KIAS-99101 hep-th/9910199
arXiv:hep-th/9910199v2 3 Nov 1999
Superconformal Symmetry in Three-dimensions
Jeong-Hyuck Park∗ School of Physics, Korea Institute for Advanced Study 207-43 Cheongryangri-dong, Dongdaemun-gu Seoul 130-012, Korea
Abstract Three-dimensional N -extended superconformal symmetry is studied within the superspace formalism. A superconformal Killing equation is derived and its solutions are classified in terms of supertranslations, dilations, Lorentz transformations, Rsymmetry transformations and special superconformal transformations. Superconformal group is then identified with a supermatrix group, OSp(N |2, R), as expected from the analysis on simple Lie superalgebras. In general, due to the invariance under supertranslations and special superconformal transformations, superconformally invariant n-point functions reduce to one unspecified (n − 2)-point function which must transform homogeneously under the remaining rigid transformations, i.e. dilations, Lorentz transformations and R-symmetry transformations. After constructing building blocks for superconformal correlators, we are able to identify all the superconformal invariants and obtain the general form of n-point functions. Superconformally covariant differential operators are also discussed.
PACS : 11.30.Pb; 11.25.Hf Keywords: Superconformal symmetry; Correlation functions; Superconformally covariant differential operators ∗
E-mail address:
[email protected]
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 LC ⊕ LR ,
(1.1)
where LC is the Lie algebra of the conformal group and LR is a R-symmetry algebra acting on the superspace Grassmann variables. Explicitly for Minkowskian spacetime o(2, 3) ⊕ o(N ) ,
d = 3;
d = 4;
d = 5; d = 6;
o(2, 4) ⊕ u(N ) , N 6= 4 o(2, 4) ⊕ su(4)
, (1.2)
o(2, 5) ⊕ su(2) , o(2, 6) ⊕ sp(N ) ,
where N or the number appearing in R-symmetry part is related to the number of supercharges. On six-dimensional Minkowskian spacetime it is possible to define Weyl spinors of opposite chiralities and so the general six-dimensional supersymmetry may be denoted by ˜ ), where N and N˜ are the numbers of chiral and anti-chiral supertwo numbers, (N , N charges. The R-symmetry group is then Sp(N ) × Sp(N˜ ). The analysis of Nahm shows that to admit a superconformal algebra either N or N˜ should be zero. Although both (1, 1) and (2, 0) supersymmetry give rise N = 4 four-dimensional supersymmetry after dimensional reduction, only (2, 0) supersymmetry 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 Poincar´e symmetry, 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 N 6= 4 extended superconformal group with a supermatrix group, SU(2, 2|N ), having dimensions (15 + N 2 |8N ), while for N = 4 case we pointed out that an equivalence relation must be imposed on the supermatrix group and so the four-dimensional N = 4 superconformal group is isomorphic to a quotient group of the supermatrix group. In fact N = 4 superconformal group is a semidirect product of U(1) and a simple Lie supergroup 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 (N , 0) superconformal group is identified with a supermatrix group, ˜ ), N , N˜ > 0 suOSp(2, 6|N ), having dimensions (28 + N (2N + 1)|16N ), while for (N , N persymmetry, we verified that although dilations may be introduced, there exist 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 1
Similarly 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 N = 1, 2, 3 [7–11] and in three-dimensions for N = 1, 2, 3, 4 [12–19]. In fact within the formalism Osborn elaborated the analysis of N = 1 superconformal symmetry for fourdimensional quantum field theories [20], and recently Kuzenko and Theisen determine the general structure of two- and three- point functions of the supercurrent and the flavour current of N = 2 superconformal field theories [21]. A common result contained in Refs. [20,21] is that the three-point functions of the conserved supercurrents in both N = 1 and N = 2 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 N -extended Majorana superconformal symmetry with an arbitrary natural number, N . In section 3, we first define the three-dimensional N -extended superconformal group 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(N ) as in eq.(1.2). We also present a definition of superinversion in three-dimensions through which supertranslations and special 3
superconformal transformations are dual to each other. The three-dimensional N -extended superconformal group is then identified with a supermatrix group, OSp(N |2, R), having dimensions (10 + 12 N (N − 1)|4N ) as expected from the analysis on simple Lie superalgebras [2, 22]. 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 quasiprimary 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, η µν = diag(+1, −1, −1), the 2 × 2 gamma matrices, γ µ , µ = 0, 1, 2, satisfy γ µ γ ν = η µν + iǫµνρ γρ .
(2.1)
γ 0 γ µ γ 0 = γ µ† .
(2.2)
The hermiticity condition is Charge conjugation matrix, ǫ, satisfies2 [23] ǫγ µ ǫ−1 = −γ µt , †
t
ǫ = −ǫ ,
(2.3) −1
ǫ =ǫ
.
γ µ forms a basis for 2 × 2 traceless matrices with the completeness relation γ µα β γµ γ δ = 2δ α δ δβ γ − δ α β δ γ δ .
2.2
(2.4)
Three-dimensional Superspace
The three-dimensional supersymmetry algebra has the standard form with Pµ = (H, −P) ¯ jβ } = 2δ i j γ µα β Pµ , {Qiα , Q ¯ iα ] = {Qiα , Qjβ } = {Q ¯ iα , Q ¯ jβ } = 0 , [Pµ , Pν ] = [Pµ , Qiα ] = [Pµ , Q
(2.5)
¯ j satisfy where 1 ≤ α ≤ 2, 1 ≤ i ≤ N and Qi , Q ¯ i = Qi† γ 0 . Q Now we define for 1 ≤ a ≤ 2N, 1 ≤ i, j ≤ N
2
Qa =
√1 (Qi 2
i √12 (Qj
¯t) + ǫ−1 Q i ¯t ) −ǫ Q j −1
(2.6)
,
(2.7)
To 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 ≡ Qa† γ 0 = Q
¯i √1 (Q 2
¯ b satisfy the Majorana condition Qa , Q
¯ j + Qjt ǫ) . − i √12 (Q
− Qit ǫ),
¯ a = Qa† γ 0 = −Qat ǫ , Q ¯t . Q =ǫ Q a a
(2.8)
(2.9)
−1
With this notation we note that the three-dimensional N-extended supersymmetry algebra (2.5) is equivalent to the 2N-extended Majorana supersymmetry algebra ¯ bβ } = 2δ a b γ µα β Pµ , {Qaα , Q
(2.10)
aα
[Pµ , Pν ] = [Pµ , Q ] = 0 . This can be generalized by replacing 2N with an arbitrary natural number, N , and hence N -extended Majorana supersymmetry algebra. Pµ , Qaα , 1 ≤ a ≤ N generate a supergroup, GT , with parameters, z M = (xµ , θaα ), which are coordinates on superspace. The general element of GT is written in terms of these coordinates as ¯ a g(z) = ei(x·P +Qa θ ) . (2.11) Corresponding to eq.(2.9) θa also satisfies the Majorana condition θ¯a = θa† γ 0 = −θat ǫ , so that
¯ a θa = θ¯a Qa , Q
(2.12)
g(z)† = g(z)−1 = g(−z) .
(2.13)
The Baker-Campbell-Haussdorff formula with the supersymmetry algebra (2.10) gives g(z1 )g(z2 ) = g(z3 ) , where
xµ3 = xµ1 + xµ2 + iθ¯1a γ µ θ2a ,
(2.14) θ3a = θ1a + θ2a .
(2.15)
Letting z1 → −z2 we may get the supertranslation invariant one forms, eM = (eµ , dθaα ), where eµ (z) = dxµ − iθ¯a γ µ dθa . (2.16) 6
The exterior derivative, d, on superspace is defined as d ≡ dz M
∂ = eM DM = eµ ∂µ − dθaα Daα , ∂z M
(2.17)
where DM = (∂µ , −Daα ) are covariant derivatives ∂µ =
∂ , ∂xµ
Daα = −
∂ ¯a γ µ )α ∂ . + i( θ ∂θaα ∂xµ
(2.18)
We also define
¯ aα = ǫ−1αβ Daβ = ∂ − i(γ µ θa )α ∂ , D ∂xµ ∂ θ¯aα satisfying the anti-commutator relations ¯ aα , Dbβ } = 2iδ a b γ µα β ∂µ . {D
(2.19)
(2.20)
Under an arbitrary superspace coordinate transformation, z −→ z ′ , eM and DM transform as ′ eM (z ′ ) = eN (z)RN M (z) , DM = R−1 M N (z)DN , (2.21) so that the exterior derivative is left invariant ′ eM (z)DM = eM (z ′ )DM ,
(2.22)
where RM N (z) is a (3 + 2N ) × (3 + 2N ) supermatrix of the form N
RM (z) =
Rµ ν (z) ∂µ θ′bβ µ −Baα (z) −Daα θ′bβ
!
