Superconformal Symmetry in Three-Dimensions and Lies in the Same Framework As Our Sequent Work on Superconformal Symmetry in Other Dimensions, D = 4, 6 [4–6]

Home , Superconformal algebra

arXiv:hep-th/9910199v2 3 Nov 1999 ieeta operators differential Keywords PACS ∗ -alades [email protected] address: E-mail omlgopi hnietfidwt uemti ru,OSp group, supermatrix a with identified then tran Lo is superconformal dilations, group special formal supertranslations, and of d transformations terms is symmetry equation in Killing classified superconformal are A formalism. perspace Three-dimensional rmteaayi nsml i ueagba.I eea,d transfo general, superconformal In invariant special superalgebras. and Lie supertranslations simple der on analysis the from oain ieeta prtr r lodiscussed. also are of operators form differential general covariant the obtain and abl invariants are we formal correlators, superconformal for blocks building tra and rigid transformations remaining Lorentz the tions, under homogeneously transform must 13.b 11.25.Hf 11.30.Pb; : uecnomlSmer nThree-dimensions in Symmetry Superconformal uecnomlsmer;Creainfntos Superc functions; Correlation symmetry; Superconformal : n pitfntosrdc ooeuseie ( unspecified one to reduce functions -point colo hsc,KraIsiuefrAvne Study Advanced for Institute Korea Physics, of School 0-3Cenragidn,Dongdaemun-gu Cheongryangri-dong, 207-43 N etne uecnomlsmer ssuidwti h su the within studied is symmetry superconformal -extended en-yc Park Jeong-Hyuck eu 3-1,Korea 130-012, Seoul R Abstract smer rnfrain.Atrconstructing After transformations. -symmetry n pitfntos Superconformally functions. -point ∗ oietf l h supercon- the all identify to e n mtos superconformally rmations, − et transformations, rentz et h naineun- invariance the to ue )pitfnto which function 2)-point frain.Supercon- sformations. rvdadissolutions its and erived somtos ..dila- i.e. nsformations, nomlycovariant onformally ( | N 2 , R ,a expected as ), hep-th/9910199 KIAS-99101 R - - 1 Introduction and Summary

Based on the classiﬁcation of simple Lie superalgebras [1], Nahm analyzed all possible superconformal algebras [2]. According to Ref. [2], not all spacetime dimensions allow the corresponding supersymmetry algebra to be extended to a superconformal algebra contrary to the ordinary conformal symmetry. The standard supersymmetry algebra admits an extension to a superconformal algebra only if d 6. Namely the highest dimension admitting superconformal algebra is six, and in d =≤ 3, 4, 5, 6 dimensions the bosonic part of the superconformal algebra has the form

, (1.1) LC ⊕LR where C is the Lie algebra of the conformal group and R is a R-symmetry algebra acting on theL superspace Grassmann variables. L Explicitly for Minkowskian spacetime

d =3; o(2, 3) o( ) , ⊕ N o(2, 4) u( ) , =4 ⊕ N N 6 d = 4;  ,  o(2, 4) su(4) (1.2) ⊕  d =5; o(2, 5) su(2) , ⊕ d =6; o(2, 6) sp( ) , ⊕ N where or the number appearing in R-symmetry part is related to the number of super- N charges.

On six-dimensional Minkowskian spacetime it is possible to define Weyl spinors of op- posite chiralities and so the general six-dimensional supersymmetry may be denoted by two numbers, ( , ˜ ), where and ˜ are the numbers of chiral and anti-chiral super- N N N N charges. The R-symmetry group is then Sp( ) Sp( ˜ ). The analysis of Nahm shows that to admit a superconformal algebra eitherN ×or Ñ should be zero. Although both N N (1, 1) and (2, 0) supersymmetry give rise = 4 four-dimensional supersymmetry after dimensional reduction, only (2, 0) supersymmetryN theories can be superconformal [3]. On five-dimensional Minkowskian spacetime Nahm’s analysis seems to imply a certain restriction on the number of supercharges as the corresponding R-symmetry algebra is to be su(2).

1 The above analysis is essentially based on the classification of simple Lie superalgebras and identification of the bosonic part with the usual spacetime conformal symmetry rather than Poincarésymmetry, since the former forms a simple group, while the latter does not. This approach does not rely on any definition of superconformal transformations on superspace.

The present paper deals with superconformal symmetry in three-dimensions and lies in the same framework as our sequent work on superconformal symmetry in other dimensions, d = 4, 6 [4–6]. Namely we analyze superconformal symmetry directly in terms of coordinate transformations on superspace. We first define the superconformal group on superspace and derive the superconformal Killing equation. Its general solutions are identified in terms of supertranslations, dilations, Lorentz transformations, R-symmetry transformations and special superconformal transformations. Based on the explicit form of the solutions the superconformal group is independently identified to agree with Nahm’s analysis and some representations are obtained.

Specifically, in Ref. [4] we identified four-dimensional = 4 extended superconformal group with a supermatrix group, SU(2, 2 ), having dimensionsN 6 (15 + 2 8 ), while for |N N | N = 4 case we pointed out that an equivalence relation must be imposed on the super- matrixN group and so the four-dimensional = 4 superconformal group is isomorphic to a N quotient group of the supermatrix group. In fact = 4 superconformal group is a semi- direct product of U(1) and a simple Lie supergroupN containing SU(4). The U(1) factor can be removed by imposing tracelessness condition on the supermatrix group so that the dimension reduces from (31 32) to (30 32) and the R-symmetry group shrinks from U(4) to Nahm’s result, SU(4)1. In Ref.| [6] by| solving the superconformal Killing equation we show that six-dimensional ( , 0) superconformal group is identified with a supermatrix group, N OSp(2, 6 ), having dimensions (28 + (2 + 1) 16 ), while for ( , ˜ ), , ˜ > 0 supersymmetry,|N we verified that althoughN dilationsN may| N be introduced,N thereN existN N no special superconformal transformations as expected from Nahm’s result.

The main advantage of our formalism is that it enables us to write general expression for two-point, three-point and n-point correlation functions of quasi-primary superfields which transform simply under superconformal transformations. In Refs. [4–6] we explicitly constructed building blocks for superconformal correlators in four- and six-dimensions, and 1Similarly if five-dimensional superconformal group is not simple, this will be a way out from the puzzling restriction on the number of supercharges in five-dimensional superconformal theories, as the corresponding R-symmetry group can be bigger than Nahm’s result, SU(2). However this is at the level of speculation at present.

2 proved that these building blocks actually generate the general form of correlation functions. In general, due to the invariance under supertranslations and special superconformal transformations, n-point functions reduce to one unspecified (n 2)-point function which must transform homogeneously under the rigid transformations− only - dilations, Lorentz transformations and R-symmetry transformations [4]. This feature of superconformally invariant correlation functions is universal for any spacetime dimension if there exists a well defined superinversion in the corresponding dimension, since superinversion plays a crucial role in its proof. For non-supersymmetric case, contrary to superinversion, the inversion map is defined of the same form irrespective of the spacetime dimension and hence n-point functions reduce to one unspecified (n 2)-point function in any dimension which transform − homogeneously under dilations and Lorentz transformations.

The formalism is powerful for applications whenever there exist off-shell superfield formulations for superconformal theories, and such formulations are known in four-dimensions for =1, 2, 3 [7–11] and in three-dimensions for =1, 2, 3, 4 [12–19]. In fact within the N N formalism Osborn elaborated the analysis of = 1 superconformal symmetry for four- dimensional quantum field theories [20], and recentlyN Kuzenko and Theisen determine the general structure of two- and three- point functions of the supercurrent and the flavour cur- rent of = 2 superconformal field theories [21]. A common result contained in Refs. [20,21] is thatN the three-point functions of the conserved supercurrents in both = 1 and =2 N N superconformal theories allow two linearly independent structures and hence there exist two numerical coefficients which can be calculated in specific perturbation theories using supergraph techniques.

The contents of the present paper are as follows. In section 2 we review supersymmetry in three-dimensions. In particular, we verify that supersymmetry algebra with N Dirac supercharges is equivalent to 2N-extended Majorana supersymmetry algebra, so that in the present paper we consider -extended Majorana superconformal symmetry with an N arbitrary natural number, . N In section 3, we first define the three-dimensional -extended superconformal group N in terms of coordinate transformations on superspace as a generalization of the definition of ordinary conformal transformations. We then derive a superconformal Killing equation, which is a necessary and sufficient condition for a supercoordinate transformation to be superconformal. The general solutions are identified in terms of supertranslations, dilations, Lorentz transformations, R-symmetry transformations and special superconformal transformations, where R-symmetry is given by O( ) as in eq.(1.2). We also present a definition of superinversion in three-dimensions throughN which supertranslations and special

3 superconformal transformations are dual to each other. The three-dimensional -extended superconformal group is then identiﬁed with a supermatrix group, OSp( 2N, R), having 1 N| dimensions (10 + 2 ( 1) 4 ) as expected from the analysis on simple Lie superalgebras [2,22]. N N − | N

In section 4, we obtain an explicit formula for the finite non-linear superconformal transformations of the supercoordinates, z, parameterizing superspace and discuss several representations of the superconformal group. We also construct matrix or vector valued functions depending on two or three points in superspace which transform covariantly under superconformal transformations. For two points, z1 and z2, we find a matrix, I(z1, z2), which transforms covariantly like a product of two tensors at z1 and z2. For three points, z1, z2, z3, we find ‘tangent’ vectors, Zi, which transform homogeneously at zi, i = 1, 2, 3. These variables serve as building blocks of obtaining two-point, three-point and general n-point correlation functions later.

In section 5, we discuss the superconformal invariance of correlation functions for quasi- primary superﬁelds and exhibit general forms of two-point, three-point and n-point functions. Explicit formulae for two-point functions of superﬁelds in various cases are given. We also identify all the superconformal invariants.

In section 6, superconformally covariant differential operators are discussed. The conditions for superfields, which are formed by the action of spinor derivatives on quasi-primary superfields, to remain quasi-primary are obtained. In general, the action of differential operator on quasi-primary fields generates an anomalous term under superconformal transformations. However, with a suitable choice of scale dimension, we show that the anomalous term may be cancelled. We regard this analysis as a necessary step to write superconformally invariant actions on superspace, as the kinetic terms in such theories may consist of superfields formed by the action of spinor derivatives on quasi-primary superfields.

In the appendix, the explicit form of superconformal algebra and a method of solving the superconformal Killing equation are exhibited.

