A _D )ZaA ZbB c _C d _D = 􀀀 i 3 facbd AB _C _D ZaA ZbB c _C d _D : (4.21) In the second line we have used f(abc)d = 0. In the third line we have used the SU(2) SU(2) identity (A.26). It can be seen that the nal expression of (4.21) is indeed SU(2) 26 SU(2) invariant. Similarly, one can combine the second term of (4.17) and the second term of the second line of (4.20) to form an SU(2) SU(2) invariant expression: 􀀀 i 3 fa0c0b0d0 _A _B CDZa0 _A Zb0 _B c0 C d0D ; (4.22) where we have used the reality condition (A.23). Now only the last term of (4.17), i.e., the mixed term, is not SU(2) SU(2) invariant. Its structure suggests that if a Yukawa term of the form ifabc0d0 D _B y _C AZaA Zc0 _C b _B d0D (4.23) arises from a to-be-determined superpotential, then they will add up to be SU(2) SU(2) invariant by the reality condition (A.23) and the identity (A.26). It is therefore natural to try W2( ) = fabc0d0 B _D y _C A a A b B c0 _C d0 _D ; (4.24) where is a constant, to be determined later. The corresponding Lagrangian is LW2 = i fabc0d0( AC BDZaA ZbB c0 C d0D + _A _C _B _D a _A b _B Zc0 _C Zd0 _D + 2 AC _B _D ZaA Zd0 _D b _B c0 C) +2i fabc0d0 D _B y _C AZaA Zc0 _C b _B d0D 􀀀2 fabc0d0 B _D y _C AZbB Zc0 _C Zd0 _D FaA 􀀀 2 fabc0d0 B _D y _C AZaA ZbBZd0 _D Fc0 _C : (4.25) Note that the rst line is SU(2) SU(2) invariant by itself. Comparing the second line with (4.23) gives = 1 2 . Combining the last term of (4.17) and the second line of (4.25), we obtain ifabc0d0 AD _B _C ZaA Zc0 _C b _B d0D ; (4.26) which is the desired result. Now all Yukawa terms are invariant under the SU(2) SU(2) global symmetry transformation. Put all Yukawa terms (the rst line of (4.20), (4.21), (4.22), (4.26) and the rst line of (4.25)) together: LY = 􀀀 i 2 (facbdZaA ZAb c _B _B d + fa0c0b0d0Za0 _A Z _ Ab0 c0B Bd0) + i 2 fabc0d0(ZaA ZbB Ac0 Bd0 + Zc0 _A Zd0 _B A_a _B b + 4ZaA Z _B d0 b _B Ac0): To calculate the bosonic potential, we rst integrate out the auxiliary elds FaA and Fa0 _A from (4.13), (4.20) and (4.25): FA a = 1 3 fabcdZbB ZBcZAd + fabc0d0 B _D y _C AZbB Zc0 _C Zd0 _D WA 1a +WA 2a; F _A a0 = 1 3 fa0b0c0d0Zb0 _B Z _B c0Z A_d0 + fa0b0cd y _B D C _A Zb0 _B ZcC ZdD W _A 1a0 +W _A 2a0 : (4.27) 27 The bosonic potential is 􀀀V = 􀀀 1 2 ( FA a FaA + F _A a0Fa0 _A ); (4.28) which is not manifestly SU(2) SU(2) invariant due to the presence of the sigma matrices. However, by using the fundamental identities (4.9) and a method rst introduced in GW theory [33] (see also [34]), we are able to rewrite (4.28) so that it has a manifest SU(2) SU(2) global symmetry. For example, let us consider 􀀀WA 1aWa 2A = 􀀀 1 3 fabcdfa ec0d0 A _C C _D ZbB ZBcZdA ZeC Zc0 _C Zd0 _D = 􀀀 1 3ffcda(bfa e)c0d0 + fcda[bfa e]c0d0g A _C C _D ZbB ZBcZdA ZeC Zc0 _C Zd0 _D S + A: (4.29) The antisymmetric part can be written as A = 1 6 fcdabfa ec0d0 A _C C _D ZBbZeB ZcC ZdA Zc0 _C Zd0 _D : (4.30) Applying the constraint condition f(cdb)a = 0 to the above potential term, we obtain A = 􀀀 1 3 fcdaefa bc0d0 A _C C _D ZbB ZBcZdA ZeC Zc0 _C Zd0 _D : (4.31) Combining this with 􀀀WA 1aWa 2A (the rst line of (4.29)) gives 􀀀WA 1aWa 2A + A = 2S: (4.32) Solving for 􀀀WA 1aWa 2A, we obtain 􀀀WA 1aWa 2A = 􀀀 1 2 fcda(bfa e)c0d0 A _C C _D ZbB ZBcZdA ZeC Zc0 _C Zd0 _D : (4.33) Let us now consider another term of (4.28): 􀀀 1 2 W _A 2a0Wa0 2 _A = 1 2 fcdb0a0fa0 e0fg D _B A _F Zb0 _B Ze0 _F ZcC ZdD ZCfZg A (4.34) = 1 2 (fcda0(b0fa0 e0)fg + fcda0[b0fa0 e0]fg) D _B A _F Zb0 _B Ze0 _F ZcC ZdD ZCfZg A: Combining this equation with (4.33), the symmetric part cancels (4.33) by the second equation of the fundamental identities (4.9), while the antisymmetric part is SU(2) SU(2) invariant by the identity (A.26). The nal result is 􀀀WA 1aWa 2A 􀀀 1 2 W _A 2a0Wa0 2 _A = 􀀀 1 4 fabc0g0fg0 d0efZ A_c0Zd0 _A ZbD ZDfZaC ZCe: (4.35) 28 One can apply the same method to the other terms of (4.28). The nal expression for the N = 4 bosonic potential is 􀀀V = + 1 12 (fabcgfg defZAaZbB ZB(cZd) C ZCeZf A + fa0b0c0g0fg0 d0e0f0Z A_a0Zb0 _B Z _B (c0Zd0) _C Z _C e0Zf0 _A ) 􀀀 1 4 (fabc0g0fg0 d0efZ A_c0Zd0 _A ZbD ZDfZaC ZCe + fa0b0cgfg de0f0ZAcZdA Zb0 _D Z _D f0Za0 _C Z _C e0) (4.36) In summary, the full N = 4 Lagrangian is given by L = 1 2 (􀀀D ZA a D ZaA 􀀀 D Z _A a0D Za0 _A + i _A a D a _A + i A a0 D a0 A ) 􀀀 i 2 (facbdZaA ZAb c _B _B d + fa0c0b0d0Za0 _A Z _ Ab0 c0B Bd0) + i 2 fabc0d0(ZaA ZbB Ac0 Bd0 + Zc0 _A Zd0 _B A_a _B b + 4ZaA Z _B d0 b _B Ac0) + 1 2 (fabcdAab @ Acd + 2 3 fabc gfgdefAab Acd Aef ) + 1 2 (fa0b0c0d0Aa0b0 @ Ac0d0 + 2 3 fa0b0c0 g0fg0d0e0f0Aa0b0 Ac0d0 Ae0f0 ) + (fabc0d0Aab @ Ac0d0 + fabc gfgde0f0Aab Acd Ae0f0 + fabc0 g0fg0d0e0f0Aab Ac0d0 Ae0f0 ) + 1 12 (fabcgfg defZAaZbB ZB(cZd) C ZCeZf A + fa0b0c0g0fg0 d0e0f0Z A_a0Zb0 _B Z _B (c0Zd0) _C Z _C e0Zf0 _A ) 􀀀 1 4 (fabc0g0fg0 d0efZ A_c0Zd0 _A ZbD ZDfZaC ZCe + fa0b0cgfg de0f0ZAcZdAZb0 _D Z _D f0Za0 _C Z _C e0): (4.37) Using the same argument given in section 3.1, we may promote the N = 1 supersymmetry transformations to N = 4: ZaA = i A _A a _A ; Za0 _A = i y_A A a0 A ; a0 A = 􀀀 D Za0 _B A _B 􀀀 1 3 fa0 b0c0d0Zb0 _B Z _B c0Zd0 _C A _C + fa0 b0cdZb0 _A ZBcZdA B _A ; a _A = 􀀀 D ZaB y_A B 􀀀 1 3 fa bcdZbB ZBcZdC y_A C + fa bc0d0ZbA Z _B c0Zd0 _A y_B A; A~ c d = i A _B b _BZaA fab c d + i y _ AB b0B Za0 _A fa0b0 c d; A~ c0 d0 = i A _B b _B ZaA fab c0 d0 + i y _ AB b0B Za0 _A fa0b0 c0 d0 ; (4.38) where the parameter satis es the reality condition y _A B = 􀀀 BC _A _B C _B : (4.39) It is still necessary to verify the closure of the N = 4 superalgebra; this will be done in the next subsection. The ordinary Lie algebra counterparts of the Lagrangian (4.37) and the 29 supersymmetry transformations (4.38) are rst constructed in Ref. [34]. If fabc0d0 = fabcd, then fabc0d0 also satisfy the constraint equation, i.e., f(abc0)d0 = 0. In this special case, the N = 4 supersymmetry will be enhanced to N = 5. If one sets the `twisted' multiplet to be zero, i.e., a0 _A = 0, then (4.37) and (4.38) be- come the the Lagrangian and the supersymmetry law of the GW theory [33], respectively, in the 3-algebra approach: L = 1 2 (􀀀D ZA a D ZaA + i _A a D a _A ) 􀀀 i 2 facbdZaA ZAb c _B _B d + 1 2 (fabcdAab @ Acd + 2 3 fabc gfgdefAab Acd Aef ) + 1 12 fabcgfg defZAaZbB ZB(cZd) C ZCeZf A; (4.40) and ZaA = i A _A a _A ; a _A = 􀀀 D ZaB y_A B 􀀀 1 3 fa bcdZbB ZBcZdC y_A C; A~ c d = i A _B b _B ZaA fab c d: (4.41) 4.2 Closure of the N=4 Algebra The closure of the algebra of the GW theory was checked in [33]. To our knowledge, there is no explicit check in the literature for the closure of the N = 4 algebra after adding the twisted multiplets into the GW theory. Here we present such a check by starting with the supersymmetry transformation of the scalar elds: [ 1; 2]ZaA = v D ZaA + 1 3 fa bcdZbB ZcC ZdD AE BCuED +if a bc0d0ZBbZ _B c0Z A_d0( 2A _A y 1 _B B 􀀀 1A _A y 2 _B B ); (4.42) where v i y 1 _AB 2B _A ; uED i( E _A 1 y 2 _A D 􀀀 E _A 2 y 1 _A D): (4.43) By using the identity AE BC = 􀀀( A B E C 􀀀 E B A C), the second term of the RHS of (4.42) can be written as 􀀀 1 3 fa bcdZbA ZcC ZdD uCD + 1 3 fa bcdZbB ZcA ZdD uBD: (4.44) 30 The second term is equal to the rst term minus the second term by the constraint condition fa (bcd) = 0: 1 3 fa bcdZbB ZcA ZdD uBD = 􀀀 1 3 fa bcdZbA ZcC ZdD uCD 􀀀 1 3 fa bcdZbB ZcA ZdD uBD: (4.45) Therefore the second term of the RHS of (4.42) is equal to 􀀀 1 2 fa bcdZcC ZdD uCDZbA : (4.46) By using the fourth equation of (A.27), the second line of the RHS of (4.42) becomes 􀀀 1 2 fa bc0d0Zc0 _A Zd0 _B u _A _B ZbA ; (4.47) where u _A _B i( yA_C 1 2C _B 􀀀 y A_C 2 1C _B ): (4.48) In summary, we have [ 1; 2]ZaA = v D ZaA + ~ a bZbA : (4.49) While the rst is the familiar covariant derivative, the second term is a gauge transfor- mation by a parameter ~ a b 􀀀 1 2 fa bcdZcC ZdD uCD 􀀀 1 2 fa bc0d0Zc0 _A Zd0 _B u _A _B : (4.50) Similarly, we have [ 1; 2]Za0 _A = v D Za0 _A + ~ a0 b0Zb0 _A ; (4.51) where the parameter ~ a0 b0 is de ned as ~ a0 b0 􀀀 1 2 fa0 b0c0d0Zc0 _C Zd0 _D u _C _D 􀀀 1 2 fa0 b0cdZcA ZdB uAB: (4.52) Let us now examine the supersymmetry transformation of the gauge elds: [ 1; 2]A~ a b = v ~ F a b 􀀀 D ~ a b +v f ~ F a b 􀀀 " [(ZcA D ZAd 􀀀 i 2 _B c d _B )fcd a b +(Zc0 _A D Z A_d0 􀀀 i 2 Bc0 d0 B)fc0d0 a b]g +O(Z4): (4.53) 31 The last term O(Z4), which is fourth order in the scalar elds Z, vanishes by the FI (4.9). The second line and the third line must be the equations of motion for the gauge elds: ~ F a b = " [(ZcA D ZAd 􀀀 i 2 _B c d _B )fcd a b +(Zc0 _A D Z A_d0 􀀀 i 2 Bc0 d0 B)fc0d0 a b]; (4.54) while the rst line remains: [ 1; 2]A~ a b = v ~ F a b 􀀀 D ~ a b: (4.55) The rst term is a covariant translation; the second term