A Detailed Investigation of First and Second Order Supersymmetries for Off-Shell N = 2 and N = 4 Supermultiplets
S. James Gates Jr., James Parker, Vincent G. J. Rodgers, Leo Rodriguez, Kory Stiffler
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November 19, 2018 UMDEPP-11-0091106.5475 [hep-th]
A Detailed Investigation ofFirst and Second Order Supersymmetries forOff-Shell N = 2 and N = 4 Supermultiplets
S. James Gates, Jr. † , James Parker † , Vincent G. J. Rodgers ∗ ,Leo Rodriguez ∗∗ , and Kory Stiffler †† Department of Physics, University of Maryland, College Park, MD 20742-4111 ∗ Department of Physics and Astronomy, The University of Iowa, Iowa City, IA 52242 ∗∗ Department of Physics, Grinnell College, Grinnell, IA 50112-1690
ABSTRACT
This paper investigates the d = 4, N = 4 Abelian, global Super-Yang Millssystem (SUSY-YM). It is shown how the N = 2 Fayet Hypermultiplet (FH) and N = 2 vector multiplet (VM) are embedded within. The central charges andinternal symmetries provide a plethora of information as to further symmetriesof the Lagrangian. Several of these symmetries are calculated to second order.It is hoped that investigations such as these may yield avenues to help solvethe auxiliary field closure problem for d = 4, N = 4, SUSY-YM and the d = 4, N = 2 Fayet-Hypermultiplet, without using an infinite number of auxiliaryfields. [email protected] [email protected] [email protected] [email protected] corresponding author: kstiffl[email protected] Introduction
The N = 4 Super-Yang Mills (SUSY-YM) system is a very active area of study, andhas become even more so over the past decade with the emergence of the AdS/CF T correspondence [1]. One very powerful aspect of this correspondence is that it relatesa perturbation theory to a strongly coupled system. As N = 4 SUSY-YM is aconformal field theory, an important undertaking has been to find dualities betweenstring theory and theories that are more QCD-like . Klebanov and Strassler tooka step in this direction in [2], where they unveiled a background which breaks thesupersymmetry to N = 1, while regulating the IR divergence behavior. Following thiswork, several other supersymmetry breaking backgrounds were discovered [3, 4, 5, 6].In parallel to the unveiling of these duality backgrounds, specific calculations weredone showing duality to confining gauge theory calculations. Herzog and Klebanovshowed duality in the tree level energy calculations between branes on the supergravityside and confining strings on the gauge theory side [7, 8]. In this newly emerginggauge/gravity picture, Regge trajectories were resurrected from the old dual resonancemodels and reinvestigated by Pando Zayas, Sonnenschein, and Vaman in [9], includingsome one loop level calculations. Most recently, one loop corrections to the k -stringenergy has been investigated, the so-called L¨uscher term. This emerges on the stringtheory side through the bosonic part of the D-brane energy, although in additiondifferent one loop information of the fermionic part has also been unveiled [10, 11,12, 13]. So we see a nice picture developing showing dualities between objects on thestring theory and gauge theory sides.In this paper, we take a step back from this picture. Even though this is thebest understood of the gauge/gravity dualities, the d = 4, N = 4 SUSY-YM theorypart of the correspondence itself still has unknown attributes. Most glaring is theauxiliary field closure problem: it is still unknown how to augment this theory withfinite numbers of auxiliary fields such that the charges satisfy the following algebra : { Q Ia , Q Jb } = 2 i δ IJ ( γ µ ) ab ∂ µ (1.1)This is a problem which has been well known for at least thirty years. In 1981, Siegeland Rocek (SR) investigated a solution within the known framework that existed atthe time and found a no-go theorem [14]. This result has been interpreted as thedefinitive statement on this issue.However, there are some loose ends that challenge this conventional wisdom aboutthe SR no-go theorem. The first of these is contained within the SR work itself. In1n often overlooked final commentary in the work, the authors state a possible wayto avoid the SR no-go theorem. It is also often overlooked that the derivation of theSR no-go theorem is based on a particular assumption of dynamics. In particular,the authors assume the gauge field is subject to the dynamics of the usual Yang-Millsaction. It is simple to consider a different starting point. It is easy to negate thisassumption.Though mostly unknown, the action for the ABJM model [15] together with adiscussion of 3D, N = 6 superconformal invariance first appeared in works written inthe period of 1991-1995 on the importance of Chern-Simons models [16, 17, 18, 19]. Soinstead of considering the fields of a vector multiplet in 4D hypermultiplet in 4D thatrealizes N = 2 SUSY, one could attempt to construct respective 3D Chern-Simonsmodels with N = 8 SUSY or N = 4 SUSY that are based on the dimensional reductionof 4D multiplets. The SR no-go theorem cannot be applied to such constructions!Thus, the study of 3D Chern-Simons theories provides a new way to attack this veryold problem.The methods in harmonic [20, 21] or projective [22, 23] superspace absolutelyoffer solutions, however these add an infinite number of auxiliary fields. In this paperwe offer an in-depth analysis of the Lagrangian symmetries generated by the centralcharges and internal symmetries of the algebra as a possible window into algebraicclosure with a finite number of auxiliary fields. To the knowledge of the authors, thesesymmetries have never been discussed in this detail; almost certainly not in the 4-DMajorana component notation that is used in this paper. In short, we are trying topush the bounds of understanding further as to precisely how the algebra fails to closewith a finite number of auxiliary fields. Furthermore, this paper analyzes the centralcharges and internal symmetries, or lack thereof, of other SUSY systems embeddedinto the overarching d = 4 N = 4 SUSY-YM system.This paper is structured as follows. We begin by showing how the Abelian d = 4, N = 4 super Yang-Mills (SUSY-YM) system can be made to split into the N = 2vector multiplet (VM), which closes, and the N = 2 Fayet Hypermultiplet (FH)systems, which doesn’t [24]. Then we quote the main result: the recovery of many firstand second order supersymmetries from the central charges and internal symmetriesof this algebra.Unless otherwise specified throughout the document, our notation convention is asfollows. Capital Latin indices are euclidean and go from one to three: I, J, K, M, · · · =1 , ,
3. Lower case Latin indices i, j, k, m, · · · = 1 , a, b, c, d, · · · = 1 , , ,
4, ranging from one to four. Greek indices are fourdimensional Minkowski space-time indices and go from zero to three: µ, ν, α, β, · · · =0 , , ,
3. Symmetrization and antisymmetrization are defined without normalization:Λ ( µν ) = Λ µν + Λ νµ (1.2)Λ [ µν ] = Λ µν − Λ νµ (1.3) N = 4 SUSY-YM to N = 2 FH andVM
In this section, the algebra for N = 4 is laid out in component notation. TheLagrangian is presented which is globally invariant to these transformations. Next,the algebra is uncovered, which of course does not close. Finally, it is shown howthis algebra splits into both the N = 2 FH and N = 2 VM multiplets, the latter ofwhich closes, the former which does not. It is commented on how after reduction tothe FH system, certain central charges and internal symmetries are removed from thealgebra. Of course, all central charges and internal symmetries are removed from thealgebra under reduction to the N = 2 VM multiplet. N = 4 Transformation Laws