4 2 Preliminary

2.1 Gamma Matrices With the three-dimensional Minkowskian metric, ηµν = diag(+1, 1, 1), the 2 2 gamma matrices, γµ, µ =0, 1, 2, satisfy − − ×

µ ν µν µνρ γ γ = η + iǫ γρ . (2.1)

The hermiticity condition is 0 µ 0 µ γ γ γ = γ † . (2.2) Charge conjugation matrix, ǫ, satisﬁes2 [23]

µ 1 µt ǫγ ǫ− = γ , − (2.3) t 1 ǫ = ǫ , ǫ† = ǫ− . − γµ forms a basis for 2 2 traceless matrices with the completeness relation × γµα γ γ =2δα δ γ δα δγ . (2.4) β µ δ δ β − β δ 2.2 Three-dimensional Superspace The three-dimensional supersymmetry algebra has the standard form with P =(H, P) µ − iα, ¯ =2δi γµα P , {Q Qjβ} j β µ (2.5) [P ,P ] = [P , iα] = [P , ¯ ]= iα, jβ = ¯ , ¯ =0 , µ ν µ Q µ Qiα {Q Q } {Qiα Qjβ} where 1 α 2, 1 i N and i, ¯ satisfy ≤ ≤ ≤ ≤ Q Qj i 0 ¯ = †γ . (2.6) Qi Q Now we deﬁne for 1 a 2N, 1 i, j N ≤ ≤ ≤ ≤ 1 i 1 ¯t √2 ( + ǫ− i) a Q Q Q =   , (2.7) 1 j 1 ¯t  i √ ( ǫ− j)   2 Q − Q  2To emphasize the anti-symmetric property of the 2 2 charge conjugation matrix in three-dimensions we adopt the symbol, ǫ, instead of the conventional one,×C.

5 and a 0 1 it 1 jt Q¯ Q †γ = ( ¯ ǫ), i ( ¯ + ǫ) . (2.8) a ≡ √2 Qi − Q − √2 Qj Q a Q , Q¯b satisfy the Majorana condition

a 0 at Q¯ = Q †γ = Q ǫ , a − (2.9) a 1 ¯t Q = ǫ− Qa . With this notation we note that the three-dimensional N-extended supersymmetry algebra (2.5) is equivalent to the 2N-extended Majorana supersymmetry algebra

Qaα, Q¯ =2δa γµα P , { bβ} b β µ (2.10) aα [Pµ,Pν] = [Pµ, Q ]=0 .

This can be generalized by replacing 2N with an arbitrary natural number, , and hence -extended Majorana supersymmetry algebra. N N P , Qaα, 1 a generate a supergroup, G , with parameters, zM = (xµ, θaα), µ ≤ ≤ N T which are coordinates on superspace. The general element of GT is written in terms of these coordinates as a i(x P +Q¯aθ ) g(z)= e · . (2.11) Corresponding to eq.(2.9) θa also satisﬁes the Majorana condition

a 0 at θ¯ = θ †γ = θ ǫ , (2.12) a − so that a a 1 Q¯ θ = θ¯ Q , g(z)† = g(z)− = g( z) . (2.13) a a − The Baker-Campbell-Haussdorﬀ formula with the supersymmetry algebra (2.10) gives

g(z1)g(z2)= g(z3) , (2.14) where µ µ µ ¯ µ a a a a x3 = x1 + x2 + iθ1aγ θ2 , θ3 = θ1 + θ2 . (2.15) M µ aα Letting z1 z2 we may get the supertranslation invariant one forms, e =(e , dθ ), where → − eµ(z) = dxµ iθ¯ γµdθa . (2.16) − a

6 The exterior derivative, d, on superspace is deﬁned as ∂ d dzM = eM D = eµ∂ dθaαD , (2.17) ≡ ∂zM M µ − aα where D =(∂ , D ) are covariant derivatives M µ − aα ∂ ∂ ∂ ∂ = ,D = + i(θ¯ γµ) . (2.18) µ ∂xµ aα −∂θaα a α ∂xµ We also deﬁne ¯ aα 1αβ ∂ µ a α ∂ D = ǫ− Daβ = i(γ θ ) µ , (2.19) ∂θ¯aα − ∂x satisfying the anti-commutator relations

D¯ aα,D =2iδa γµα ∂ . (2.20) { bβ} b β µ

M Under an arbitrary superspace coordinate transformation, z z′, e and DM transform as −→ M N M 1 N e (z′)= e (z) (z) ,D′ = − (z)D , (2.21) RN M R M N so that the exterior derivative is left invariant

M M e (z)DM = e (z′)DM′ , (2.22)

where N (z) isa(3+2 ) (3+2 ) supermatrix of the form RM N × N ν bβ N Rµ (z) ∂µθ′ M (z)= µ bβ , (2.23) R B (z) Daαθ′ − aα − ! with ν a ν ∂x′ ν ∂θ′ R (z)= iθ¯′ γ , (2.24) µ ∂xµ − a ∂xµ

µ µ ¯ µ b Baα(z)= Daαx′ + iθb′ γ Daαθ′ . (2.25)

For Majorana spinors it is useful to note from eqs.(2.2,2.3,A.3a)

a a ε¯aρ =ρ ¯aε , (2.26a)

7 a a a ρ ε¯a + ε ρ¯a +ρ ¯aε 1=0 , (2.26b)

ρ¯ γµ1 γµ2 γµn εa =( 1)nε¯ γµn γµ2 γµ1 ρa , (2.26c) a ··· − a ···

µ1 µ2 µn a µn µ2 µ1 a (¯ρ γ γ γ ε )∗ =ε ¯ γ γ γ ρ . (2.26d) a ··· a ··· In particular θaθ¯ = 1 θ¯ θa 1 . (2.27) a − 2 a 3 Superconformal Symmetry in Three-dimensions

In this section we ﬁrst deﬁne the three-dimensional superconformal group on superspace and then discuss its superconformal Killing equation along with the solutions.

3.1 Superconformal Group & Killing Equation The superconformal group is deﬁned here as a group of superspace coordinate transforma- g 2 µ ν tions, z z′, that preserve the inﬁnitesimal supersymmetric interval length, e = ηµν e e , up to a−→ local scale factor, so that

2 2 2 2 e (z) e (z′) = Ω (z; g)e (z) , (3.1) → where Ω(z; g) is a local scale factor. µ This requires Baα(z)=0 µ ¯ µ b Daαx′ + iθb′ γ Daαθ′ =0 . (3.2) and

µ ν µ e (z′)= e (z)Rν (z; g) , (3.3)

λ ρ 2 3 Rµ (z; g)Rν (z; g)ηλρ = Ω (z; g)ηµν , det R(z; g) = Ω (z; g) . (3.4) Hence N in eq.(2.23) is of the form3 RM ν bβ N Rµ (z; g) ∂µθ′ M (z; g)= bβ . (3.5) R 0 Daαθ′ ! 3 N − More explicit form of M is obtained later in eq.(4.40). R 8 Infinitesimally z′ z + δz, eq.(3.2) gives ≃ µ µ Daαh =2i(λ¯aγ )α , (3.6) or equivalently D¯ aαhµ = 2i(γµλa)α , (3.7) − where we define a a λ = δθ , λ¯a = δθ¯a , (3.8) hµ = δxµ iθ¯ γµδθa . − a ν Infinitesimally from eq.(2.24) Rµ is of the form

R ν δ ν + ∂ hν , (3.9) µ ≃ µ µ so that the condition (3.4) reduces to the ordinary conformal Killing equation

∂ h + ∂ h η . (3.10) µ ν ν µ ∝ µν We note that eq.(3.10) follows from eq.(3.6). Using the anti-commutator relation for Daα (2.20) we get from eqs.(3.6,3.7,A.1a)

δa ∂ h = 1 D¯ aα(λ¯ γ γ ) (γ γ D λa)α , (3.11) b ν µ 2 b µ ν α − ν µ bα and hence δ b(∂ h + ∂ h )=(D¯ aαλ¯ D λaα)η , (3.12) a µ ν ν µ bα − bα µν which implies eq.(3.10). Thus eq.(3.6) or eq.(3.7) is a necessary and suﬃcient condition for a supercoordinate transformation to be superconformal.

aα From eq.(3.6,3.7) λ , λ¯aα are given by

λaα = i 1 D¯ aβhα , λ¯ = i 1 D hβ , (3.13) 6 β aα − 6 aβ α where α µ α h β = h γµ β . (3.14) Substituting these expressions back into eqs.(3.6,3.7) gives using eqs.(2.1,2.4)

D hµ = i 1 ǫµ D hνγλβ , aα − 2 νλ aβ α (3.15) ¯ aα µ 1 µ λα ¯ aβ ν D h = i 2 ǫ νλγ βD h .

9 or equivalently D hβ = 2 δ βD hδ 1 δβ D hδ , aα γ 3 α aδ γ − 3 γ aδ α (3.16) D¯ aαhβ = 2 δα D¯ aδhβ 1 δβ D¯ aδhα . γ 3 γ δ − 3 γ δ Eq.(3.15) or eq.(3.16) may therefore be regarded as the fundamental superconformal Killing equation and its solutions give the generators of extended superconformal transformations in three-dimensions. The general solution is4

hµ(z)= 2x b xµ (x2 1 (θ¯ θa)2)bµ + ǫµ xν bλθ¯ θa · − − 4 a νλ a µ a µ ν 1 µ νλ a µ 2¯ρax γ θ + w νx + ǫ νλw θ¯aθ + λx (3.17) − − 4 b µ a µ a µ +ita θ¯bγ θ +2iε¯aγ θ + a , where aµ, bµ,λ,w = w are real, εa, ρa satisfy the Majorana condition (2.9) and µν − νµ t so( ) satisfying ∈ N t t† = t = t . (3.18) − We also set µ 1 a x= x γ , x = x i θ¯aθ 1 . (3.19) ± ± 2 Eq.(3.17) gives

λa = x b γθa ix ρa + 2(¯ρ θa)θb +(w + 1 λ)θa θbt a + εa (3.20) + · − + b 2 − b satisfying the Majorana condition

a 0 at λ¯ = λ †γ = λ ǫ , (3.21) a − where we put 1 µ ν w = 4 wµν γ γ . (3.22) For later use it is worth to note

0 0 1 t γ wγ = w† , ǫwǫ− = w . (3.23) − − 4A method of obtaining the solution (3.17) is demonstrated in Appendix B.

10 3.2 Extended Superconformal Transformations In summary, the generators of superconformal transformations in three-dimensions act- 3 2 M µ aα ing on the three-dimensional superspace, R | N , with coordinates, z = (x , θ ), can be classiﬁed as

1. Supertranslations, a, ε

δxµ = aµ iθ¯ γµεa , δθa = εa . (3.24) − a This is consistent with eq.(2.15).

2. Dilations, λ µ µ a 1 a δx = λx , δθ = 2 λθ . (3.25) 3. Lorentz transformations, w

µ µ ν a a δx = w νx , δθ = wθ . (3.26)

4. R-symmetry transformations, t

δxµ =0 , δθa = θbt a . (3.27) − b where t so( ) of dimension 1 ( 1). ∈ N 2 N N − 5. Special superconformal transformations, b, ρ

δxµ =2x b xµ (x2 + 1 (θ¯ θa)2)bµ ρ¯ x γµθa , · − 4 a − a + (3.28) δθa = x b γθa ix ρa + 2(¯ρ θa)θb . + · − + b As we consider inﬁnitesimal transformations we obtain SO( ) as R-symmetry group. However ﬁnitely R-symmetry group can be extended to O( ) whichN leaves the supertrans- N lation invariant one form (2.16) invariant manifestly.