,
(2.23)
with Rµ ν (z) =
′a ∂x′ν ¯′ γ ν ∂θ , − i θ a ∂xµ ∂xµ
µ Baα (z) = Daα x′µ + iθ¯b′ γ µ Daα θ′b .
(2.24) (2.25)
For Majorana spinors it is useful to note from eqs.(2.2,2.3,A.3a) ε¯a ρa = ρ¯a εa , 7
(2.26a)
ρa ε¯a + εa ρ¯a + ρ¯a εa 1 = 0 ,
(2.26b)
ρ¯a γ µ1 γ µ2 · · · γ µn εa = (−1)n ε¯a γ µn · · · γ µ2 γ µ1 ρa ,
(2.26c)
(¯ ρa γ µ1 γ µ2 · · · γ µn εa )∗ = ε¯a γ µn · · · γ µ2 γ µ1 ρa .
(2.26d)
θa θ¯a = − 12 θ¯a θa 1 .
(2.27)
In particular
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 transformag tions, z −→ z ′ , that preserve the infinitesimal supersymmetric interval length, e2 = ηµν eµ eν , up to a local scale factor, so that e2 (z) → e2 (z ′ ) = Ω2 (z; g)e2 (z) ,
(3.1)
where Ω(z; g) is a local scale factor. µ This requires Baα (z) = 0 Daα x′µ + iθ¯b′ γ µ Daα θ′b = 0 .
(3.2)
and eµ (z ′ ) = eν (z)Rν µ (z; g) , Rµλ (z; g)Rνρ (z; g)ηλρ = Ω2 (z; g)ηµν ,
det R(z; g) = Ω3 (z; g) .
Hence RM N in eq.(2.23) is of the form3 N
RM (z; g) = 3
Rµν (z; g) ∂µ θ′bβ 0 −Daα θ′bβ
More explicit form of RM N is obtained later in eq.(4.40).
8
(3.3)
!
.
(3.4)
(3.5)
Infinitesimally z ′ ≃ z + δz, eq.(3.2) gives or equivalently where we define
¯ a γ µ )α , Daα hµ = 2i(λ
(3.6)
¯ aα hµ = −2i(γ µ λa )α , D
(3.7)
¯ a = δ θ¯a , λ
λa = δθa ,
(3.8)
hµ = δxµ − iθ¯a γ µ δθa .
Infinitesimally from eq.(2.24) Rµ ν is of the form
Rµ ν ≃ δµν + ∂µ hν ,
(3.9)
so that the condition (3.4) reduces to the ordinary conformal Killing equation ∂µ hν + ∂ν hµ ∝ ηµν .
(3.10)
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)
(3.11)
¯ bα − Dbα λaα )ηµν , ¯ aα λ δab (∂µ hν + ∂ν hµ ) = (D
(3.12)
δ a b ∂ν hµ = and hence
1 2
¯ b γµ γν )α − (γν γµ Dbα λa )α , ¯ aα (λ D
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 are given by From eq.(3.6,3.7) λaα , λ ¯ aα = −i 1 Daβ hβ α , λ 6
¯ aβ hα β , λaα = i 16 D
(3.13)
where hα β = hµ γµ α β .
(3.14)
Substituting these expressions back into eqs.(3.6,3.7) gives using eqs.(2.1,2.4) Daα hµ = −i 12 ǫµ νλ Daβ hν γ λβ α , ¯ aα hµ = i 1 ǫµ νλ γ λα β D ¯ aβ hν . D 2 9
(3.15)
or equivalently
Daα hβ γ = 32 δα β Daδ hδ γ − 13 δ β γ Daδ hδ α , ¯ aα hβ γ = 2 δ α γ D ¯ aδ hβ δ − 1 δ β γ D ¯ aδ hα δ . D 3 3
(3.16)
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 − 14 (θ¯a θa )2 )bµ + ǫµ νλ xν bλ θ¯a θa −2¯ ρa x− γ µ θa + w µ ν xν + 41 ǫµ νλ w νλθ¯a θa + λxµ
(3.17)
+ita b θ¯b γ µ θa + 2i¯ εa γ µ θa + aµ , where aµ , bµ , λ, wµν = −wνµ are real, εa , ρa satisfy the Majorana condition (2.9) and t ∈ so(N ) satisfying t† = tt = −t . (3.18) We also set
x± = x ± i 12 θ¯a θa 1 .
x = xµ γ ,
(3.19)
Eq.(3.17) gives λa = x+ b·γθa − ix+ ρa + 2(¯ ρb θa )θb + (w + 12 λ)θa − θb tb a + εa
(3.20)
satisfying the Majorana condition ¯ a = λa† γ 0 = −λat ǫ , λ
(3.21)
w = 41 wµν γ µ γ ν .
(3.22)
where we put For later use it is worth to note γ 0 wγ 0 = −w † , 4
ǫwǫ−1 = −w t .
A method of obtaining the solution (3.17) is demonstrated in Appendix B.
10
(3.23)
3.2
Extended Superconformal Transformations
In summary, the generators of superconformal transformations in three-dimensions acting on the three-dimensional superspace, R3|2N , with coordinates, z M = (xµ , θaα ), can be classified as 1. Supertranslations, a, ε δxµ = aµ − iθ¯a γ µ εa ,
δθa = εa .
(3.24)
This is consistent with eq.(2.15). 2. Dilations, λ δθa = 12 λθa .
δxµ = λxµ ,
(3.25)
3. Lorentz transformations, w δxµ = w µ ν xν ,
δθa = wθa .
(3.26)
4. R-symmetry transformations, t δxµ = 0 ,
δθa = −θb tb a .
(3.27)
where t ∈ so(N ) of dimension 12 N (N − 1). 5. Special superconformal transformations, b, ρ δxµ = 2x·b xµ − (x2 + 41 (θ¯a θa )2 )bµ − ρ¯a x+ γ µ θa , a
a
a
a
(3.28)
b
δθ = x+ b·γθ − ix+ ρ + 2(¯ ρb θ )θ . As we consider infinitesimal transformations we obtain SO(N ) as R-symmetry group. However finitely R-symmetry group can be extended to O(N ) which leaves the supertranslation invariant one form (2.16) invariant manifestly.
11
3.3
Superinversion i
s z ′M = (x′µ , θ′aα ) ∈ R3|2N , by In three-dimensions we define superinversion, z M −→
x′± = −x−1 ± ,
a θ′a = ix−1 + θ .
(3.29)
As a consistency check we note from x+ x− = (x2 + 14 (θ¯a θa )2 ) 1 θ¯a′ = θ′a† γ 0 = −θ′at ǫ ,
x′+ − x′− = iθ¯a′ θ′a 1 .
(3.30)
It is easy to verify that superinversion is idempotent
Using
i2s = 1 .
(3.31)
e(z) = eµ (z)γµ = dx+ + 2idθa θ¯a ,
(3.32)
−1 e(z ′ ) = x−1 + e(z)x− .
(3.33)
we get under superinversion and hence e2 (z ′ ) = Ω2 (z; is )e2 (z) ,
Ω(z; is ) =
1 x2
+
1 ¯ a 2 (θ θ ) 4 a
.
(3.34)
Eq.(3.33) can be rewritten as µ −1 Rν µ (z; is ) = 21 tr(γν x−1 − γ x+ ) .
eµ (z ′ ) = eν (z)Rν µ (z; is ) ,
(3.35)
Explicitly Rν µ (z; is ) =
1 (x2 +
1 ¯ a 2 2 (θ θ ) ) 4 a
2xν xµ − (x2 − 41 (θ¯a θa )2 )δν µ − ǫν µλ xλ θ¯a θa .
(3.36)
Note that µ −1 γ ν Rν µ (z; is ) = x−1 − γ x+ ,
−1 Rν µ (z; is )γµ = x−1 + γν x− .
(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 − 41 (θ¯a θa )2 )aµ + ǫµ νλ xν aλ θ¯a θa −2¯ εa x− γ µ θa + w µ ν xν + 41 ǫµ νλ w νλ θ¯a θa − λxµ +ita b θ¯b γ µ θa + 2i¯ ρa γ µ θa + bµ . 12
(3.38)
Hence, under superinversion, the superconformal transformations are related by
K≡
aµ bµ εa ρa λ wµν ta b
−→
bµ aµ ρa εa −λ wµν ta b
.
(3.39)
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, L, is given by L = hµ ∂µ − λaα Daα .