The Lagrangian for the Abelian d = 4, N = 4 SUSY-YM system L = − ( ∂ µ A J )( ∂ µ A J ) − ( ∂ µ B J )( ∂ µ B J )+ i ( γ µ ) ab ψ Ja ∂ µ ψ Jb + ( F J ) + ( G J ) − F µν F µν + i ( γ µ ) cd λ c ∂ µ λ d + d (2.1)is invariant with respect to the global supersymmetric transformationsD a A J = ψ Ja , D a B J = i ( γ ) ab ψ Jb , D a ψ Jb = i ( γ µ ) a b ∂ µ A J − ( γ γ µ ) a b ∂ µ B J − i C a b F J + ( γ ) a b G J , D a F J = ( γ µ ) ab ∂ µ ψ Jb , D a G J = i ( γ γ µ ) ab ∂ µ ψ Jb . (2.2)3 a A µ = ( γ µ ) ab λ b , D a λ b = − ( σ µν ) ab F µν + ( γ ) a b d , D a d = i ( γ γ µ ) ab ∂ µ λ b . (2.3)D Ia A J = δ IJ λ a − ǫ IJK ψ Ka , D Ia B J = i ( γ ) ab [ δ IJ λ b + ǫ IJK ψ Kb ] , D Ia ψ Jb = δ IJ [ ( σ µν ) ab F µν + ( γ ) a b d ] − ǫ IJK [ − i ( γ µ ) a b ∂ µ A K − ( γ γ µ ) a b ∂ µ B K + i C a b F K + ( γ ) a b G K ] , D Ia F J = ( γ µ ) ab ∂ µ [ δ IJ λ b − ǫ IJK ψ Kb ] , D Ia G J = i ( γ γ µ ) ab ∂ µ [ − δ IJ λ b + ǫ IJK ψ Kb ] . (2.4)D Ia A µ = − ( γ µ ) ab ψ Ib , D Ia λ b = i ( γ µ ) a b ∂ µ A I − ( γ γ µ ) a b ∂ µ B I − i C a b F I − ( γ ) a b G I , D Ia d = i ( γ γ µ ) ab ∂ µ ψ Ib . (2.5)where σ µν = i [ γ µ , γ ν ] , F µν = ∂ µ A ν − ∂ ν A µ . (2.6)and our conventions for the gamma matrices are as in Appendix A of [25].These transformations are known as zeroth order symmetries of the Lagrangian.The main result of this paper will be the first and second order symmetries of theLagrangian, and how they can be recovered from the algebra. In this section, we will discover the central charges and internal symmetries of thisalgebra which will lead us to the Lagrangian symmetries in section 3. Using theshorthand χ = ( A I , B I , F I , G I , d , ψ Jc , λ c ) , (2.7)4he algebra can be written { D a , D b } χ = 2 i ( γ µ ) ab ∂ µ χ, { D a , D b } A ν = 2 i ( γ µ ) ab F µν (2.8)and { D Ia , D Jb } A K =2 iδ IJ ( γ µ ) ab ∂ µ A K − ǫ IJK ( γ ) ab d+ − Z IJKM [ iC ab F M + ( γ ) ab G M ] , { D Ia , D Jb } B K =2 iδ IJ ( γ µ ) ab ∂ µ B K + 2 iǫ IJK C ab d , { D Ia , D Jb } F K =2 iδ IJ ( γ µ ) ab ∂ µ F K + 2 ǫ IJK ( γ γ µ ) ab ∂ µ d++ 2 Z IJKM [ − iC ab (cid:3) A M + ( γ γ µ ) ab ∂ µ G M ] { D Ia , D Jb } G K =2 iδ IJ ( γ µ ) ab ∂ µ G K − ǫ IJK ( γ γ µ ) ab ∂ ν F µν + − Z IJKM [( γ ) ab (cid:3) A M + ( γ γ µ ) ab ∂ µ F M ] (2.9) { D Ia , D Jb } d =2 iδ IJ ( γ µ ) ab ∂ µ d++ 2 ǫ IJK (( γ ) ab (cid:3) A K − iC ab (cid:3) B K + ( γ γ µ ) ab ∂ µ F K ) { D Ia , D Jb } A ν =2 iδ IJ ( γ µ ) ab F µν ++ 2 ǫ IJK ( iC ab ∂ ν A K + ( γ ) ab ∂ ν B K − ( γ γ ν ) ab G K ) { D Ia , D Jb } λ c =2 iδ IJ ( γ µ ) ab ∂ µ λ c + iǫ IJK [ − C ab ( γ µ ) dc + ( γ ) ab ( γ γ µ ) dc ++ ( γ γ ν ) ab ( γ γ ν γ µ ) dc ] ∂ µ ψ Kd { D Ia , D Jb } ψ Kc =2 iδ IJ ( γ µ ) ab ∂ µ ψ Kc − iǫ IJK [ − C ab ( γ µ ) dc + ( γ ) ab ( γ γ µ ) dc ++ ( γ γ ν ) ab ( γ γ ν γ µ ) dc ] ∂ µ λ d + − iZ IJKM [ C ab ( γ µ ) dc + ( γ ) ab ( γ γ µ ) dc ++ ( γ γ ν ) ab ( γ γ ν γ µ ) dc ] ∂ µ ψ Md (2.10)and for the cross terms { D a , D Ib } A J =2 iǫ IJK C ab F K { D a , D Ib } B J =2 iǫ IJK C ab G K { D a , D Ib } F J =2 iǫ IJK C ab (cid:3) A K { D a , D Ib } G J =2 iǫ IJK C ab (cid:3) B K { D a , D Ib } λ c =0 (2.11)5 D a , D Ib } d =0 { D a , D Ib } A ν =2 iC ab ∂ ν A I − γ ) ab ∂ ν B I { D a , D Ib } ψ Jc =2 iǫ IJK C ab ( γ µ ) dc ∂ µ ψ Kd (2.12)where Z IJKM ≡ δ IM δ JK − δ IK δ JM (2.13) We will use the notation ( A J , F K ) to indicate, for instance, the presence of a non-zeroterm involving the field F K on the right hand side of the anti-commutator { D Ia , D Jb } A K and vice-versa. In this notation, we list the following fields which are coupled througha central charge or internal symmetry:( A J , F K ) , ( A J , G K ) , ( B J , G K ) , ( A J , d) , ( B J , d) , ( G J , A µ ) , ( F J , G K ) , ( F J , d) , ( ψ Ja , λ b ) , ( ψ Ja , ψ Kb ) fields coupled by a central chargeor internal symmetry (2.14)In addition, the algebra couples the following fields through a U (1) gauge symmetry( A µ , A K ) , ( A µ , B K ) , fields coupled through a gauge symmetry (2.15)In section 3, we will show how these central charges and internal symmetries canbe used to uncover several first and second order Lagrangian symmetries. We notethat this algebra is absent of central charges and internal symmetries between( F J , A µ ) , ( A µ , d) , ( B J , F K ) , ( B J , A K ) , ( G J , d) fields not coupled througha central charge or internal symmetry (2.16) N = 2 Systems
Before we fully investigate the first and second order Lagrangian symmetries, we willinvestigate how to split the N = 4 system into the N = 2 FH and VM systems.When we do this, some of the central charges and internal symmetries vanish. Infact, in the case of the N = 2 VM system all of these vanish, and the algebra has no information on first and second order Lagrangian symmetries. This is of coursebecause the N = 2 VM algebra closes. 6irst making the following definitions˜D a ≡ D a , ˜D a ≡ D a (2.17)where i = 1 , N = 2VM system is composed of half of the fields of the N = 4 system: A ≡ A , B ≡ B , F ≡ F , G ≡ G ,A µ , d , ζ a ≡ ψ a , ζ a ≡ λ a (2.18)and the embedded N = 2 FH system is composed of the other half˜ A ≡ A , ˜ A ≡ A , ˜ B ≡ B , ˜ B ≡ B , ˜ F ≡ F , ˜ F ≡ F , ˜ G ≡ G , ˜ G ≡ G , ˜ ψ a ≡ ψ a , ˜ ψ a ≡ ψ a (2.19) N = 2 VM The resulting N = 2 VM algebra is˜D ia A = ζ ia , ˜D ia B = i ( γ ) ba ζ ib , ˜D ia F = ( γ µ ) ba ∂ µ ζ ib , ˜D ia G = i ( σ ) ij ( γ γ µ ) ba ∂ µ ζ jb , ˜D ia A µ = i ( σ ) ij ( γ µ ) ba ζ jb , ˜D ia d = i ( σ ) ij ( γ γ µ ) ba ∂ µ ζ jb , ˜D ia ζ jb = δ ij ( i ( γ µ ) ab ∂ µ A − ( γ γ µ ) ab ∂ µ B − iC ab F ) + ( σ ) ij ( γ ) ab G + − i ( σ ) ij ( σ µν ) ab F µν + ( σ ) ij ( γ ) ab d , (2.20)where( σ ) ij = ! , ( σ ) ij = − ii ! , ( σ ) ij = − ! , (2.21)and ζ b = ψ b , ζ b = λ b . (2.22)7he algebra reduces to { ˜D ia , ˜D jb }V = 2 iδ ij ( γ µ ) ab ∂ µ V (2.23) { ˜D ia , ˜D jb } A ν = 2 iδ ij ( γ µ ) ab F µν + i ( σ ) ij (2 iC ab ∂ ν A − γ ) ab ∂ ν B ) . (2.24)where V = ( A, B, F, G, d , ψ c , λ c ) . (2.25)So this algebra closes up to gauge transformations and all the central charges andinternal symmetries from the overarching N = 4 algebra have vanished, aside fromthe U (1) gauge symmetries. The algebra, therefore, contains no information on extrasymmetries of the Lagrangian. N = 2 FH The transformation laws for the embedded N = 2 FH system are˜D ia ˜ A j = δ ij ˜ ψ a + i ( σ ) ij ˜ ψ a , ˜D ia ˜ B j = i ( γ ) ba [ ( σ ) ij ˜ ψ b + ( σ ) ij ˜ ψ b ] , ˜D ia ˜ F j = ( γ µ ) ba ∂ µ [ δ ij ˜ ψ b + i ( σ ) ij ˜ ψ b ] , ˜D ia ˜ G j = i ( γ γ µ ) ba ∂ µ [ ( σ ) ij ˜ ψ b + ( σ ) ij ˜ ψ b ] , ˜D ia ˜ ψ b = i ( γ µ ) ab ∂ µ ˜ A i − iC ab ˜ F i + ( σ ) ij [( γ ) ab ˜ G j − ( γ γ µ ) ab ∂ µ ˜ B j ] , ˜D ia ˜ ψ b = ( σ ) ij [ − ( γ µ ) ab ∂ µ ˜ A j + C ab ˜ F j ] + ( σ ) ij [( γ ) ab ˜ G j − ( γ γ µ ) ab ∂ µ ˜ B j ] (2.26)with algebra { ˜D ia , ˜D jb } ˜ A k = 2 iδ ij ( γ µ ) ab ∂ µ ˜ A k − i ˜ Z ijkm C ab ˜ F m , { ˜D ia , ˜D jb } ˜ B k = 2 iδ ij ( γ µ ) ab ∂ µ ˜ B k − i ˜ Z ijkm C ab ˜ G m , { ˜D ia , ˜D jb } ˜ F k = 2 iδ ij ( γ µ ) ab ∂ µ ˜ F k − i ˜ Z ijkm C ab (cid:3) ˜ A m , { ˜D ia , ˜D jb } ˜ G k = 2 iδ ij ( γ µ ) ab ∂ µ ˜ G k − i ˜ Z ijkm C ab (cid:3) ˜ B m , { ˜D ia , ˜D jb } ˜ ψ kc = 2 iδ ij ( γ µ ) ab ∂ µ ˜ ψ c − i ˜ Z ijkm C ab ( γ µ ) dc ∂ µ ˜ ψ md (2.27)where ˜ Z ijkm ≡ δ im δ jk − δ ik δ jm , i, j, k, m = 1 , . (2.28)So only the couplings ( A J , G K ) and ( F J , G K ) have vanished from the overarching N = 4 theory. Couplings still remain between ( ˜ A j , ˜ F k ) and ( ˜ B j , ˜ G k ) and ( ˜ ψ ia , ˜ ψ jb ).8 Extra Symmetries of the Lagrangian