11 3.3 Superinversion

M is M µ aα 3 2 In three-dimensions we deﬁne superinversion, z z′ =(x′ , θ′ ) R | N , by −→ ∈ 1 a 1 a x′ = x− , θ′ = ix+− θ . (3.29) ± − ± 2 1 ¯ a 2 As a consistency check we note from x+x =(x + (θaθ ) )1 − 4 ¯ a 0 at ¯ a θa′ = θ′ †γ = θ′ ǫ , x+′ x′ = iθa′ θ′ 1 . (3.30) − − − It is easy to verify that superinversion is idempotent

2 is =1 . (3.31) Using µ a e(z)= e (z)γµ = dx+ +2idθ θ¯a , (3.32) we get under superinversion 1 1 e(z′) = x+− e(z)x− . (3.33) − and hence 1 e2(z ) = Ω2(z; i )e2(z) , Ω(z; i )= . ′ s s 2 1 ¯ a 2 (3.34) x + 4 (θaθ ) Eq.(3.33) can be rewritten as

µ ν µ µ 1 1 µ 1 e (z′)= e (z)Rν (z; is) , Rν (z; is)= 2 tr(γνx− γ x+− ) . (3.35) − Explicitly

µ 1 µ 2 1 ¯ a 2 µ µλ ¯ a Rν (z; is)= 2 1 a 2 2 2xνx (x 4 (θaθ ) )δν ǫν xλθaθ . (3.36) (x + (θ¯aθ ) ) − − − 4 Note that ν µ 1 µ 1 µ 1 1 γ Rν (z; is) = x− γ x+− , Rν (z; is)γµ = x+− γνx− . (3.37) − − is g is ◦ ◦ If we consider a transformation, z z′, where g is a three-dimensional superconformal transformation, then we get −−−−→ hµ(z)= 2x a xµ (x2 1 (θ¯ θa)2)aµ + ǫµ xν aλθ¯ θa · − − 4 a νλ a µ a µ ν 1 µ νλ a µ 2¯εax γ θ + w νx + ǫ νλw θ¯aθ λx (3.38) − − 4 − b µ a µ a µ +ita θ¯bγ θ +2iρ¯aγ θ + b .

12 Hence, under superinversion, the superconformal transformations are related by

aµ bµ µ µ  b   a  εa ρa      a   a   ρ   ε  . (3.39) K ≡   −→    λ   λ       µ   −µ   w ν   w ν   a   a   t b   t b          In particular, special superconformal transformations (3.28) can be obtained by

is (b,ρ) is ◦ ◦ z z′ , (3.40) −−−−−−−→ where (b, ρ) is a supertranslation.

3.4 Superconformal Algebra The generator of inﬁnitesimal superconformal transformations, , is given by L = hµ∂ λaαD . (3.41) L µ − aα If we write the commutator of two generators, , , as L1 L2 [ , ]= = hµ∂ λaαD , (3.42) L2 L1 L3 3 µ − 3 aα µ aα then h3 , λ3 are given by

hµ = hν∂ hµ hν∂ hµ +2iλ¯ γµλa , 3 2 ν 1 − 1 ν 2 1a 2 (3.43) λa = λa λa , 3 L2 1 −L1 2

13 µ a and h3 , λ3 satisfy eq.(3.7) verifying the closure of the Lie algebra. Explicitly with eqs.(3.17 ,3.20) we get

aµ = wµ aν + λ aµ + iε¯ γµεa (1 2) , 3 1 ν 2 1 2 1a 2 − ↔ εa = w εa + 1 λ εa ia γρa εb t a (1 2) , 3 1 2 2 1 2 − 2· 1 − 2 1b − ↔ λ =2a b 2¯ρ εa (1 2) , 3 2· 1 − 1a 2 − ↔ wµν = wµ wλν + 2(aµbν aνbµ)+2¯ρ γ[µγν]εa (1 2) , (3.44) 3 1 λ 2 2 1 − 2 1 1a 2 − ↔ bµ = wµ bν λ bµ + iρ¯ γµρa (1 2) , 3 1 ν 2 − 1 2 1a 2 − ↔ ρa = w ρa 1 λ ρa ib γεa ρb t a (1 2) , 3 1 2 − 2 1 2 − 2· 1 − 2 1b − ↔ t b =(t t ) b + 2(¯ρ εb ε¯ ρb ) (1 2) . 3a 1 2 a 2a 1 − 1a 2 − ↔ From eq.(3.44) we can read oﬀ the explicit forms of three-dimensional superconformal algebra as exhibited in Appendix C.

If we deﬁne a (4 + 2 ) (4+2 ) supermatrix, M, as N × N w + 1 λ ia γ √2εb 2 · M = ib γ w 1 λ √2ρb , (3.45)  · − 2  √2¯ρ √2¯ε t b  a a a   − −  then the relation above (3.44) agrees with the matrix commutator

[M1, M2]= M3 . (3.46)

In general, M can be deﬁned as a (4, 2 ) supermatrix subject to N 0 γ0 0 1 0 BMB− = M † , B =  γ 0 0  , (3.47a) − 0 0 1     0 ǫ 0 1 t CMC− = M , C =  ǫ 0 0  . (3.47b) − 0 0 1    

14 Supermatrix of the form (3.45) is the general solution of these two equations.

The 4 4 matrix appearing in M, × w + 1 λ ia γ 2 · , (3.48) ib γ w 1 λ · − 2 ! R corresponds to a generator of SO(2, 3) ∼= Sp(2, ) as demonstrated in Appendix D. Thus, the -extended Majorana superconformal group in three-dimensions may be identiﬁed withN the supermatrix group generated by supermatrices of the form M (3.45), which is OSp( 2, R) G having dimensions (10 + 1 ( 1) 4 ). N| ≡ S 2 N N − | N

15 4 Coset Realization of Transformations

In this section, we ﬁrst obtain an explicit formula for the ﬁnite non-linear superconformal transformations of the supercoordinates and discuss several representations of the superconformal group. We then construct matrix or vector valued functions depending on two or three points in superspace which transform covariantly under superconformal transformations. These variables serve as building blocks of obtaining two-point, three-point and general n-point correlation functions later.

4.1 Superspace as a Coset To obtain an explicit formula for the finite non-linear superconformal transformations, we 3 2 first identify the superspace, R | N , as a coset, GS/G0, where G0 GS is the subgroup µ a ⊂ generated by matrices, M0, of the form (3.45) with a = 0, ε = 0 and depending on µ a b parameters b , ρ , λ, wµν, ta . The group of supertranslations, GT , parameterized by coor- M 3 2 dinates, z R | N , has been defined by general elements as in eq.(2.11) with the group ∈ property given by eqs.(2.14, 2.15). Now we may represent it by supermatrices5

0 ix √2θb 1 ix √2θb − GT (z) = exp  0 0 0  =  0 1 0  . (4.1) 0 √2θ¯ 0 0 √2θ¯ δ b  a   a a   −   −  1 Note G (z)− = G ( z). T T − 1 In general an element of GS can be uniquely decomposed as GT G0− . Thus for any g element G(g) G we may deﬁne a superconformal transformation, z z′, and an ∈ S −→ associated element G (z; g) G by 0 ∈ 0 1 G(g)− GT (z)G0(z; g)= GT (z′) . (4.2) If G(g) G then clearly G (z; g) = 1. Inﬁnitesimally eq.(4.2) becomes ∈ T 0 δG (z)= MG (z) G (z)Mˆ (z) , (4.3) T T − T 0

where M is given by eq.(3.45) and Mˆ 0(z), the generator of G0, has the form

1 ˆ wˆ(z)+ 2 λ(z) 0 0 Mˆ (z)= 1 ˆ √ b . (4.4) 0  ib γ wˆ(z) 2 λ(z) 2ˆρ (z)  · − b  √2ρˆ¯ (z) 0 tˆa (z)   − a  5The subscript, T , denotes supertranslations. 

16 The components depending on z are given by

wˆ(z)+ 1 λˆ(z)= w + 1 λ + x b γ +2θaρ¯ , 2 2 + · a

1 1 a wˆ(z) λˆ(z)= w λ b γx 2ρ θ¯a , − 2 − 2 − · − −

a λˆ(z)= λ +2x b 2θ¯aρ , · − (4.5) ρˆa(z)= ρa + ib γθa , · a 0 a t ρˆ¯ (z)=¯ρ iθ¯ b γ = (ˆρ (z))†γ = ρˆ (z) ǫ , a a − a · − tˆ b(z)= t b +2iθ¯ b γθb +2θ¯ ρb 2¯ρ θb . a a a · a − a 1 µ ν wˆ(z) can be also written asw ˆ(z)= 4 wˆµν (z)γ γ with

wˆ (z)= w + 2(x b x b )+ ǫ (bλθ¯ θa +2iρ¯ γλθa) . (4.6) µν µν µ ν − ν µ µνλ a a Writing δG (z)= G (z) we may verify that is identical to eq.(3.41). T L T L The deﬁnitions (4.5) can be summarized by

D λbβ(z)= 1 δ bδ βλˆ(z) δ bwˆβ (z)+ δ βtˆ b(z) , (4.7a) aα − 2 a α − a α α a

∂ν hµ(z)=w ˆµν(z)+ ηµν λˆ(z) , (4.7b)

and they give

[D , ]= D λbβD =( 1 δ bδ βλˆ(z)+ δ bwˆβ (z) δ βtˆ b(z))D . (4.8) aα L − aα bβ 2 a α a α − α a bβ For later use we note Daαwˆµν (z)=2(ρˆ¯a(z)γ[µγν])α ,

ˆ Daαλ(z)=2ρˆ¯aα(z) , (4.9)