(3.41)
If we write the commutator of two generators, L1 , L2 , as [L2 , L1 ] = L3 = hµ3 ∂µ − λaα 3 Daα ,
(3.42)
then hµ3 , λaα 3 are given by ¯ 1a γ µ λa , hµ3 = hν2 ∂ν hµ1 − hν1 ∂ν hµ2 + 2iλ 2 λa3
=
L2 λa1
−
L1 λa2
,
13
(3.43)
and hµ3 , λa3 satisfy eq.(3.7) verifying the closure of the Lie algebra. Explicitly with eqs.(3.17 ,3.20) we get aµ3 = w1µ ν aν2 + λ1 aµ2 + i¯ ε1a γ µ εa2 − (1 ↔ 2) , εa3 = w1 εa2 + 21 λ1 εa2 − ia2 ·γρa1 − εb2 t1b a − (1 ↔ 2) , λ3 = 2a2 ·b1 − 2¯ ρ1a εa2 − (1 ↔ 2) , w3µν = w1µ λ w2λν + 2(aµ2 bν1 − aν2 bµ1 ) + 2¯ ρ1a γ [µ γ ν] εa2 − (1 ↔ 2) ,
(3.44)
bµ3 = w1µ ν bν2 − λ1 bµ2 + i¯ ρ1a γ µ ρa2 − (1 ↔ 2) , ρa3 = w1 ρa2 − 21 λ1 ρa2 − ib2 ·γεa1 − ρb2 t1b a − (1 ↔ 2) , t3a b = (t1 t2 )a b + 2(¯ ρ2a εb1 − ε¯1a ρb2 ) − (1 ↔ 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 + 2N ) × (4 + 2N ) supermatrix, M, as √ b w + 21 λ ia·γ √ 2εb 1 M = 2ρ , w√ − 2λ ib·γ √ ρa − 2¯ εa ta b − 2¯
(3.45)
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, 2N ) supermatrix subject to
0 γ0 0 0 0 0 B= γ , 0 0 1
BMB −1 = −M † ,
0 ǫ 0 C= ǫ 0 0 . 0 0 1
CMC −1 = −M t ,
14
(3.47a)
(3.47b)
Supermatrix of the form (3.45) is the general solution of these two equations. The 4 × 4 matrix appearing in M, w + 21 λ ia·γ ib·γ w − 21 λ
!
,
(3.48)
corresponds to a generator of SO(2, 3) ∼ = Sp(2, R) as demonstrated in Appendix D. Thus, the N -extended Majorana superconformal group in three-dimensions may be identified with the supermatrix group generated by supermatrices of the form M (3.45), which is OSp(N |2, R) ≡ GS having dimensions (10 + 21 N (N − 1)|4N ).
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 first identify the superspace, R3|2N , as a coset, GS /G0 , where G0 ⊂ GS is the subgroup generated by matrices, M0 , of the form (3.45) with aµ = 0, εa = 0 and depending on parameters bµ , ρa , λ, wµν , ta b . The group of supertranslations, GT , parameterized by coordinates, z M ∈ R3|2N , 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 √ b √ b 0 ix 2θ 2θ 1 ix− 0 0 0 0 1 0 GT (z) = exp (4.1) . = √ √ 0 − 2θ¯a 0 − 2θ¯a δa b 0 Note GT (z)−1 = GT (−z).
In general an element of GS can be uniquely decomposed as GT G−1 0 . Thus for any g element G(g) ∈ GS we may define a superconformal transformation, z −→ z ′ , and an associated element G0 (z; g) ∈ G0 by G(g)−1GT (z)G0 (z; g) = GT (z ′ ) .
(4.2)
If G(g) ∈ GT then clearly G0 (z; g) = 1. Infinitesimally eq.(4.2) becomes ˆ 0 (z) , δGT (z) = MGT (z) − GT (z)M
(4.3)
ˆ 0 (z), the generator of G0 , has the form where M is given by eq.(3.45) and M
5
ˆ 0 0 w(z) ˆ + 21 λ(z) √ b 1 ˆ . ˆ M0 (z) = ib·γ w(z) ˆ − λ(z) 2ˆ ρ (z) 2 √ ˆ¯a (z) − 2ρ 0 tˆa b (z)
The subscript, T , denotes supertranslations.
16
(4.4)
The components depending on z are given by ˆ w(z) ˆ + 21 λ(z) = w + 21 λ + x+ b·γ + 2θa ρ¯a , ˆ w(z) ˆ − 12 λ(z) = w − 12 λ − b·γx− − 2ρa θ¯a , ˆ λ(z) = λ + 2x·b − 2θ¯a ρa , a
a
(4.5)
a
ρˆ (z) = ρ + ib·γθ , ρˆ¯a (z) = ρ¯a − iθ¯a b·γ = (ˆ ρa (z))† γ 0 = −ˆ ρa (z)t ǫ , tˆa b (z) = ta b + 2iθ¯a b·γθb + 2θ¯a ρb − 2¯ ρa θb . w(z) ˆ can be also written as w(z) ˆ = 41 wˆµν (z)γ µ γ ν with wˆµν (z) = wµν + 2(xµ bν − xν bµ ) + ǫµνλ (bλ θ¯a θa + 2i¯ ρa γ λ θa ) .
(4.6)
Writing δGT (z) = LGT (z) we may verify that L is identical to eq.(3.41). The definitions (4.5) can be summarized by ˆ Daα λbβ (z) = − 21 δa b δα β λ(z) − δa b wˆ β α (z) + δα β tˆa b (z) ,
(4.7a)
ˆ , ∂ν hµ (z) = wˆµν (z) + ηµν λ(z)
(4.7b)
ˆ [Daα , L] = −Daα λbβ Dbβ = ( 21 δa b δα β λ(z) + δa b wˆ β α (z) − δα β tˆa b (z))Dbβ .
(4.8)
and they give
For later use we note
Daα wˆµν (z) = 2(ρˆ¯a (z)γ[µ γν] )α , ˆ Daα λ(z) = 2ρˆ¯aα (z) , Daα tˆb c (z) = 2(δab δ cd − δa c δb d )ρˆ¯dα .
17
(4.9)
The above analysis can be simplified by reducing G0 (z; g). To achieve this we let
0 0 Z0 = 1 0 , 0 1
and then M0 Z0 = Z0 H0 , Now if we define
(4.10) √
w − 21 λ 0
H0 =
2ρb ta b
√
!
.
(4.11)
2θb 0 , δa b
(4.12)
δZ(z) = LZ(z) = MZ(z) − Z(z)H(z) ,
(4.13)
ix− Z(z) ≡ GT (z)Z0 = √1 − 2θ¯a
then Z(z) transforms under infinitesimal superconformal transformations as where H(z) is given by
√
ˆ w(z) ˆ − 21 λ(z) 0
ˆ 0 (z)Z0 = Z0 H(z) , H(z) = M
2ˆ ρb (z) b tˆa (z)
!
.
(4.14)
From eqs.(3.42, 3.46) considering [L2 , L1 ]Z(z) = L3 Z(z) ,
(4.15)
we get H3 (z) = L2 H1 (z) − L1 H2 (z) + [H1 (z), H2 (z)] , ˆ ρˆ and tˆa b , thus λ ˆ 3 = L2 λ ˆ 1 − L1 λ ˆ 2 , etc. which gives separate equations for w, ˆ λ, ¯ As a conjugate of Z(z) we define Z(z) by ¯ Z(z) = This satisfies
γ0 0 0 1
!
ǫ−1 0 0 1
†
Z(z) B =
!
t
Z(z) C =
√ b ! 2θ 1 −ix − + √ . ¯ 2θa δa b 0
−1 ¯ ¯ Z(z) = Z(0)G , T (z)
¯ H(z) =
ˆ 0 w(z) ˆ √+ 21 λ(z) ˆ¯a (z) tˆa b (z) − 2ρ 18
(4.17)
(4.18)
and corresponding to eq.(4.13) we have ¯ ¯ ¯ Z(z) ¯ ¯ δ Z(z) = LZ(z) = H(z) − Z(z)M , where
(4.16)
!
.
(4.19) (4.20)
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 zt = LM (zt ) , dt
z1 = z ′ ,
z0 = z ,
(4.21)
where, with L given in eq.(3.41), LM (z) is defined by L = LM (z)∂M .
(4.22)
d Z(zt ) = MZ(zt ) − Z(zt )H(zt ) , dt
(4.23)
Z(zt ) = etM Z(z)K(z, t) ,
(4.24)
From eq.(4.13) we get
which integrates to where K(z, t) satisfies d K(z, t) = −K(z, t)H(zt ) , dt
1 0 0 1
K(z, 0) =
!
.
(4.25)
g
Hence for t = 1 with K(z, 1) ≡ K(z; g), z −→ z ′ , eq.(4.24) becomes Z(z ′ ) = G(g)−1Z(z)K(z; g) ,
G(g)−1 = eM .
(4.26)
G0 (z; g) in eq.(4.2) is related to K(z; g) from eq.(4.26) by G0 (z; g)Z0 = Z0 K(z; g) .
(4.27)
In general K(z; g) is of the form 1
K(z; g) =
Ω(z; g) 2 L(z; g) 0
√
2Σb (z; g) Ua b (z; g)
!