is a gauge transformation, as expected. Similarly, we have [ 1; 2]A~ a0 b0 = v ~ F a0 b0 􀀀 D ~ a0 b0 ; (4.56) and ~ F a0 b0 = " [(Zc0 _A D Z A_d0 􀀀 i 2 Bc0 d0 B)fc0d0 a0 b0 + (ZcA D ZAd 􀀀 i 2 _B c d _B )fcd a0 b0 ]: (4.57) Finally we examine the fermion supersymmetry transformation: [ 1; 2] a _A = v D a _A + ~ a b b _A 􀀀 i 2 ( y _C B 1 2B _A 􀀀 y _C B 2 1B _A )Ea _C 􀀀 1 2 v Ea _A ; (4.58) where Ea _A = D a _A + fcdb aZbB ZBc d _A 􀀀 fc0d0b aZc0 _A Zd0 _C _C b + 2fc0d0b aZbB Zc0 _A Bd0 : (4.59) In order to achieve the closure of the algebra, we must impose the equations of motion for the fermionic elds: Ea _A = 0: (4.60) As a result, only the rst line of (4.58) remains. Similarly, we obtain [ 1; 2] a0 A = v D a0 A + ~ a0 b0 b0A ; and 0 = Ea0 A = D a0 A + fc0d0b0 a0Zb0 _B Z _B c0 d0 A 􀀀 fcdb0 a0ZcA ZdC Cb0 + 2fcdb0 a0Zb0 _B ZcA _B d: (4.61) One can derive all the equations of motion as the Euler-Lagrangian equations from the Lagrangian (4.37). CHAPTER 5 3-ALGEBRAS, LIE SUPERALGEBRAS AND EMBEDDING TENSORS 5.1 3-algebras and Lie Superalgebras In this section, we will demonstrate that the symplectic 3-algebra can be realized in terms of a super Lie algebra [44]. Recall that in section 3.1, we note that fIJKL can be speci ed as kmn m IJ nK L (up to an unimportant constant), i.e., fIJKL = kmn m IJ nK L; (5.1) where the set of matrices m IK is in the fundamental representation of Sp(2L) or its subalgebra, and kmn is the Killing-Cartan metric. Later we will prove that (5.1) is an explicit solution of the FI (2.6) (see section 5.2). Further more, the constraint condition f(IJK)L = 0 implies that f(IJK)L = kmn m (IJ nK )L = 0. As GW pointed out [33], the constraint equation kmn m (IJ nK )L = 0 can be solved in terms of the Jacobi identity for following super Lie algebra: 1 [Mm;Mn] = Cmn sMs; [Mm;QI ] = 􀀀 m IJ!JKQK; fQI ;QJg = m IJkmnMn: (5.2) Namely, the QQQ Jacobi identity [fQI ;QJg;QK] + [fQJ ;QKg;QI ] + [fQK;QIg;QJ ] = 0 (5.3) is equivalent to the constraint equation kmn m (IJ nK )L = 0. Therefore GW's approach suggests that the symplectic 3-algebra can be realized in terms of the super Lie algebra 1This is not the D = 3 super-Pioncare algebra. 33 (5.2), if we think of the 3-algebra generator TI as the fermionic generator QI . Comparing the 3-bracket [TI ; TJ ; TK] = fIJK LTl with [fQI ;QJg;QK] = kmn m IJ nK LQL; (5.4) and taking account of (5.1), we see that the 3-bracket may be realized in terms of the double graded commutator [TI ; TJ ; TK] := [fQI ;QJg;QK]: (5.5) Here the RHS is also obviously symmetric in IJ. It is instructive to examine the FI (2.5) with the 3-brackets replaced by the double graded commutators: [fQI ;QJg; [fQM;QNg;QK]] = [f[fQI ;QJg;QM];QNg;QK] + [fQM; [fQI ;QJg;QN]g;QK] +[fQM;QNg; [fQI ;QJg;QK]]: (5.6) By using the super Lie algebra (5.2), we obtain m IJ nM N([Mn; [Mm;QK]] 􀀀 [Mm; [Mn;QK]] + [[Mm;Mn];QK]) = 0; (5.7) which is equivalent to the MMQ Jacobi identity of the super Lie algebra (5.2). It is not di cult to prove that kmn m IJ nK L also enjoy the symmetry properties (2.18). So indeed the symplectic 3-algebra can be realized in terms of the super Lie algebra. Now recall the component formalism of the basic de nition of the global transformation ~ XK = IJfIJ K LXL: (5.8) Replacing fIJ K L by kmn m IJ nK L gives ~ XK = IJkmn m IJ nK LXL: (5.9) From the ordinary Lie group point of view, this is a transformation with parameters IJkmn m IJ and generators nK L. On the other hand, the second equation of (5.2) indicates that the fermionic generators furnish a representation of the bosonic part of the super Lie algebra (5.2), i.e., the matrix m IJ is a quaternion representation of Mm. Therefore, the gauge group generated by the 3-algebra can be determined as follows: its Lie algebra is just the bosonic part of the super Lie algebra (5.2), which must be Sp(2L) 34 or its subalgebras. The representation of the matter elds is determined by the fermionic generators of the super Lie algebra (5.2). For a more mathematical approach, see Ref. [37, 42, 46], in which the relations between the 3-algebras and Lie superalgebras are discussed by using Lie algebra repre- sentation theories. 5.2 Three-algebras and Lie Algebras It is less obvious that one can also prove that (5.1) is an explicit solution of the FI (2.6) by using the QQM Jacobi identity of the super Lie algebra, which reads [fQI ;QJg;Mm] 􀀀 f[QJ ;Mm];QIg + f[Mm;QI ];QJg = 0: (5.10) After some algebraic steps we obtain n IJknp[Mp;Mm] 􀀀 mK J nK IknpMp 􀀀 mK I nK JknpMp = 0: (5.11) Since the matrix m IJ is a representation of Mm, the above equation implies n IJknp[ p; m]MN 􀀀 mK J nK Iknp p MN 􀀀 mK I nK Jknp p MN = 0; (5.12) where [ p; m]MN = p MO mO N 􀀀 m MO pO N: (5.13) Multiplying both sides by kmq q KL gives knp n IJkmq q KL[ p; m]MN 􀀀 kmq q KL mK J nK Iknp p MN 􀀀 kmq q KL mK I nK Jknp p MN = 0: (5.14) Rearranging the above equation veri es explicitly that (5.1) satis es the FI (2.6). Appli- cation of the commutator [ m; n]IJ = Cmn p p IJ (5.15) to Eq. (5.14) gives (knpkqmCpm s + kqmkspCpm n) n IJ q KL sM N = 0: (5.16) Here the equation in the bracket is simply the statement that the structure constants ~ Cnqs = knpkqmCpm s (5.17) 35 are totally antisymmetric if the three adjoint indices nqs are on equal footing. Note that kmn is an invariant bilinear form on the bosonic subalgebra, since Eq. (5.16) or (5.17) also implies [k;Cm] = 0: (5.18) Here the matrices (Cm)p n = Cmp n furnish the usual adjoint representation of the bosonic subalgebra. In this way, we see that the FI of the 3-algebra can be converted into two ordinary commutators (5.15) and (5.18) (this is rst discovered in the second paper of Ref. [15] with a di erent approach). Eq. (5.9) indicates that fIJKL = kmn m IJ nK L also furnish a quaternion represen- tation of the bosonic subalgebra. In fact, if we write fIJKL as (fIJ )KL, then fIJ is a set of matrices, and corresponding matrix elements are (fIJ )KL. If nK L furnish a quaternion of representation of Mn, then (fIJ )KL furnish a quaternion representation of MIJ = kmn m IJMn, since the operator MIJ is a linear combination of Mn. With this understanding, we are able to rewrite the FI (2.6) as a commutator [fIJ ; fKL]MN = CIJ;KL OP (fOP )MN = (fIJK O P L + fIJL O PK )(fOP )MN = 􀀀[fIJ ; fMN]KL: (5.19) The third equation says that the quantity [fIJ ; fKL]MN are totally antisymmetric in the 3 pairs of indices. Eq. (5.19) is equivalent to Eq. (5.15). Also, the matrices (fIJ )KL satisfy the conventional Jacobi identity as a result of the MMM Jacobi identity of the superal- gebra of (5.2). We now must check whether ~ CIJ;KL;MN = kMN;OPCIJ;KL OP are totally antisymmetric or not. To be consistent with the transformation MIJ = kmn m IJMn, we must transform the Killing-Cartan metric kmn as kmn ! kIJ;KL = kqm q IJkpn p KLkmn = kmn m IJ nK L = fIJKL: (5.20) Namely the structure constants fIJKL also play a role of the Killing-Cartan metric kIJ;KL. So we must use fMNOP to lower the OP indices of CIJ;KL OP : 2 ~ CIJ;KL;MN = fMNOPCIJ;KL OP = [fMN; fIJ ]KL: (5.21) 2This is a comment by E. Witten, quoted in the second paper of Ref. [15]. 36 By the third equation of (5.19), the structure constants ~ CIJ;KL;MN are indeed totally antisymmetric in the 3 pairs of indices. Therefore Eq. (5.18) now takes the following form [f;CIJ ] = 0 or [fMN; fIJ ]KL + [fKL; fIJ ]MN = 0; (5.22) which is nothing but the third equation of Eq. (5.19). Namely both Eq. (5.15) and Eq. (5.18) can be written as the third equation of Eq. (5.19), if we express everything in terms of the 3-algebra structure constants fIJKL. Note that we use kmn to lower an adjoint index, while use !IJ to lower a fundamental index. If Eq. (5.1) holds, then Eq. (2.8) implies a compatible condition between kmn and !IJ . Eq. (2.8) is equivalent to knm mK I !KJ + knm mK J !IK = 0, i.e., ~ nIJ 􀀀 knm!IK mK J = 0; (5.23) where ~ nIJ knm m IJ . 