Here begins the main result of the paper. We list the first order bosonic symmetriesunveiled directly by the central charges and internal symmetries. We next calculatefrom these symmetries first order fermionic and second order bosonic symmetries ofthe Lagrangian. We will notice that more symmetries exist which are not revealedby this algebra. We discuss the N = 4 SUSY-YM system first and the N = 2 FHsystem last. Contracting the coupling from the anticommutator on A J and F J in Eq. (2.11) withthe Grassmann spinors ε a and χ bI results in the first order bosonic symmetry of theLagrangian δ (1) BS a A J F J ! ≡ ε a χ bI i { D a , D Ib } A J F J ! = ε a χ bI ǫ IJK C ab F K (cid:3) A K ! . (3.1)Interestingly, contracting the coupling from the anticommutators on A K and F K in Eq. (2.9) with the Grassmann spinors ε aI and χ bJ results in a very similar first orderbosonic symmetry of the Lagrangian δ (1) BS b A K F K ! ≡ ε aI χ bJ Z IJKM C ab F M (cid:3) A M ! . (3.2)In fact, these two symmetries are identical, and we can define them succinctly as: δ (1) BS ( T ) A K F K ! ≡ T KM F M (cid:3) A M ! . (3.3)where T KM ≡ ε aI χ bJ Z IJKM C ab or ε a χ bJ ǫ JKM C ab (3.4)The unique first order bosonic symmetries revealed by all the central charges andinternal symmetries in this way are: δ (1) BS ( P ) A K d ! ≡ P K − d (cid:3) A K ! , δ (1) BS ( Q ) B K d ! ≡ Q K − d (cid:3) B K ! (3.5)9 (1) BS ( T ) A K F K ! ≡ T KM F M (cid:3) A M ! (3.6) δ (1) BS ( T ) B K G K ! ≡ T KM G M (cid:3) B M ! , δ (1) BS ( W ) A J G J ! ≡ W JK G K (cid:3) A K ! (3.7) δ (1) BS ( V ) F J G J ! ≡ ( V µ ) JK ∂ µ G K − F K ! (3.8) δ (1) BS ( U ) F K d ! ≡ ( U µ ) K ∂ µ d F K ! (3.9) δ (1) BS ( U ) G K A ν ! ≡ ( U µ ) K ∂ ν F µν η µν G K ! (3.10) δ (1) BS ( Q ) λ c ψ Kc ! ≡ Q K ( γ µ ) dc ∂ µ ψ Kd − λ d ! (3.11) δ (1) BS ( U ) λ c ψ Kc ! ≡ ( U ν ) K ( γ γ ν γ µ ) dc ∂ µ ψ Kd − λ d ! (3.12) δ (1) BS ( P ) λ c ψ Kc ! ≡ P K ( γ γ µ ) dc ∂ µ ψ Kd − λ d ! (3.13) δ (1) BS ( W ) ψ Kc ≡ W KM ( γ γ µ ) dc ∂ µ ψ Md (3.14) δ (1) BS ( V ) ψ Kc ≡ ( V ν ) KM ( γ γ ν γ µ ) dc ∂ µ ψ Md (3.15) δ (1) BS ( T ) ψ Kc ≡ T KM ( γ µ ) dc ∂ µ ψ Md (3.16)along with the U (1) gauge symmetries δ G A ν ≡ Q K ∂ ν A K , δ G A ν ≡ P K ∂ ν B K ,δA ν ≡ ε a χ bI C ab ∂ ν A I , δA ν ≡ ε a χ bI ( γ ) ab ∂ ν B I (3.17)where P K ≡ ε aI χ bJ ǫ IJK ( γ ) ab Q K ≡ ε aI χ bJ ǫ IJK C ab ,T KM ≡ ε aI χ bJ Z IJKM C ab or ε a χ bJ ǫ JKM C ab , ( U µ ) K ≡ ε aI χ bJ ǫ IJK ( γ γ µ ) ab ,W KM ≡ ε a χ bJ ǫ JKM ( γ ) ab or ε aI χ bJ Z IJKM ( γ ) ab , ( V µ ) KM ≡ ε a χ bJ ǫ JKM ( γ γ µ ) ab or ε aI χ bJ Z IJKM ( γ γ µ ) ab (3.18)10he following identity proves useful in directly verifying these as Lagrangian symme-tries: ( γ γ ( µ γ α γ ν ) ) ( ab ) = 0 (3.19)where ( ) denotes symmetrization, i.e., ( γ µ ) ( ab ) = ( γ µ ) ab + ( γ µ ) ba .It is interesting to note here that because of the absence of B J to F J couplingin the algebra, this method fails to uncover the first order bosonic symmetry of theLagrangian δ (1) BS ( T ) B K F K ! ≡ T KM F M (cid:3) B M ! (3.20)In addition, Lagrangian symmetries such as δ (1) BS ( U ) G K d ! ≡ ( U µ ) K ∂ µ d G K ! (3.21) δ (1) BS ( U ) F K A ν ! ≡ ( U µ ) K ∂ ν F µν η µν F K ! (3.22)also are not manifest in the algebra. We will leave all such symmetries not manifestedby the algebra out of the remaining calculations of second order bosonic and first orderfermionic symmetries, as we are investigating how the absence of these symmetriesfails to uncover further symmetries down the line. By taking the commutators of each of the first order bosonic symmetries with eachother, we reveal second order bosonic symmetries. This procedure will sometimeslead to redundant symmetries as in δ (2) BS a ( P , P ) A K ≡ [ δ (1) BS ( P ) , δ (1) BS ( P )] A K = Λ KJ , ( P , P ) (cid:3) A J δ (2) BS b ( T , T ) A K ≡ [ δ (1) BS ( T ) , δ (1) BS ( T )] A K = Λ JK , ( T , T ) (cid:3) A J δ (2) BS c ( W , W ) A J ≡ [ δ (1) BS ( W ) , δ (1) BS ( W )] A J = Λ IJ , ( W , W ) (cid:3) A I (3.23)where Λ KJ , ( P , P ) ≡ P K [1 P J , Λ KJ , ( T , T ) ≡ T KM [1 T MJ , Λ IJ , ( W , W ) ≡ W KI [1 W JK (3.24)We can succinctly write these three redundant symmetries as one δ (2) BS (Λ ) A K ≡ Λ [ KJ ]1 (cid:3) A J (3.25)11here (Λ ) KJ is an arbitrary 3 × ) [ KJ ] = (Λ ) KJ − (Λ ) JK . (3.26)In Appendix A.1.1, we list all the second order bosonic symmetries which are calcu-lated in this way, including their redundancies. Here, we list only the unique symme-tries, written in terms of the arbitrary matrices (Λ ) KJ , (Λ µν ) JK , (Λ ) IJ , (Λ µ ) K , Λ K ,and (Λ µν ) J : δ (2) BS (Λ ) A K ≡ Λ [ KJ ]1 (cid:3) A J , δ (2) BS (Λ ) B K ≡ Λ [ KJ ]1 (cid:3) B J δ (2) BS (Λ ) F K ≡ Λ [ KJ ]1 (cid:3) F J , δ (2) BS (Λ ) G K ≡ Λ [ KJ ]1 (cid:3) G K δ (2) BS (Λ ) F J ≡ (Λ µν ) [ IJ ] ∂ µ ∂ ν F I , δ (2) BS (Λ ) G J ≡ (Λ µν ) [ IJ ] ∂ µ ∂ ν G I δ (2) BS (Λ ) A ν ≡ η νβ (Λ [ µβ ]2 ) JJ ∂ α F µα (3.27) δ (2) BS (Λ ) A K B K ! ≡ Λ IJ δ IK (cid:3) B J − δ JK (cid:3) A I ! (3.28) δ (2) BS (Λ ) A K F K ! ≡ (Λ µ ) IJ δ IK ∂ µ F J δ JK ∂ µ (cid:3) A I ! (3.29) δ (2) BS (Λ ) B K F K ! ≡ (Λ µ ) IJ δ IK ∂ µ F J δ KJ ∂ µ (cid:3) B I ! (3.30) δ (2) BS (Λ ) A J G J ! ≡ (Λ µ ) IK δ IJ ∂ µ G K δ JK ∂ µ (cid:3) A I ! (3.31) δ (2) BS (Λ ) A K d ! ≡ (Λ µ ) K ∂ µ d ∂ µ (cid:3) A K ! (3.32) δ (2) BS (Λ ) A J A ν ! ≡ (Λ µ ) J ∂ ν F µν η µν (cid:3) A J ! (3.33) δ (2) BS (Λ ) B J A ν ! ≡ (Λ µ ) J ∂ ν F µν η µν (cid:3) B J ! (3.34) δ (2) BS (Λ ) F K d ! ≡ Λ K (cid:3) d − (cid:3) F K ! (3.35) δ (2) BS (Λ ) G K d ! ≡ Λ K (cid:3) d − (cid:3) G K ! (3.36)12 (2) BS (Λ ) G J d ! ≡ (Λ µν ) J ∂ µ ∂ ν d − ∂ µ ∂ ν G J ! (3.37) δ (2) BS (Λ ) F J G J ! ≡ Λ IK δ IJ (cid:3) G K − (cid:3) δ JK F I ! (3.38) δ (2) BS (Λ ) F J A α ! ≡ (Λ µν ) J ( U, V ) ∂ ν ∂ α F µα − η µα ∂ ν F J ! (3.39)and δ (2) BS (Λ ) ψ Kc ≡ Λ [ JK ]1 (cid:3) ψ Jc (3.40) δ (2) BS (Λ ) ψ Kc ≡ [(Λ ρσ ) KJ − (Λ σρ ) JK ]( γ ρ γ µ γ σ γ ν ) dc ∂ µ ∂ ν ψ Jd (3.41) δ (2) BS (Λ ) λ c ≡ (Λ [ µν ]2 ) KK ( γ µ γ α γ ν γ β ) dc ∂ α ∂ β λ d , (3.42) δ (2) BS (Λ ) ψ Kc ≡ (Λ µ ) [ JK ] ( γ γ µ ) dc (cid:3) ψ Jd (3.43) δ (2) BS (Λ ) λ c ≡ (Λ ν ) KK ( γ γ µ ) dc ∂ µ ∂ ν λ d (3.44) δ (2) BS (Λ ) ψ Kc ≡ (Λ µ ) KJ ( γ γ ν ) dc ∂ µ ∂ ν ψ Jd (3.45) δ (2) BS (Λ ) ψ Kc ≡ (Λ µ ) [ JK ] ( γ µ ) dc (cid:3) ψ Jd + 2(Λ µ ) KJ ( γ ν ) dc ∂ µ ∂ ν ψ Jd (3.46) δ (2) BS (Λ ) λ c ≡ (Λ µ ) KK ( γ ν ) dc ∂ µ ∂ ν λ d (3.47) δ (2) BS (Λ ) ψ Kc ≡ Λ KJ ( γ ) dc (cid:3) ψ Jd (3.48) δ (2) BS (Λ ) λ c ≡ Λ KK ( γ ) dc (cid:3) λ d (3.49) δ (2) BS (Λ ) λ c ψ Kc ! ≡ Λ K ( γ ) dc (cid:3) ψ Kd (cid:3) λ d ! (3.50) δ (2) BS (Λ ) λ c ψ Kc ! ≡ Λ K (cid:3) ψ Kc − (cid:3) λ c ! (3.51) δ (2) BS (Λ ) λ c ψ Kc ! ≡ (Λ α ) K ( γ γ ν γ α γ ν ) dc ∂ µ ∂ ν ψ Kd ( γ γ α ) dc (cid:3) λ d ! (3.52) δ (2) BS (Λ ) λ c ψ Kc ! ≡ (Λ µ ) K ( γ γ µ ) dc (cid:3) ψ Kd ( γ γ α γ µ γ β ) dc ∂ α ∂ β λ d ! (3.53) δ (2) BS (Λ ) λ c ψ Kc ! ≡ (Λ µ ) K ( γ µ ) dc (cid:3) ψ Kd ( γ ν γ µ γ α ) dc ∂ ν ∂ α λ d ! (3.54) δ (2) BS (Λ ) λ c ψ Kc ! ≡ (Λ µ ) K ( γ α γ µ γ β ) dc ∂ α ∂ β ψ Kd ( γ µ ) dc (cid:3) λ d ! , (3.55)13 (2) BS (Λ ) λ c ψ Kc ! ≡ (Λ µν ) K ( γ ν γ α γ µ γ β ) dc ∂ α ∂ β ψ Kd − ( γ µ γ α γ ν γ β ) dc ∂ α ∂ β λ d ! , (3.56)This analysis seems to not miss any second order bosonic symmetries which acton the fermions λ a and ψ Ja . However, the missing first order bosonic symmetriesalluded to previously which act on the bosons clearly manifest themselves here inmissing second order bosonic symmetries. Basically, as the fields A J and B J enterthe Lagrangian in the same way, they should have the same first and second ordersymmetries. The same should hold for F J and G J . But clearly since, for example,the algebra is not symmetric between exchange of A J ↔ B J or F J ↔ G J , Lagrangiansymmetries involving these field pairs will be missed when generated from the algebrain the manner presented here. Analogous to how we found the second order bosonic symmetries, we can uncoverfirst order fermionic symmetries through calculations such as: δ (1) F S ( P ) d ψ Jb ! ≡ − ε a [D a , δ (1) BS ( P )] d ψ Jb ! = ε a P J (cid:3) ψ Ja i ( γ µ ) ab ∂ µ d ! (3.57)All such possible calculations are listed in the Appendix A.1.2, some of which areredundant as in the second order bosonic case. Here is listed only the unique sym-metries. δ (1) F S ( P ) A K ψ Jb ! ≡ ε aJ P K i ( γ γ µ ) ba ∂ µ ψ Jb ( γ ) ab (cid:3) A K ! (3.58) δ (1) F S ( P ) A J λ b ! ≡ ε a P J i ( γ γ µ ) ba ∂ µ λ b ( γ ) ab (cid:3) A J ! (3.59) δ (1) F S ( Q ) B J λ b ! ≡ ε a Q J i ( γ γ µ ) ba ∂ µ λ b ( γ ) ab (cid:3) B J ! (3.60) δ (1) F S ( Q ) B K ψ Ib ! ≡ ε aI Q K i ( γ γ µ ) ba ∂ µ ψ Ib ( γ ) ab (cid:3) B K ! (3.61) δ (1) F S ( P ) B K λ b ! ≡ ε a P K i ( γ µ ) ba ∂ µ λ b C ab (cid:3) B K ! (3.62) δ (1) F S ( P ) B J ψ Kb ! ≡ ε aJ P K i ( γ µ ) ba ∂ µ ψ Kb C ab (cid:3) B J ! (3.63)14 (1) F S ( Q ) A J λ b ! ≡ ε a Q J ( γ µ ) ba ∂ µ λ b − iC ab (cid:3) A J ! (3.64) δ (1) F S ( Q ) A I ψ Kb ! ≡ ε aI Q K ( γ µ ) ba ∂ µ ψ Kb − iC ab (cid:3) A I ! (3.65)and δ (1) F S ( P ) F K λ b ! ≡ ε a P K ( γ ) ba (cid:3) λ b i ( γ γ µ ) ab ∂ µ F K ! (3.66) δ (1) F S ( P ) F J ψ Kb ! ≡ ε aJ P K ( γ ) ba (cid:3) ψ Kb i ( γ γ µ ) ab ∂ µ F J ! (3.67) δ (1) F S ( Q ) G J λ b ! ≡ ε a Q J ( γ ) ba (cid:3) λ b i ( γ γ µ ) ab ∂ µ G J ! (3.68) δ (1) F S ( Q ) G I ψ Kb ! ≡ ε aI Q K ( γ ) ba (cid:3) ψ Kb i ( γ γ µ ) ab ∂ µ G I ! (3.69) δ (1) F S ( Q ) d ψ Jb ! ≡ ε a Q J i ( γ ) ba (cid:3) ψ Jb − ( γ γ µ ) ab ∂ µ d ! (3.70) δ (1) F S ( Q ) d λ b ! ≡ ε aI Q I i ( γ ) ba (cid:3) λ b − ( γ γ µ ) ab ∂ µ d ! (3.71)and δ (1) F S ( Q ) F I ψ Kb ! ≡ ε aI Q K (cid:3) ψ Ka i ( γ µ ) ab ∂ µ F I ! (3.72) δ (1) F S ( Q ) F J λ b ! ≡ ε a Q J (cid:3) λ a i ( γ µ ) ab ∂ µ F J ! (3.73) δ (1) F S ( P ) G K λ b ! ≡ ε a P K (cid:3) λ a i ( γ µ ) ab ∂ µ G K ! (3.74) δ (1) F S ( P ) G J ψ Kb ! ≡ ε aJ P K (cid:3) ψ Ka i ( γ µ ) ab ∂ µ G J ! (3.75) δ (1) F S ( P ) d ψ Jb ! ≡ ε a P J (cid:3) ψ Ja i ( γ µ ) ab ∂ µ d ! (3.76) δ (1) F S ( P ) d λ b ! ≡ ε aI P I (cid:3) λ a i ( γ µ ) ab ∂ µ d ! (3.77)15nd δ (1) F S ( T ) A J ψ Jb ! ≡ ε a T JM ( γ µ ) ba ∂ µ ψ Mb iC ab (cid:3) A M ! (3.78) δ (1) F S ( W ) B J ψ Jb ! ≡ ε a W JM ( γ µ ) ba ∂ µ ψ Mb iC ab (cid:3) B M ! (3.79) δ (1) F S ( T ) A K λ b ! ≡ ε aI T IK i ( γ µ ) ba ∂ µ λ b C ab (cid:3) A K ! (3.80) δ (1) F S ( T ) A M ψ Jb ! ≡ ε aI ǫ IJK T KM i ( γ µ ) ba ∂ µ ψ Jb C ab (cid:3) A M ! (3.81) δ (1) F S ( T ) A J ψ Mb ! ≡ ε aI ǫ IJK T KM i ( γ µ ) ba ∂ µ ψ Mb C ab (cid:3) A J ! (3.82) δ (1) F S ( W ) B J ψ Mb ! ≡ ε aI ǫ IJK W KM i ( γ µ ) ba ∂ µ ψ Mb C ab (cid:3) B J ! (3.83)and δ (1) F S ( W ) A J ψ Jb ! ≡ ε a W JM ( γ γ µ ) ba ∂ µ ψ Mb i ( γ ) ab (cid:3) A M ! (3.84) δ (1) F S ( T ) B J ψ Jb ! ≡ ε a T JM ( γ γ µ ) ba ∂ µ ψ Mb i ( γ ) ab (cid:3) B M ! (3.85) δ (1) F S ( W ) A M λ b ! ≡ ε aI W IM i ( γ γ µ ) ba ∂ µ λ b ( γ ) ab (cid:3) A M ! (3.86) δ (1) F S ( W ) A J ψ Mb ! ≡ ε aI ǫ IJK W KM i ( γ γ µ ) ba ∂ µ ψ Mb ( γ ) ab (cid:3) A J ! (3.87) δ (1) F S ( W ) A M ψ Jb ! ≡ ε aI ǫ IJK W KM i ( γ γ µ ) ba ∂ µ ψ Jb ( γ ) ab (cid:3) A M ! (3.88) δ (1) F S ( T ) B K λ b ! ≡ ε aI T IK i ( γ γ µ ) ba ∂ µ λ b ( γ ) ab (cid:3) B K ! (3.89) δ (1) F S ( T ) B J ψ Mb ! ≡ ε aI ǫ IJK T KM i ( γ γ µ ) ba ∂ µ ψ Mb ( γ ) ab (cid:3) B J ! (3.90) δ (1) F S ( T ) B M ψ Jb ! ≡ ε aI ǫ IJK T KM i ( γ γ µ ) ba ∂ µ ψ Jb ( γ ) ab (cid:3) B M ! (3.91)16nd δ (1) F S ( T ) G J ψ Jb ! ≡ ε a T JM i ( γ ) ba (cid:3) ψ Mb ( γ γ µ ) ab ∂ µ G M ! (3.92) δ (1) F S ( W ) F J ψ Jb ! ≡ ε a W JM i ( γ ) ba (cid:3) ψ Mb ( γ γ µ ) ab ∂ µ F M ! (3.93) δ (1) F S ( W ) F J ψ Mb ! ≡ ε aI ǫ IJK W KM ( γ ) ba (cid:3) ψ Mb i ( γ γ µ ) ab ∂ µ F J ! (3.94) δ (1) F S ( T ) G K λ b ! ≡ ε aI T IK ( γ ) ba (cid:3) λ b i ( γ γ µ ) ab ∂ µ G K ! (3.95) δ (1) F S ( T ) G K ψ Jb ! ≡ ε aI ǫ IJK T KM ( γ ) ba (cid:3) ψ Nb i ( γ γ µ ) ab ∂ µ G M ! (3.96) δ (1) F S ( T ) G J ψ Mb ! ≡ ε aI ǫ IJK T KM ( γ ) ba (cid:3) ψ Mb i ( γ γ µ ) ab ∂ µ G J ! (3.97) δ (1) F S ( T ) d ψ Mb ! ≡ ε aI T IM ( γ ) ba (cid:3) ψ Mb i ( γ γ µ ) ab ∂ µ d ! (3.98)and δ (1) F S ( V ) F J ψ Jb ! ≡ ε a ( V µ ) JM i ( γ γ ν ) ba ∂ µ ∂ ν ψ Mb ( γ ) ab ∂ µ F M ! (3.99) δ (1) F S ( V ) F J ψ Jb ! ≡ ε a ( V ρ ) JM ( γ γ ν γ ρ γ µ ) ba ∂ µ ∂ ν ψ Mb i ( γ γ ρ γ µ ) ba ∂ µ F M ! (3.100) δ (1) F S ( V ) F M λ b ! ≡ ε aI ( V µ ) IM ( γ γ ν ) ba ∂ µ ∂ ν λ b i ( γ ) ab ∂ µ F M ! (3.101) δ (1) F S ( V ) F M ψ Jb ! ≡ ε aI ǫ IJK ( V µ ) KM ( γ γ ν ) ba ∂ µ ∂ ν ψ Jb i ( γ ) ab ∂ µ F M ! (3.102) δ (1) F S ( U ) F J λ b ! ≡ ε a ( U µ ) J ( γ γ ν ) ba ∂ µ ∂ ν λ b i ( γ ) ab ∂ µ F J ! (3.103) δ (1) F S ( U ) F K ψ Ib ! ≡ ε aI ( U µ ) K ( γ γ ν ) ba ∂ µ ∂ ν ψ Ib i ( γ ) ab ∂ µ F K ! (3.104) δ (1) F S ( V ) F J ψ Mb ! ≡ ε aI ǫ IJK ( V ρ ) KM ( γ γ µ γ ρ γ ν ) ba ∂ µ ∂ ν ψ Mb − i ( γ γ ρ γ µ ) ba ∂ µ F J ! (3.105)17 (1) F S ( U ) F K λ b ! ≡ ε a ( U ρ ) K ( γ γ µ γ ρ γ ν ) ba ∂ µ ∂ ν λ b − i ( γ γ ρ γ µ ) ba ∂ µ F K ! (3.106) δ (1) F S ( U ) F J ψ Kb ! ≡ ε aJ ( U ρ ) K ( γ γ µ γ ρ γ ν ) ba ∂ µ ∂ ν ψ Kb − i ( γ γ ρ γ µ ) ba ∂ µ F J ! (3.107)and δ (1) F S ( U ) G J λ b ! ≡ ε a ( U µ ) J ∂ ν ∂ [ µ ( γ ν ] ) ba λ b ( σ νµ ) ab ∂ ν G J ! (3.108) δ (1) F S ( U ) G K ψ Ib ! ≡ ε aI ( U µ ) K ∂ ν ∂ [ µ ( γ ν ] ) ba ψ Ib ( σ νµ ) ab ∂ ν G K ! (3.109) δ (1) F S ( U ) G K λ b ! ≡ ε a ( U ρ ) K i ( γ µ γ ρ γ ν ) ba ∂ µ ∂ ν λ b ( γ ρ γ µ ) ba ∂ µ G K ! (3.110) δ (1) F S ( U ) d ψ Jb ! ≡ ε a ( U µ ) J ( γ ν ) ba ∂ µ ∂ ν ψ Jb iC ab ∂ µ d ! (3.111) δ (1) F S ( U ) d λ b ! ≡ ε aI ( U µ ) I ( γ ν ) ba ∂ µ ∂ ν λ b iC ab ∂ µ d ! (3.112) δ (1) F S ( V ) G J ψ Jb ! ≡ ε a ( V ρ ) JM i ( γ µ γ ρ γ ν ) ba ∂ µ ∂ ν ψ Mb − ( γ ρ γ µ ) ba ∂ µ G M ! (3.113) δ (1) F S ( V ) G J ψ Jb ! ≡ ε a ( V µ ) JM ( γ ν ) ba ∂ µ ∂ ν ψ Mb − iC ab ∂ µ G M ! (3.114) δ (1) F S ( U ) G J ψ Kb ! ≡ ε aJ ( U ρ ) K i ( γ µ γ ρ γ ν ) ba ∂ µ ∂ ν ψ Kb ( γ ρ γ µ ) ba ∂ ν G J ! (3.115) δ (1) F S ( V ) G M λ b ! ≡ ε aI ( V µ ) IM ( γ ν ) ba ∂ µ ∂ ν λ b iC ab ∂ µ G M ! (3.116) δ (1) F S ( V ) G M ψ Jb ! ≡ ε aI ǫ IJK ( V µ ) KM ( γ ν ) ba ∂ µ ∂ ν ψ Na iC ab ∂ µ G M ! (3.117) δ (1) F S ( V ) G J ψ Mb ! ≡ ε aI ǫ IJK ( V ρ ) KM i ( γ µ γ ρ γ ν ) ba ∂ µ ∂ ν ψ Mb ( γ ρ γ µ ) ba ∂ µ G J ! (3.118) δ (1) F S ( U ) d λ b ! ≡ ε aI ( U ρ ) I i ( γ µ γ ρ γ ν ) ba ∂ µ ∂ ν λ b ( γ ρ γ µ ) ba ∂ µ d ! (3.119)18 (1) F S ( V ) d ψ Mb ! ≡ ε aI ( V ρ ) IM i ( γ µ γ ρ γ ν ) ba ∂ µ ∂ ν ψ Mb ( γ ρ γ µ ) ba ∂ µ d ! (3.120)and δ (1) F S ( P ) A µ λ b ! ≡ ε aI P I ( γ γ µ γ ν ) ba ∂ ν λ b − i ( γ γ β ) ab ∂ α F αβ ! (3.121) δ (1) F S ( P ) A µ ψ Kb ! ≡ ε a P K ( γ γ µ γ ν ) ba ∂ ν ψ Kb − i ( γ γ β ) ab ∂ α F αβ ! (3.122) δ (1) F S ( Q ) A µ ψ Jb ! ≡ ε a Q J − ( γ µ γ ν ) ba ∂ ν ψ Jb ( γ α σ µν ) ba ∂ α F µν ! (3.123) δ (1) F S ( Q ) A µ λ b ! ≡ ε aI Q I − ( γ µ γ ν ) ba ∂ ν λ b ( γ α σ µν ) ba ∂ α F µν ! (3.124) δ (1) F S ( T ) A µ ψ Mb ! ≡ ε aI T IM ( γ µ γ ν ) ba ∂ ν ψ Mb − ( γ α σ µν ) ba ∂ α F µν ! (3.125) δ (1) F S ( W ) A µ ψ Mb ! ≡ ε aI W IM − ( γ γ µ γ ν ) ba ∂ ν ψ Mb − ( γ γ α σ µν ) ba ∂ α F µν ! (3.126) δ (1) F S ( U ) A µ ψ Jb ! ≡ ε a ( U µ ) J i ( γ γ ν ) ba ∂ ν ψ Jb − ( γ ) ab ∂ ν F µν ! (3.127) δ (1) F S ( U ) A µ λ b ! ≡ ε aI ( U µ ) I − i ( γ γ ν ) ba ∂ ν λ b ( γ ) ab ∂ ν F µν ! (3.128) δ (1) F S ( U ) A µ λ b ! ≡ ε aI ( U ρ ) I ( γ γ µ γ ρ γ ν ) ba ∂ ν λ b ( γ γ ρ γ ν σ αβ ) ba ∂ ν F αβ ! (3.129) δ (1) F S ( V ) A µ ψ Mb ! ≡ ε aI ( V ρ ) IM ( γ γ µ γ ρ γ ν ) ba ∂ ν ψ Mb ( γ γ ρ γ ν σ αβ ) ba ∂ ν F αβ ! (3.130)and δ (1) F S ( U ) A J ψ Kb ! ≡ ε aJ ( U ρ ) K − ( γ γ ρ γ µ ) ba ∂ µ ψ Kb i ( γ γ ρ ) ab (cid:3) A J ! (3.131) δ (1) F S ( V ) A J ψ Mb ! ≡ ε aI ǫ IJK ( V ρ ) KM − ( γ γ ρ γ µ ) ba ∂ µ ψ Mb i ( γ γ ρ ) ab (cid:3) A J ! (3.132) δ (1) F S ( V ) A J ψ Jb ! ≡ ε a ( V ρ ) JM ( γ γ ρ γ µ ) ba ∂ µ ψ Mb i ( γ γ ρ ) ab (cid:3) A M ! (3.133)19 (1) F S ( U ) A K λ b ! ≡ ε a ( U ρ ) K ( γ γ ρ γ µ ) ba ∂ µ λ b − i ( γ γ ρ ) ab (cid:3) A K ! (3.134) δ (1) F S ( U ) B K λ b ! ≡ ε a ( U ρ ) K i ( γ ρ γ µ ) ba ∂ µ λ b ( γ ρ ) ab (cid:3) B K ! (3.135) δ (1) F S ( V ) B J ψ Mb ! ≡ ε aI ǫ IJK ( V ρ ) KM i ( γ ρ γ µ ) ba ∂ µ ψ Mb ( γ ρ ) ab (cid:3) B J ! (3.136) δ (1) F S ( U ) B J ψ Kb ! ≡ ε aJ ( U ρ ) K i ( γ ρ γ µ ) ba ∂ µ ψ Kb ( γ ρ ) ab (cid:3) B J ! (3.137) δ (1) F S ( V ) B J ψ Jb ! ≡ ε a ( V ρ ) JM − i ( γ ρ γ µ ) ba ∂ µ ψ Mb ( γ ρ ) ab (cid:3) B M ! (3.138)and δ (1) F S ( T ) F J ψ Jb ! ≡ ε a T JM i (cid:3) ψ Ma ( γ µ ) ab ∂ µ F M ! (3.139) δ (1) F S ( W ) G J ψ Jb ! ≡ ε a W JM i (cid:3) ψ Ma ( γ µ ) ab ∂ µ G M ! (3.140) δ (1) F S ( T ) F K λ b ! ≡ ε aI T IK (cid:3) λ a i ( γ µ ) ab ∂ µ F K ! (3.141) δ (1) F S ( T ) F J ψ Mb ! ≡ ε aI ǫ IJK T KM (cid:3) ψ Ma i ( γ µ ) ab ∂ µ F J ! (3.142) δ (1) F S ( T ) F M ψ Jb ! ≡ ε aI ǫ IJK T KM (cid:3) ψ Ja i ( γ µ ) ab ∂ µ F M ! (3.143) δ (1) F S ( W ) G M λ b ! ≡ ε aI W IM (cid:3) λ a i ( γ µ ) ab ∂ µ G M ! (3.144) δ (1) F S ( W ) G J ψ Mb ! ≡ ε aI ǫ IJK W KM (cid:3) ψ Ma i ( γ µ ) ab ∂ µ G J ! (3.145) δ (1) F S ( W ) G M ψ Jb ! ≡ ε aI ǫ IJK W KM (cid:3) ψ Ja i ( γ µ ) ab ∂ µ G M ! (3.146) δ (1) F S ( W ) d ψ Mb ! ≡ ε aI W IM (cid:3) ψ Ma i ( γ µ ) ab ∂ µ d ! (3.147)20 .4 Symmetries of the N = 2 FH Lagrangian