,
(4.28)
where Ω(z; g) is identical to the local scale factor in eq.(3.1), U(z; g) ∈ SO(N ) U −1 = U † = U t ,
det U = 1 ,
19
(4.29)
and L(z; g) satisfies det L(z; g) = 1 ,
(4.30a)
L−1 (z; g) = ǫ−1 L(z; g)t ǫ = γ 0 L(z; g)† γ 0 .
(4.30b)
¯ From eq.(4.26) Z(z) transforms as ¯ ′ ) = K(z; ¯ g)Z(z)G(g) ¯ Z(z ,
(4.31)
where γ0 0 0 1
¯ g) = K(z;
!
γ0 0 0 1
K(z)†
1 2
!
=
−1
Ω(z; 0 √g) L (z; g) −1 b ¯ 2Σa (z; g) U a (z; g)
=
ǫ−1 0 0 1 !
!
K(z)t
ǫ 0 0 1
!
(4.32) .
i
s z ′ , (3.29) If we define for superinversion, z −→
G(is )−1
ǫ 0 0 = 0 ǫ 0 , 0 0 1
with
K(z; is ) =
−i(ǫx− )−1 0
√
b 2ix−1 + θ b Va (z)
b Va b (z) = δa b + 2iθ¯a x−1 + θ ,
!
,
(4.33)
(4.34)
an analogous formula to eq.(4.26) can be obtained for superinversion
1 √ 0 ′t −1 ′t ¯ ′ )t . 2θ¯a = Z(z −ix+ G(is ) Z(z)K(z; is ) = √ 2θ′bt δ b a Similarly we have
′ t ¯ is )Z(z)G(i ¯ K(z; s ) = Z(z ) ,
where ¯ is ) = K(z;
−1 0 √iǫx+ −1 − 2iθ¯a x− V −1 a b (z)
20
(4.35)
(4.36) !
.
(4.37)
Note that ¯ −1 θ , V −1 (z) = V † = V (z)t = V (−z) = 1 − 2iθx −
(4.38a)
Va b (z)θ¯b = θ¯a x−1 + x− ,
(4.38b)
b θa Va b (z) = x− x−1 + θ ,
Rµ ν (z; g)γν = Ω(z; g)L−1 (z; g)γµL(z; g) ,
(4.38c)
γ ν Rν µ (z; g) = Ω(z; g)L(z; g)γ µ L−1 (z; g) .
(4.38d)
where Rµ ν (z; g) is identical to the definition (3.3). We may normalize Rµ ν (z; g) as ˆ µ ν (z; g) = Ω(z; g)−1 Rµ ν (z; g) = 1 tr(γµ L(z; g)γ ν L−1 (z; g)) ∈ SO(1, 2) . R 2
4.3
(4.39)
Representations
Based on the results in the previous subsection, it is easy to show that the matrix, RM N (z; g), given in eq.(3.5) is of the form N
RM (z; g) =
ˆ µ ν (z; g) iΩ(z; g) 21 (L−1 (z; g)γµΣb (z; g))β Ω(z; g)R 1 0 Ω(z; g) 2 L−1β α (z; g)Ua b (z; g)
!
.
(4.40)
Since RM N (z; g) is a representation of the three-dimensional superconformal group, each of the following also forms a representation of the group, though it is not a faithful representation ˆ g) ∈ SO(1, 2) , Ω(z; g) ∈ D , R(z; (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 ′′ ) ,
4.4
and so on.
(4.42)
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. If F (z) is defined for z ∈ R3|2N by
√
2θb δa b
√ix− − 2θ¯a
¯ F (z) = Z(0)G T (z)Z(0) =
!
,
(4.43)
then F (z) satisfies F (−z) =
=
γ0 0 0 1 −ix √ + 2θ¯a
!
F (z)
†
γ0 0 0 1
!
=
!
ǫ−1 0 0 1
F (z)
ǫ 0 0 1
t
!
(4.44)
! √ − 2θb , δa b
and the superdeterminant of F (z) is given by sdet F (z) = − det x+ = x2 + 14 (θ¯a θa )2 .
(4.45)
We also note √ 1 −1 0 −i 2θ¯a x− 1
!
! √ b θ 1 i 2x−1 − = 0 1
F (z)
ix− 0 0 Va b (−z)
!
,
(4.46)
where Va b (−z) is identical to eq.(4.38a) and from eqs.(4.45, 4.46) it is evident that det V (z) = 1 .
(4.47)
Hence, with eq.(4.38a), V (z) ∈ SO(N ). Now with the supersymmetric interval for R3|2N defined by M a ¯ M z12 = (xµ12 , θ12 , θ12a ) = −z21 ,
GT (z2 )−1 GT (z1 ) = GT (z12 ) , xµ12
=
xµ1
−
xµ2
− iθ¯2a γ µ θ1a ,
a θ12
we may write ¯ 2 )Z(z1 ) = F (z12 ) = Z(z and
√ix12− − 2θ¯12a
a 2 sdet F (z12 ) = x212 + 41 (θ¯12a θ12 ) ,
22
=
√
θ1a
b 2θ12 δa b
−
θ2a
!
(4.48)
,
,
det V (z12 ) = 1 ,
(4.49) (4.50)
where
a x12− = x1− − x2+ − 2iθ2a θ¯1a = x12 − i 21 θ¯12a θ12 1,
x12+ = x1+ − x2− +
2iθ1a θ¯2a
= x12 +
a i 12 θ¯12a θ12
(4.51)
1.
From eqs.(4.26, 4.31) F (z12 ) transforms as ′ ¯ 2 ; g)F (z12 )K(z1 ; g) . F (z12 ) = K(z
(4.52)
′a Explicitly with eqs.(4.28, 4.32) we get the transformation rules for x′12± and θ12 1
1
1 2
1 2
x′12− = Ω(z1 ; g) 2 Ω(z2 ; g) 2 L−1 (z2 ; g)x12− L(z1 ; g) , (4.53a) x′12+ = Ω(z1 ; g) Ω(z2 ; g) L−1 (z1 ; g)x12+ L(z2 ; g) , 1
′a b θ12 = Ω(z1 ; g) 2 L−1 (z1 ; g)(θ12 Ub a (z2 ; g) + ix12+ Σa2 ) ,
(4.53b) 1 2
b ′a Ub a (z1 ; g) − ix12− Σa1 ) . θ21 = Ω(z2 ; g) L−1 (z2 ; g)(θ21
In particular
1 ¯′ 1 ¯ ′a 2 2 a 2 x′2 12 + 4 (θ12a θ12 ) = Ω(z1 ; g)Ω(z2 ; g)(x12 + 4 (θ12a θ12 ) ) .
(4.54)
From eqs.(4.38d, 4.53a) tr(γ µ x12− γ ν x12+ ) transforms covariantly as tr(γ µ x′12− γ ν x′12+ ) = tr(γ λ x12− γ ρ x12+ )Rλ µ (z2 ; g)Rρν (z1 ; g) .
(4.55)
From eq.(4.53b) we get ! √ c 1 −i 2x−1 12− θ12 K(z1 ; g) 0 δa c
=
1 2
0 Ω(z1 ; g) L(z1 ; g) 0 U(z1 ; g)
√ 1′ ′−1 0c −i 2θ¯12a x12− δa =
! √ ′b 1 i 2x′−1 12− θ12 0 δd b
1 2
!
¯ 2 ; g) K(z
−1
(4.56a) ,
1
√ i 2θ¯12d x−1 12−
0 Ω(z2 ; g) L (z2 ; g) −1 0 U (z2 ; g) 23
!
!
0 δd b
!
(4.56b) .
Using this and eq.(4.46) we can rederive eq.(4.53a) and obtain ′ V (z12 ) = U −1 (z1 ; g)V (z12 )U(z2 ; g) ,
(4.57) ′ V (z21 ) = U −1 (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, Zi , which transform homogeneously at zi , i = 1, 2, 3. is is (z31 )′ , we define Z1M = (X1µ , Θa1 ) ∈ R3|2N by (z21 )′ , z31 −→ With z21 −→ GT ((z31 )′ )−1 GT ((z21 )′ ) = GT (Z1 ) . Explicit expressions for Z1M can be obtained by calculating ¯ 31 )′ )Z((z21 )′ ) = F (Z1 ) = Z((z
√iX1− ¯ 1a − 2Θ
(4.58)
√
2Θb1 δa b
!
.