5.3 Three-algebras and Embedding Tensors In Ref. [31, 32], the authors derive some extended superconformal gauge theories by taking a conformal limit of D = 3 gauged supergravity theories. In their approach, the embedding tensor plays a crucial role. By de nition, the embedding tensor mn = nm acts as a projector [32]: D = @ 􀀀 Am mntn; (5.24) where tn is a set of independent generators. The above equation says that mn projects tn onto another set of generators ~tm = mntn, whose symmetries are gauged. Let us now consider the commutator [~tm; ~tn] = mp nsCps qtq: (5.25) Since we expect that [~tm; ~tn] = ~ Cmn r~tr, we must set mp nsCps q = ~ Cmn r rq: (5.26) It is necessary to examine the Jacobi identity [[~tm; ~tn]; ~tp] + [[~tn; ~tp]; ~tm] + [[~tp; ~tm]; ~tn] = ( ~ Cmn s ~ Csp r + ~ Cnp s ~ Csm r + ~ Cpm s ~ Csn r) rqtq = (Clq rCrs t + Cqs rCrl t + Csl rCrq t) ml nq pstt = 0: (5.27) 37 In the last line we have used (5.26). The last line is nothing but the Jacobi identity satis ed by Cmn p. So Eq. (5.27) is indeed the desired result. To construct a physical theory, the embedding tensor is required to be invariant under the transformations which are gauged. Since the embedding tensor mn carries two adjoint indices, we have to set ~ Cnq r rs + ~ Cns r qr = 0: (5.28) Taking account of (5.26), the above equation is equivalent to np qmCpm s + np smCpm q = 0: (5.29) This quadratic constraint takes the same form for all extended supergravity theories. We will focus on the N = 5 case. If we represent the adjoint index m as a pair of fundamental indices IJ, the embedding tensor becomes IJ;KL, satisfying the same symmetry properties as fIJKL do (see (2.18)) [31]. To construct N = 5 supergravity theories, the embedding tensor is required to satisfy the linear constraint: (IJ;K)L = 0; (5.30) and the structure constants in (5.29) are required to be those of Sp(2L) [31]. We observe that if one identi es the embedding tensor mn with the Killing-Cartan metric kmn, Eq. (5.29) is precisely the same as Eq. (5.16), which is the FI satis ed by the 3-algebra structure constants fIJKL = kmn m IJ nK L. Recall that fIJKL also play the role of the Killing-Cartan metric (see section 5.2). So identifying the embedding tensor with the Killing-Cartan metric is equivalent to identifying the embedding tensor with the 3-algebra structure constants. With this identi cation, Eq. (5.30) is also solved since it is nothing but f(IJK)L = 0. We are therefore led to the conclusion that fIJKL also play the role of the embedding tensor. It is straightforward to generalize the discussion of this section to the cases with other values of N. In summary, if we realize the symplectic 3-algebra in terms of the superalgebra (5.2), we nd that fIJKL = kmn m IJ nK L play four roles simultaneously: fIJKL are the structure constants of the symplectic 3-algebra or the double graded commutator (5.4); fIJKL furnish a quaternion representation of the bosonic part of the superalgebra; fIJKL play the role of the Killing-Cartan metric; 38 fIJKL are the components of the embedding tensor used to construct the D = 3 extended supergravity theories. CHAPTER 6 N=4, 5 THEORIES IN TERMS OF THE BOSONIC PARTS OF SUPERALGEBRAS The N = 4; 5 theories in Chapter 3 and 4 are constructed in terms of 3-algebras. After the discussions of the last section, we are ready to derive their ordinary Lie Algebra constructions by the solution (5.1) [44]. 6.1 N = 5 Theories in Terms of the Bosonic Parts of Superalgebras With the solution fIJKL = kmn m IJ nK L; [ m; n]IJ = Cmn p p IJ ; (6.1) the gauge eld becomes A~ K L = AIJ fIJ K L = AIJ m IJkmn nK L Am kmn nK L: (6.2) Following Ref. [33], we de ne the `momentum map' and `current ' operator as follows m AB m IJZIA ZJB ; jm AB m IJZIA JB : (6.3) Here A = 1; ; 4 is the fundamental index of the R-symmetry group Sp(4). Substituting the (6.1) and (6.2) into the Lagrangian (3.38) gives L = 1 2 (􀀀D ZA I D ZIA + i A I D IA ) 􀀀 i 2 !AB!CDkmn(jm ACjnB D 􀀀 2jm ACjnD B) + 1 2 (kmnAm @ An + 1 3 ~ CmnpAm An Ap ) (6.4) + 1 30 ~ Cmnp mA B nB C pC A + 3 20 kmpkns( m n)IJZAIZJA pB C sC B: 40 Similarly, with the solution (6.1), the supersymmetry transformation law becomes ZIA = i A B IB ; IA = D ZIB B A + 1 3 kmn mI J!BCZJB n CD D A 􀀀 2 3 kmn mI J!BDZJC n DA C B; Am = i AB jm AB: (6.5) Here the parameter A B obeys the traceless condition and the reality condition (3.46). The N = 5 Lagrangian (6.4) and supersymmetry transformation law (6.5) are in agree- ment with those given in Ref. [35], which were derived directly in terms of ordinary Lie algebra. In section (5.1), we have demonstrated that if the structure constants of the 3-algebra are speci ed as (6.1), then the Lie algebra of the gauge group generated by the 3-algebra is just the bosonic part of the superalgebra (5.2). The following classical super-Lie algebras: U(MjN); OSp(Mj2N); OSp(2j2N); F(4); G(3); D(2j1; ); (6.6) (with a continuous parameter) are of the same form as that of the superalgebra (5.2). Therefore their bosonic parts can be selected to be the Lie algebras of the gauge groups of the N = 5 theories. Especially, if we choose the U(MjN) or OSp(2j2N), whose bosonic part is in the two conjugate representations (R R ), then the supersymmetry will get enhanced to N = 6 [35]. In Appendix C.1, we work out the details of the N = 5; Sp(2N) O(M) CSM theory. 6.2 N = 4 Theories in Terms of the Bosonic Parts of Superalgebras 6.2.1 Sp(2N1) SO(N2) Sp(2N3) Example In order to realize the 3-algebra used to construct the N = 4 theories, one needs to `fuse' two simple super Lie algebras into a single superalgebra, by requiring that the bosonic parts of these two simple superalgebras to share one simple factor. The general structure of this `fused' will be constructed in a forthcoming paper [45]. In this section, we demonstrate how to `fuse' a pair of simple superalgebras into one superalgebra by presenting an explicit example. Suppose that the untwisted multiplets are in the bifundamental representation of Sp(2N1) SO(N2) (the bosonic part of OSp(N2j2N1)), while the twisted multiplets are in the bifundamental representation of Sp(2N3) SO(N4) (the bosonic part of OSp(N4j2N3)). Without loss of generality, we assume that N2 = N4 41 and N1 6= N3, i.e., the two superalgebras share one simple factor so(N2). So the gauge group is Sp(2N1) SO(N2) Sp(2N3). Let's work out the details. The super Lie algebra OSp(N2j2N1) reads [M i j ;M k l ] = j kM i l 􀀀 i kM j l + i l M j k 􀀀 j l M i k; [M^i ^j ;M^k^l ] = !^j ^kM^i ^l + !^i ^kM^j ^l + !^i ^l M^j ^k + !^j ^l M^i ^k; [M i j ;Q k^k] = j kQ i ^k 􀀀 i kQ j ^k; [M^i ^j ;Q k^k] = !^j ^kQ k^i + !^i ^kQ k^j ; [Q i ^i ;Q j ^j ] = k(!^i ^j M i j + i j M^i ^j ); (6.7) where i = 1; ;N2 is an SO(N2) fundamental index, and ^i = 1; ; 2N1 is an Sp(2N1) fundamental index, Qa = Q i ^i and !ab = ! i ^i; j ^j = i j !^i ^j . The super Lie algebra OSp(N2j2N3) has similar expressions. We denote the fermionic generators of OSp(N2j2N3) as Qa0 = Q ii0 , where i0 = 1; ; 2N3 is an Sp(2N3) fundamental index. Since Q ii0 also carries an SO(N2) fundamental index, the anticommutator fQ i^i;Q j j0g cannot vanish. Actually, if fQ i ^i ;Q j j0g = 0, then the Q i ^i Q j j0Q kk0 Jacobi identity implies that [M j k;Q i ^i ] = 0, which is contradictory with the third equation of (6.7). Namely, if these two superalgebras share one simple factor, then we indeed have fQ i ^i ;Q j j0g 6= 0, i.e., this anticommutator must equal to a bosonic operator. On the other hand, since ^i and j0 are independent indices (recall N1 6= N3), it is natural to de ne fQ i ^i ;Q j j0g = k i j M^ij0 ; [M^i ^j ;Q kl0 ] = [Mi0j0 ;Q k^l ] = 0; [M^ii0 ;Q j j0 ] = !i0j0Q i ^i ; [M^ii0 ;Q j ^j ] = !^i ^j Q j i0 : (6.8) It is not di cult (though a little tedious) to verify that every Jacobi identity is satis ed. So the ve graded commutators in (6.8) must be the correct ones; they are explicit examples of the last ve graded commutators of the new superalgebra in Ref. [45] `fused' by two simple superalgebras. Since the structure constants of the double graded commutator are also the struc- ture constants of the symplectic 3-algebra, let us consider the following double graded commutator: [fQ i ^i ;Q j ^j g;Q kk0 ] = k!^i ^j ( j kQ ik0 􀀀 i kQ j k0): (6.9) It is not di cult to read o the structure constants of 3-algebra fabc0d0 : fabc0d0 = f i ^i; j ^j ; kk0; l l0 = 􀀀k!^i ^j !k0l0( i k j l 􀀀 i l j k): (6.10) 42 Note that fabc0d0 = f i ^i; j ^j ; kk0; l l0 are not subjected to any linear constraint such as (4.12). To see this, let us consider the QaQbQc0 or Q i ^i Q j ^j Q kk0 Jacobi identity [fQ i ^i ;Q j ^j g;Q kk0 ] + [fQ i ^i ;Q kk0g;Q j ^j ] + [fQ kk0 ;Q j ^j g;Q i ^i ] = 0; (6.11) which can be converted into k[􀀀!^i ^j !k0l0( i k j l 􀀀 i l j k) + !^i ^j !k0l0( i k j l 􀀀 i l j k)]Q l l0 = 0: (6.12) This equation is merely a statement that f i ^i; j ^j ; kk0; l l0 are antisymmetric in i j or in ^i ^j . Therefore the Q i ^i Q j ^j Q kk0 Jacobi identity does not impose a linear constraint on f i ^i; j ^j ; kk0; l l0 , since the two simple superalgebras share only one simple factor, as we claimed in the last section. Similarly, the Q ii0Q j j0Q k^k Jacobi identity also does not impose a linear constraint on f ii0; j j0; k^k; l ^l . One can obtain fabcd by considering [fQ i ^i ;Q j ^j g;Q k^k]. A short calculation gives fabcd = f i ^i; j ^j ; k^k; l ^l = 􀀀k[( i k j l 􀀀 i l j k)!