D tˆ c(z)=2(δ δcd δ cδ d)ρˆ¯ . aα b ab − a b dα

17 The above analysis can be simplified by reducing G0(z; g). To achieve this we let 0 0 Z0 =  1 0  , (4.10) 0 1     and then 1 √ b w 2 λ 2ρ M0Z0 = Z0H0 , H0 = − b . (4.11) 0 ta ! Now if we define ix √2θb − Z(z) G (z)Z = 1 0 , (4.12) ≡ T 0   √2θ¯ δ b  − a a  then Z(z) transforms under infinitesimal superconformal transform ations as δZ(z)= Z(z)= MZ(z) Z(z)H(z) , (4.13) L − where H(z) is given by 1 ˆ √ b ˆ wˆ(z) 2 λ(z) 2ˆρ (z) M0(z)Z0 = Z0H(z) , H(z)= − b . (4.14) 0 tâ (z) ! From eqs.(3.42, 3.46) considering [ , ]Z(z)= Z(z) , (4.15) L2 L1 L3 we get H (z)= H (z) H (z) + [H (z),H (z)] , (4.16) 3 L2 1 −L1 2 1 2 which gives separate equations forw, ˆ λ,ˆ ρˆ and tˆ b, thus λˆ = λˆ λˆ , etc. a 3 L2 1 −L1 2 As a conjugate of Z(z) we define Z¯(z) by 0 1 √ b ¯ γ 0 ǫ− 0 t 1 ix+ 2θ Z(z)= Z(z)†B = Z(z) C = − − b . (4.17) 0 1 ! 0 1 ! 0 √2θ¯a δa ! This satisfies 1 Z¯(z)= Z¯(0)GT (z)− , (4.18) and corresponding to eq.(4.13) we have δZ¯(z)= Z¯(z)= H¯ (z)Z¯(z) Z¯(z)M , (4.19) L − where 1 ˆ ¯ wˆ(z)+ 2 λ(z) 0 H(z)= b . (4.20) √2ρˆ¯a(z) tâ (z) ! −

18 4.2 Finite Transformations Finite superconformal transformations can be obtained by exponentiation of inﬁnitesimal g transformations. To obtain a superconformal transformation, z z′, we therefore solve the diﬀerential equation −→

d M M z = (z ) , z = z , z = z′ , (4.21) dt t L t 0 1 where, with given in eq.(3.41), M (z) is deﬁned by L L = M (z)∂ . (4.22) L L M From eq.(4.13) we get d Z(z )= MZ(z ) Z(z )H(z ) , (4.23) dt t t − t t which integrates to tM Z(zt)= e Z(z)K(z, t) , (4.24) where K(z, t) satisﬁes

d 1 0 K(z, t)= K(z, t)H(zt) , K(z, 0) = . (4.25) dt − 0 1 !

g Hence for t = 1 with K(z, 1) K(z; g), z z′, eq.(4.24) becomes ≡ −→ 1 1 M Z(z′)= G(g)− Z(z)K(z; g) , G(g)− = e . (4.26)

G0(z; g) in eq.(4.2) is related to K(z; g) from eq.(4.26) by

G0(z; g)Z0 = Z0K(z; g) . (4.27)

In general K(z; g) is of the form

1 Ω(z; g) 2 L(z; g) √2Σb(z; g) K(z; g)= b , (4.28) 0 Ua (z; g) !

where Ω(z; g) is identical to the local scale factor in eq.(3.1), U(z; g) SO( ) ∈ N 1 t U − = U † = U , det U =1 , (4.29)

19 and L(z; g) satisﬁes

det L(z; g)=1 , (4.30a)

1 1 t 0 0 L− (z; g)= ǫ− L(z; g) ǫ = γ L(z; g)†γ . (4.30b)

From eq.(4.26) Z¯(z) transforms as

Z¯(z′)= K¯ (z; g)Z¯(z)G(g) , (4.31)

where

0 0 1 γ 0 γ 0 ǫ− 0 t ǫ 0 K¯ (z; g) = K(z)† = K(z) 0 1 ! 0 1 ! 0 1 ! 0 1 ! (4.32) 1 1 Ω(z; g) 2 L− (z; g) 0 = 1 b . √2Σ¯ a(z; g) U − a (z; g) !

is If we deﬁne for superinversion, z z′, (3.29) −→

ǫ 0 0 1 1 b 1 i(ǫx )− √2ix+− θ G(i ) = 0 ǫ 0 , K(z; i )= − , (4.33) s −   s − 0 V b(z) 0 0 1 a !     with b b ¯ 1 b Va (z)= δa +2iθax+− θ , (4.34) an analogous formula to eq.(4.26) can be obtained for superinversion

1 0 1 t t t G(i )− Z(z)K(z; i )= ix′ √2θ¯′ = Z¯(z′) . (4.35) s s  − + a  √2θ bt δb  ′ a    Similarly we have t K¯ (z; is)Z¯(z)G(is)= Z(z′) , (4.36) where 1 ¯ iǫx+− 0 K(z; is)= ¯ 1 1 b . (4.37) √2iθax− V − a (z) ! − −

20 Note that

1 t 1 V − (z)= V † = V (z) = V ( z)=1 2iθ¯x− θ , (4.38a) − − −

a b 1 b b ¯ ¯ 1 θ Va (z) = x x+− θ , Va (z)θb = θax+− x , (4.38b) − −

ν 1 Rµ (z; g)γν = Ω(z; g)L− (z; g)γµL(z; g) , (4.38c)

ν µ µ 1 γ Rν (z; g)=Ω(z; g)L(z; g)γ L− (z; g) . (4.38d)

ν ν where Rµ (z; g) is identical to the deﬁnition (3.3). We may normalize Rµ (z; g) as

ν 1 ν 1 ν 1 Rˆ (z; g)=Ω(z; g)− R (z; g)= tr(γ L(z; g)γ L− (z; g)) SO(1, 2) . (4.39) µ µ 2 µ ∈ 4.3 Representations Based on the results in the previous subsection, it is easy to show that the matrix, N (z; g), RM given in eq.(3.5) is of the form

1 ˆ ν 2 1 b β N Ω(z; g)Rµ (z; g) iΩ(z; g) (L− (z; g)γµΣ (z; g)) 1 M (z; g)= 1β b . (4.40) R 0 Ω(z; g) 2 L− α(z; g)Ua (z; g) !

Since N (z; g) is a representation of the three-dimensional superconformal group, each RM of the following also forms a representation of the group, though it is not a faithful representation Ω(z; g) D , Rˆ(z; g) SO(1, 2) , ∈ ∈ (4.41) L(z; g) , U(z; g) O( ) , ∈ N where D is the one dimensional group of dilations. g g′ Under the successive superconformal transformations, g′′ : z z′ z′′, they satisfy −→ −→

L(z; g)L(z′; g′)= L(z; g′′) , and so on. (4.42)

4.4 Functions of Two Points

In this subsection, we construct matrix valued functions depending on two points, z1 and z2, in superspace which transform covariantly like a product of two tensors at z1 and z2

21 under superconformal transformations.

3 2 If F (z) is deﬁned for z R | N by ∈ b ¯ ix √2θ F (z)= Z(0)GT (z)Z(0) = − b , (4.43) √2θ¯a δa ! − then F (z) satisﬁes

0 0 1 γ 0 γ 0 ǫ− 0 t ǫ 0 F ( z) = F (z)† = F (z) − 0 1 ! 0 1 ! 0 1 ! 0 1 ! (4.44) b ix+ √2θ = − ¯ − b , √2θa δa ! and the superdeterminant of F (z) is given by

sdet F (z)= det x = x2 + 1 (θ¯ θa)2 . (4.45) − + 4 a We also note

1 b 1 0 1 i√2x− θ ix 0 ¯ 1 F (z) − = − b , (4.46) i√2θax− 1 ! 0 1 ! 0 Va ( z) ! − − − where V b( z) is identical to eq.(4.38a) and from eqs.(4.45, 4.46) it is evident that a − det V (z)=1 . (4.47)

Hence, with eq.(4.38a), V (z) SO( ). ∈ N 3 2 Now with the supersymmetric interval for R | N deﬁned by

1 M µ a M G (z )− G (z )= G (z ) , z =(x , θ , θ¯ )= z , T 2 T 1 T 12 12 12 12 12a − 21 (4.48) xµ = xµ xµ iθ¯ γµθa , θa = θa θa , 12 1 − 2 − 2a 1 12 1 − 2 we may write √ b ¯ ix12 2θ12 Z(z2)Z(z1)= F (z12)= ¯ − b , (4.49) √2θ12a δa ! − and 2 1 ¯ a 2 sdet F (z12)= x12 + 4 (θ12aθ12) , det V (z12)=1 , (4.50)

22 where a ¯ 1 ¯ a x12 = x1 x2+ 2iθ2 θ1a = x12 i θ12aθ12 1 , − − − − − 2 (4.51) a ¯ 1 ¯ a x12+ = x1+ x2 +2iθ1 θ2a = x12 + i θ12aθ12 1 . − − 2 From eqs.(4.26, 4.31) F (z12) transforms as ¯ F (z12′ )= K(z2; g)F (z12)K(z1; g) . (4.52)

a Explicitly with eqs.(4.28, 4.32) we get the transformation rules for x12′ and θ12′ ± 1 1 2 2 1 x12′ = Ω(z1; g) Ω(z2; g) L− (z2; g)x12 L(z1; g) , − − (4.53a) 1 1 2 2 1 x12+′ = Ω(z1; g) Ω(z2; g) L− (z1; g)x12+L(z2; g) ,

1 a 2 1 b a a θ12′ = Ω(z1; g) L− (z1; g)(θ12Ub (z2; g)+ ix12+Σ2) , (4.53b) 1 a 2 1 b a a θ21′ = Ω(z2; g) L− (z2; g)(θ21Ub (z1; g) ix12 Σ1) . − − In particular 2 1 ¯ a 2 2 1 ¯ a 2 x12′ + 4 (θ12′ aθ12′ ) = Ω(z1; g)Ω(z2; g)(x12 + 4 (θ12aθ12) ) . (4.54) µ ν From eqs.(4.38d, 4.53a) tr(γ x12 γ x12+) transforms covariantly as − µ ν λ ρ µ ν tr(γ x12′ γ x12+′ ) = tr(γ x12 γ x12+)Rλ (z2; g)Rρ (z1; g) . (4.55) − − From eq.(4.53b) we get

1 c 1 b 1 i√2x12− θ12 1 i√2x12′− θ12′ − c − K(z1; g) b− 0 δa ! 0 δd ! (4.56a) 1 Ω(z ; g) 2 L(z ; g) 0 = 1 1 , 0 U(z1; g) !

1 0 ¯ 1 0 ¯ 1 c K(z2; g) ¯ 1 b i√2θ12′ ax12′− δa ! i√2θ12dx12− δd ! − − − (4.56b) 1 1 Ω(z2; g) 2 L− (z2; g) 0 = 1 . 0 U − (z2; g) !