(4.59)
We get −1 X1− = x−1 31+ x23− x21− ,
Θa1
=
Using one can assure
a i(x−1 21+ θ21
−
−1 ¯ ¯ 1a = −i(θ¯21a x−1 Θ 21− − θ31a x31− ) .
a x−1 31+ θ31 ) ,
a ¯ x23− = x21− − x31+ − 2iθ31 θ21a , −1 X1+ = γ 0 X†1− γ 0 = −ǫ−1 Xt1− ǫ = x−1 21+ x23+ x31− ,
X1+ − X1− =
¯ 1a −2iΘa1 Θ
¯ 1a Θa1 1 . = iΘ
(4.60)
(4.61)
(4.62)
g ¯ 1a transform From eq.(4.53a) under superconformal transformations, z −→ z ′ , X1± , Θa1 , Θ as
X′1± = Ω(z1 ; g)−1 L−1 (z1 ; g)X1± L(z1 ; g) , 1
(4.63a)
− 2 −1 L (z1 ; g)Θb1Ub a (z1 ; g) , Θ′a 1 = Ω(z1 ; g)
(4.63b)
¯ ′ = Ω(z1 ; g)− 21 U −1 a b (z1 ; g)Θ ¯ 1b L(z1 ; g) , Θ 1a
(4.63c)
24
so that
ˆ ν µ (z1 ; g) . X1′µ = Ω(z1 ; g)−1 X1ν R
(4.64)
Thus Z1 transforms homogeneously at z1 , as ‘tangent’ vectors do. Eq.(4.63a) can be summarized as 1
F (Z1′ )
=
Ω(z1 ; g)− 2 L−1 (z1 g) 0 −1 0 U (z1 ; g)
!
1
F (Z1 )
Ω(z1 ; g)− 2 L(z1 g) 0 0 U(z1 ; g)
!
.
(4.65) Direct calculation using eq.(4.38b) shows that V (Z1 ) = V (z12 )V (z23 )V (z31 ) .
(4.66)
Similarly for Rµ ν (z; is ) given in eq.(3.35) we obtain from eqs.(3.37,4.60)
a 2 ) R(Z1 ; is ) = x212 + 14 (θ¯12a θ12
2
a 2 x231 + 14 (θ¯31a θ31 )
2
R(z12 ; is )R(z23 ; is )R(z31 ; is ) . (4.67)
From eqs.(4.55, 4.57) Va b (Z1 ), Rµ ν (Z1 ; is ) transform homogeneously at z1 under supercong formal transformation, z −→ z ′ , V (Z1′ ) = U −1 (z1 ; g)V (Z1 )U(z1 ; g) ,
(4.68a)
R(Z1′ ; is ) = Ω(z1 ; g)2R−1 (z1 ; g)R(Z1 ; is )R(z1 ; g) .
(4.68b)
It is useful to note det X1± =
−X12
−
1 ¯ (Θ1a Θa1 )2 4
=
−
a 2 ) x223 + 14 (θ¯23a θ23 a 2 x212 + 14 (θ¯12a θ12 )
a 2 x231 + 41 (θ¯31a θ31 )
.
(4.69)
By taking cyclic permutations of z1 , z2 , z3 in eq.(4.60) we may define Z2 , Z3 . We find Z2 , Z3 are related to Z1 as a
b a f = ix Θ 21+ Θ1 Vb (z12 ) , 2
f = −x X 21+ X1+ x12+ , 2−
b
−1 f X3− = x−1 31− X1+ x13− ,
a f Θa3 = ix−1 31− Θ1 Vb (z13 ) ,
is e f Θ) e is defined by superinversion, Z −→ Z. where Ze = (X,
25
(4.70a) (4.70b)
5
Superconformal Invariance of Correlation Functions
In this section we discuss the superconformal invariance of correlation functions for quasiprimary 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 superg conformal transformation, z −→ z ′ , transform as ΨI −→ Ψ′I ,
Ψ′I (z ′ ) = ΨJ (z)DJI (z; g) .
(5.1)
D(z; g) obeys the group property so that under the successive superconformal transformag
g′
tions, g ′′ : z −→ z ′ −→ z ′′ , it satisfies D(z; g)D(z ′ ; g ′) = D(z; g ′′ ) ,
(5.2)
D(z; g)−1 = D(z ′ ; g −1) .
(5.3)
and hence We choose here D(z; g) to be a representation of SO(1, 2) × O(N ) × D, which is a subgroup 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(N ) index, r, as ΨI ≡ Ψρ r . Now DJ I (z; g) is factorized as DJ I (z; g) = D ρ σ (L(z; g))Dr s (U(z; g))Ω(z; g)−η ,
(5.4)
where D ρ σ (L), Dr s (U) are representations of SO(1, 2) and O(N ) respectively, while η is the scale dimension of Ψρ r . Infinitesimally r r α σ ˆ δΨρ r (z) = −(L + η λ(z))Ψ ˆ β α (z) − Ψρ s (z) 21 (sab )s r tˆab (z) , ρ (z) − Ψσ (z)(s β ) ρ w
(5.5)
where tˆab (z) = δ ac tˆc b (z), and sα β , sab satisfy [sα β , sγ δ ] = δ α δ sγ β − δβ γ sα δ , [sab , scd] = −ηac sbd + ηad sbc + ηbc sad − ηbd sac . 26
(5.6)
sab is the generator of O(N ), while sα β is connected to the generator of SO(1, 2), sµν , through sµν ≡ 21 sα β (γ[µγν] )β α , sα β = − 21 sµν (γ [µ γ ν] )α β , [sµν , sλρ ] = −ηµλ sνρ + ηµρ sνλ + ηνλ sµρ − ηνρ sµλ ,
(5.7)
sα β wˆ β α (z) = 12 sµν wˆ µν (z) . From eqs.(4.15, 4.16) using eq.(5.6) we have δ3 Ψρ r = [δ2 , δ1 ]Ψρ r .
(5.8)
¯ ρ r (z), which transforms as It is useful to consider the conjugate superfield of Ψρ r , Ψ ¯ ′ρ r (z ′ ) = Ω(z; g)−η D ρ σ (L−1 (z; g))Dr s (U −1 (z; g))Ψ ¯ σ s (z) . Ψ
(5.9)
Superconformal invariance for a general n-point function requires hΨ′I1 1 (z1 )Ψ′I2 2 (z2 ) · · · Ψ′In n (zn )i = hΨI11 (z1 )ΨI22 (z2 ) · · · ΨInn (zn )i .
5.2
(5.10)
Two-point Correlation Functions
¯ ρr , has the The solution for the two-point function of the quasi-primary superfields, Ψρ r , Ψ general form ρ s ¯ ρ r (z1 )Ψσ s (z2 )i = CΨ I σ (ˆx12+ )Ir (V (z12)) hΨ (5.11) η , a 2 x212 + 41 (θ¯12a θ12 )
where we put
x ˆ12+ =
x12+ x212
+
1 ¯ (θ θ a )2 4 12a 12
1
,
(5.12)
2
and I ρ σ (ˆx12+ ), Ir s (V (z12 )) are tensors transforming covariantly according to the appropriate representations of SO(1, 2), O(N ) which are formed by decomposition of tensor products of xˆ12+ , V (z12 ). Under superconformal transformations, I ρ σ (ˆx12+ ) and Ir s (V (z12 )) satisfy from eqs.(4.53a, 4.57) D(L−1 (z1 ; g))I(ˆx12+ )D(L(z2 ; g)) = I(ˆx′12+ ) ,
(5.13a)
′ D(U −1 (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, S(z), φα (z), φ¯α (z), ζ a (z), ζa (z). They satisfy S(z) = S(z)∗ , φ¯α (z) = ǫ−1αβ φβ (z) = (γ 0 φ(z)† )α ,
(5.14)
ζ a(z) = ζ a (z)∗ = ζa (z) , and transform as
S ′ (z ′ ) = Ω(z; g)−η S(z) , φ′α (z ′ ) = Ω(z; g)−η φβ (z)Lβ α (z; g) , φ¯′α (z ′ ) = Ω(z; g)−η L−1α β (z; g)φ¯β (z) ,
(5.15)
ζ ′a (z ′ ) = Ω(z; g)−η ζ b(z)Ub a (z; g) , ζa′ (z ′ ) = Ω(z; g)−η U −1 a b (z; g)ζb (z) . The two-point functions of them are 1 η , hS(z1 )S(z2 )i = CS 1 a 2 x212 + 4 (θ¯12a θ12 ) hφ¯α (z1 )φβ (z2 )i = iCφ
(x12+ )α β
x212
+
1 ¯ (θ θ a )2 4 12a 12
(5.16)
η+ 1
,
(5.17)
2
Va b (z12 ) η . hζa (z1 )ζ b(z2 )i = Cζ a 2 ) x212 + 14 (θ¯12a θ12
(5.18) (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 ˆ µ (z; g), we have R I µν (z12 ) η , hJ µ (z1 )J ν (z2 )i = CV (5.20) x212 + 1 (θ¯12a θa )2 ν
4
28
12
where
ˆ µν (z; is ) = 1 tr(γ µ xˆ+ γ ν x I µν (z) = R ˆ− ) . 2
(5.21)
From eq.(3.37) we note I µν (z) = I νµ (−z) ,
I µν (z)Iλν (z) = δ µ λ .