^i ^j !^k^l 􀀀 i j k l (!^i ^k!^j ^l + !^i ^l !^j ^k)]: (6.13) And fa0b0c0d0 have a similar expression: fa0b0c0d0 = f ii0; j j0; kk0; l l0 = 􀀀k[( i k j l 􀀀 i l j k)!i0j0!k0l0 􀀀 i j k l (!i0k0!j0l0 +!i0l0!j0k0 )]: (6.14) Eqs. (6.10) (6.14) satisfy the symmetry conditions (4.10), the reality conditions (4.11) and the FIs (4.9). Eqs. (6.13) and (6.14) also satisfy the constraint equations (4.12). Substituting Eqs. (6.10) (6.14) into (4.37) and (4.38) gives the N = 4 CSM theory with gauge group Sp(2N1) SO(N2) Sp(2N3). Alternatively, one can read o kuv and u ab from (6.7) by comparing (6.7) with [Mu;Qa] = 􀀀 u ab!bcQc and fQa;Qbg = u abkuvMv: For instance, ( m n) i ^i; j ^j = !^i^j( m i n j 􀀀 m j n i ); (6.15) k m n; p q = k 4 ( m p n q 􀀀 m q n p): (6.16) Similarly, we have (t p q) kk0; l l0 = 􀀀!k0l0( p k q l 􀀀 p l q k): (6.17) Combining Eqs. (6.15) (6.17) gives (6.10): fabc0d0 = kuv u abtvc0d0 = k m n; p q( m n) i ^i; j ^j (t p q) kk0; l l0 = f i ^i; j ^j ; kk0; l l0 = 􀀀k!^i ^j !k0l0( i k j l 􀀀 i l j k): (6.18) In this way, one also calculate fabcd = kuv u ab v cd and fa0b0c0d0 = ku0v0 u0 a0b0 v0 c0d0 ; they are the same as (6.13) and (6.14), respectively. 43 Analogously, we may set N1 = N3 but N2 6= N4. In this case, the common simple factor is sp(2N1), and the gauge group is SO(N2) Sp(2N1) SO(N4). However, if N1 = N3 and N2 = N4, then the two simple superalgebras are identical, or v cd = v0 c0d0 . As a result, the structure constants fabc0d0 = fabcd, and fabc0d0 also satisfy the constraint equation f(abc0)d0 . In this case, the gauge group is Sp(2N1) SO(N2), and the N = 4 supersymmetry is enhanced to N = 5. For details, see [35]. If N1 6= N3 and N2 6= N4, then fQa;Qc0g = 0 and fabc0d0 = 0. Or equivalently, if the gauge group of the N = 4 theory is Sp(2N1) SO(N2) Sp(2N3) SO(N4), then the untwisted multiplets furnish a trivial representation of Sp(2N3) SO(N4) and a fundamental representation of Sp(2N1) SO(N2), while the twisted multiplets furnish a trivial representation of Sp(2N1) SO(N2) and a fundamental representation of Sp(2N3) SO(N4). In this case, the N = 4 Lagrangian becomes two uncoupled GW Lagrangians (see section 6.2.3). The theories of this section are rst constructed in Ref. [34], using a di erent approach. We will rederive the general N = 4 CSM theory in terms of ordinary Lie algebra in a forthcoming paper [45]. 6.2.2 Examples of N = 4 Quiver Gauge Theories After working out the example in section 6.2.1, it is not di cult to nd out the other gauge groups. We can consider the following pairs of super Lie algebras [35, 37]: (G1;G2) = (U(N1jN2); (U(N2jN3)); (OSp(N1j2N2); (OSp(N1j2N3)); (OSp(N1j2N2); (OSp(N3j2N2)); (OSp(N1j2N2); (OSp(2j2N2)); (OSp(2j2N1); (OSp(2j2N1)): (6.19) For every pair, the even parts share at least one common simple factor, hence can be chosen as the Lie algebras of the gauge groups. It is straightforward to generalize the construction of section 4 by decomposing the set of 3-algebra generators TI as three sets of generators, and decomposing one N = 5 multiplet as three N = 4 multiplets. Then the gauge group must be the even parts of (G1;G2;G3), where Gi (i = 1; 2; 3) is a super Lie algebra selected from the list (6.6). Here we assume that the even parts of G1 and G2 share at least one common simple factor, while the even parts of G2 and G3 share at least one common simple factor. For 44 example, one can choose (G1;G2;G3) as (OSp(N1j2N2); (OSp(N1j2N3); (OSp(N4j2N3)). The resulting quiver diagram for gauge groups is Sp(2N2) 􀀀 SO(N1) 􀀀 Sp(2N3) 􀀀 SO(N4): (6.20) Or we can set (G1;G2;G3)= (U(N1jN2); (U(N2jN3)); (U(N3jN4), and the resulting quiver diagram for gauge groups is U(N1) 􀀀 U(N2) 􀀀 U(N3) 􀀀 U(N4): (6.21) In the general case, one can choose the even parts of (G1; ;Gn), where Gi (i = 1; ; n) is a super Lie algebra selected from the list (6.6); the even parts of Gi and Gi+1 (i = 1; ; n 􀀀 1) share one common simple factor [34, 37]. If the even parts of G1 and Gn (with n an even number) also share one common simple factor, then the linear quiver becomes a closed loop. The linear quiver gauge theories described in this paragraph exhaust all known examples of N = 4 superconformal CMS theories. As [37] pointed out, if one also takes account of the exceptional super Lie algebras, and the isomorphisms of the Lie algebras, there are additional possibilities. We will elaborate these ideas by constructing some N = 4 theories with new gauge groups. Let us rst consider the exceptional Lie algebras. The even parts of the super groups F(4), G(3) and D(2j1; ) are SO(7) SU(2) (SO(7) is in the spinor representation), G2 SU(2) and SO(4) Sp(2), respectively. We therefore may have (G1;G2) = (F(4); (SU(2jN2)); (G(4); (SU(2jN2)); (G(4); F(4)); (OSp(7j2N); F(4); (OSp(4j2N);D(2j1; )): (6.22) Their even parts can be selected as the Lie algebras of the gauge groups. It also is possible to construct some new N = 4 CMS theories by using the four isomorphisms of the Lie algebras: so(3) = su(2) = sp(2); so(5) = sp(4); so(6) = su(4): (6.23) The pairs of the super Lie algebras can be chosen as (G1;G2) = (OSp(3j2N1); (OSp(N2j2)); (OSp(3jN1); (SU(2jN2)); (OSp(3j2N1); F4); (OSp(3j2N1);D(2j1; )); (OSp(3j2N1);G3); (OSp(N1j2); (SU(2jN2)); (OSp(N1j2); F4); (OSp(N1j2);G3); (G3;D(2j1; )); (F4;D(2j1; )); (OSp(5j2N1); (OSp(N2j4)); (OSp(6jN1); (SU(4jN2)); (6.24) and their even parts can be selected as the Lie algebras of the gauge groups. 45 Clearly, one can use the constructions of the last two paragraphs in the general case (G1; ;Gn), where we select Gi (i = 1; ; n) from the list of the super Lie algebra (6.6). The even parts of Gi and Gi+1 (i = 1; ; n􀀀1) share one common simple factor; or one simple factor of the even part of Gi is isomorphic to one simple factor of the even part of Gi+1. Finally, one can obtain new gauge groups by noting that the even parts of m (m > 2) super Lie algebras can share one simple factor. One therefore can construct `meshy' N = 4 quiver gauge theories. For example, the even parts of the super Lie algebras G1 G4 can share one common simple factor. For instance, if we set (G1;G2;G3;G4) = (OSp(Nj2N1);OSp(Nj2N2);OSp(Nj2N3);OSp(Nj2N4)); (6.25) then the resulting quiver diagram for gauge groups is Sp(2N1) j Sp(2N2) 􀀀 SO(N) 􀀀 Sp(2N4) (6.26) j Sp(2N3) The four multiplets are in the bifundamental representations of Sp(2Ni) SO(N) (i = 1; ; 4), respectively. It can be seen that (6.26) is `meshy', while (6.20) or (6.21) is `linear'. In summary, by using (6.19), (6.22) and (6.24), one can construct a general N = 4 quiver gauge theory by requiring that the even parts of a (a 2) adjacent super Lie algebras share one simple factor. (If two simple factors are isomorphic to each other, we also consider them as the same simple factor, even they may be in di erent representations.) The total number of the super Lie algebras is n (n a), and the n super Lie algebras are selected from (6.6). The Lie algebras of the gauge groups are just the even parts of the super Lie algebras, and the multiplets are in the bifundamental representations. 6.2.3 N = 4 GW Theory in Terms of Lie Algebras Here we consider the N = 4 GW theory without the `twisted' hyper multiplets, i.e., setting a0 _A= 0. Then with the solution for structure constants of the 3-algebra given by fabcd = kmn m ab n cd; [ m; n]ab = Cmn p p ab; (6.27) 46 which satisfy the FI as well as appropriate constraint and symmetry conditions, the gauge elds of the GW theory become A~ c d = Aab fab c d = Aab m abkmn nc d Am kmn nc d: (6.28) Following Ref. [33], we de ne the `momentum map' and `current ' operators as follows m AB m abZaA ZbB ; jm A _B m abZaA b _B : (6.29) With Eqs (6.27) (6.29), Eqs. (4.40) and (4.41) become the Lagrangian and the supersymmetry law of the GW theory in Ref. [33], respectively: L = 1 2 (kmnAm @ An + 1 3 ~ CmnpAm An Ap ) + 1 2 (􀀀D ZA a D ZaA + i _A a D a _A ) 􀀀 i 2 kmnjm A _B jnA _B 􀀀 1 24 ~ Cmnp mA B nB C pC A; (6.30) with ~ Cmnp = kmrknsCrs p and ZaA = i A _A a _A ; a _A = 􀀀 D ZaB y_A B 􀀀 1 3 kmn ma bZbB nB C y_A C; Am = i A _B jm A _B : (6.31) Since we derived the GW theory by decomposing the N = 5 theory and setting the twisted mulitplets to zero, so the classical superalgebras, which are used to realize the 3-algebra, must be the same as those used in the N = 5 case, i.e., U(MjN); OSp(Mj2N); OSp(2j2N); F(4); G(3); D(2j1; ): (6.32) Indeed, they are of the same form as that of the superalgebra (5.2). Therefore their bosonic parts can be selected to be the Lie algebras of the gauge groups of the GW theory [33, 34, 37]; and the corresponding representations are determined by the fermionic generators. CHAPTER 7 N = 6; 8 CSM THEORIES AND 3-ALGEBRAS In Ref. [35], the N = 6 theories are derived from the N = 5 theories by enhancing the R-symmetry from Sp(4) to SU(4). In this section we will implement the same idea in the context of 3-algebras [40]. We will call the symplectic 3-algebras presented in Ref. [40] and in [38], respectively, to construct the N = 5, N = 6 theories as the \N = 5, N = 6 three-algebra", respectively. We will see that the symplectic 3-algebra provides a framework unifying the N = 4; 5; 6; 8 CSM theories. 