The symmetries of the N = 2 FH system follow analogously from the N = 4 calcu-lations. The first order bosonic symmetries of the N = 2 FH system calculated fromthe central charges and internal symmetries are˜ δ (1) BS ( ˜ T ) ˜ A k ˜ F k ! ≡ ˜ T km ˜ F m (cid:3) ˜ A m ! (3.148)˜ δ (1) BS ( ˜ T ) ˜ B k ˜ G k ! ≡ ˜ T km ˜ G m (cid:3) ˜ B m ! (3.149)˜ δ (1) BS ( ˜ T ) ˜ ψ kc ≡ ˜ T km ( γ µ ) dc ∂ µ ˜ ψ md (3.150)with ˜ T km ≡ ˜ R ijkm C ab ε ai χ bj (3.151)where i, j, k, m = 1 ,
2, and ε ai and χ bj are once again infinitesimal Grassmann spinors.Here, we clearly notice the absence of symmetries between A J ↔ B J , A J ↔ G J , B J ↔ F J , and G J ↔ F J . As in the N = 4 case, this is a direct result of the absenceof coupling terms between these fields in the algebra.Interestingly, we find that the second order bosonic symmetries calculated fromthese first order symmetries all vanish identically˜ δ (2) BS ( ˜ T , ˜ T ) ˜ A k ≡ [˜ δ (1) BS ( ˜ T ) , ˜ δ (1) BS ( ˜ T )] ˜ A k = ˜Λ jk ( ˜ T , ˜ T ) (cid:3) ˜ A j = 0 (3.152)˜ δ (2) BS ( ˜ T , ˜ T ) ˜ B k ≡ [˜ δ (1) BS ( ˜ T ) , ˜ δ (1) BS ( ˜ T )] ˜ B k = ˜Λ jk ( ˜ T , ˜ T ) (cid:3) ˜ B j = 0 (3.153)˜ δ (2) BS ( ˜ T , ˜ T ) ˜ F k ≡ [˜ δ (1) BS ( ˜ T ) , ˜ δ (1) BS ( ˜ T )] ˜ F k = ˜Λ jk ( ˜ T , ˜ T ) (cid:3) ˜ F j = 0 (3.154)˜ δ (2) BS ( ˜ T , ˜ T ) ˜ G k ≡ [˜ δ (1) BS ( T ) , ˜ δ (1) BS ( ˜ T )] ˜ G k = ˜Λ jk ( ˜ T , ˜ T ) (cid:3) ˜ G j = 0 (3.155)˜ δ (2) BS ( ˜ T , ˜ T ) ˜ ψ kc ≡ [˜ δ (1) BS ( ˜ T ) , ˜ δ (1) BS ( ˜ T )] ˜ ψ kc = ˜Λ jk ( ˜ T , ˜ T ) (cid:3) ˜ ψ Jc = 0 (3.156)as ˜Λ jk , ( ˜ T , ˜ T ) ≡ ˜ T jm [1 ˜ T mk = 0 , j, k, m = 1 , jk ,˜ δ (2) BS ˜ χ jC ≡ ˜Λ [ jk ] (cid:3) ˜ χ kC , ˜ χ jC ≡ ( ˜ A j , ˜ B j , ˜ F j , ˜ G j , ˜ ψ jc ) (3.158)is still a symmetry of the N = 2 FH Lagrangian.21n the other hand, several first order fermionic symmetries still remain afterreduction to the N = 2 FH system:˜ δ (1) F S ( ˜ T ) ˜ A k ˜ ψ b ! ≡ ε ai ˜ T ik − ( γ µ ) ba ∂ µ ˜ ψ b iC ab (cid:3) ˜ A k ! ˜ δ (1) F S ( ˜ T ) ˜ F k ˜ ψ b ! ≡ ε ai ˜ T ik (cid:3) ˜ ψ a i ( γ µ ) ab ∂ µ ˜ F k ! ˜ δ (1) F S ( ˜ T ) ˜ A k ˜ ψ b ! ≡ ε ai ( σ ) ij ˜ T jk i ( γ µ ) ba ∂ µ ˜ ψ b C ab (cid:3) ˜ A k ! ˜ δ (1) F S ( ˜ T ) ˜ F k ˜ ψ b ! ≡ ε ai ( σ ) ij T jk − i (cid:3) ˜ ψ a ( γ µ ) ab ∂ µ ˜ F k ! (3.159)˜ δ (1) F S ( ˜ T ) ˜ B k ˜ ψ b ! ≡ ε ai ( σ ) ij ˜ T jk i ( γ γ µ ) ba ∂ µ ˜ ψ b ( γ ) ab (cid:3) ˜ B k ! ˜ δ (1) F S ( ˜ T ) ˜ G k ˜ ψ b ! ≡ ε ai ( σ ) ij ˜ T jk − i ( γ ) ba (cid:3) ˜ ψ b ( γ γ µ ) ab ∂ µ ˜ G k ! ˜ δ (1) F S ( ˜ T ) ˜ B k ˜ ψ b ! ≡ ε ai ( σ ) ij ˜ T jk i ( γ γ µ ) ba ∂ µ ˜ ψ b ( γ ) ab (cid:3) ˜ B k ! ˜ δ (1) F S ( ˜ T ) ˜ G k ˜ ψ b ! ≡ ε ai ( σ ) ij ˜ T jk − i ( γ ) ba (cid:3) ˜ ψ b ( γ γ µ ) ab ∂ µ ˜ G k ! (3.160)These are only the unique symmetries uncovered via this method, the redundantcalculations being shown once again in Appendix A.2. Here we notice as in thebosonic case, that these fermionic symmetries are not themselves symmetric withrespect to A J ↔ B J and F J ↔ G J . Again, this is a direct result of the absence ofthe corresponding central charge or internal symmetry in the algebra. The d = 4, N = 4 SUSY-YM system is important to many theoretical models inphysics today. As it is a conformal field theory, it’s possible that its study can leadto further understanding of ‘walking’ theories such as technicolor. In string theory,the AdS/CFT correspondence relates calculations of d = 4, N = 4 SUSY-YM toclassical supergravity calculations on AdS × S , where the correspondence is weak tostrong and vice versa. In an effort to more accurately describe the standard model,22his has been taken further to include correspondences to gauge theories with runningcouplings. Even so, the problem of how to augment the dynamical theory of d = 4, N = 4 SUSY-YM with a finite number of auxiliary fields such that the algebra closeshas been unsolved for quite some time. A solution to this problem would be helpfulto more fully understand these aforementioned theories relating to conformal fieldtheories.In this paper, we chose a particular set of auxiliary fields for d = 4, N = 4SUSY-YM and catalogued the Lagrangian symmetries manifest in the central chargesand internal symmetries of the resulting algebra. It was noted how not all possibleLagrangian symmetries can be uncovered this way, as certain central charges andinternal symmetries are missing from the algebra. We reinforce here that all resultspresented are from straightforward, actual calculations with no assumptions of cen-trality. For instance, we have directly calculated that the SUSY-YM Lagrangian inEq. (2.1) is invariant with respect to the transformation laws in Eqs. (2.2), (2.3),(2.4), and (2.5). We have directly calculated that these transformation laws satisfythe anti-commutation relations in Eqs. (2.8), (2.9), (2.10), (2.11), and (2.12). Themain result of this paper is how these transformation laws and anti-commutators leadby direct calculation to the first and second order Lagrangian symmetries presentedin section 3.Furthermore, reduction of this particular N = 4 system to the N = 2 Fayethypermultiplet and N = 2 vector multiplet was shown to follow from our directcalculations. Here it was noticed how in this reduction, central charges and internalsymmetries are lost from the algebra. In the case of the vector multiplet, all chargesand internal symmetries are lost as the algebra closes. In the case of the Fayethypermultiplet, some central charges and internal symmetries remain, as this algebradoes not close.Finally, we make a note on quantization of non-closed systems such as the N = 4SUSY-YM system investigated in detail in this paper. In general, non-closure ofan algebra leads to an added difficulty in the quantization procedure. Perhaps themost ubiquitous example is the criticality of string theory. For quantum non-criticalstrings, one must solve the Liouville theory. This is not necessary in the case of criticalstrings [26, 27]. In the case of our results of the N = 4 SUSY-YM system, we havelaid out our results in the hopes of eventually obtaining a closed system, in the senseof Eq. (1.1), without an infinite number of auxiliary fields. For instead quantizationof the non-closed system presented, the specific forms of the non-closure terms wecalculated are important in the same vein as the Liouville theory for non-criticalstrings. We leave this quantization as a future project.23 It is while you are patiently toiling at the little tasks oflife that the meaning and shape of the great whole of lifedawn on you.” - Phillips Brooks
Acknowledgements
This research was supported in part by the endowment of the John S. Toll Profes-sorship, the University of Maryland Center for String & Particle Theory, NationalScience Foundation Grant PHY-0354401. SJG & KS offer additional gratitude to theM. L. K. Visiting Professorship program and to the M. I. T. Center for TheoreticalPhysics for support and hospitality extended during the undertaking of this work.We would also like to thank Michael Faux for pointing out an incorrect wording withrespect to the definition of central charges.
AppendixA Explicit Calculation of First Order FermionicSymmetries
In this appendix, we explain in more detail the procedure which led us to the sym-metries presented in the body of the paper. Many symmetries found in this mannerare redundant, and those presented in the paper are the unique symmetries foundthrough this procedure.
A.1 N = 4 SUSY-YM
A.1.1 Second Order Bosonic Symmetries