23 Using this and eq.(4.46) we can rederive eq.(4.53a) and obtain

1 V (z12′ )= U − (z1; g)V (z12)U(z2; g) , (4.57) 1 V (z21′ )= U − (z2; g)V (z21)U(z1; g) .

4.5 Functions of Three Points

In this subsection, for three points, z1, z2, z3 in superspace, we construct ‘tangent’ vectors, i, which transform homogeneously at zi, i =1, 2, 3. Z is is M µ a 3 2 With z (z )′, z (z )′, we deﬁne =(X , Θ ) R | N by 21 −→ 21 31 −→ 31 Z1 1 1 ∈ 1 G ((z )′)− G ((z )′)= G ( ) . (4.58) T 31 T 21 T Z1 Explicit expressions for M can be obtained by calculating Z1 √ b ¯ iX1 2Θ1 Z((z31)′)Z((z21)′)= F ( 1)= ¯ − b . (4.59) Z √2Θ1a δa ! − We get

1 1 X1 = x31+− x23 x21− , − − − (4.60) a 1 a 1 a ¯ ¯ 1 ¯ 1 Θ1 = i(x21+− θ21 x31+− θ31) , Θ1a = i(θ21ax21− θ31ax31− ) . − − − − − Using a ¯ x23 = x21 x31+ 2iθ31θ21a , (4.61) − − − − one can assure 0 0 1 t 1 1 X1+ = γ X1† γ = ǫ− X1 ǫ = x21+− x23+x31− , − − − − (4.62) a ¯ ¯ a X1+ X1 = 2iΘ1Θ1a = iΘ1aΘ1 1 . − − − g a ¯ From eq.(4.53a) under superconformal transformations, z z′, X1 , Θ1, Θ1a transform ± as −→

1 1 X1′ = Ω(z1; g)− L− (z1; g)X1 L(z1; g) , (4.63a) ± ±

1 a 2 1 b a Θ1′ = Ω(z1; g)− L− (z1; g)Θ1Ub (z1; g) , (4.63b)

1 ¯ 2 1 b ¯ Θ1′ a = Ω(z1; g)− U − a (z1; g)Θ1bL(z1; g) , (4.63c)

24 so that µ 1 ν ˆ µ X1′ = Ω(z1; g)− X1 Rν (z1; g) . (4.64)

Thus 1 transforms homogeneously at z1, as ‘tangent’ vectors do. Eq.(4.63a)Z can be summarized as

1 1 2 1 2 Ω(z1; g)− L− (z1g) 0 Ω(z1; g)− L(z1g) 0 F ( 1′ )= 1 F ( 1) . Z 0 U − (z1; g) ! Z 0 U(z1; g) ! (4.65) Direct calculation using eq.(4.38b) shows that

V ( )= V (z )V (z )V (z ) . (4.66) Z1 12 23 31 ν Similarly for Rµ (z; is) given in eq.(3.35) we obtain from eqs.(3.37,4.60)

2 2 R( ; i )= x2 + 1 (θ¯ θa )2 x2 + 1 (θ¯ θa )2 R(z ; i )R(z ; i )R(z ; i ) . (4.67) Z1 s 12 4 12a 12 31 4 31a 31 12 s 23 s 31 s b ν From eqs.(4.55, 4.57) Va ( 1), Rµ ( 1; is) transform homogeneously at z1 under supercon- Zg Z formal transformation, z z′, −→ 1 V ( ′ )= U − (z ; g)V ( )U(z ; g) , (4.68a) Z1 1 Z1 1

2 1 R( ′ ; i )=Ω(z ; g) R− (z ; g)R( ; i )R(z ; g) . (4.68b) Z1 s 1 1 Z1 s 1 It is useful to note x2 + 1 (θ¯ θa )2 2 1 ¯ a 2 23 4 23a 23 det X1 = X1 (Θ1aΘ1) = . (4.69) ± − − 4 − 2 1 ¯ a 2 2 1 ¯ a 2 x12 + 4 (θ12aθ12) x31 + 4 (θ31aθ31) By taking cyclic permutations of z , z , z in eq.(4.60) we may define , . We find 1 2 3 Z2 Z3 , are related to as Z2 Z3 Z1 a b a X2 = x21+X1+x12+ , Θ2 = ix21+Θ1Vb (z12) , (4.70a) − − f f 1 1 a 1 b a X3 = x31− X1+x13− , Θ3 = ix31− Θ1 Vb (z13) , (4.70b) − − − − where =(X, Θ) is defined byf superinversion, is . f Z Z −→ Z e f e e

25 5 Superconformal Invariance of Correlation Functions

In this section we discuss the superconformal invariance of correlation functions for quasi- primary superﬁelds and exhibit general forms of two-point, three-point and n-point functions without proof, as the proof is essentially identical to those in our earlier work [4,5].

5.1 Quasi-primary Superfields We first assume that there exist quasi-primary superfields, ΨI (z), which under the super- g conformal transformation, z z′, transform as −→ I I I J I Ψ Ψ′ , Ψ′ (z′)=Ψ (z)D (z; g) . (5.1) −→ J D(z; g) obeys the group property so that under the successive superconformal transforma- g g′ tions, g′′ : z z′ z′′, it satisfies −→ −→

D(z; g)D(z′; g′)= D(z; g′′) , (5.2)

and hence 1 1 D(z; g)− = D(z′; g− ) . (5.3) We choose here D(z; g) to be a representation of SO(1, 2) O( ) D, which is a subgroup × N × of the stability group at z = 0, and so we decompose the spin index, I, of superﬁelds into SO(1, 2) index, ρ, and O( ) index, r, as ΨI Ψ r. Now D I (z; g) is factorized as N ≡ ρ J I ρ s η DJ (z; g)= D σ(L(z; g))Dr (U(z; g))Ω(z; g)− , (5.4)

ρ s where D σ(L),Dr (U) are representations of SO(1, 2) and O( ) respectively, while η is the r N scale dimension of Ψρ .

Inﬁnitesimally

δΨ r(z)= ( + ηλˆ(z))Ψ r(z) Ψ r(z)(sα )σ wˆβ (z) Ψ s(z) 1 (s ) rtˆab(z) , (5.5) ρ − L ρ − σ β ρ α − ρ 2 ab s ab ac b α where tˆ (z)= δ tˆc (z), and s β, sab satisfy

[sα ,sγ ]= δα sγ δ γ sα , β δ δ β − β δ (5.6) [s ,s ]= η s + η s + η s η s . ab cd − ac bd ad bc bc ad − bd ac 26 α sab is the generator of O( ), while s β is connected to the generator of SO(1, 2), sµν, through N s 1 sα (γ γ )β , sα = 1 s (γ[µγν])α , µν ≡ 2 β [µ ν] α β − 2 µν β [s ,s ]= η s + η s + η s η s , (5.7) µν λρ − µλ νρ µρ νλ νλ µρ − νρ µλ α β 1 µν s βwˆ α(z)= 2 sµνwˆ (z) . From eqs.(4.15, 4.16) using eq.(5.6) we have

r r δ3Ψρ = [δ2, δ1]Ψρ . (5.8)

r ρ It is useful to consider the conjugate superﬁeld of Ψρ , Ψ¯ r(z), which transforms as

ρ η ρ 1 s 1 σ Ψ¯ ′ r(z′)=Ω(z; g)− D σ(L− (z; g))Dr (U − (z; g))Ψ¯ s(z) . (5.9)

Superconformal invariance for a general n-point function requires

I1 I2 In I1 I2 In Ψ′ (z )Ψ′ (z ) Ψ′ (z ) = Ψ (z )Ψ (z ) Ψ (z ) . (5.10) h 1 1 2 2 ··· n n i h 1 1 2 2 ··· n n i 5.2 Two-point Correlation Functions

r ρ The solution for the two-point function of the quasi-primary superﬁelds, Ψρ , Ψ¯ r, has the general form ρ s ρ s I σ(ˆx12+)Ir (V (z12)) Ψ¯ r(z1)Ψσ (z2) = CΨ η , (5.11) h i 2 1 ¯ a 2 x12 + 4 (θ12aθ12) where we put x12+ ˆx12+ = 1 , (5.12) 2 1 ¯ a 2 2 x12 + 4 (θ12aθ12) ρ s and I σ(ˆx12+), Ir (V (z12)) are tensors transforming covariantly according to the appropriate representations of SO(1, 2), O( ) which are formed by decomposition of tensor products N ρ s of ˆx12+, V (z12). Under superconformal transformations, I σ(ˆx12+) and Ir (V (z12)) satisfy from eqs.(4.53a, 4.57)

1 D(L− (z1; g))I(ˆx12+)D(L(z2; g)) = I(ˆx12+′ ) , (5.13a)

1 D(U − (z1; g))I(V (z12))D(U(z2; g)) = I(V (z12′ )) . (5.13b)

27 As examples, we ﬁrst consider real scalar, spinorial and gauge superﬁelds, α a S(z), φα(z), φ¯ (z), ζ (z), ζa(z). They satisfy

S(z) = S(z)∗ ,

α 1αβ 0 α φ¯ (z)= ǫ− φβ(z)=(γ φ(z)†) , (5.14)

a a ζ (z)= ζ (z)∗ = ζa(z) , and transform as η S′(z′) =Ω(z; g)− S(z) ,

η β φα′ (z′)=Ω(z; g)− φβ(z)L α(z; g) ,

α η 1α β φ¯′ (z′)=Ω(z; g)− L− β(z; g)φ¯ (z) , (5.15)

a η b a ζ′ (z′)=Ω(z; g)− ζ (z)Ub (z; g) ,

η 1 b ζa′ (z′)=Ω(z; g)− U − a (z; g)ζb(z) . The two-point functions of them are 1 S(z1)S(z2) = CS η , (5.16) h i 2 1 ¯ a 2 x12 + 4 (θ12aθ12) α ¯α (x12+) β φ (z )φβ(z ) = iCφ 1 , (5.17) 1 2 η+ h i 2 1 ¯ a 2 2 x12 + 4 (θ12aθ12) b b Va (z12) ζa(z1)ζ (z2) = Cζ η . (5.18) h i 2 1 ¯ a 2 x12 + 4 (θ12aθ12) (5.19) Note that to have non-vanishing two-point correlation functions, the scale dimensions, η, of the two ﬁelds must be equal.