7.1 General N=6 CSM Theories The enhancement of R-symmetry from Sp(4) to SU(4) in Ref. [35] is based on the following observation: The reality condition (3.2) implies that the complex conjugates of the matter elds can be obtained by similarity transformations, i.e., ZA I = !AB!IJZJB ; A I = !AB!IJ JB : (7.1) Therefore the matter elds actually furnish a pseudo-real presentation of the gauge group. If we decompose this pseudo-real representation into a complex representation and its conjugate representation, then the Sp(4) R-symmetry will be enhanced to SU(4), and the global N = 5 SUSY will get enhanced to N = 6. In this section, we will show that this enhancement can be implemented exclusively in the framework of symplectic 3-algebra, which thus provides a uni ed framework for both N = 5 and N = 6 theories. Since in our approach the ordinary Lie algebra of the gauge groups is generated by the FI and the 3-brackets, the challenge we face is to derive the N = 6 three-algebra from the 3-algebra proposed in Ref. [40]. Following Ref. [35], we rst decompose an N = 5 scalar eld as a direct sum of an N = 6 scalar eld and its complex conjugate (See Eq. (7.6)): ZIA = Za A = ZaA 1 + !ABZB a 2 = ZaA 1 + !ABZB a 2 ; (7.2) 48 where the right hand side of the arrow contains N = 6 elds. Here the index I runs from 1 to 2L, while the index a runs from 1 to L. And 1 and 2 are \spin up" and \spin down" spinor, respectively, i.e., 1 1 = 1 0 ; 2 = 0 1 : (7.3) To make the N = 5 SUSY transformation law (3.45) consistent with that of N = 6 (see below the rst two equations of Eq. (7.22)), we have to decompose the N = 5 fermion elds as follows: IA = a A = !AB Ba 1 􀀀 Aa 2 ; (7.4) where the right hand side contains N = 6 fermion elds. We further decompose the antisymmetric tensor !IJ and its inverse as !IJ = !a ;b = a b 1 2 􀀀 a b 2 1 ; !IJ = !a ;b = a b 2 1 􀀀 a b 1 2 : (7.5) Then the reality condition (7.1) reads Z A a = ZaA ; Aa = Aa; (7.6) in agreement with those for N = 6 theories. This justi es the above decomposition (7.5) of the antisymmetric tensor of the N = 5 three-algebra to derive the N = 6 three-algebra. To be compatible with the decomposition of scalar and fermion elds, one has to decompose the gauge elds as A~ I J = A~ a b = A~ a b 1 1 􀀀 A~ b a 2 2 ; (7.7) where the right hand side is a direct sum of an N = 6 gauge eld and its complex conjugate. Since our gauge elds A~ K L are de ned in terms of the structure constants of a 3-algebra, i.e., A~ K L = AIJ fIJ K L; (7.8) we have to decompose its structure constants properly to result in the desired decompo- sition Eq. (7.7). We nd that Eq. (7.7) indeed follows from the decomposition of the structure constants given by fIJKL = fa ;b ;c ;d = fac db 2 1 2 1 + fad cb 2 1 1 2 +fbc da 1 2 2 1 + fbd ca 1 2 1 2 ; (7.9) 1Here the index is not an index of a spacetime spinor. We hope this will not cause any confusion. 49 combined with the decomposition of AIJ given by AIJ = Aa ;b = 􀀀 1 2 (A a b 1 2 + A b a 2 1 ): (7.10) With these decompositions, the N = 6 gauge elds become: (see the right side of Eq. (7.7)) A~ c d = A b afca bd: (7.11) Later we will identify the above fca bd in the right side of Eq. (7.9) as the structure constants of the N = 6 three-algebra. With Eq. (7.5) and (7.9), the reality condition of the structure constants (2.15) reduces to f ab cd = fcd ab; (7.12) as desired for the N = 6 three-algebra [38, 39]. Eq. (7.5) motivates us to decompose the generators of the 3-algebra as follows: TI = Ta = !a ;b Tb = Ta2 1 􀀀 Ta1 2 : (7.13) Since we decompose a matter eld as a direct sum of a N = 6 matter eld and its complex conjugate, it is necessary to decompose a generator of the 3-algebra as a direct sum of a generator of a 3-algebra and its complex conjugate. This can be accomplished by setting ta = Ta1; ta t a = Ta2; (7.14) where ta is a generator of the 3-algebra, and ta its complex conjugate. The hermitian bilinear form of two N = 5 elds will be (for instance): Z I 1AZI 2A = !IJ!ABZJ 1BZI 2A = !a ;b !ABZb 1BZa 2A = Za 2AZA 1a + Za 1AZA 2a: (7.15) Namely, it becomes a sum of the hermitian bilinear form of two N = 6 elds and its complex conjugate. Generally speaking, the hermitian bilinear form of two arbitrary N = 6 three-algebra valued elds will become h(X; Y ) = X aYa X aYa: (7.16) 50 The reality condition (7.12) and Eq. (7.9) imply that the N = 5 three-bracket (2.1) can be decomposed as a direct sum of N = 6 brackets and their complex conjugates as follows: [TI ; TJ ; TK] = [Ta ; Tb ; Tc ] = [ta; tc; tb] 2 1 2 + [ta; tc; tb] 1 2 1 +[tb; tc; ta] 1 2 2 + [tb; tc; ta] 2 1 1 : (7.17) Here the 3-brackets [ta; tc; tb] = fac bdtd: (7.18) are those for the N = 6 three-algebra. Such 3-brackets were rst proposed by Bagger and Lambert [38] for a N = 6 CSM theory. An unusual feature of the 3-brackets is that it involves complex conjugate for the third generator. Our above decomposition from the N = 5 three-algebra reveals clearly the origin of the need for complex conjugation of the third generator. Later we will see that the structure constants de ned in Eq. (7.18) are indeed anti- symmetric in the rst two indices. (See Eq. (7.20).) With Eq. (7.17), the fundamental identity (2.6) reduces to ffc dgfag eb 􀀀 faf gbfgc de + fcf egfag db 􀀀 fac gbfgf ed = 0; (7.19) as desired. Also the constraint condition (2.17) on the structure constants and the symmetry properties (2.18) of the structure constants reduce to fab cd = 􀀀fba cd = fba dc: (7.20) One easily recognizes that eqs. (7.16), (7.18), (7.19), (7.12), and (7.20) are those de ning the N = 6 three-algebra used in Ref. [38]. (The relation between the N = 6 three-algebra and super Lie algebra was discussed in Ref. [46].) Substituting Eq. (7.2), (7.4), (7.9), and (7.10) into the N = 5 Lagrangian (3.38) and the SUSY transformation law (3.45), and using the Sp(4) identity (A.34) and (A.35), we reproduce the N = 6 Lagrangian 51 L = 􀀀D ZaA D ZA a 􀀀 i Aa D Aa 􀀀if ab cd Ad AaZB b ZcB + 2if ab cd Ad BaZB b ZcA 􀀀 i 2 "ABCDfab cd Ac BdZC a ZD b 􀀀 i 2 "ABCDfcd ab Ac Bd ZaC ZbD + 1 2 " (fab cdA c b@ A d a + 2 3 fac dgfge fbA b aA d cA f e) (7.21) 􀀀 2 3 (fab cdfed fg 􀀀 1 2 feb cdfad fg) ZcA ZA e Zf BZB a Zg DZD b ; and the N = 6 SUSY transformation law reads ZA d = 􀀀i AB Bd ZdA = 􀀀i AB Bd Bd = D ZA d AB + fab cdZC a ZA b ZcC AB + fab cdZC a ZD b ZcB CD Bd = D ZdA AB + fcd ab ZaC ZbA ZC c AB + fcd ab ZaC ZbD ZB c CD (7.22) A~ c d = 􀀀i AB ZA a Bbfca bd + i AB ZaA Bbfcb ad: Here the SUSY transformation parameters AB satisfy AB = 􀀀 BA (7.23) AB = AB = 1 2 "ABCD CD (7.24) Now the parameters AB transform as the 6 of SU(4). It is in this sense that the global N = 5 SUSY gets enhanced to N = 6. The Lagrangian (7.21) and the transformation law (7.22) are the same as the ones obtained in the 3-algebra approach for N = 6 theories in Ref. [38]. The N = 6 superconformal CSM theories in three dimensions can be classi ed by super Lie algebras [33, 35, 49] or by using group theory [48]. Two primary types are allowed: with gauge group U(M) U(N) and Sp(2N) U(1), respectively. In section 7.4, we will drive these two theories by specifying the structure constants of the N = 6 three-algebra. 