(5.22)
If we define J α β (z) = J µ (z)(γµ )α β ,
(5.23)
then from eqs.(2.4,5.20) and Daω (z1 )(x12+ )α β = 2iδω α θ¯12aβ , Daω (z1 )(x12− )α β = 2i(δω α θ¯12aβ − δ α β θ¯12aω ) ,
(5.24)
a 2 Daω (z1 )(x212 + 41 (θ¯12a θ12 ) ) = 2i(θ¯12a x12− )α ,
we get
(θ¯12a γ ν x12− )β Daα (z1 )hJ α β (z1 )J ν (z2 )i = 2iCV (2 − η) η+1 . a 2 x212 + 41 (θ¯12a θ12 )
(5.25)
Hence hJ α β (z1 )J ν (z2 )i is conserved if η = 2
Daα (z1 )hJ α β (z1 )J ν (z2 )i = 0
if η = 2 .
(5.26)
The anti-commutator relation for Daα (2.20) implies also ∂ hJ µ (z1 )J ν (z2 )i = 0 ∂xµ1
if η = 2 .
(5.27)
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
The solution for the three-point correlation function of the quasi-primary superfields, Ψρ r , has the general form hΨ1ρ r (z1 )Ψ2σ s (z2 )Ψ3τ t (z3 )i ′
′
′
′
Hρ r σ′ s τ ′ t (Z1 )I σ σ (ˆx12+ )I τ τ (ˆx13+ )Is′ s (V (z12 ))It′ t (V (z13 )) η2 η3 , = x212 + 1 (θ¯12a θa )2 x213 + 1 (θ¯13a θa )2 4
12
4
29
13
(5.28)
where Z1 M = (X1µ , Θa1 ) ∈ R3|2N is given by eq.(4.58). Superconformal invariance (5.10) is now equivalent to ′
′
′
Hρ′ r σ′ s τ ′ t (Z)D ρ ρ (L)D σ σ (L)D τ τ (L) = Hρ r σ s τ t (Z ′ ) , (5.29a) Z ′
′
′M
ˆ ν µ (L), L−1 Θa ) , = (X R ν
′
Hρ r σ s τ t (Z)Dr′ r (U)Ds′ s (U)Dt′ t (U) = Hρ r σ s τ t (Z ′′ ) , (5.29b) Z ′′M = (X µ , Θb Ub a ) , Hρ r σ s τ t (Z) = λη2 +η3 −η1 Hρ r σ s τ t (Z ′′′ ) , (5.29c) Z
′′′M
1 2
µ
a
= (λX , λ Θ ) ,
where U ∈ O(N ), λ ∈ R and 2 × 2 matrix, L, satisfies L−1 = γ 0 L† γ 0 = ǫ−1 Lt ǫ ,
det L = 1 , (5.30)
ˆ ν µ (L) = 1 tr(γν Lγ µ L−1 ) . R 2 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
The solution for n-point correlation functions of the quasi-primary superfields, Ψρ r , has the general form hΨ1ρ1 r1 (z1 ) · · · Ψnρn rn (zn )i = Hρ1
r1
ρ′2
r2′
· · · ρ′n
′ rn
′
I ρk ρk (ˆx1k+ )Irk′ rk (V (z1k )) ηk , (Z1(1) , · · · , Z1(n−2) ) x2 + 1 (θ¯1ka θa )2 k=2 n Y
1k
i
4
(5.31)
1k
s zg where, in a similar fashion to eq.(4.58), with zk1 −→ k1 , k ≥ 2, Z1(1) , · · · , Z1(n−2) are given by −1 GT (zg j = 2, 3, · · · , n − 1 . (5.32) n1 ) GT (zf j1 ) = GT (Z1(j−1) ) ,
30
We note that all of them are ‘tangent’ vectors at z1 . Superconformal invariance (5.10) is equivalent to Hρ′1 r1 · · · ρ′n rn (Z(1) , · · · , Z(n−2) )
n Y
′
′ ′ , · · · , Z(n−2) ), D ρk ρk (L) = Hρ1 r1 · · · ρn rn (Z(1)
k=1
(5.33a)
′M ν ˆ µ Z(j) = (X(j) Rν (L), L−1 Θa(j) ) ,
′ rn
r1′
Hρ1 · · · ρn (Z(1) , · · · , Z(n−2) )
n Y
′′ ′′ Drk′ rk (U) = Hρ1 r1 · · · ρn rn (Z(1) , · · · , Z(n−2) ),
k=1
(5.33b)
µ ′′M Z(j) = (X(j) , Θb(j) Ub a ) , ′′′ ′′′ Hρ1 r1 · · · ρn rn (Z(1) , · · · , Z(n−2) ) = λ−η1 +η2 +···+ηn Hρ1 r1 · · · ρn rn (Z(1) , · · · , Z(n−2) ), ′′′M Z(j)
=
µ , (λX(j)
λ
1 2
(5.33c)
Θa(j) ) .
Thus n-point functions reduce to one unspecified (n − 2)-point function which must transform homogeneously under the rigid transformations, SO(1, 2) × O(N ) × D. From −1 X1(j−1)+ = x−1 j1+ xjn+ xn1− ,
we get and hence
−1 X1(j−1)− = x−1 n1+ xjn− xj1− ,
(5.34)
¯ 1(m−1)a = x−1 xlm+ x−1 , X(l,m)+ ≡ X1(l−1)+ − X1(m−1)− + 2iΘa1(l−1) Θ l1+ m1−
Now if we define
det X(l,m)± = −
a 2 x2lm + 41 (θ¯lma θlm ) a 2 ) x21l + 41 (θ¯1la θ1l
1 ∆lm = − 2(n−1)(n−2)
n X
ηi +
a 2 x21m + 41 (θ¯1ma θ1m )
1 (η 2(n−2) l
(5.35)
.
(5.36)
+ ηm ) ,
(5.37)
i=1
then using the following identity which holds for any matrix, Slm , and number, λ,
Y
l6=m
(Slm )
∆lm
n Y
(S1k Sk1 )
k=2
− 12 ηk
!
=λ
− 12 (−η1 +η2 +···+ηn )
Y
2≤l6=m
31
λSlm Sl1 S1m
!∆lm
,
(5.38)
we can rewrite the n-point correlation functions (5.31) as hΨ1ρ1 r1 (z1 ) · · · Ψnρn rn (zn )i ′
=
′
Kρ1 r1 ρ′2 r2 · · · ρ′n rn (Z1(1) , · · · , Z1(n−2) ) Q
l6=m
where
Qn
k=2 I
ρ′k
a 2 x2lm + 41 (θ¯lma θlm )
x1k+ )Irk′ ρk (ˆ
rk
∆lm
Kρ1 r1 · · · ρn rn (Z1(1) , · · · , Z1(n−2) ) = Hρ1 r1 · · · ρn rn (Z1(1) , · · · , Z1(n−2) )
(V (z1k ))
Y
(5.39) ,
(− det X(l,m)± )∆lm .
2≤l6=m
(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 K
r′ ρ′1 1
′ rn
· · · ρ′n (Z(1) , · · · , Z(n−2) )
n Y
′
′ ′ D ρk ρk (L)Drk′ rk (U) = Kρ1 r1 · · · ρn rn (Z(1) , · · · , Z(n−2) ),
k=1
′M ν ˆ µ Z(j) = (λX(j) Rν (L), λ 2 L−1 Θb(j) Ub a ) . 1
(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(Z1(1) , · · · , Z1(n−2) ) is a function of the superconformal invariants and furthermore that all of the superconformal invariants can be generated by contracting the indices µ M of Z1(j) , Θaα = (X1(j) 1(j) ) to make them SO(1, 2) ×O(N ) ×D invariant according to the recipe µ by Weyl [25]. To do so we first normalize Z1(j) as µ ˆµ , Θ ˆa ), Zˆ1(j) = (X 1(j) 1(j)
ˆµ = X 1(j)
µ X1(j) 1
2 (X1(1) )2
ˆa = Θ 1(j)
,
32
(5.42)
Θa1(j) 1
2 (X1(1) )4
.
By virtue of eqs.(A.3a,A.7a) all the SO(1, 2) × O(N ) × D invariants or three-dimensional superconformal invariants are ˆ¯ ˆ ˆa Θ 1(j)a X1(k) ·γ Θ1(l) ,
ˆ¯ ˆ b ˆ¯ ˆa Θ 1(j)a Θ1(k) Θ1(l)b Θ1(m) ,
ˆ¯ ˆa Θ 1(j)a Θ1(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] a 2 a 2 x2rs + 41 (θ¯rsa θrs x2tu + 14 (θ¯tua θtu ) ) . (5.44) x2 + 1 (θ¯rua θa )2 x2ts + 1 (θ¯tsa θa )2 ˆ 1(j) ·X ˆ 1(k) , X
ru
ru
4
4
ts
If we restrict the R-symmetry group to be SO(N ) instead of O(N ) then the followings are also superconformal invariants in the case of even N , according to Weyl [25] N
a1
ǫ
b1
···
aN /2
bN /2
2 Y
Tjaj bj ,
j=1
ˆ¯ ˆ bj Tjaj bj = Θ 1(j1 )aj Θ1(j2 )
ˆ¯ ˆ ˆ bj or Θ 1(j1 )aj X1(j2 ) ·γ Θ1(j3 ) , (5.45)
which we may call pseudo-invariants.