For a real vector superﬁeld, J µ(z), where the representation of SO(1, 2) is given by ν Rˆµ (z; g), we have µν µ ν I (z12) J (z1)J (z2) = CV η , (5.20) h i 2 1 ¯ a 2 x12 + 4 (θ12aθ12) 28 where µν µν 1 µ ν I (z)= Rˆ (z; is)= tr(γ ˆx+γ ˆx ) . (5.21) 2 − From eq.(3.37) we note Iµν(z)= Iνµ( z) , Iµν(z)I (z)= δµ . (5.22) − λν λ If we deﬁne α µ α J β(z)= J (z)(γµ) β , (5.23) then from eqs.(2.4,5.20) and

α α Daω(z1)(x12+) β =2iδω θ¯12aβ ,

α α ¯ α ¯ Daω(z1)(x12 ) β =2i(δω θ12aβ δ βθ12aω) , (5.24) − − 2 1 ¯ a 2 ¯ Daω(z1)(x12 + (θ12aθ12) )=2i(θ12ax12 )α , 4 − we get ¯ ν α ν (θ12aγ x12 )β − Daα(z1) J β(z1)J (z2) =2iCV (2 η) η+1 . (5.25) h i − 2 1 ¯ a 2 x12 + 4 (θ12aθ12) Hence J α (z )J ν(z ) is conserved if η =2 h β 1 2 i D (z ) J α (z )J ν(z ) =0 if η =2 . (5.26) aα 1 h β 1 2 i The anti-commutator relation for Daα (2.20) implies also

∂ µ ν µ J (z1)J (z2) =0 if η =2 . (5.27) ∂x1 h i This agrees with the non-supersymmetric general result that two-point correlation function of vector ﬁeld in d-dimensional conformal theory is conserved if the scale dimension is d 1 [24]. − 5.3 Three-point Correlation Functions

r The solution for the three-point correlation function of the quasi-primary superﬁelds, Ψρ , has the general form Ψ r(z )Ψ s(z )Ψ t(z ) h 1ρ 1 2σ 2 3τ 3 i r s′ t′ σ′ τ ′ s t (5.28) Hρ σ′ τ ′ ( 1)I σ(ˆx12+)I τ (ˆx13+)Is′ (V (z12))It′ (V (z13)) = Z η2 η3 , 2 1 ¯ a 2 2 1 ¯ a 2 x12 + 4 (θ12aθ12) x13 + 4 (θ13aθ13) 29 M µ a 3 2 where =(X , Θ ) R | N is given by eq.(4.58). Z1 1 1 ∈ Superconformal invariance (5.10) is now equivalent to r s t ρ′ σ′ τ ′ r s t H ′ ′ ′ ( )D (L)D (L)D (L)= H ( ′) , ρ σ τ Z ρ σ τ ρ σ τ Z (5.29a) M ν µ 1 a ′ =(X Rˆ (L), L− Θ ) , Z ν

r′ s′ t′ r s t r s t Hρ σ τ ( )Dr′ (U)Ds′ (U)Dt′ (U)= Hρ σ τ ( ′′) , Z Z (5.29b) M µ b a ′′ =(X , Θ U ) , Z b

r s t η2+η3 η1 r s t H ( )= λ − H ( ′′′) , ρ σ τ Z ρ σ τ Z (5.29c) M µ 1 a ′′′ =(λX , λ 2 Θ ) , Z where U O( ), λ R and 2 2 matrix, L, satisfies ∈ N ∈ × 1 0 0 1 t L− = γ L†γ = ǫ− L ǫ , det L =1 , (5.30) ˆ µ 1 µ 1 Rν (L)= 2 tr(γνLγ L− ) . In general there are a finite number of linearly independent solutions of eq.(5.29a), and this number can be considerably reduced by taking into account the symmetry properties, superfield conservations and the superfield constraints [5,20,21].

5.4 n-point Correlation Functions - in general

r The solution for n-point correlation functions of the quasi-primary superﬁelds, Ψρ , has the general form Ψ r1 (z ) Ψ rn (z ) h 1ρ1 1 ··· nρn n i ′ n ρk ′ rk (5.31) ′ ′ I ρk (ˆx1k+)Ir (V (z1k)) r1 ′ r2 ′ rn k = Hρ1 ρ ρ ( 1(1), , 1(n 2)) η , 2 ··· n Z ··· Z − x2 + 1 (θ¯ θa )2 k kY=2 1k 4 1ka 1k is where, in a similar fashion to eq.(4.58), with zk1 zk1, k 2, 1(1), , 1(n 2) are given −→ ≥ Z ··· Z − by 1 g GT (zn1)− GT (zj1)= GT ( 1(j 1)) , j =2, 3, , n 1 . (5.32) Z − ··· − g f 30 We note that all of them are ‘tangent’ vectors at z1.

Superconformal invariance (5.10) is equivalent to

n ′ ρ r1 r ′ r1 ′ rn k n Hρ1 ρn ( (1), , (n 2)) D ρk (L)= Hρ1 ρn ( (1)′ , , (′n 2)) , ··· Z ··· Z − ··· Z ··· Z − kY=1 (5.33a)

M ν µ 1 a ′ =(X Rˆ (L), L− Θ ) , Z(j) (j) ν (j)

n ′ ′ 1 r1 rn ′ rk r rn Hρ1 ρn ( (1), , (n 2)) Dr (U)= Hρ1 ρn ( (1)′′ , , (′′n 2)) , ··· Z ··· Z − k ··· Z ··· Z − kY=1 (5.33b) M µ b a ′′ =(X , Θ U ) , Z(j) (j) (j) b

r1 rn η1+η2+ +ηn r1 rn Hρ1 ρn ( (1), , (n 2))= λ− ··· Hρ1 ρn ( (1)′′′ , , (′′′n 2)) , ··· Z ··· Z − ··· Z ··· Z − (5.33c) M µ 1 a ′′′ =(λX , λ 2 Θ ) . Z(j) (j) (j) Thus n-point functions reduce to one unspeciﬁed (n 2)-point function which must trans- − form homogeneously under the rigid transformations, SO(1, 2) O( ) D. × N × From 1 1 1 1 X1(j 1)+ = xj−1+xjn+xn−1 , X1(j 1) = xn−1+xjn xj−1 , (5.34) − − − − − − we get a ¯ 1 1 X(l,m)+ X1(l 1)+ X1(m 1) +2iΘ1(l 1)Θ1(m 1)a = xl−1+xlm+xm−1 , (5.35) ≡ − − − − − − − and hence 2 1 ¯ a 2 xlm + 4 (θlmaθlm) det X(l,m) = . (5.36) ± − 2 1 ¯ a 2 2 1 ¯ a 2 x1l + 4 (θ1laθ1l) x1m + 4 (θ1maθ1m) Now if we deﬁne n 1 1 ∆lm = 2(n 1)(n 2) ηi + 2(n 2) (ηl + ηm) , (5.37) − − − − Xi=1 then using the following identity which holds for any matrix, Slm, and number, λ,

n ∆lm 1 1 ∆ η ( η1+η2+ +ηn) λSlm (S ) lm (S S )− 2 k = λ− 2 − ··· , (5.38)  lm  1k k1 l=m k=2 ! 2 l=m Sl1S1m ! Y6 Y ≤Y6   31 we can rewrite the n-point correlation functions (5.31) as

Ψ r1 (z ) Ψ rn (z ) h 1ρ1 1 ··· nρn n i ′ ′ ′ 1 n r ′ r2 ′ rn ρk ′ rk (5.39) Kρ1 ρ2 ρ ( 1(1), , 1(n 2)) k=2 I ρk (ˆx1k+)Ir (V (z1k)) n − k = ··· Z ··· Z ∆ , 2 1 ¯ Q a 2 lm l=m xlm + 4 (θlmaθlm) 6 Q where

r1 rn r1 rn ∆lm Kρ1 ρn ( 1(1), , 1(n 2))= Hρ1 ρn ( 1(1), , 1(n 2)) ( det X(l,m) ) . − − ± ··· Z ··· Z ··· Z ··· Z 2 l=m − ≤Y6 (5.40) Note the diﬀerence in eq.(5.31) and eq.(5.39), namely the denominator in the latter is written in a democratic fashion.

Superconformal invariance (5.33a) is equivalent to

n ′ ′ ′ ρ r r1 r ′ r1 ′ rn k ′ k n Kρ1 ρn ( (1), , (n 2)) D ρk (L)Dr (U)= Kρ1 ρn ( (1)′ , , (′n 2)) , ··· Z ··· Z − k ··· Z ··· Z − kY=1

1 M ν µ 1 b a ′ =(λX Rˆ (L), λ 2 L− Θ U ) . Z(j) (j) ν (j) b (5.41) In particular, K is invariant under dilations contrary to H.

5.5 Superconformal Invariants In the case of correlation functions of quasi-primary scalar superﬁelds, eqs.(5.10,5.39,5.41) imply that K( 1(1), , 1(n 2)) is a function of the superconformal invariants and further- − more that all ofZ the··· superconformalZ invariants can be generated by contracting the indices of M =(Xµ , Θaα ) to make them SO(1, 2) O( ) D invariant according to the recipe Z1(j) 1(j) 1(j) × N × by Weyl [25]. To do so we ﬁrst normalize µ as Z1(j) ˆµ =(Xˆ µ , Θˆ a ) , Z1(j) 1(j) 1(j) Xµ Θa (5.42) ˆ µ 1(j) ˆ a 1(j) X1(j) = 1 , Θ1(j) = 1 . 2 2 2 4 (X1(1)) (X1(1))

32 By virtue of eqs.(A.3a,A.7a) all the SO(1, 2) O( ) D invariants or three-dimensional superconformal invariants are × N ×

Xˆ Xˆ , Θ¯ˆ Xˆ γΘˆ a , Θ¯ˆ Θˆ b Θ¯ˆ Θˆ a , Θ¯ˆ Θˆ a . 1(j)· 1(k) 1(j)a 1(k)· 1(l) 1(j)a 1(k) 1(l)b 1(m) 1(j)a 1(k) (5.43) In particular, from eq.(5.36) we note that they produce cross ratio type invariants depending on four points, zr, zs, zt, zu, the non-supersymmetric of which are well known - see e.g. Ref. [26] 2 1 ¯ a 2 2 1 ¯ a 2 xrs + (θrsaθrs) xtu + (θtuaθtu) 4 4 . (5.44) 2 1 ¯ a 2 2 1 ¯ a 2 xru + 4 (θruaθru) xts + 4 (θtsaθts)

If we restrict the R-symmetry group to be SO( ) instead of O( ) then the followings are also superconformal invariants in the case of evenN , accordingN to Weyl [25] N N 2 b b ǫa1 aN /2 bj , bj = Θ¯ˆ Θˆ j or Θ¯ˆ Xˆ γΘˆ j , b1 ··· bN /2 Tjaj Tjaj 1(j1)aj 1(j2) 1(j1)aj 1(j2)· 1(j3) jY=1 (5.45) which we may call pseudo-invariants.

6 Superconformally Covariant Operators

r In general acting on a quasi-primary superfield, Ψρ (z), with the spinor derivative, Daα, 6 r does not lead to a quasi-primary field . For a superfield, Ψρ , from eqs.(4.8, 4.9 ,5.5) we have D δΨ r = ( +(η + 1 )λˆ)D Ψ r aα ρ − L 2 aα ρ r β r β γ σ DaβΨρ wˆ α DaαΨσ (s γ wˆ β) ρ − − (6.1) D Ψ rtˆb D Ψ s 1 (s tˆbc) r − bα ρ a − aα ρ 2 bc s bβ r +2ρˆ¯bβ(ΨY aα)ρ , bβ where Y aα is given by

Y bβ =2sb δβ + δb sβ ηδb δβ . (6.2) aα a α a α − a α 6For conformally covariant diﬀerential operators in non-supersymmetric theories, see e.g. [27,28].