7.2 N=8 CSM Theory If the inner product (7.16) becomes the standard inner product in the Euclidian space h(X; Y ) = XaYa or h(ta; tb) = ab; (7.25) 52 then there is no di erence between a lower index a and an upper index a, i.e., ta = ta. As a result, the 3-bracket (7.18) becomes [ta; tc; tb] = facb dtd: (7.26) If the rst 3 indices of facb d are antisymmetric, then Eq. (7.26) becomes the famous Nambu bracket. And the reality condition (7.12) becomes fab cd = fcd ab: (7.27) The symmetry properties of the structure constants (7.20) imply that fabcd defabc e (7.28) are totally antisymmetric. Now the FI (7.19) can be converted into fafe gfcdg b 􀀀 fcda gfgfe b 􀀀 fcdf gfage b 􀀀 fcde gfafg b = 0: (7.29) The 3-algebra de ned by Eq. (7.26) (7.29) is nothing but the Nambu 3-algebra. Substituting Eq. (7.28) into (7.21) and (7.22) gives the BLG theory, since the N = 6 supersymmetry is promoted to N = 8 if the structure constants are totally antisymmetric [47]. To demonstrate that the Nambu 3-algebra does generate an SO(4) gauge group, we choose the following 4 -matrices (the rst three are Pauli matrices) a = ( 1; 2; 3; iI) (7.30) to realize the generators of the Nambu 3-algebra [47], i.e., ta := a and ta := ay; (7.31) where ay is the hermitian conjugate of a. It is well known that one can establish a connection between the SU(2) SU(2) and SO(4) group by (7.30). These -matrices satisfy the Cli ord algebra: a by + b ay = 2 ab and ay b + by a = 2 ab: (7.32) The inner product (7.25) is de ned as: h( a; b) = 1 2 Tr( ay b) = ab; (7.33) 53 where we have normalized the trace by a factor 1 2 . We specify the 3-bracket as [38]: [ a; b; cy] = k( a cy b 􀀀 b cy a) = 􀀀2k"abcd d; where "abcd is the familiar Levi-Civita tensor. So in this realization, the structure constants fabcd are nothing but "abcd (up to an unimportant constant). And from the point of view of ordinary Lie algebra, a eld valued in the Nambu 3-algebra ZA _ = ZA a a _ (a = 1; ; 4; ; _ = 1; 2:) is indeed in the bifundamental representation of SU(2) SU(2). The N = 8 BLG theory is essentially unique, since the Nambu 3-algebra with a symmetric and positive de ne metric can only generate an SO(4) gauge symmetry [20, 21]. In section (8.2), we will demonstrate that the Nambu 3-algebra can be realized in terms of a super Lie algebra PSU(2j2). 7.3 Closure of the N = 6 Algebra We require the on-shell closure of the supersymmetry algebra. Namely, after imposing equations of motion, the commutator of two supersymmetry transformations must be equal to a translation plus a gauge term. The commutator of two supersymmetry transformations acting on the scalar elds reads [38, 39] [ 1; 2]ZA d = v @ ZA d + (~ a d 􀀀 v A~ a d)ZA a ; (7.34) where v = i 2 CD 2 1CD; (7.35) ~ a d = c bfab cd; (7.36) c b = i( DE 2 1CE 􀀀 DE 1 2CE) ZcD ZC b : (7.37) The rst term of Eq. (7.34) is a translation, and the second represents a gauge transfor- mation, as expected. In deriving (7.34), we have used Eq. (7.20): fab cd = 􀀀fba cd. 54 For the gauge eld, using the FI (7.19) and some identities in Appendix A.5, we obtain [ 1; 2]A~ c d = v @ A~ c d + D (~ c d 􀀀 v A~ c d) (7.38) +v ~ F c d + " D ZA a ZbA 􀀀 ZA a D ZbA 􀀀 i Ab Aa fac bd : where ~ F c d = @ A~ c d 􀀀 @ A~ c d + [A~ ;A~ ]c d is the eld strength. We recognize the rst term as a translation, and the second a gauge transformation. To achieve the closure, we need to impose the following equation of motion for the gauge eld: ~ F c d = 􀀀" D ZA a ZbA 􀀀 ZA a D ZbA 􀀀 i Ab Aa fac bd: (7.39) As BL discovered [38], the FI implies D fca bd = 0, if one writes A~ c d = A b afca bd in the expression of the covariant derivative. We have used this important equation to derive the second term in Eq. (7.38): fca bdD b a = D ~ c d. The commutator of two supersymmetry transformations acting on the fermionic elds reads [ 1; 2] Dd = v @ Dd + (~ a d 􀀀 v A~ a d) Da 􀀀 i 2 ( AC 1 2AD 􀀀 AC 2 1AD)ECd + i 4 ( AB 1 2AB) EDd; (7.40) where ECd = D Cd + fab cd CaZD b ZcD 􀀀 2 DaZD b ZcC 􀀀 "CDEF DcZE a ZF b : (7.41) Again, the rst two term are a translation and a gauge transformation, respectively. To achieve the closure of the supersymmetry algebra, we have to impose the following equations of motion for the fermionic elds: 0 = ECd = D Cd + fab cd CaZD b ZcD 􀀀 2 DaZD b ZcC 􀀀 "CDEF DcZE a ZF b : (7.42) To derive the equations of motion of the scalar elds, we take the super-variation of the equations of motion of the fermionic elds: ECd = 0. Two equations are obtained: One is 0 = D D ZB c 􀀀 if ab cd( Ad AaZB b 􀀀 2 Bd AaZA b 􀀀 "ABCD Aa Cb ZdD ) (7.43) + 1 3 (fae fdfbd cg 􀀀 2fab cdfed fg 􀀀 2fdb gcfae fd + 2fab fdfed cg 􀀀 4feb fdfad cg) ZB e Zf AZA a Zg DZD b : The other equation is equivalent to the equation of motion of the gauge eld (7.39). 55 The equations of motion of the gauge, fermion and scalar elds, Eqs. (7.39), (7.42) and (7.43), respectively, can be derived from the Lagrangian (7.21). 7.4 Examples of the N = 6 Theories 7.4.1 N = 6, Sp(2N) U(1) We rst specify the structure constants as 2 [39] fa􀀀;b􀀀;c+;d+ = 􀀀k[(!ab!cd + !ac!bd)h􀀀+h􀀀+ + (!ad 􀀀+)(!bc 􀀀+)]; (7.44) where k is a real constant, !ab an antisymmetric bilinear form (a; b = 1; 2; ; 2N), h+􀀀 = h􀀀+ = 1 and +􀀀 = 􀀀 􀀀+ = ih+􀀀. Here a; b are the Sp(2N) indices while +;􀀀 the SO(2) indices. We use the gauge invariant antisymmetric tensor !a+;b􀀀 !abh+􀀀 to raise the rst two pairs of indices of the structure constants (7.44): fa+b+ c+d+ = k[(!ab!cd 􀀀 a c b d) + + + + 􀀀 ( a d)(􀀀i + +)( b c)(􀀀i + +)]: (7.45) Suppressing the SO(2) indices gives fab cd = k(!ab!cd + a d b c 􀀀 a c b d): (7.46) It is not too di cult to check that the structure constants satisfy the FI (7.19) and the reality condition (7.12), and also have the desired symmetry properties (7.20). In fact, in accordance with Eq. (7.11) and (7.46), the gauge elds can be decomposed into two parts: A~ c d = A b afca bd (7.47) = 􀀀(A d c + A c d) + (A a a) c d B c d + A c d: It is natural to identify the trace part A A a a as the U(1) part of the gauge potential, and B c d as the Sp(2N) part. The reason is that we can recast B c d as Aab (tab)c d, where (tab)c d is in the fundamental representation of the Lie algebra of Sp(2N). Therefore the gauge group is nothing but Sp(2N) U(1), whose Lie algebra is the bosonic part of the super Lie algebra OSp(2j2N). In section 8.1, we will use OSp(2j2N) to realize the 3-algebra used in this subsection. 2In the Lagrangian (7.21) of section 7.1, the index a runs from 1 to L. In this subsection, we split it into two indices: a ! a , and set L = 4N. We hope this will not cause any confusion. 56 We substitute the structure constants (7.46) into (7.22). We then obtain the N = 6 (on-shell) SUSY transformation law in the theory (see Appendix C.2.1). The equations of motion can be derived from the Lagrangian obtained by substituting Eq. (7.46) into the Lagrangian (7.21) and replacing A b a by 1 kA b a (see Appendix C.2.1). The SUSY transformation law (C.23) and the Lagrangian (C.22) are indeed in agreement with the N = 6; Sp(2M) U(1) superconformal CSM theory derived from the ordinary Lie algebra in Ref. [35]. 7.4.2 N = 6, U(M) U(N) The Lagrangian of this theory has been constructed in Ref. [38]. For this thesis to be self-contained, it is worth presenting the Lagrangian and SUSY transformation law of D = 3;N = 6, U(M) U(N) theory in this subsection. To generate a direct gauge group such as U(M) U(N), we split up a lower 3-algebra index a into two indices: a ! n^n, where n = 1; :::;M is a fundamental index of U(M), ^n = 1; :::;N an antifundamental index of U(N). With this decomposition, the hermitian inner product (7.16) can be written as a trace: X aYa ! X n ^nYn^n = X t ^nnYn^n Tr(XyY ); (7.48) where the superscript \t" stands for the usual transpose. On the other hand, according to the de nition (7.16), the hermitian inner product can be also written as: X aYa X aYa, which leads us to decompose an upper index a as a ! ^nn. Thus the hermitian inner product can be written as X aYa X aYa ! X ^nnYn^n Tr( X Y ) = Tr(XyY ): (7.49) We then specify the 3-bracket (7:18) to be [t ^kk; t ^l l; tm^m] = k( ^k ^m l mt ^l k 􀀀 ^l ^m k mt ^kl): (7.50) The structure constants can be easily read o as f ^kk;^l l m^ m;n^n = k( ^k ^m ^l ^n k n l m 􀀀 ^k ^n ^l ^m k m l n): (7.51) It is straightforward to check that the structure constants f ^kk;^l l m^ m;n^n satisfy the FI (7.19) and the reality conditions (7.12), and has the symmetry properties (7.20). The structure constants are rst discovered by BL [38] (though they did not write down Eq. (7.51) 57 explicitly), and they are also the same as the components of an embedding tensor in Ref. [32]. Now let us show that the 3-bracket (7.50) is indeed equivalent to Bagger and Lambert's 3-bracket [38]. Writing X = Xk^kt^kk, and Z = Z ^mm tm^m, by Eq. (7.50), one can get [X; Y ; Z] = k(X ZY 􀀀 Y ZX)n^nt^nn: (7.52) The right hand side is the ordinary matrix multiplication. It is exactly the same as eqn. (53) of Ref. [38]. In accordance with eq. (7.51), the gauge elds can be decomposed as A~ ^kk n^n = A ^mm l^l f ^kk;^l l m^ m;n^n = A ^kl l^n k n 􀀀 A ^l k n^l ^k ^n A^ ^k ^n k n + A k n ^k ^n: (7.53) So the 3-bracket (7.52) and the FI (7.19) generate a U(M) U(N) gauge group [38], with A^ ^k ^n the U(M) part and A k n the U(N) part of the gauge potential. The Lie algebra of the gauge group is the bosonic part of the super Lie algebra U(MjN). Indeed, the super Lie algebra U(MjN) can be used to realize the 3-algebra used in this subsection (see section 8.1). The supersymmetry transformation law and the Lagrangian in this theory can be obtained by substituting the expression (7.51) of the structure constants into Eqs. (7.22) and (7.21), and replacing A b a by 1 kA b a. To make the paper self-contained, we include the results in Appendix C.2.2. The SUSY transformation law (C.27) and the Lagrangian (C.24) are in agreement with the D = 3;N = 6 U(M) U(N) CSM theory, which has been derived from the ordinary Lie algebra approach in Ref. [35] and from the 3-algebra approach in Ref. [38]. This theory is conjectured to be the dual gauge theory of M2-branes a C4=Zk singu- larity. If M = N, this theory becomes the well-known ABJM theory [25, 29, 30]. CHAPTER 8 N=6, 8 THEORIES IN TERMS OF THE BOSONIC PARTS OF SUPERALGEBRAS In this chapter, we rst try to nd a super Lie algebra which can be used to realize the hermitian 3-algebra and the Nambu 3-algebra. We then derive the ordinary Lie algebra constructions of the N = 6; 8 theories by using the super-Lie-algebra realization of 3-algebras. 