In this section of the appendix, we explicitly show how the second order bosonicsymmetries are discovered through the N = 4 algebra. Several are redundant, and inthe body of the paper, only the unique symmetries were listed. δ (2) BS ( P , P ) A K ≡ [ δ (1) BS ( P ) , δ (1) BS ( P )] A K = Λ KJ , ( P , P ) (cid:3) A J (A.1) δ (2) BS ( T , T ) A K ≡ [ δ (1) BS ( T ) , δ (1) BS ( T )] A K = Λ JK , ( T , T ) (cid:3) A J (A.2) δ (2) BS ( Q , Q ) B K ≡ [ δ (1) BS ( Q ) , δ (1) BS ( Q )] B K = Λ KJ , ( Q , Q ) (cid:3) B J (A.3) δ (2) BS ( T , T ) B K ≡ [ δ (1) BS ( T ) , δ (1) BS ( T )] B K = Λ JK , ( T , T ) (cid:3) B J (A.4)24 (2) BS ( T , T ) F K ≡ [ δ (1) BS ( T ) , δ (1) BS ( T )] F K = Λ JK , ( T , T ) (cid:3) F J (A.5) δ (2) BS ( T , T ) G K ≡ [ δ (1) BS ( T ) , δ (1) BS ( T )] G K = Λ JK , ( T , T ) (cid:3) G J (A.6) δ (2) BS ( U , U ) F K ≡ [ δ (1) BS ( U ) , δ (1) BS ( U )] F K = (Λ µν , ) JK ( U , U ) ∂ µ ∂ ν F J (A.7) δ (2) BS ( U , U ) G K ≡ [ δ (1) BS ( U ) , δ (1) BS ( U )] G K + η µν (Λ µν , ) JK ( U , U ) (cid:3) G K = (Λ µν , ) JK ( U , U ) ∂ µ ∂ ν G J (A.8) δ (2) BS ( U , U ) A ν ≡ [ δ (1) BS ( U ) , δ (1) BS ( U )] A ν = η νβ (Λ µβ , ) JJ ( U , U ) ∂ α F µα (A.9)with Λ KJ , ( P , P ) ≡ P K [1 P J , Λ KJ , ( T , T ) = Λ KJ , ( T , T ) ≡ T KM [1 T MJ , Λ KJ , ( Q , Q ) ≡ Q K [1 Q J , (Λ µν , ) JK ( U , U ) = (Λ µν , ) JK ( U , U ) ≡ ( U µ [1 ) J ( U ν ) K , (A.10)and δ (2) BS ( P, Q ) A K B K ! ≡ [ δ (1) BS ( P ) , δ (1) BS ( Q )] A K B K ! =Λ IJ , ( P, Q ) − δ IK (cid:3) B J δ JK (cid:3) A I ! (A.11) δ (2) BS ( P, U ) A K F K ! ≡ [ δ (1) BS ( P ) , δ (1) BS ( U )] A K F K ! = − (Λ µ , ) IJ ( P, U ) δ IK ∂ µ F J δ JK ∂ µ (cid:3) A I ! (A.12) δ (2) BS ( T, U ) A K d ! ≡ [ δ (1) BS ( T ) , δ (1) BS ( U )] A K d ! = − (Λ µ , ) K ( T, U ) ∂ µ d ∂ µ (cid:3) A K ! (A.13) δ (2) BS ( T, U ) B K A ν ! ≡ [ δ (1) BS ( T ) , δ (1) BS ( U )] B K A ν ! = − (Λ µ , ) K ( T, U ) ∂ ν F µν η µν (cid:3) B K ! (A.14)25 (2) BS ( Q, U ) B K F K ! ≡ [ δ (1) BS ( Q ) , δ (1) BS ( U )] B K F K ! =(Λ µ , ) IJ ( Q, U ) δ IK ∂ µ F J δ KJ ∂ µ (cid:3) B I ! (A.15) δ (2) BS ( P, T ) F K d ! ≡ [ δ (1) BS ( P ) , δ (1) BS ( T )] F K d ! =Λ K , ( P, T ) − (cid:3) d (cid:3) F K ! (A.16) δ (2) BS ( Q, T ) G K d ! ≡ [ δ (1) BS ( Q ) , δ (1) BS ( T )] G K d ! =Λ K , ( Q, T ) (cid:3) d − (cid:3) G K ! (A.17)with Λ JK , ( P, Q ) = − Λ KJ , ( Q, P ) ≡ P J Q K , (Λ µ , ) JK ( P, U ) = − (Λ µ , ) KJ ( U, P ) ≡ P J ( U µ ) K , (Λ µ , ) K ( T, U ) = − (Λ µ , ) K ( U, T )= (Λ µ , ) K ( T, U ) = − (Λ µ , ) K ( U, T ) ≡ T KM ( U µ ) M , (Λ µ , ) JK ( Q, U ) = − (Λ µ , ) KJ ( U, Q ) ≡ Q J ( U µ ) K , Λ K , ( P, T ) = − Λ K , ( T, P ) ≡ P M T MK , Λ K , ( Q, T ) = − Λ K , ( T, Q )= Λ K , ( Q, T ) = − Λ K , ( T, Q ) ≡ Q M T MK , (A.18)and δ (2) BS ( Q, W ) λ c ψ Kc ! ≡ [ δ (1) BS ( Q ) , δ (1) BS ( W )] λ c ψ Kc ! =Λ K , ( Q, W )( γ ) dc (cid:3) ψ Kc (cid:3) λ d ! (A.19) δ (2) BS ( Q, V ) λ c ψ Kc ! ≡ [ δ (1) BS ( Q ) , δ (1) BS ( V )] λ c ψ Kc ! =(Λ α , ) K ( Q, V ) ( γ γ µ γ α γ ν ) dc ∂ µ ∂ ν ψ Kd ( γ γ α ) dc (cid:3) λ d ! (A.20)26 (2) BS ( Q, T ) λ c ψ Kc ! ≡ [ δ (1) BS ( Q ) , δ (1) BS ( T )] λ c ψ Kc ! =Λ K , ( Q, T ) − (cid:3) ψ Kc (cid:3) λ c ! (A.21) δ (2) BS ( W, V ) ψ Kc ≡ [ δ (1) BS ( W ) , δ (1) BS ( V )] ψ Kc =(Λ µ , ) [ JK ] ( γ µ ) dc (cid:3) ψ Jd + 2(Λ µ , ) KJ ( γ ν ) dc ∂ µ ∂ ν ψ Jd (A.22) δ (2) BS ( W, T ) ψ Kc ≡ [ δ (1) BS ( W ) , δ (1) BS ( T )] ψ Kc = − Λ KJ , ( W, T )( γ ) dc (cid:3) ψ Jd (A.23) δ (2) BS ( V, T ) ψ Kc ≡ [ δ (1) BS ( V ) , δ (1) BS ( T )] ψ Kc + δ (2) BS ( V, T ) ψ Kc =(Λ α , ) [ JK ] ( V, T )( γ γ α ) dc (cid:3) ψ Jd (A.24) δ (2) BS ( Q , Q ) ψ Kc ≡ [ δ (1) BS ( Q ) , δ (1) BS ( Q )] ψ Kc = − Λ JK , ( Q , Q ) (cid:3) ψ Jc (A.25) δ (2) BS ( W , W ) ψ Kc ≡ [ δ (1) BS ( W ) , δ (1) BS ( W )] ψ Kc = Λ KJ , ( W , W ) (cid:3) ψ Jc (A.26) δ (2) BS ( V , V ) ψ Kc ≡ [ δ (1) BS ( V ) , δ (1) BS ( V )] ψ Kc = − (Λ ρσ , ) KJ ( V , V )( γ ρ γ µ γ σ γ ν ) dc ∂ µ ∂ ν ψ Jd (A.27) δ (2) BS ( T , T ) ψ Kc ≡ [ δ (1) BS ( T ) , δ (1) BS ( T )] ψ Kc = − Λ KJ , ( T , T ) (cid:3) ψ Jc (A.28)with Λ JK , ( Q , Q ) ≡ Q J [1 Q K , Λ K , ( Q, W ) = − Λ K , ( W, Q ) ≡ Q M W MK , (Λ µ , ) K ( Q, V ) = − Λ K , ( V, Q ) ≡ Q M ( V µ ) MK , Λ JK , ( W , W ) ≡ W JM [1 W MK , (Λ µ , ) JK ( W, V ) = − (Λ µ , ) KJ ( V, W ) ≡ W JM ( V µ ) MK , Λ JK , ( W, T ) = − Λ KJ , ( T, W ) ≡ W M ( J T K ) M , (A.29)(Λ ρσ , ) KJ ( V , V ) ≡ ( V ρ [1 ) KM ( V σ ) MJ , (Λ W , ) JK ( V, T ) = − (Λ W , ) KJ ( T, V ) ≡ ( V W ) JM T MK , Λ KJ , ( T , T ) ≡ T KM [1 T MJ (A.30)and δ (2) BS ( W , W ) A J ≡ [ δ (1) BS ( W ) , δ (1) BS ( W )] A J = Λ IJ , ( W , W ) (cid:3) A I (A.31) δ (2) BS ( W , W ) G J ≡ [ δ (1) BS ( W ) , δ (1) BS ( W )] G J = Λ IJ , ( W , W ) (cid:3) G I (A.32) δ (2) BS ( V , V ) F J ≡ [ δ (1) BS ( V ) , δ (1) BS ( V )] F J = − (Λ µν , ) IJ ( V , V ) ∂ µ ∂ ν F I (A.33) δ (2) BS ( V , V ) G J ≡ [ δ (1) BS ( V ) , δ (1) BS ( V )] G J = − (Λ µν , ) IJ ( V , V ) ∂ µ ∂ ν G I (A.34)27 (2) BS ( P, W ) G J d ! ≡ [ δ (1) BS ( P ) , δ (1) BS ( W )] G J d ! =Λ J , ( P, W ) − (cid:3) d (cid:3) G J ! (A.35) δ (2) BS ( T, W ) F J G J ! ≡ [ δ (1) BS ( T ) , δ (1) BS ( W )] F J G J ! =Λ IK , ( T, W ) − δ IJ (cid:3) G K (cid:3) δ JK F I ! (A.36) δ (2) BS ( T, W ) A J B J ! ≡ [ δ (1) BS ( T ) , δ (1) BS ( W )] A J B J ! =Λ IK , ( T, W ) δ JK (cid:3) B I − δ IJ (cid:3) A K ! (A.37) δ (2) BS ( U, W ) A J A ν ! ≡ [ δ (1) BS ( U ) , δ (1) BS ( W )] A J A ν ! = − (Λ µ , ) J ( U, W ) ∂ ν F µν η µν (cid:3) A J ! (A.38) δ (2) BS ( W, V ) A J F J ! ≡ − [ δ (1) BS ( W ) , δ (1) BS ( V )] A J F J ! = − (Λ µ , ) IK ( W, V ) δ IJ ∂ µ F K δ JK ∂ µ (cid:3) A I ! (A.39) δ (2) BS ( T, V ) A J G J ! ≡ − [ δ (1) BS ( T ) , δ (1) BS ( V )] A J G J ! =(Λ µ , ) IK ( T, V ) δ IJ ∂ µ G K δ JK ∂ µ (cid:3) A I ! (A.40) δ (2) BS ( T, V ) B J F J ! ≡ − [ δ (1) BS ( T ) , δ (1) BS ( V )] B J F J ! = − Λ µ , ) IK ( T, V ) δ IJ ( ∂ µ F K δ JK ∂ µ (cid:3) B I ! (A.41)28 (2) BS ( U, V ) G J d ! ≡ − [ δ (1) BS ( U ) , δ (1) BS ( V )] G J d ! =(Λ µν , ) J ( U, V ) − ∂ µ ∂ ν d ∂ µ ∂ ν G J ! (A.42) δ (2) BS ( U, V ) F J A α ! ≡ − [ δ (1) BS ( U ) , δ (1) BS ( V )] F J A α ! =(Λ µν , ) J ( U, V ) ∂ ν ∂ α F µα − η µα ∂ ν F J ! (A.43)with Λ IJ , ( W , W ) ≡ W KI [1 W JK , Λ IJ , ( V , V ) ≡ ( V µ [1 ) KI ( V ν ) JK , Λ J , ( P, W ) ≡ P K W KJ , Λ IJ , ( T, W ) = Λ IJ , ( T, W ) ≡ T IK W KJ , (Λ µ , ) J ( U, W ) ≡ ( U µ ) K W KJ , (Λ µ , ) JI ( W, V ) ≡ W JK ( V µ ) KI , (Λ µ , ) JI ( T, V ) = (Λ µ , ) JI ( T, V ) ≡ T JK ( V µ ) KI , (Λ µν , ) J ( U, V ) = (Λ µν , ) J ( U, V ) ≡ ( U µ ) K ( V ν ) KJ (A.44)and δ (2) BS ( Q, U ) λ c ≡ [ δ (1) BS ( Q ) , δ (1) BS ( U )] λ c = − µ , ) KK ( Q, U )( γ γ ν ) dc ∂ µ ∂ ν λ d , (A.45) δ (2) BS ( Q, U ) ψ Kc ≡ [ δ (1) BS ( Q ) , δ (1) BS ( U )] ψ Kc = − µ , ) KJ ( Q, U )( γ γ ν ) dc ∂ µ ∂ ν ψ Jd , (A.46) δ (2) BS ( W, U ) λ c ψ Kc ! ≡ [ δ (1) BS ( W ) , δ (1) BS ( U )] λ c ψ Kc ! = − (Λ µ , ) K ( W, U ) ( γ µ ) dc (cid:3) ψ Kd ( γ ν γ µ γ α ) dc ∂ ν ∂ α λ d ! (A.47) δ (2) BS ( V, U ) λ c ψ Kc ! ≡ [ δ (1) BS ( V ) , δ (1) BS ( U )] λ c ψ Kc ! = (Λ µν , ) K ( V, U ) − ( γ ν γ α γ µ γ β ) dc ∂ α ∂ β ψ Kd ( γ µ γ α γ ν γ β ) dc ∂ α ∂ β λ d ! (A.48)29 (2) BS ( T, U ) λ c ψ Kc ! ≡ [ δ (1) BS ( T ) , δ (1) BS ( U )] λ c ψ Kc ! = − (Λ µ , ) K ( T, U ) ( γ γ µ ) dc (cid:3) ψ Kd ( γ γ α γ µ γ β ) dc ∂ α ∂ β λ d ! (A.49) δ (2) BS ( U , U ) λ c ≡ [ δ (1) BS ( U ) , δ (1) BS ( U )] λ c =(Λ µν , ) KK ( U , U )( γ µ γ α γ ν γ β ) dc ∂ α ∂ β λ d (A.50) δ (2) BS ( U , U ) ψ Kc ≡ [ δ (1) BS ( U ) , δ (1) BS ( U )] ψ Kc =(Λ µν , ) KJ ( U , U )( γ µ γ α γ ν γ β ) dc ∂ α ∂ β ψ Jd (A.51) δ (2) BS ( U, P ) λ c ≡ [ δ (1) BS ( U ) , δ (1) BS ( P )] λ c = 2(Λ µ , ) KK ( U, P )( γ ν ) dc ∂ µ ∂ ν λ d (A.52) δ (2) BS ( U, P ) ψ Kc ≡ [ δ (1) BS ( U ) , δ (1) BS ( P )] ψ Kc =(Λ µ , ) JK ( U, P )( γ α γ µ γ β ) dc ∂ α ∂ β ψ Jd + (Λ µ , ) KJ ( U, P )( γ µ ) dc (cid:3) ψ Jd (A.53) δ (2) BS ( Q, P ) λ c ≡ [ δ (1) BS ( Q ) , δ (1) BS ( P )] λ c = − Λ KK , ( Q, P )( γ ) dc (cid:3) λ d (A.54) δ (2) BS ( Q, P ) ψ Kc ≡ [ δ (1) BS ( Q ) , δ (1) BS ( P )] ψ Kc = − Λ KJ , ( Q, P )( γ ) dc (cid:3) ψ Jd (A.55) δ (2) BS ( W, P ) λ c ψ Kc ! ≡ [ δ (1) BS ( W ) , δ (1) BS ( P )] λ c ψ Kc ! =Λ K , ( W, P ) (cid:3) ψ Kc − (cid:3) λ d ! (A.56) δ (2) BS ( V, P ) λ c ψ Kc ! ≡ [ δ (1) BS ( V ) , δ (1) BS ( P )] λ c ψ Kc ! = (Λ µ , ) K ( V, P ) ( γ α γ µ γ β ) dc ∂ α ∂ β ψ Kd ( γ µ ) dc (cid:3) λ d ! (A.57) δ (2) BS ( T, P ) λ c ψ Kc ! ≡ [ δ (1) BS ( T ) , δ (1) BS ( P )] λ c ψ Kc ! = − (Λ µ , ) K ( T, P ) ( γ ) dc (cid:3) ψ Kd ( γ ) dc (cid:3) λ d ! (A.58) δ (2) BS ( P , P ) ψ Kc ≡ [ δ (1) BS ( P ) , δ (1) BS ( P )] ψ Kc = − Λ KJ , ( P , P ) (cid:3) ψ Jc , (A.59)30ith(Λ µ , ) JK ( Q, U ) ≡ Q J ( U µ ) K , (Λ µ , ) K ( W, U ) = W KM ( U µ ) M , (Λ µν , ) K ( V, U ) ≡ ( V µ ) KM ( U ν ) M , (Λ µ , ) K ( T, U ) = T KM ( U µ ) M , (Λ µν , ) KJ ( U , U ) ≡ ( U µ [1 ) K ( U ν ) J , (Λ µ , ) KM ( U, P ) ≡ ( U µ ) K P M , Λ KJ , ( Q, P ) ≡ Q ( K P J ) , Λ K , ( W, P ) ≡ W KM P M , (Λ µ , ) K ( V, P ) ≡ ( V µ ) KM P M ) , Λ K , ( T, P ) ≡ T KM P M , Λ KM , ( P , P ) ≡ P K [1 P M (A.60)where [ ] denotes antisymmetry, i.e. U J [1 U K = U J U K − U J U K . A.1.2 Fermionic Symmetries