33 r To ensure that DaαΨρ is quasi-primary it is necessary that the terms proportional to ρˆ¯ r vanish and this can be achieved by restricting DaαΨρ to an irreducible representation of SO(1, 2) O( ) and choosing a particular value of η so that ΨY = 0. The change of the scale dimension,× N η η + 1 , in eq.(6.1) is also apparent from eq.(2.21) → 2 1 2 1β b Daα = Ω(z; g) L− α(z; g)Ua (z; g)Dbβ′ . (6.3)

As an illustration we consider tensorial ﬁelds, Ψa1 amα1 αn , which transform as ··· ··· ˆ δΨa1 amα1 αn = ( + ηλ)Ψa1 amα1 αn ··· ··· − L ··· ··· m n (6.4) ˆb β Ψa1 b amα1 αn t ap Ψa1 amα1 β αn wˆ αq . ··· ··· ··· ··· ··· ··· −p − q X=1 X=1 Note that spinorial indices and gauge indices, α, a may be raised or lowered by 1αβ ab ǫ− , ǫαβ, δ , δab.

For Ψa1 amα1 αn we have ··· ··· β 1 β (ΨYb aα)a1 amα1 αn = (η + n)δbaδ αΨa1 amα1 αn ··· ··· − 2 ··· ··· m β +δ α (δaap Ψa1 b amα1 αn δbap Ψa1 a amα1 αn ) ··· ··· ··· − ··· ··· ··· (6.5) pX=1 n β +δba δ αq Ψa1 amα1 α αn . ······ ··· ··· q X=1 In particular, eq.(6.5) shows that the following are quasi-primary

1 D[b(βΨa1 am]α1 αn) if η = m + n , (6.6a) ··· ··· 2

β 3 D[b β Ψa1 am] if η = m , (6.6b) | | ··· − 2 where ( ), [ ] denote the usual symmetrization, anti-symmetrization of the indices respectively and obviously eq.(6.6a) is nontrivial if 1 m +1 . Note that due to the term ≤ ≤ N containing δaap in eq.(6.5) one should anti-symmetrize the gauge indices.

34 Now we consider the case where more than one spinor derivative, Daα, act on a quasi- primary superﬁeld. In this case, it is useful to note

D[a[αDb]β] =0 , (6.7) and D ρˆ¯ = iδ (ǫb γ) . (6.8) aα bβ − ab · αβ From eq.(6.5) one can derive

D[b1(β1 Db β δΨa1 am]α1 αn) ··· l l ··· ··· 1 3 ˆ =2l( η + m + n + (l 1))ρ¯[b1(β1 Db2β2 Dblβl Ψa1 am]α1 αn) + homogeneous terms . − 2 4 − ··· ··· ··· (6.9) Hence the following is quasi-primary

1 3 D[b1(β1 Db β Ψa1 am]α1 αn) if η = m + n + (l 1) . (6.10) ··· l l ··· ··· 2 4 −

Acknowledgments

I am deeply indebted to Hugh Osborn for introducing me the subject in this paper.

35 Appendix

A Useful Equations

Some useful identities relevant to the present paper are

1 µ ν µν 2 tr(γ γ )= η , (A.1a)

γµγνγρ = ηµν γρ ηµργν + ηνργµ + iǫµνρ , (A.1b) −

i 1 ǫµνργ γ = γµ . (A.1c) − 2 ν ρ

ǫµνκǫ = δµ δν δµ δν , (A.2a) λρκ λ ρ − ρ λ

µνκ µ ǫ ǫλνκ =2δ λ , (A.2b)

µνκ ǫ = ǫµνκ . (A.2c)

ρε¯ = 1 (¯εγµργ +¯ερ 1) , (A.3a) − 2 µ

1 t t 1 µ ǫ− ε¯ ρ ǫ = (¯εγ ργ ερ¯ 1) . (A.3b) − 2 µ − For Majorana spinors D θbβ = δ bδ β ,D θ¯ = δ ǫ , (A.4a) aα − a α aα bβ ab αβ

aα bβ ab 1αβ aα a α D¯ θ = δ ǫ− , D¯ θ¯ = δ δ , (A.4b) − bβ b β

b Daαθ¯bθ =2θ¯aα , (A.4c)

β β Daαx+ γ =2iδα θ¯aγ , (A.4d)

β β β Daαx γ =2i(δα θ¯aγ δ γ θ¯aα) . (A.4e) − − 36 0 0 γ x γ = x† , (A.5a) ± ∓

1 t ǫx ǫ− = x , (A.5b) ± − ∓

2 1 a 2 det x+ = det x = x (θ¯aθ ) . (A.5c) − − − 4

δ γ γ δ 1γδ δ δ δ δ = ǫ ǫ− . (A.6) α β − α β αβ

µ θγ¯ θ′ψγ¯ ψ′ = 2θψ¯ ′ ψθ¯ ′ θθ¯ ′ ψψ¯ ′ , (A.7a) µ − −

ǫµ1 µ ǫν1 ν = sign(p) δµ1νp δµ νp p : permutations , (A.7b) ··· d ··· d 1 ··· d d pX=1 µ1 µ d i1 id ǫµ1 µd x x d = ǫ ··· x(1) x(i1) x(d) x(id) . (A.7c) ··· (1) ··· ( ) ± · ··· · q B Solution of Superconformal Killing Equation

From the well known solution of the ordinary conformal Killing equation (3.10) [26], we may write the general solution of the superconformal Killing equation (3.15) as

hµ(z)= 2x b(θ) xµ (x2 1 (θ¯ θa)2)bµ(θ)+ ǫµ xνbλ(θ)θ¯ θa · − − 4 a νλ a (B.1) µ ν 1 µ νλ ¯ a µ µ +w ν(θ)x + 4 ǫ νλw (θ)θaθ + λ(θ)x + a (θ) . Substituting this expression into eq.(3.15) leads three independent equations corresponding to the second, ﬁrst and zeroth order in x. Considering the quadratic terms or the coeﬃcients of xρxλ, we get

ηµρD bλ(θ)+ ηµλD bρ(θ) ηρλD bµ(θ) aα aα − aα (B.2) = i 1 ǫµ γκβ (ηνρD bλ(θ)+ ηνλD bρ(θ) ηρλD bν(θ)) . − 2 νκ α aα aα − aα 37 Contracting this with ηρλ gives

D bµ(θ)= i 1 ǫµ γκβ D bν (θ) , (B.3) aα − 2 νκ α aβ

while contraction with ηµλ leads

ρ 1 ρ κβ ν Daαb (θ)= i 3 ǫ νκγ αDaβb (θ) . (B.4) Thus µ Daαb (θ)=0 , (B.5) µ µ 2 1 ¯ a 2 µ b (θ) is constant. Straightforward calculation shows that 2x b x (x 4 (θaθ ) )b + µ ν λ a · − − ǫ νλx b θ¯aθ is a solution of the superconformal Killing equation (3.15).

Now the linear in x terms become

D (ǫµνκv (θ)+ ηµνλ(θ)) = i 1 D (ηµνv(θ) γ vµ(θ)γν ǫµν λ(θ)γκ)β , (B.6) aα κ 2 aβ · − − κ α where vκ(θ) is the dual form of wµν(θ)

κ 1 κ µν µν µνκ v = 2 ǫ µν w , w = ǫ vκ . (B.7)

Contracting eq.(B.6) with ηµν gives

1 β β µ β Daαλ(θ)= i 3 Daβv α(θ) , v α(θ)= v (θ)γµ α . (B.8) Substituting this back into eq.(B.6) leads

0=5ǫµνρD v (θ) iηµν D vβ (θ)+4iD vµ(θ)γνβ iD vν(θ)γµβ . (B.9) aα ρ − aβ α aβ α − aβ α κ µ Contraction with ǫ µν shows that v (θ) satisﬁes the superconformal Killing equation (3.15)

D vκ(θ)= i 1 ǫκ D vµ(θ)γνβ . (B.10) aα − 2 µν aβ α Eqs.(B.8,B.10) are actually equivalent to eq.(B.6), since from eq.(B.10) successively

D vκ(θ)γρα = i 1 ǫκρµD v (θ)+ 1 ηκρD vα (θ) 1 D vρ(θ)γκα , aα β 2 aβ µ 2 aα β − 2 aα β (κ ρ)α 1 κρ α Daαv (θ)γ β = 3 η Daαv β(θ) , (B.11) [κ ρ]α κρµ Daαv (θ)γ β = iǫ Daβvµ(θ) ,

κ ρα κρµ 1 κρ α Daαv (θ)γ β = iǫ Daβvµ(θ)+ 3 η Daαv β(θ) ,

38 and the last expression makes eq.(B.9) hold.

To solve eq.(B.10) we ﬁrst note from

D vβ (θ)= 2 δ βD vδ (θ) 1 δβ D vδ (θ) , (B.12) aα γ 3 α aδ γ − 3 γ aδ α that D D vγ (θ) = 2 δ γ D D vω (θ)+ 1 δγ D D vω (θ) bβ aα δ − 3 α aω bβ δ 3 δ aω bβ α = 4 δ γD D vω (θ) 2 δ γ D D vω (θ) 2 δγ D D vω (θ) (B.13) 9 α bω aβ δ − 9 α bω aδ β − 9 δ bω aβ α 1 γ ω + 9 δ δDbωDaαv β(θ) .

β Contraction with δγ gives

D D vβ (θ)= D D vβ (θ) , (B.14) bγ aα δ − bγ aδ α so that eq.(B.13) becomes

D D vγ (θ)= 2 δ γD D vω (θ) 1 δγ D D vω (θ) , (B.15) bβ aα δ 3 α bω aβ δ − 3 δ bω aβ α which is in fact equivalent to eq.(B.13).

From eq.(B.15) and D D vγ (θ)= D D vγ (θ) we get bβ aα δ − aα bβ δ γ ω γ ω γ ω ω 2δα DbωDaβv δ(θ)+2δβ DaωDbαv δ(θ)= δ δ(DaωDbαv β(θ)+ DbωDaβv α(θ)) . (B.16)

α Contracting with δγ gives

3D D vω (θ)= 2D D vω (θ)+ D D vω (θ) . (B.17) bω aβ δ − aω bβ δ aω bδ β Hence from eq.(B.14) we can put

ω DbωDaαv β =Γab(θ)ǫαβ , (B.18) 1 1 βα Γ (θ)= D D (v(θ)ǫ− ) = Γ (θ) , ab 2 bβ aα − ba so that eq.(B.15) becomes with eq.(A.6)

D D vγ (θ)= 1 (2δ γǫ ǫ δγ )Γ (θ) . (B.19) bβ aα δ 3 α βδ − βα δ ab 39 Thus 1 1 βα DcγΓab(θ) = 2 DbβDaαDcγ(v(θ)ǫ− )

1 = 2 DbγΓca(θ) (B.20)

=0 .