8.1 N = 6 Theories in Terms of the Bosonic Parts of Superalgebras In this section, we derive the super Lie algebra used to realize the hermitian 3-algebra and the Nambu 3-algebra by decomposing the super Lie algebra (5.2), and classify the gauge groups of the N = 6 theories. For convenience, we cite the super Lie algebra (5.2) used to realize the symplectic 3-algebra here: [Mm;Mn] = Cmn sMs; [Mm;QI ] = 􀀀 m IJ!JKQK; fQI ;QJg = m IJkmnMn: (8.1) If we use the fermionic generators QI of (8.1) to realize the 3-algebra generators TI in Eq. (7.13), i.e., TI := QI , then Eq. (7.13) becomes QI = Qa = !a ;b Qb = Qa2 1 􀀀 Qa1 2 : (8.2) Recall that QI furnish a pseudo-real (quaternion) representation of the bosonic part of the super Lie algebra (8.1), and we decompose this pseudo-real representation into a complex representation and its complex conjugate representation for promoting the N = 5 supersymmetry to N = 6. So, with the decomposition (8.2), if the fermionic generators Qa1 furnish a complex representation of the bosonic part of (8.1), then Qa2 59 must furnish a complex conjugate representation of the bosonic part of (8.1). Namely, if we de ne Qa = Qa1 and Q a = Qa2; (8.3) we must have [Mm;Qa] = 􀀀 ma bQb and [Mm; Q a] = mb a Q b; (8.4) where ma b are anti-hermitian, i.e., ma b = 􀀀 mb a: (8.5) Substituting QI = Q a 1 􀀀 Qa 2 (8.6) into the LHS of the second equation of (8.1) gives [Mm; Q a 1 􀀀 Qa 2 ] = mb a Q b 1 + ma bQb 2 : (8.7) Comparing the RHS with the RHS of the second equation of (8.1), we obtain mJ I = mb a 1 1 􀀀 ma b 2 2 : (8.8) By (8.5), the RHS is a direct sum of mb a and its complex conjugate. So the pseudo- real representation is indeed decomposed into a complex representation and its complex conjugate representation. Substituting (8.6) and (8.8) into the LHS and RHS of the third equation of (8.1), respectively, we obtain f Q a 1 􀀀 Qa 2 ; Q b 1 􀀀 Qb 2 g = 􀀀( mb akmnMn 1 2 + ma bkmnMn 2 1 ); (8.9) where we have used Eq. (7.5). The anticommutators can be easily read o from the above equation: fQb; Q ag = mb akmnMn; f Q a; Q bg = fQa;Qbg = 0: (8.10) In summary, the super Lie algebra used to realize the N = 6 (hermitian) 3-algebra is the following: [Mm;Mn] = Cmn sMs; [Mm;Qa] = 􀀀 ma bQb; [Mm; Q a] = mb a Q b; fQa; Q bg = ma bkmnMn; f Q a; Q bg = fQa;Qbg = 0: (8.11) In this way, we rederive the above super Lie algebra by decomposing the super Lie algebra (8.1) properly. The super Lie algebras OSp(2j2N) and U(MjN) (or its cousins SU(MjN) and PSU(MjN)) take the form of (8.11). 60 With these decompositions, the double graded commutator (see section 5.1) [fQI ;QJg;QK] = kmn m IJ nK LQL (8.12) is decomposed into two sets: [fQb; Q ag;Qc] = 􀀀kmn mb a nc dQd; [fQa; Q bg; Q c] = kmn ma b nd c Q d: (8.13) However, their structure constants are related by a reality condition (see Eqs. (8.16) and (8.21)). So we need only to consider the rst equation. Recall that we use the the dou- ble commutator to realize the symplectic 3-bracket, i.e., [TI ; TJ ; TK] := [fQI ;QJg;QK]. Comparing the decomposition of (8.12) with (7.17), we are led to the following equations: [tb; tc; ta] := [fQb; Q ag;Qc] = 􀀀kmn mb a nc dQd; (8.14) [tb; tc; ta] := 􀀀[fQa; Q bg; Q c] = 􀀀kmn ma b nd c Q d: (8.15) where the LHS of the rst equation is the 3-bracket of the hermitian 3-algebra, and ta are the generators of the hermitian 3-algebra (see section 7.1). The structure constants can be read o immediately: fbc ad = 􀀀kmn mb a nc d: (8.16) It is straightforward to verify that the above tensor product is a solution of the FI (7.19) of the hermitian 3-algebra (for convenience, we cite it here): ffc dgfag eb 􀀀 faf gbfgc de + fcf egfag db 􀀀 fac gbfgf ed = 0: (8.17) The solution (8.16) is rst discovered by BL [38], using a di erent approach. Similarly, the QIQJQK Jacobi identity is decomposed into two sets: the QbQc Q a Jacobi identity and the Q b Q cQa Jacobi identity. Let us examine the QbQc Q a Jacobi identity: [fQb; Q ag;Qc] + [f Q a Qcg;Qb] + [fQc;Qbg; Q a] = 0: (8.18) By fQc;Qbg = 0, the last term of the LHS vanishes. The equation for the remaining two terms implies that kmn mb a nc d + kmn mc a nb d = 0: (8.19) Namely, the structure constants fbc ad are antisymmetric in the rst two indices and in the last two indices: fbc ad = 􀀀fcb ad = fcb da: (8.20) 61 Also, the reality condition (8:5) implies that the structure constants satisfy the reality condition: f ab cd = fcd ab: (8.21) Eqs (8.20) and (8.21) are nothing but Eqs (7.20) and (7.12), respectively. Here we would like to demonstrate that the FI (8.17), satis ed by the structure constants, is equivalent to the MMQ or MM Q Jacobi identity of the super Lie algebra (8.11). With Eq. (8.6), the FI (5.6) is decomposed into eight sets; one of them reads [f Q a;Qbg; [f Q e;Qf g;Qc]] (8.22) = [f[f Q a;Qbg Q e];Qf g;Qc] + [f Q e; [f Q a;Qbg;Qf ]g;Qc] + [f Q e;Qf g; [f Q a;Qbg;Qc]]: Substituting (8.13) into this equation shows that it precisely coincides with the FI (8.17). The rest (seven) sets can be also converted into the FI (8.17). So it is su cient to examine Eq. (8.22). On the other hand, by using the super Lie algebra (8.11), one can convert Eq. (8.22) into the following equation: mb a nf e([Mn; [Mm;Qc]] 􀀀 [Mm; [Mn;Qc]] + [[Mm;Mn];Qc]) = 0; (8.23) which is the MMQ Jacobi identity of the super Lie algebra (8.11). One can also derive the above equation by decomposing Eq. (5.7). With Qc replaced by Q c, Eq. (8.22) becomes another set FI decomposed from (5.6). It can be converted into MM Q Jacobi identity of (8.11). Therefore, the FI (8.17) is indeed equivalent to the MMQ or MM Q Jacobi identity of the super Lie algebra (8.11). For a more mathematical approach, see Ref. [37, 42, 46], in which the relations between the hermitian 3-algebras and Lie superalgebras are discussed by using Lie algebra representation theories. Substituting Eq. (8.16) into the Lagrangian (7.21) and the SUSY law (7.22) gives the ordinary Lie algebra constructions of the N = 6 theories. The bosonic parts of the super Lie algebras OSp(2j2N) and U(MjN) (or its cousins SU(MjN) and PSU(MjN)) can be selected as the Lie algebras of the gauge groups of the N = 6 theories (see (8.11)). In particular, if the super Lie algebra is PSU(2j2), then (8.14) becomes the Nambu bracket, and the N = 6 supersymmetry gets enhanced to N = 8. This will be the topic of the next section. 