6
Superconformally Covariant Operators
In general acting on a quasi-primary superfield, Ψρ r (z), with the spinor derivative, Daα , does not lead to a quasi-primary field6 . For a superfield, Ψρ r , from eqs.(4.8, 4.9 ,5.5) we have ˆ aα Ψρ r Daα δΨρ r = −(L + (η + 21 )λ)D − Daβ Ψρ r wˆ β α − Daα Ψσ r (sβ γ wˆ γ β )σ ρ
(6.1)
− Dbα Ψρ r tˆb a − Daα Ψρ s 12 (sbc tˆbc )s r + 2ρˆ¯bβ (ΨY bβ aα )ρ r , where Y bβ aα is given by
6
Y bβ aα = 2sb a δ β α + δ b a sβ α − ηδ b a δ β α . For conformally covariant differential operators in non-supersymmetric theories, see e.g. [27, 28].
33
(6.2)
To ensure that Daα Ψρ r is quasi-primary it is necessary that the terms proportional to ρˆ¯ vanish and this can be achieved by restricting Daα Ψρ r to an irreducible representation of SO(1, 2) × O(N ) and choosing a particular value of η so that ΨY = 0. The change of the scale dimension, η → η + 12 , in eq.(6.1) is also apparent from eq.(2.21) 1
′ Daα = Ω(z; g) 2 L−1β α (z; g)Ua b (z; g)Dbβ .
(6.3)
As an illustration we consider tensorial fields, Ψa1 ···am α1 ···αn , which transform as ˆ a ···a α ···α δΨa1 ···am α1 ···αn = −(L + η λ)Ψ m 1 n 1 −
m X
p=1
Ψa1 ···b···am α1 ···αn tˆb ap −
n X
(6.4) Ψa1 ···am α1 ···β···αn wˆ β αq .
q=1
Note that spinorial indices and gauge indices, α, a may be raised or lowered by ǫ−1αβ , ǫαβ , δ ab , δab . For Ψa1 ···am α1 ···αn we have (ΨYb β aα )a1 ···am α1 ···αn = −(η + 12 n)δba δ β α Ψa1 ···am α1 ···αn +δ β α
m X
p=1
+δba
n X
q=1
(δaap Ψa1 ···b···am α1 ···αn − δbap Ψa1 ···a···am α1 ···αn )
(6.5)
δ β αq Ψa1 ······am α1 ···α···αn .
In particular, eq.(6.5) shows that the following are quasi-primary D[b(β Ψa1 ···am ]α1 ···αn )
if η = m + 21 n ,
(6.6a)
D[b|β| Ψa1 ···am ] β
if η = m − 32 ,
(6.6b)
where ( ), [ ] denote the usual symmetrization, anti-symmetrization of the indices respectively and obviously eq.(6.6a) is nontrivial if 1 ≤ m + 1 ≤ N . Note that due to the term 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 quasiprimary superfield. In this case, it is useful to note D[a[α Db]β] = 0 ,
(6.7)
ˆ¯bβ = −iδab (ǫb·γ)αβ . Daα ρ
(6.8)
and From eq.(6.5) one can derive D[b1 (β1 · · · Dbl βl δΨa1 ···am ]α1 ···αn ) = 2l(−η + m + 12 n + 43 (l − 1))ρˆ¯[b1 (β1 Db2 β2 · · · Dbl βl Ψa1 ···am ]α1 ···αn ) + homogeneous terms . (6.9) Hence the following is quasi-primary if η = m + 21 n + 43 (l − 1) .
D[b1 (β1 · · · Dbl βl Ψa1 ···am ]α1 ···αn )
Acknowledgments I am deeply indebted to Hugh Osborn for introducing me the subject in this paper.
35
(6.10)
Appendix A
Useful Equations
Some useful identities relevant to the present paper are 1 tr(γ µ γ ν ) 2
= η µν ,
(A.1a)
γ µ γ ν γ ρ = η µν γ ρ − η µρ γ ν + η νρ γ µ + iǫµνρ ,
(A.1b)
−i 21 ǫµνρ γν γρ = γ µ .
(A.1c)
ǫµνκ ǫλρκ = δ µ λ δ ν ρ − δ µ ρ δ ν λ ,
(A.2a)
ǫµνκ ǫλνκ = 2δ µ λ ,
(A.2b)
ǫµνκ = ǫµνκ .
(A.2c)
εγ µ ργµ + ε¯ρ 1) , ρ¯ ε = − 21 (¯
(A.3a)
ǫ−1 ε¯t ρt ǫ = − 21 (¯ εγ µ ργµ − ε¯ρ 1) .
(A.3b)
For Majorana spinors Daα θbβ = −δa b δα β ,
Daα θ¯bβ = δab ǫαβ ,
(A.4a)
¯ aα θbβ = −δ ab ǫ−1αβ , D
¯ aα θ¯bβ = δ a b δ α β , D
(A.4b)
Daα θ¯b θb = 2θ¯aα ,
(A.4c)
Daα x+ β γ = 2iδα β θ¯aγ ,
(A.4d)
Daα x− β γ = 2i(δα β θ¯aγ − δ β γ θ¯aα ) .
(A.4e)
36
γ 0 x± γ 0 = x†∓ ,
(A.5a)
ǫx± ǫ−1 = −xt∓ ,
(A.5b)
det x+ = det x− = −x2 − 41 (θ¯a θa )2 .
(A.5c)
δα δ δβ γ − δα γ δβ δ = ǫαβ ǫ−1γδ .
(A.6)
¯ µ θ′ ψγ ¯ µ ψ ′ = −2θψ ¯ ′ ψθ ¯ ′ − θθ ¯ ′ ψψ ¯ ′, θγ
(A.7a)
ǫµ1 ···µd ǫν1 ···νd =
d! X
p=1
sign(p) δµ1 νp1 · · · δµd νpd
p : permutations ,
q
ǫµ1 ···µd xµ(1)1 · · · xµ(d)d = ± ǫi1 ···id x(1) ·x(i1 ) · · · x(d) ·x(id ) .
B
(A.7b) (A.7c)
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 − 14 (θ¯a θa )2 )bµ (θ) + ǫµ νλ xν bλ (θ)θ¯a θa +w µ ν (θ)xν + 41 ǫµ νλ w νλ (θ)θ¯a θa + λ(θ)xµ + aµ (θ) .
(B.1)
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 xρ xλ , we get η µρ Daα bλ (θ) + η µλ Daα bρ (θ) − η ρλ Daα bµ (θ) =
−i 21 ǫµ νκ γ κβ α (η νρ Daα bλ (θ)
νλ
ρ
(B.2) ρλ
ν
+ η Daα b (θ) − η Daα b (θ)) . 37
Contracting this with ηρλ gives Daα bµ (θ) = −i 12 ǫµ νκ γ κβ α Daβ bν (θ) ,
(B.3)
while contraction with ηµλ leads Daα bρ (θ) = i 31 ǫρ νκ γ κβ α Daβ bν (θ) .
(B.4)
Daα bµ (θ) = 0 ,
(B.5)
Thus bµ (θ) is constant. Straightforward calculation shows that 2x·b xµ − (x2 − 41 (θ¯a θa )2 )bµ + ǫµ νλ xν bλ θ¯a θa is a solution of the superconformal Killing equation (3.15). Now the linear in x terms become Daα (ǫµνκ vκ (θ) + η µν λ(θ)) = i 12 Daβ (η µν v(θ)·γ − v µ (θ)γ ν − ǫµν κ λ(θ)γ κ )β α ,
(B.6)
where v κ (θ) is the dual form of w µν (θ) v κ = 21 ǫκ µν w µν ,
w µν = ǫµνκ vκ .
(B.7)
vβ α (θ) = v µ (θ)γµ β α .
(B.8)
Contracting eq.(B.6) with ηµν gives Daα λ(θ) = i 13 Daβ vβ α (θ) , Substituting this back into eq.(B.6) leads 0 = 5ǫµνρ Daα vρ (θ) − iη µν Daβ vβ α (θ) + 4iDaβ v µ (θ)γ ν β α − iDaβ v ν (θ)γ µβ α .
(B.9)
Contraction with ǫκ µν shows that v µ (θ) satisfies the superconformal Killing equation (3.15) Daα v κ (θ) = −i 21 ǫκ µν Daβ v µ (θ)γ ν β α .