Therefore Γab(θ) is independent of θ and v(θ) is at most quadratic in θ. From eq.(B.19) we get γ DbβDaγ v δ(θ)= ǫβδΓab . (B.21) Integrating this gives γ b Daγ v α(θ)=6i(ta θ¯bα +ρ ¯aα) , (B.22) b where 6ita =Γab so that tt = t , (B.23) − and the spinor,ρ ¯aα, appears as a constant of integration. Now eq.(B.12) becomes with eq.(2.4)

β b µ µ β Daαv γ (θ)=2i(ta θ¯bγ +ρ ¯aγ )αγµ γ . (B.24)

Integrating this gives µ b µ a µ a µ v (θ)= ita θ¯bγ θ +2iρ¯aγ θ + v . (B.25) Eq.(B.8) becomes D λ(θ)= 2t bθ¯ 2¯ρ , (B.26) aα − a bα − aα so that D D λ(θ)= i 1 Γ ǫ . (B.27) bβ aα − 3 ab αβ However from DbβDaαλ(θ)+ DaαDbβλ(θ)=0 we note Γab = 0. Hence

wµν(θ)=¯ρ (γµγν γνγµ)θa + wµν , (B.28a) a −

λ(θ)= 2¯ρ θa + λ . (B.28b) − a µ ν 1 µ νλ ¯ a With these expressions straightforward calculation shows that w ν(θ)x + 4 ǫ νλw (θ)θaθ + λ(θ)xµ is a solution of the superconformal Killing equation (3.15).

The remaining terms are

D aµ(θ)= i 1 ǫµ D aλ(θ)γρβ , (B.29) aα − 2 λρ aβ α 40 the general solution of which we already obtained. From eq.(B.25) µ b µ a µ a µ a (θ)= ita θ¯bγ θ +2iε¯aγ θ + a . (B.30) For aµ(θ) to be real t must be anti-hermitian and hence with eq.(B.23) t o( ). ∈ N All together, we obtain the general solution of the superconformal Killing equation (3.17).

C Basis for Superconformal Algebra

We write the superconformal generators in general as = aµP +¯ε Qa + λD + 1 wµνM + bµK +ρ ¯ Sa + 1 tabA , (C.1) K·P µ a 2 µν µ a 2 ab for =(aµ, bµ,εa, ρa,λ,wµν, t b) , (C.2a) K a

=(P ,K , Qa,Sa,D,M , A b) , (C.2b) P µ µ µν a ab ac b c where we put t = δ tc and the R-symmetry generators, Aab = Aa δcb, satisfy the o( ) t N condition, A† = A = A. − The superconformal algebra can now be obtained by imposing [ , ]= i , (C.3) K1·P K2·P − K3·P where 3 is given by eq.(3.44). From this expression, we can read oﬀ the following super- conformalK algebra. Poincar´ealgebra • [P ,P ]=0 , [M ,P ]= i(η P η P ) , µ ν µν λ µλ ν − νλ µ (C.4) [M , M ]= i(η M η M η M + η M ) . µν λρ µλ νρ − µρ νλ − νλ µρ νρ µλ Supersymmetry algebra • Qaα, Q¯ =2δa γµα P , { bβ} b β µ a 1 a [Mµν , Q ]= i 2 γ[µγν]Q , (C.5)

aα [Pµ, Q ]=0 .

41 Special superconformal algebra • [K ,K ]=0 , [M ,K ]= i(η K η K ) , µ ν µν λ µλ ν − νλ µ Saα, S¯ =2δa γµα K , { bβ} b β µ (C.6) a 1 a [Mµν ,S ]= i 2 γ[µγν]S ,

aα [Kµ,S ]=0 .

Cross terms between (P, Q) and (K,S) •

[Pµ,Kν]=2i(Mµν + ηµν D) ,

[P ,Sa]= γ Qa , µ − µ (C.7) [K , Qa]= γ Sa , µ − µ Qaα, S¯ = iδa (2δα D +(γ[µγν])α M )+2iδα Aa . { bβ} − b β β µν β b Dilations • [D,P ]= iP , [D,K ]= iK , µ − µ µ µ [D, Qa]= i 1 Qa , [D,Sa]= i 1 Sa , (C.8) − 2 2 b [D,D] = [D, Mµν] = [D, Aa ]=0 .

R-symmetry, o( ) • N [A , A ]= i(δ A δ A δ A + δ A ) , ab cd ac bd − ad bc − bc ad bd ac [A , Qc]= i(δ cδ δ cδ )Qd , ab a bd − b ad (C.9) [A ,Sc]= i(δ cδ δ cδ )Sd , ab a bd − b ad b b b [Aa ,Pµ] = [Aa ,Kµ] = [Aa , Mµν]=0 .

42 R D Realization of SO(2, 3) ∼= Sp(2, ) structure in M We exhibit explicitly the relation of the three-dimensional conformal group to SO(2, 3) = Sp(2, R) by introducing ﬁve-dimensional gamma matrices, ΓA, A =0, 1, , 4 ∼ ··· γµ 0 0 i 0 i Γµ = , Γ3 = , Σ4 = . (D.1) 0 γµ i 0 i 0 − ! ! − ! They satisfy with GAB = diag(+1, 1, 1, 1, +1) − − − ΓAΓB +ΓBΓA =2GAB , (D.2) and

0 0 1 0 γ A 0 γ A 0 ǫ A 0 ǫ− At 0 Γ 0 = Γ † , Γ 1 =Γ . (D.3) γ 0 ! γ 0 ! − ǫ 0 ! ǫ− 0 ! For the supermatrix, M, given in eq.(3.45), we may now express the 4 4 part in terms of AB 1 A B × Γ 4 [Γ , Γ ] as ≡ 1 w + 2 λ ia γ 1 AB m · = wABΓ , (D.4) ≡ ib γ w 1 λ 2 · − 2 ! where w34, wµ3, wµ4 are given by w = λ, w = a b , w = a + b . (D.5) 34 µ3 µ − µ µ4 µ µ ΓAB generates the Lie algebra of SO(2, 3)

[ΓAB, ΓCD]= GAC ΓBD + GADΓBC + GBC ΓAD GBDΓAC . (D.6) − − In general, m can be deﬁned as a 4 4 matrix subject to two conditions × 0 γ0 bm + m†b =0 , b = 0 , (D.7a) γ 0 ! 0 ǫ cm + mtc =0 , c = , (D.7b) ǫ 0 !

R 0 To show SO(2, 3) ∼= Sp(2, ) we take, without loss of generality, γ = iǫ and ǫ to be real. Now if we deﬁne

1 1 0 m˜ = pmp− , p = , (D.8) 0 ǫ !

43 then from 0 1 p 1 = p = pt , pcp 1 = = j , (D.9) − † − 1 0 − ! we note that eq.(D.7a) is equivalent to the sp(2, R) condition

t m˜ ∗ =m ˜ , jm˜ +m ˜ j =0 . (D.10)

References

[1] V. Kac. A Sketch of Lie Superalgebra Theory. Comm. Math. Phys., 53: 31, 1977.

[2] W. Nahm. Supersymmetries and Their Representations. Nucl. Phys., B135: 149, 1978.

[3] N. Seiberg. Notes on Theories with 16 Supercharges. Proceedings of The Trieste Spring School, 1997. hep-th/9705117.

[4] J-H Park. Superconformal Symmetry and Correlation Functions. Nucl. Phys., B 559: 455, 1999. hep-th/9903230.

[5] J-H Park. N=1 Superconformal Symmetry in Four Dimensions. Int. J. Mod. Phys. A, 13: 1743, 1998. hep-th/9703191.

[6] J-H Park. Superconformal Symmetry in Six-dimensions and Its Reduction to Four. Nucl. Phys., B 539: 599, 1999. hep-th/9807186.

[7] M. Sohnius. Introducing Supersymmetry. Phys. Rep., 128: 39–204, 1985.

[8] J. Wess and J. Bagger. Supersymmetry and Supergravity, volume second edition. Princeton University Press, 1991.

[9] I. L. Buchbinder and S. M. Kuzenko. Ideas and Methods of Supersymmetry and Su- pergravity or a Walk through Superspace. Institute of Physics Publishing Ltd., 1995.

[10] P. West. Introduction to Supersymmetry and Supergravity. World Scientiﬁc, Singapore, 1990.

[11] A. Galperin, E. Ivanov, S. Kalitsyn, V. Ogievetsky and E. Sokatchev. Unconstrained Oﬀ-shell N = 3 Supersymmetric Yang-Mills Theory. Class.Quant.Grav. , 2: 155, 1985.

[12] S. Gates, M. Grisaru, M. Roˇcek and W. Siegel. Superspace, or One Thousand and One Lessons in Supersymmetry. Benjamin/Cummins, Reading MA, 1983.

44 [13] C. Lee, K. Lee and E. Weinberg. Supersymmetry and Self-dual Chern-Simons Systems. Phys. Lett., B243: 105, 1990.

[14] E. A. Ivanov. Chern-Simons Matter Systems with Manifest N = 2 Supersymmetry. Phys. Lett., B268: 203, 1991.

[15] S. Gates Jr. and H. Nishino. Remarks on N = 2 Supersymmetric Chern-Simons Theories. Phys. Lett., B281: 72, 1992.

[16] H. Nishino and S. Gates Jr. Chern-Simons Theories with Supersymmetries in Three- dimensions. Int. J. Mod. Phys. A, Vol.8 No.19: 3371, 1993.

[17] E. Sokatchev. Oﬀ-shell Supersingletons. Class. Quant. Grav., 6: 93, 1989.

[18] B. Zupnik. Harmonic Superspaces For Three-Dimensional Theories. talk at ”Super- symmetries and Quantum Symmetries” (Dubna, July 22-26, 1997). hep-th/9804167.

[19] B. Zupnik. Harmonic Superpotentials and Symmetries in Gauge Theories with Eight Supercharges. Nucl. Phys., B554: 365, 1999. hep-th/9902038.

[20] H. Osborn. N = 1 Superconformal Symmetry in Four Dimensional Quantum Field Theory. Ann. Phys., 272:243, 1999. hep-th/9808041.

[21] S. M. Kuzenko and S. Theisen. Correlation Functions of Conserved Currents in =2 Superconformal Theory. hep-th/9907107. N

[22] A. Van Proeyen. Tools for Supersymmetry. hep-th/9910030, Lectures in the spring school Cˇalimˇane¸sti, Romania, April 1998.

[23] T. Kugo and P. Townsend. Supersymmetry and the Division Algebras. Nucl. Phys., B221: 357, 1983.

[24] H. Osborn and A. Petkou. Implications of Conformal Invariance in Field Theories for General Dimensions. Ann. Phys., 231: 311–361, 1994.

[25] H. Weyl. The Classical Groups. Princeton University Press, 1946.

[26] P. Ginsparg. Applied Conformal Field Theory. Champs. Cordes et Phénomènes Cri- tiques (E. Brézin and J. Zinn-Justin, Eds.), page 1, 1989.

[27] J. Erdmenger. Conformally Covariant Diﬀerential Operators: Properties and Applica- tions. Class. Quantum Grav., 14: 2061, 1997.

45 [28] J. Erdmenger and H. Osborn. Conformally Covariant Diﬀerential Operators: Sym- metric Tensor Fields. Class. Quantum Grav., 15: 273, 1998.