Taking the commutators or D a and D Ia with the first order bosonic symmetries forthe N = 4 SUSY-YM system, we find several first order fermionic symmetries, someof which are redundant. The symmetries calculated below which involve ε aI P K ǫ IJK , ε aI Q K ǫ IJK , and ε aI ( U ρ ) K ǫ IJK are redefined through ε aI P K ǫ IJK → ε a P J ε aI Q K ǫ IJK → ε a Q J ε aI ( U ρ ) K ǫ IJK → ε a ( U ρ ) J (A.61)as symmetries defined either way are equivalent for the Lagrangian. In section 3.3,all symmetries are listed using this redefinition where applicable. δ (1) F S ( P ) d ψ Jb ! ≡ ε a P J (cid:3) ψ Ja i ( γ µ ) ab ∂ µ d ! = − ε a [D a , δ (1) BS ( P )] d ψ Jb ! (A.62) δ (1) F S ( Q ) d ψ Jb ! ≡ ε a Q J i ( γ ) ba (cid:3) ψ Jb − ( γ γ µ ) ab ∂ µ d ! = ε a [D a , δ (1) BS ( Q )] d ψ Jb ! (A.63) δ (1) F S ( U ) d ψ Jb ! ≡ ε a ( U µ ) J ∂ µ ( γ ν ) ba ∂ ν ψ Jb iC ab d ! = ε a [D a , δ (1) BS ( U )] d ψ Jb ! (A.64) δ (1) F S ( U ) A µ ψ Jb ! ≡ ε a ( U µ ) J ∂ ν i ( γ γ ν ) ba ψ Jb − ( γ ) ab F µν ! = ε a [D a , δ (1) BS ( U )] A µ ψ Jb ! (A.65) δ (1) F S ( P ) A J λ b ! ≡ ε a P J i ( γ γ µ ) ba ∂ µ λ b ( γ ) ab (cid:3) A J ! = ε a [D a , δ (1) BS ( P )] A J λ b ! (A.66)31 (1) F S ( Q ) B J λ b ! ≡ ε a Q J i ( γ γ µ ) ba ∂ µ λ b ( γ ) ab (cid:3) B J ! = − ε a [D a , δ (1) BS ( Q )] B J λ b ! (A.67) δ (1) F S ( U ) G J λ b ! ≡ ε a ( U µ ) J ∂ ν ∂ [ µ ( γ ν ] ) ba λ b ( σ νµ ) ab ∂ ν G J ! = ε a [D a , δ (1) BS ( U )] G J λ b ! (A.68) δ (1) F S ( U ) F J λ b ! ≡ ε a ( U µ ) J ( γ γ ν ) ba ∂ µ ∂ ν λ b i ( γ ) ab ∂ µ F J ! = − iε a [D a , δ (1) BS ( U )] F J λ b ! (A.69)and from [D a , δ (1) BS ( T )] we have δ (1) F S ( T ) A J ψ Jb ! ≡ ε a T JM ( γ µ ) ba ∂ µ ψ Mb iC ab (cid:3) A M ! δ (1) F S ( T ) F J ψ Jb ! ≡ ε a T JM (cid:3) ψ Ma − i ( γ µ ) ab ∂ µ F M ! (A.70)and from [D a , δ (1) BS ( T )] δ (1) F S ( T ) B J ψ Jb ! ≡ ε a T JM ( γ γ µ ) ba ∂ µ ψ Mb i ( γ ) ab (cid:3) B M ! δ (1) F S ( T ) G J ψ Jb ! ≡ ε a T JM i ( γ ) ba (cid:3) ψ Mb ( γ γ µ ) ab ∂ µ G M ! (A.71)and from [D a , δ (1) BS ( Q )] δ (1) F S ( Q ) A µ ψ Jb ! ≡ ε a Q J − ( γ µ γ ν ) ba ∂ ν ψ Jb ( γ α σ µν ) ba ∂ α F µν ! δ (1) F S ( Q ) dψ Jb ! ≡ ε a Q J i ( γ ) ba (cid:3) ψ Jb − ( γ γ µ ) ab ∂ µ d ! (A.72)and from [D a , δ (1) BS ( W )] δ (1) F S ( W ) A J ψ Jb ! ≡ ε a W JM ( γ γ µ ) ba ∂ µ ψ Mb i ( γ ) ab (cid:3) A M ! δ (1) F S ( W ) B J ψ Jb ! ≡ ε a W JM ( γ µ ) ba ∂ µ ψ Mb C ab (cid:3) B M ! δ (1) F S ( W ) F J ψ Jb ! ≡ ε a W JM i ( γ ) ba (cid:3) ψ Mb ( γ γ µ ) ab ∂ µ F M ! δ (1) F S ( W ) G J ψ Jb ! ≡ ε a W JM i (cid:3) ψ Ma ( γ µ ) ab ∂ µ G M ! (A.73)32nd from [D a , δ (1) BS ( V )] δ (1) F S ( V ) A J ψ Jb ! ≡ ε a ( V ρ ) JM ( γ γ ρ γ µ ) ba ∂ µ ψ Mb i ( γ γ ρ ) ab (cid:3) A M ! δ (1) F S ( V ) B J ψ Jb ! ≡ ε a ( V ρ ) JM − i ( γ ρ γ µ ) ba ∂ µ ψ Mb ( γ ρ ) ab (cid:3) B M ! δ (1) F S ( V ) F J ψ Jb ! ≡ ε a ( V ρ ) JM ( γ γ ν γ ρ γ µ ) ba ∂ µ ∂ ν ψ Mb i ( γ γ ρ γ µ ) ba ∂ µ F M ! δ (1) F S ( V ) G J ψ Jb ! ≡ ε a ( V ρ ) JM i ( γ µ γ ρ γ µ ) ba ∂ µ ∂ ν ψ Mb − ( γ ρ γ µ ) ba ∂ µ G M ! (A.74)and from [D a , δ (1) BS ( T )] δ (1) F S ( T ) A J ψ Jb ! ≡ ε a T JM ( γ µ ) ba ∂ µ ψ Mb iC ab (cid:3) A M ! δ (1) F S ( T ) B J ψ Jb ! ≡ ε a T JM ( γ γ µ ) ba ∂ µ ψ Mb i ( γ ) ab (cid:3) B M ! (A.75) δ (1) F S ( T ) F J ψ Jb ! ≡ ε a T JM i (cid:3) ψ Ma ( γ µ ) ab ∂ µ F M ! δ (1) F S ( T ) G J ψ Jb ! ≡ ε a T JM i ( γ ) ba (cid:3) ψ Mb ( γ γ µ ) ab ∂ µ G M ! (A.76)and from [D a , δ (1) BS ( W )] δ (1) F S ( W ) A J ψ Jb ! ≡ ε a W JM ( γ γ µ ) ba ∂ µ ψ Mb i ( γ ) ab (cid:3) A M ! δ (1) F S ( W ) G J ψ Jb ! ≡ ε a W JM i (cid:3) ψ Ma ( γ µ ) ab ∂ µ G M ! (A.77)and from [D a , δ (1) BS ( V )] δ (1) F S ( V ) F J ψ Jb ! ≡ ε a ( V µ ) JM i ( γ γ ν ) ba ∂ µ ∂ ν ψ Mb ( γ ) ab ∂ µ F M ! δ (1) F S ( V ) G J ψ Jb ! ≡ ε a ( V µ ) JM ( γ ν ) ba ∂ µ ∂ ν ψ Mb − iC ab ∂ µ G M ! (A.78)33nd from [D a , δ (1) BS ( Q )] δ (1) F S ( Q ) A J λ b ! ≡ ε a Q J ( γ µ ) ba ∂ µ λ b − iC ab (cid:3) A J ! δ (1) F S ( Q ) B J λ b ! ≡ ε a Q J i ( γ γ µ ) ba ∂ µ λ b ( γ ) ab (cid:3) B J ! δ (1) F S ( Q ) F J λ b ! ≡ ε a Q J (cid:3) λ a i ( γ µ ) ab ∂ µ F J ! δ (1) F S ( Q ) G J λ b ! ≡ ε a Q J i ( γ ) ba (cid:3) λ b − ( γ γ µ ) ab ∂ µ G J ! (A.79)and and from [D Ia , δ (1) BS ( P )] δ (1) F S ( P ) A K ψ Jb ! ≡ ε aI P K i ( γ γ µ ) ba ∂ µ ψ Ib δ IJ ( γ ) ab (cid:3) A K ! δ (1) F S ( P ) d λ b ! ≡ ε aI P I (cid:3) λ a i ( γ µ ) ab ∂ µ d ! (A.80) δ (1) F S ( P ) d ψ Jb ! ≡ ε aI P K ǫ IJK (cid:3) ψ Jb i ( γ µ ) ab ∂ µ d ! → ε a P J (cid:3) ψ Jb i ( γ µ ) ab ∂ µ d ! (A.81)and from [D Ia , δ (1) BS ( T )] δ (1) F S ( T ) A M ψ Jb ! ≡ ε aI ǫ IJK T KM i ( γ µ ) ba ∂ µ ψ Jb C ab (cid:3) A M ! δ (1) F S ( T ) F M ψ Jb ! ≡ ε aI ǫ IJK T KM (cid:3) ψ Ja i ( γ µ ) ab ∂ µ F M ! δ (1) F S ( T ) A K λ b ! ≡ ε aI T IK i ( γ µ ) ba ∂ µ λ b C ab (cid:3) A K ! δ (1) F S ( T ) F K λ b ! ≡ ε aI T IK (cid:3) λ a i ( γ µ ) ab ∂ µ F K ! (A.82)34nd from [D Ia , δ (1) BS ( Q )] δ (1) F S ( Q ) B K ψ Jb ! ≡ ε aI Q K i ( γ γ µ ) ba ∂ µ ψ Ib δ IJ ( γ ) ab (cid:3) B K ! δ (1) F S ( P ) d ψ Kb ! ≡ ε aI Q J ǫ IJK i ( γ ) ba (cid:3) ψ Kb − ( γ γ µ ) ab ∂ µ d ! → ε a Q K i ( γ ) ba (cid:3) ψ Kb − ( γ γ µ ) ab ∂ µ d ! δ (1) F S ( Q ) d λ b ! ≡ ε aI Q I i ( γ ) ba (cid:3) λ b − ( γ γ µ ) ab ∂ µ d ! (A.83)and from [D Ia , δ (1) BS ( T )] δ (1) F S ( T ) B M ψ Jb ! ≡ ε aI ǫ IJK T KM i ( γ γ µ ) ba ∂ µ ψ Jb ( γ ) ab (cid:3) B M ! δ (1) F S ( T ) G K ψ Jb ! ≡ ε aI ǫ IJK T KM ( γ ) ba (cid:3) ψ Nb i ( γ γ µ ) ab ∂ µ G M ! δ (1) F S ( T ) B K λ b ! ≡ ε aI T IK i ( γ γ µ ) ba ∂ µ λ b ( γ ) ab (cid:3) B K ! δ (1) F S ( T ) G K λ b ! ≡ ε aI T IK ( γ ) ba (cid:3) λ b i ( γ γ µ ) ab ∂ µ G K ! (A.84)and from [D Ia , δ (1) BS ( U )] δ (1) F S ( U ) F K ψ Ib ! ≡ ε aI ( U µ ) K ∂ µ ( γ γ ν ) ba ∂ ν ψ Ib i ( γ ) ab F K ! δ (1) F S ( U ) d ψ Jb ! ≡ ε aI ( U µ ) K ǫ IJK ( γ ν ) ba ∂ µ ∂ ν ψ Jb iC ab ∂ µ d ! → ε a ( U µ ) J ( γ ν ) ba ∂ µ ∂ ν ψ Jb iC ab ∂ µ d ! δ (1) F S ( U ) d λ b ! ≡ ε aI ( U µ ) I ∂ µ ( γ ν ) ba ∂ ν λ b iC ab d ! (A.85)35nd from [D Ia , δ (1) BS ( U )] δ (1) F S ( U ) G K ψ Ib ! ≡ ε aI ( U µ ) K ∂ ν ∂ [ µ ( γ ν ] ) ba ψ Ib ( σ νµ ) ab ∂ ν G K ! δ (1) F S ( U ) A ν ψ Jb ! ≡ ε aI ( U ν ) K ǫ IJK i ( γ γ µ ) ba ∂ µ ψ Jb − ( γ ) ab ∂ µ F νµ ! → ε a ( U ν ) J i ( γ γ µ ) ba ∂ µ ψ Jb − ( γ ) ab ∂ µ F νµ ! δ (1) F S ( U ) A ν λ b ! ≡ ε aI ( U ν ) I − i ( γ γ µ ) ba ∂ µ λ b ( γ ) ab ∂ µ F νµ ! (A.86)and from [D Ia , δ (1) BS ( Q )] δ (1) F S ( Q ) A I ψ Kb ! ≡ ε aI Q K ( γ µ ) ba ∂ µ ψ Kb − iC ab (cid:3) A I ! δ (1) F S ( Q ) B I ψ Kb ! ≡ ε aI Q K i ( γ γ µ ) ba ∂ µ ψ Kb ( γ ) ab (cid:3) B I ! δ (1) F S ( Q ) F I ψ Kb ! ≡ ε aI Q K (cid:3) ψ Ka i ( γ µ ) ab ∂ µ F I ! δ (1) F S ( Q ) G I ψ Kb ! ≡ ε aI Q K ( γ ) ba (cid:3) ψ Kb i ( γ γ µ ) ab ∂ µ G I ! (A.87)and from [D Ia , δ (1) BS ( Q )] δ (1) F S ( Q ) A ν λ b ! ≡ ε aI Q I − ( γ ν γ µ ) ba ∂ µ λ b ( γ α σ µν ) ba ∂ α F µν ! δ (1) F S ( Q ) d λ b ! ≡ ε aI Q I i ( γ ) ba (cid:3) λ b − ( γ γ µ ) ab ∂ µ d ! δ (1) F S ( Q ) A J λ b ! ≡ ε aI Q K ǫ IJK ( γ µ ) ba ∂ µ λ b − iC ab (cid:3) A J ! → ε a Q J ( γ µ ) ba ∂ µ λ b − iC ab (cid:3) A J ! δ (1) F S ( Q ) B J λ b ! ≡ ε aI Q K ǫ IJK i ( γ γ µ ) ba ∂ µ λ b ( γ ) ab (cid:3) B J ! → ε a Q J i ( γ γ µ ) ba ∂ µ λ b ( γ ) ab (cid:3) B J ! (A.88)36 (1) F S ( Q ) F J λ b ! ≡ ε aI Q K ǫ IJK (cid:3) λ a i ( γ µ ) ab ∂ µ F J ! → ε a Q J (cid:3) λ a i ( γ µ ) ab ∂ µ F J ! δ (1) F S ( Q ) G J λ b ! ≡ ε aI Q K ǫ IJK ( γ ) ba (cid:3) λ b i ( γ γ µ ) ab ∂ µ G J ! → ε a Q J ( γ ) ba (cid:3) λ b i ( γ γ µ ) ab ∂ µ G J ! (A.89)and from [D Ia , δ (1) BS ( W )] δ (1) F S ( W ) A J ψ Mb ! ≡ ε aI ǫ IJK W KM i ( γ γ µ ) ba ∂ µ ψ Mb ( γ ) ab (cid:3) A J ! δ (1) F S ( W ) B J ψ Mb ! ≡ ε aI ǫ IJK W KM i ( γ µ ) ba ∂ µ ψ Mb C ab (cid:3) B J ! δ (1) F S ( W ) F J ψ Mb ! ≡ ε aI ǫ IJK W KM ( γ ) ba (cid:3) ψ Mb i ( γ γ µ ) ab ∂ µ F J ! δ (1) F S ( W ) G J ψ Mb ! ≡ ε aI ǫ IJK W KM (cid:3) ψ Ma i ( γ µ ) ab ∂ µ G J ! δ (1) F S ( W ) A ν ψ Mb ! ≡ ε aI W IM − ( γ γ ν γ µ ) ba ∂ µ ψ Mb − ( γ γ µ σ αν ) ba ∂ µ F αν ! δ (1) F S ( W ) d ψ Mb ! ≡ ε aI W IM (cid:3) ψ Ma i ( γ µ ) ab ∂ µ d ! (A.90)and from [D Ia , δ (1) BS ( V ) δ (1) F S ( V ) A J ψ Mb ! ≡ ε aI ǫ IJK ( V ρ ) KM − ( γ γ ρ γ µ ) ba ∂ µ ψ Mb i ( γ γ ρ ) ab (cid:3) A J ! δ (1) F S ( V ) B J ψ Mb ! ≡ ε aI ǫ IJK ( V ρ ) KM i ( γ ρ γ µ ) ba ∂ µ ψ Mb ( γ ρ ) ab (cid:3) B J ! δ (1) F S ( V ) F J ψ Mb ! ≡ ε aI ǫ IJK ( V ρ ) KM ( γ γ µ γ ρ γ ν ) ba ∂ µ ∂ ν ψ Mb − i ( γ γ ρ γ µ ) ba ∂ µ F J ! δ (1) F S ( V ) G J ψ Mb ! ≡ ε aI ǫ IJK ( V ρ ) KM i ( γ µ γ ρ γ ν ) ba ∂ µ ∂ ν ψ Mb ( γ ρ γ µ ) ba ∂ µ G J ! (A.91)37 (1) F S ( V ) A ν ψ Mb ! ≡ ε aI ( V ρ ) IM ( γ γ ν γ ρ γ µ ) ba ∂ µ ψ Mb ( γ γ ρ γ µ σ αν ) ba ∂ µ F αν ! δ (1) F S ( V ) d ψ Mb ! ≡ ε aI ( V ρ ) IM i ( γ µ γ ρ γ ν ) ba ∂ µ ∂ ν ψ Mb ( γ ρ γ µ ) ba ∂ µ d ! (A.92)and from [D Ia , δ (1) BS ( T )] δ (1) F S ( T ) A J ψ Mb ! ≡ ε aI ǫ IJK T KM i ( γ µ ) ba ∂ µ ψ Mb C ab (cid:3) A J ! δ (1) F S ( T ) B J ψ Mb ! ≡ ε aI ǫ IJK T KM i ( γ γ µ ) ba ∂ µ ψ Mb ( γ ) ab (cid:3) B J ! δ (1) F S ( T ) F J ψ Mb ! ≡ ε aI ǫ IJK T KM (cid:3) ψ Ma i ( γ µ ) ab ∂ µ F J ! δ (1) F S ( T ) G J ψ Mb ! ≡ ε aI ǫ IJK T KM ( γ ) ba (cid:3) ψ Mb i ( γ γ µ ) ab ∂ µ G J ! δ (1) F S ( T ) A ν ψ Mb ! ≡ ε aI T IM ( γ ν γ µ ) ba ∂ µ ψ Mb − ( γ µ σ αν ) ba ∂ µ F αν ! δ (1) F S ( T ) d ψ Mb ! ≡ ε aI T IM ( γ ) ba (cid:3) ψ Mb i ( γ γ µ ) ab ∂ µ d ! (A.93)and from [D Ia , δ (1) BS ( W )] δ (1) F S ( W ) A M ψ Jb ! ≡ ε aI W KM ǫ IJK i ( γ γ µ ) ba ∂ µ ψ Jb ( γ ) ab (cid:3) A M ! (A.94) δ (1) F S ( W ) G M ψ Jb ! ≡ ε aI W KM ǫ IJK (cid:3) ψ Na i ( γ µ ) ab ∂ µ G M ! δ (1) F S ( W ) A M λ b ! ≡ ε aI W IM i ( γ γ µ ) ba ∂ µ λ b ( γ ) ab (cid:3) A M ! δ (1) F S ( W ) G M λ b ! ≡ ε aI W IM (cid:3) λ a i ( γ µ ) ab ∂ µ G M ! (A.95)38nd from [D Ia , δ (1) BS ( V )] δ (1) F S ( V ) F M ψ Jb ! ≡ ε aI ( V µ ) KM ǫ IJK ( γ γ ν ) ba ∂ µ ∂ ν ψ Jb i ( γ ) ab ∂ µ F M ! δ (1) F S ( V ) G M ψ Jb ! ≡ ε aI ( V µ ) KM ǫ IJK ( γ ν ) ba ∂ µ ∂ ν ψ Na iC ab ∂ µ G M ! δ (1) F S ( V ) F M λ b ! ≡ ε aI ( V µ ) IM ( γ γ ν ) ba ∂ µ ∂ ν λ b i ( γ ) ab ∂ µ F M ! δ (1) F S ( V ) G M λ b ! ≡ ε aI ( V µ ) IM ( γ ν ) ba ∂ µ ∂ ν λ b iC ab ∂ µ G M ! (A.96)and from [D a , δ (1) BS ( U )] δ (1) F S ( U ) d ψ Kb ! ≡ ε a ( U ρ ) K i ( γ µ γ ρ γ ν ) ba ∂ µ ∂ ν ψ Kb ( γ ρ γ µ ) ba ∂ µ d ! δ (1) F S ( U ) A µ ψ Kb ! ≡ ε a ( U ρ ) K ( γ γ µ γ ρ γ ν ) ba ∂ ν ψ Kb ( γ γ ρ γ µ σ αβ ) ba ∂ µ F αβ ! (A.97)and from [D a , δ (1) BS ( U )] δ (1) F S ( U ) A K λ b ! ≡ ε a ( U ρ ) K ( γ γ ρ γ µ ) ba ∂ µ λ b − i ( γ γ ρ ) ab (cid:3) A K ! δ (1) F S ( U ) B K λ b ! ≡ ε a ( U ρ ) K i ( γ ρ γ µ ) ba ∂ µ λ b ( γ ρ ) ab (cid:3) B K ! δ (1) F S ( U ) F K λ b ! ≡ ε a ( U ρ ) K ( γ γ µ γ ρ γ ν ) ba ∂ µ ∂ ν λ b − i ( γ γ ρ γ µ ) ba ∂ µ F K ! δ (1) F S ( U ) G K λ b ! ≡ ε a ( U ρ ) K i ( γ µ γ ρ γ ν ) ba ∂ µ ∂ ν λ b ( γ ρ γ µ ) ba ∂ µ G K ! (A.98)and from [D Ia , δ (1) BS ( U )] δ (1) F S ( U ) A I ψ Mb ! ≡ ε aI ( U ρ ) M − ( γ γ ρ γ µ ) ba ∂ µ ψ Mb i ( γ γ ρ ) ab (cid:3) A I ! δ (1) F S ( U ) B I ψ Mb ! ≡ ε aI ( U ρ ) M i ( γ ρ γ µ ) ba ∂ µ ψ Mb ( γ ρ ) ab (cid:3) B I ! (A.99)39 (1) F S ( U ) F I ψ Mb ! ≡ ε aI ( U ρ ) M ( γ γ µ γ ρ γ ν ) ba ∂ µ ∂ ν ψ Mb − i ( γ γ ρ γ µ ) ba ∂ µ F I ! δ (1) F S ( U ) G I ψ Mb ! ≡ ε aI ( U ρ ) M i ( γ µ γ ρ γ ν ) ba ∂ µ ∂ ν ψ Mb ( γ ρ γ µ ) ba ∂ µ G I ! (A.100)and from [D Ia , δ (1) BS ( U )] δ (1) F S ( U ) A J λ b ! ≡ ε aI ( U ρ ) K ǫ IJK ( γ γ ρ γ µ ) ba ∂ µ λ b − i ( γ γ ρ ) ab (cid:3) A J ! → ε a ( U ρ ) J ( γ γ ρ γ µ ) ba ∂ µ λ b − i ( γ γ ρ ) ab (cid:3) A J ! δ (1) F S ( U ) B J λ b ! ≡ ε aI ( U ρ ) K ǫ IJK i ( γ ρ γ µ ) ba ∂ µ λ b ( γ ρ ) ab (cid:3) B J ! → ε a ( U ρ ) J i ( γ ρ γ µ ) ba ∂ µ λ b ( γ ρ ) ab (cid:3) B J ! δ (1) F S ( U ) F J λ b ! ≡ ε aI ( U ρ ) K ǫ IJK − ( γ γ µ γ ρ γ ν ) ba ∂ µ ∂ ν λ b i ( γ γ ρ γ µ ) ba ∂ µ F J ! → ε a ( U ρ ) J − ( γ γ µ γ ρ γ ν ) ba ∂ µ ∂ ν λ b i ( γ γ ρ γ µ ) ba ∂ µ F J ! δ (1) F S ( U ) G J λ b ! ε aI ( U ρ ) K ǫ IJK i ( γ µ γ ρ γ ν ) ba ∂ µ ∂ ν λ b ( γ ρ γ µ ) ba ∂ µ G J ! → ε a ( U ρ ) J i ( γ µ γ ρ γ ν ) ba ∂ µ ∂ ν λ b ( γ ρ γ µ ) ba ∂ µ G J ! δ (1) F S ( U ) d λ b ! ≡ ε aI ( U ρ ) I i ( γ µ γ ρ γ ν ) ba ∂ µ ∂ ν λ b ( γ ρ γ µ ) ba ∂ µ d ! δ (1) F S ( U ) A µ λ b ! ≡ ε aI ( U ρ ) I ( γ γ µ γ ρ γ ν ) ba ∂ ν λ b ( γ γ ρ γ ν σ αβ ) ba ∂ ν F αβ ! (A.101)and from [D a , δ (1) BS ( P )] δ (1) F S ( U ) d ψ Kb ! ≡ ε a P K i (cid:3) ψ kb − ( γ µ ) ab ∂ µ d ! δ (1) F S ( U ) A µ ψ Ib ! ≡ ε a P K ( γ γ µ γ ν ) ba ∂ ν ψ Kb − i ( γ γ ν ) ab ∂ µ F µν ! (A.102)40nd from [D a , δ (1) BS ( P )] δ (1) F S ( P ) A K λ b ! ≡ ε a P K i ( γ γ µ ) ba ∂ µ λ b ( γ ) ab (cid:3) A K ! δ (1) F S ( P ) B K λ b ! ≡ ε a P K i ( γ µ ) ba ∂ µ λ b C ab (cid:3) B K ! δ (1) F S ( P ) F K λ b ! ≡ ε a P K ( γ ) ba (cid:3) λ b i ( γ γ µ ) ab ∂ µ F K ! δ (1) F S ( P ) G K λ b ! ≡ ε a P K (cid:3) λ a i ( γ µ ) ab ∂ µ G K ! (A.103)and from [D Ia , δ (1) BS ( P )] δ (1) F S ( P ) A I ψ Mb ! ≡ ε aI P M − ( γ γ µ ) ba ∂ µ ψ Mb i ( γ ) ab (cid:3) A I ! δ (1) F S ( P ) B I ψ Mb ! ≡ ε aI P M i ( γ µ ) ba ∂ µ ψ Mb C ab (cid:3) B I ! δ (1) F S ( P ) F I ψ Mb ! ≡ ε aI P M ( γ ) ba (cid:3) ψ Mb i ( γ γ µ ) ab ∂ µ F I ! δ (1) F S ( P ) G I ψ Mb ! ≡ ε aI P M (cid:3) ψ Ma i ( γ µ ) ab ∂ µ G I ! (A.104)from [D Ia , δ (1) BS ( P )] δ (1) F S ( P ) d λ b ! ≡ ε aI P I − i (cid:3) λ b ( γ µ ) ab ∂ µ d ! δ (1) F S ( P ) A µ λ b ! ≡ ε aI P I ( γ γ µ γ ν ) ba ∂ ν λ b − i ( γ γ ν ) ab ∂ µ F µν ! δ (1) F S ( P ) A J λ b ! ≡ ε aI P K ǫ IJK i ( γ γ µ ) ba ∂ µ λ b ( γ ) ab (cid:3) A J ! → ε a P J i ( γ γ µ ) ba ∂ µ λ b ( γ ) ab (cid:3) A J ! (A.105)41 (1) F S ( P ) B J λ b ! ≡ ε aI P K ǫ IJK i ( γ µ ) ba ∂ µ λ b C ab (cid:3) B J ! → ε a P J i ( γ µ ) ba ∂ µ λ b C ab (cid:3) B J ! δ (1) F S ( P ) F J λ b ! ≡ ε aI P K ǫ IJK ( γ ) ba (cid:3) λ b i ( γ γ µ ) ab ∂ µ F J ! → ε a P J ( γ ) ba (cid:3) λ b i ( γ γ µ ) ab ∂ µ F J ! δ (1) F S ( P ) G J λ b ! ≡ ε aI P K ǫ IJK − i (cid:3) λ a ( γ µ ) ab ∂ µ G J ! → ε a P J − i (cid:3) λ a ( γ µ ) ab ∂ µ G J ! (A.106) A.2 N = 2 FH In this section, we list all of the N = 2 FH fermionic first order symmetries uncoveredvia our method, including the redundant ones. Only the unique symmetries werelisted in the body of the paper. From from [ ˜D ia , ˜ δ (1) BS ( ˜ T )] we find the symmetries˜ δ (1) F S ( ˜ T ) ˜ A k ˜ ψ b ! ≡ ε ai ˜ T ik − ( γ µ ) ba ∂ µ ˜ ψ b iC ab (cid:3) ˜ A k ! ˜ δ (1) F S ( ˜ T ) ˜ F k ˜ ψ b ! ≡ ε ai ˜ T ik (cid:3) ˜ ψ a i ( γ µ ) ab ∂ µ ˜ F k ! ˜ δ (1) F S ( ˜ T ) ˜ A k ˜ ψ b ! ≡ ε ai ( σ ) ij ˜ T jk i ( γ µ ) ba ∂ µ ˜ ψ b C ab (cid:3) ˜ A k ! ˜ δ (1) F S ( ˜ T ) ˜ F k ˜ ψ b ! ≡ ε ai ( σ ) ij T jk − i (cid:3) ˜ ψ a ( γ µ ) ab ∂ µ ˜ F k ! (A.107)and from [ ˜D ia , ˜ δ (1) BS ( ˜ T )]˜ δ (1) F S ( ˜ T ) ˜ B k ˜ ψ b ! ≡ ε ai ( σ ) ij ˜ T jk i ( γ γ µ ) ba ∂ µ ˜ ψ b ( γ ) ab (cid:3) ˜ B k ! ˜ δ (1) F S ( ˜ T ) ˜ G k ˜ ψ b ! ≡ ε ai ( σ ) ij ˜ T jk − i ( γ ) ba (cid:3) ˜ ψ b ( γ γ µ ) ab ∂ µ ˜ G k ! (A.108)42 δ (1) F S ( ˜ T ) ˜ B k ˜ ψ b ! ≡ ε ai ( σ ) ij ˜ T jk i ( γ γ µ ) ba ∂ µ ˜ ψ b ( γ ) ab (cid:3) ˜ B k ! ˜ δ (1) F S ( ˜ T ) ˜ G k ˜ ψ b ! ≡ ε ai ( σ ) ij ˜ T jk − i ( γ ) ba (cid:3) ˜ ψ b ( γ γ µ ) ab ∂ µ ˜ G k ! (A.109)Calculation of [ ˜D ia , ˜ δ (1) BS ( ˜ T )] uncovers no new symmetries, just these same eight again:˜ δ (1) F S ( ˜ T ) ˜ A k ˜ ψ b ! ≡ ε ai ( σ ) ik ˜ T i ( γ µ ) ba ∂ µ ˜ ψ b C ab (cid:3) ˜ A k ! ˜ δ (1) F S ( ˜ T ) ˜ F k ˜ ψ b ! ≡ ε ai ( σ ) ik ˜ T i (cid:3) ˜ ψ a − ( γ µ ) ab ∂ µ ˜ F k ! ˜ δ (1) F S ( ˜ T ) ˜ A k ˜ ψ b ! ≡ ε ak ˜ T − ( γ µ ) ba ∂ µ ˜ ψ b iC ab (cid:3) ˜ A k ! ˜ δ (1) F S ( ˜ T ) ˜ F k ˜ ψ b ! ≡ ε ak ˜ T (cid:3) ˜ ψ a i ( γ µ ) ab ∂ µ ˜ F k ! ˜ δ (1) F S ( ˜ T ) ˜ B k ˜ ψ b ! ≡ ε ai ( σ ) ik ˜ T i ( γ γ µ ) ba ∂ µ ˜ ψ b ( γ ) ab (cid:3) ˜ B k ! ˜ δ (1) F S ( ˜ T ) ˜ G k ˜ ψ b ! ≡ ε ai ( σ ) ik ˜ T i ( γ ) ba (cid:3) ˜ ψ b − ( γ γ µ ) ab ∂ µ ˜ G k ! ˜ δ (1) F S ( ˜ T ) ˜ B k ˜ ψ b ! ≡ ε ai ( σ ) ik ˜ T i ( γ γ µ ) ba ∂ µ ˜ ψ b ( γ ) ab (cid:3) ˜ B k ! ˜ δ (1) F S ( ˜ T ) ˜ G k ˜ ψ b ! ≡ ε ai ( σ ) ik ˜ T − i ( γ ) ba (cid:3) ˜ ψ b ( γ γ µ ) ab ∂ µ ˜ G k ! (A.110)under redefinitions of ˜ T . References [1] J. M. 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