62 8.2 N = 8 Theory in Terms of the Bosonic Part of PSU(2j2) In section (7.2), we have realized the Nambu 3-algebra in terms of a set of SU(2) SU(2) -matrices. In this section, we show explicitly that the Nambu 3-algebra can be realized in term of PSU(2j2). Some useful (anti-) commutators of PSU(2j2) generators are [M ;Q _ ] = Q _ 􀀀 1 2 Q _ ; [M_ _ ;Q _ ] = 􀀀 _ _Q _ + 1 2 _ _ Q _ ; fQ _ ; Q _ g = _ _M + M_ _ ; (8.24) fQ _ ;Q _ g = 0; where ; _ = 1; 2 are SU(2) SU(2) indices. We use the antisymmetric matrix ( _ _ ) to lower undotted (dotted) indices. De ne the SU(2) SU(2) -matrices as (see section 7.2): a _ = ( 1; 2; 3; iI); ay _ = ( 1; 2; 3;􀀀iI) ab = 1 4 ( a by 􀀀 b ay) ; ab _ _ = 1 4 ( ay b 􀀀 by a) _ _ ; (8.25) where ab and ab satisfy the further `duality' conditions ab = 􀀀 1 2 "abcd cd; ab = 1 2 "abcd cd: (8.26) Since we wish to work in the vector representation of SO(4), it is useful to de ne Qa = 1 2 ay _ Q _ ; Q a = 1 2 a _ Q _ ; Mab = 􀀀( ab M + ab _ _ M_ _ ) M = Mab ab ; M _ _ = Mab ab _ _ : (8.27) After some work, we obtain [tb; tc; ta] := [fQb; Q ag;Qc] = 1 2 "bcadQd: (8.28) Namely, the double graded commutator is indeed a realization of the Nambu 3-bracket. Also, the FI satis ed by the Nambu 3-bracket is equivalent to theMMQ Jacobi identity of (8.24), as we proved in the last section. Therefore the Nambu 3-algebra is realized in terms of the super Lie algebra PSU(2j2). Hence the bosonic part of PSU(2j2), SU(2) SU(2) = SO(4), is the Lie algebra of the gauge group of the N = 8 BLG theory. And the 63 matter elds are in the bifundamental representation of SU(2) SU(2) or the vector representation of SO(4). The same theory is obtained in Ref. [34] by promoting the N = 4 supersymmetry to N = 8. Eq. (8.28) may be counterintuitive at rst sigh, since the anticommutator satis es fQb; Q ag = f Q a;Qbg, i.e., it seems that it is symmetric in ab. However, there is no clash with fact that (8.28) is antisymmetric in ab if we notice that fQb; Q ag = 1 4 "bacdMcd = 􀀀fQa; Q bg = f Q a;Qbg; (8.29) namely, the last two anticommutators are di erent. It is well known that the Nambu 3-bracket is di cult to quantize [12, 13]. However, if we promote the fermionic and bosonic generators of (8.24) as quantum mechanical operators, and promote (8.28) as a quantum mechanical double graded commutator, our approach may provide a quantization scheme for the Nambu 3-bracket. Similarly, the 3-brackets of the symplectic and hermitian 3-algebras may be quantized in the same fashion. CHAPTER 9 CONCLUSIONS In this thesis, we have combined the symplectic 3-algebra with the superspace formal- ism by letting the matter super elds take values in the symplectic 3-algebra. Based on the 3-algebra, we then have constructed the general N = 5 CMS theory by enhancing the N = 1 supersymmetry to N = 5. The N = 5 Lagrangian is same as the one derived with an on-shell approach [40]. We have constructed the general N = 4 CSM theory by decomposing one N = 5 multiplet into a N = 4 untwisted hypermultiplet and a N = 4 twisted hypermultiplet, and then proposing a new superpotential. In deriving the general N = 4 CSM theory, we have also decomposed the set of 3-algebra generators into two sets of 3-algebra generators. As a result, both the FIs and 3-brackets are decomposed into 4 sets. The resulting general N = 4 CSM theory is a quiver gauge theory based on the 3-algebra. We have also examined the closure of the N = 4 algebra. We then have realized the symplectic 3-algebra in terms of the super Lie algebra (5.2). The 3-bracket is realized in terms of a double graded bracket: [TI ; TJ ; TK] := [fQI ;QJg;QK], where QI are the fermionic generators; the structure constants of the 3-algebra are just the structure constants of the double graded bracket, i.e., fIJKL = kmn m IJ nK L. The fundamental identity of the 3-algebra is equivalent to the MMQ Jacobi identity of the super Lie algebra, where Ms are the bosonic generators in the super Lie algebra. The linear constraint equation f(IJK)L = 0, required by the enhancement of the supersymmetry, is equivalent to the QQQ Jacobi identity. We have constructed a new super Lie algebra by requiring that the bosonic subalgebras of two simple super Lie algebras share one simple factor. The bosonic part of this new super algebra can be selected as the Lie algebra of the N = 4 quiver gauge theory. We have demonstrated how to `fuse' two simple super Lie algebras into single one, by constructing an explicit example of this new super Lie algebra. 65 We have also analyzed the relations between the symplectic 3-algebra and the ordinary Lie algebra. The fundamental identity of 3-algebra can be solved in terms of a tensor product: fIJKL = kmn m IJ nK L. We have proved that the structure constants fIJKL furnish a quaternion representation of the bosonic part of the super Lie algebra (5.2), and fIJKL also play a role of Killing-Cartan metric. We found that the FI of the 3-algebra can be converted into an ordinary commutator (5.19); the structure constants of the commutator are (5.21). The FI of the 3-algebra can be understood as the statement that the structure constants of the commutator (5.21) are total antisymmetric (see Eqs. (5.22)). We have proved that the components of an embedding tensor [31, 32], used to construct the D = 3 extended supergravity theories, are just the structure constants of the 3- algebra. Hence the concepts and techniques of the 3-algebra may be used to construct new D = 3 extended supergravity theories. We have succeeded in enhancing the N = 5 supersymmetry to N = 6 by decomposing the sympelctic 3-algebra and the elds properly. At the same time, we also demonstrate that the FI and the symmetry and reality properties of the structure constants of the N = 6 hermitian 3-algebra can be derived from the N = 5 (symplectic) counterparts. In the particular case of fabcd / "abcd, the N = 6 supersymmetry is promoted to N = 8, hence the N = 6 theory becomes the N = 8 BLG theory. We have shown that the (N = 6) hermitian 3-algebra and the Nambu algebra can be also realized in terms of super Lie algebras, and introduced a scheme for quantizing the 3-brackets. The general N = 5; 6; 8 CSM theories in terms of ordinary Lie algebras are rederived. We have been able to derive all known N = 4; 5; 6; 8 superconformal Chern-Simons matter theories, as well as some new N = 4 quiver gauge theories. Thus our superspace for- mulation for the super-Lie-algebra realization of symplectic 3-algebras provides a uni ed framework of all known N = 4; 5; 6; 8 CSM theories, including new examples of N = 4 quiver gauge theories as well. It would be nice to investigate the physical signi cance of this uni ed framework. It would be nice to redrive them by brane constructions, and to nd out their gravity duals. (Most of them are not found yet.) It would also be very interesting to investigate the integrability of these theories. The `fused' super Lie algebra might be independently interesting in its own right. It would be nice to investigate its structure in detail. 66 The `meshy' quiver diagram (6.26) is just a special example. It would be nice to work out the most general structure of the `meshy' quiver diagrams. APPENDIX A CONVENTIONS AND USEFUL IDENTITIES A.1 Spinor Algebra In 1 + 2 dimensions, the gamma matrices are de ned as ( ) ( ) + ( ) ( ) = 2 : (A.1) For the metric we use the (􀀀;+; +) convention. The gamma matrices in the Majorana representation can be de ned in terms of Pauli matrices: ( ) = (i 2; 1; 3), satisfying the important identity ( ) ( ) = + " ( ) : (A.2) We also de ne " = 􀀀" . So " " = 􀀀2 . We raise and lower spinor indices with an antisymmetric matrix = 􀀀 , with 12 = 􀀀1. For example, = and = ( ) , where is a Majorana spinor. Notice that = (I;􀀀 3; 1) are symmetric in . A vector can be represented by a symmetric bispinor and vice versa: A = A ; A = 􀀀 1 2 A : (A.3) We use the following spinor summation convention: = ; = ( ) ; (A.4) where and are anticommuting Majorana spinors. In 1 + 2 dimensions the Fierz transformations are ( ) = 􀀀 1 2 ( ) 􀀀 1 2 ( ) ; (A.5) ( 1 2)( 3 4) = ( 1 2)( 4 3) = 􀀀 1 2 ( 1 3)( 4 2) 􀀀 1 2 ( 1 3)( 4 2); ( 1 2)( 3 4) = 􀀀 1 2 ( 1 3)( 4 2) 􀀀 1 2 ( 1 3)( 4 2) + 1 2 " ( 1 3)( 4 2): 68 A.2 The N = 1 Superspace In this subsection, we mainly follow the conventions of Ref. [35]. We denote the superspace coordinates as . A real scalar super eld can be expanded as = + i 􀀀 i 2 2F; (A.6) where and are Majorana spinors. The superalgebra fQ ;Q g = 􀀀2 P (A.7) can be realized in terms of superspace derivatives: Q = i@ + @ : (A.8) The supercovariant derivative must anticommute with Q ; it takes the following form: D = @ + i @ : (A.9) The supersymmetry transformation of is de ned as = 􀀀i Q + i 􀀀 i 2 2 F: (A.10) Equating powers of gives the supersymmetry transformations of the component elds: = i ; (A.11) = 􀀀@ 􀀀 F ; (A.12) F = i @ : (A.13) In the Wess-Zumino gauge, the superconnection becomes 􀀀 = i A + 2 ; (A.14) and the supersymmetry transformations for the component elds are A = 􀀀i ( ) ; (A.15) = 􀀀 1 2 F ( ) : (A.16) The Berezin integral is de ned as Z d2 2 = 􀀀4: (A.17) The superpotential is given by LW = i 2 Z d2 W( ) = 􀀀 i 2 W00( ) 2 􀀀W0( )F: (A.18) 69 A.3 SU(2) SU(2) Identities We de ne the 4 sigma matrices as a A _B = ( 1; 2; 3; iI); (A.19) by which one can establish a connection between the SU(2) SU(2) and SO(4) group. These sigma matrices satisfy the following Cli ord algebra: a A _C by _C B + b A _C ay _C B = 2 ab A B; (A.20) ay _A C b C _B + by _A C a C _B = 2 ab _A _B : (A.21) We use antisymmetric matrices AB = 􀀀 AB = 0 􀀀1 1 0 and _A _B = 􀀀 _A _B = 0 1 􀀀1 0 (A.22) to raise or lower undotted and dotted indices, respe