(B.10)
Eqs.(B.8,B.10) are actually equivalent to eq.(B.6), since from eq.(B.10) successively Daα v κ (θ)γ ρα β = i 12 ǫκρµ Daβ vµ (θ) + 21 η κρDaα vα β (θ) − 12 Daα v ρ (θ)γ κα β , Daα v (κ (θ)γ ρ)α β = 13 η κρ Daα vα β (θ) , [κ
Daα v (θ)γ
ρ]α
β
κρµ
= iǫ
Daβ vµ (θ) ,
Daα v κ (θ)γ ρα β = iǫκρµ Daβ vµ (θ) + 31 η κρDaα vα β (θ) , 38
(B.11)
and the last expression makes eq.(B.9) hold. To solve eq.(B.10) we first note from Daα vβ γ (θ) = 23 δα β Daδ vδ γ (θ) − 31 δ β γ Daδ vδ α (θ) ,
(B.12)
that Dbβ Daα vγ δ (θ) = − 32 δα γ Daω Dbβ vω δ (θ) + 31 δ γ δ Daω Dbβ vω α (θ) = 94 δα γ Dbω Daβ vω δ (θ) − 29 δα γ Dbω Daδ vω β (θ) − 92 δ γ δ Dbω Daβ vω α (θ)
(B.13)
+ 91 δ γ δ Dbω Daα vω β (θ) . Contraction with δγ β gives Dbγ Daα vβ δ (θ) = −Dbγ Daδ vβ α (θ) ,
(B.14)
Dbβ Daα vγ δ (θ) = 32 δα γ Dbω Daβ vω δ (θ) − 13 δ γ δ Dbω Daβ vω α (θ) ,
(B.15)
so that eq.(B.13) becomes
which is in fact equivalent to eq.(B.13). From eq.(B.15) and Dbβ Daα vγ δ (θ) = −Daα Dbβ vγ δ (θ) we get 2δα γ Dbω Daβ vω δ (θ) + 2δβ γ Daω Dbα vω δ (θ) = δ γ δ (Daω Dbα vω β (θ) + Dbω Daβ vω α (θ)) . (B.16) Contracting with δγ α gives 3Dbω Daβ vω δ (θ) = −2Daω Dbβ vω δ (θ) + Daω Dbδ vω β (θ) .
(B.17)
Hence from eq.(B.14) we can put Dbω Daα vω β = Γab (θ)ǫαβ , (B.18) Γab (θ) =
1 D D (v(θ)ǫ−1 )βα 2 bβ aα
= −Γba (θ) ,
so that eq.(B.15) becomes with eq.(A.6) Dbβ Daα vγ δ (θ) = 31 (2δα γ ǫβδ − ǫβα δ γ δ )Γab (θ) . 39
(B.19)
Thus
Dcγ Γab (θ) = 12 Dbβ Daα Dcγ (v(θ)ǫ−1 )βα = 12 Dbγ Γca (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 Dbβ Daγ vγ δ (θ) = ǫβδ Γab . Integrating this gives
(B.21)
Daγ vγ α (θ) = 6i(ta b θ¯bα + ρ¯aα ) ,
(B.22)
tt = −t ,
(B.23)
where 6ita b = Γab so that and the spinor, ρ¯aα , appears as a constant of integration. Now eq.(B.12) becomes with eq.(2.4)
Integrating this gives
Daα vβ γ (θ) = 2i(ta b θ¯b γ µ + ρ¯a γ µ )α γµ β γ .
(B.24)
v µ (θ) = ita b θ¯b γ µ θa + 2i¯ ρa γ µ θa + v µ .
(B.25)
Daα λ(θ) = −2ta b θ¯bα − 2¯ ρaα ,
(B.26)
Dbβ Daα λ(θ) = −i 31 Γab ǫαβ .
(B.27)
Eq.(B.8) becomes so that
However from Dbβ Daα λ(θ) + Daα Dbβ λ(θ) = 0 we note Γab = 0. Hence w µν (θ) = ρ¯a (γ µ γ ν − γ ν γ µ )θa + w µν ,
(B.28a)
λ(θ) = −2¯ ρa θa + λ .
(B.28b)
With these expressions straightforward calculation shows that w µ ν (θ)xν + 14 ǫµ νλ w νλ (θ)θ¯a θa + λ(θ)xµ is a solution of the superconformal Killing equation (3.15). The remaining terms are Daα aµ (θ) = −i 21 ǫµ λρ Daβ aλ (θ)γ ρβ α , 40
(B.29)
the general solution of which we already obtained. From eq.(B.25) aµ (θ) = ita b θ¯b γ µ θa + 2i¯ εa γ µ θa + aµ .
(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 K·P = aµ Pµ + ε¯a Qa + λD + 21 w µν Mµν + bµ Kµ + ρ¯a S a + 12 tab Aab ,
(C.1)
K = (aµ , bµ , εa , ρa , λ, w µν , ta b ) ,
(C.2a)
P = (Pµ , Kµ , Qa , S a , D, Mµν , Aa b ) ,
(C.2b)
for
where we put tab = δ ac tc b and the R-symmetry generators, Aab = Aa c δcb , satisfy the o(N ) condition, A† = At = −A. The superconformal algebra can now be obtained by imposing [K1 ·P, K2 ·P] = −iK3 ·P ,
(C.3)
where K3 is given by eq.(3.44). From this expression, we can read off the following superconformal algebra. • Poincar´e algebra
[Pµ , Pν ] = 0 ,
[Mµν , Pλ ] = i(ηµλ Pν − ηνλ Pµ ) ,
(C.4)
[Mµν , Mλρ ] = i(ηµλ Mνρ − ηµρ Mνλ − ηνλ Mµρ + ηνρ Mµλ ) . • Supersymmetry algebra
¯ bβ } = 2δ a b γ µα β Pµ , {Qaα , Q [Mµν , Qa ] = i 21 γ[µ γν] Qa , [Pµ , Qaα ] = 0 . 41
(C.5)
• Special superconformal algebra [Kµ , Kν ] = 0 ,
[Mµν , Kλ ] = i(ηµλ Kν − ηνλ Kµ ) ,
{S aα , S¯bβ } = 2δ a b γ µα β Kµ , a
[Mµν , S ] =
i 12 γ[µ γν] S a
(C.6)
,
[Kµ , S aα ] = 0 . • Cross terms between (P, Q) and (K, S) [Pµ , Kν ] = 2i(Mµν + ηµν D) , [Pµ , S a ] = −γµ Qa , a
(C.7)
a
[Kµ , Q ] = −γµ S , {Qaα , S¯bβ } = −iδ a b (2δ α β D + (γ [µ γ ν] )α β Mµν ) + 2iδ α β Aa b . • Dilations
[D, Pµ ] = −iPµ ,
[D, Kµ ] = iKµ ,
[D, Qa ] = −i 21 Qa ,
[D, S a ] = i 12 S a ,
(C.8)
[D, D] = [D, Mµν ] = [D, Aa b ] = 0 . • R-symmetry, o(N ) [Aab , Acd ] = i(δac Abd − δad Abc − δbc Aad + δbd Aac ) , [Aab , Qc ] = i(δa c δbd − δb c δad )Qd , [Aab , S c ] = i(δa c δbd − δb c δad )S d , [Aa b , Pµ ] = [Aa b , Kµ ] = [Aa b , Mµν ] = 0 .
42
(C.9)
D
Realization of SO(2, 3) ∼ = Sp(2, R) 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 −γ µ
Γµ =
!
Γ3 =
,
0 i i 0
!
Σ4 =
,
0 i −i 0
!
.
(D.1)
They satisfy with GAB = diag(+1, −1, −1, −1, +1) ΓA ΓB + ΓB ΓA = 2GAB ,
(D.2)
and 0 γ0 γ0 0
!
ΓA
0 γ0 γ0 0
!
0 ǫ ǫ 0
= −ΓA† ,
!
0
ΓA
ǫ−1
ǫ−1 0
!
= ΓAt .
(D.3)
For the supermatrix, M, given in eq.(3.45), we may now express the 4 × 4 part in terms of ΓAB ≡ 14 [ΓA , ΓB ] as ! w + 21 λ ia·γ m≡ = 12 wAB ΓAB , (D.4) ib·γ w − 21 λ where w34 , wµ3 , wµ4 are given by
wµ3 = aµ − bµ ,
w34 = λ ,
wµ4 = aµ + bµ .
(D.5)
ΓAB generates the Lie algebra of SO(2, 3) [ΓAB , ΓCD ] = −GAC ΓBD + GAD ΓBC + GBC ΓAD − GBD ΓAC .
(D.6)
In general, m can be defined as a 4 × 4 matrix subject to two conditions †
bm + m b = 0 , t
cm + m c = 0 ,
b=
0 γ0 γ0 0
c=
0 ǫ ǫ 0
!
!
,
,
(D.7a) (D.7b)
To show SO(2, 3) ∼ = Sp(2, R) we take, without loss of generality, γ 0 = iǫ and ǫ to be real. Now if we define m ˜ = pmp
−1
,
p= 43
1 0 0 ǫ
!
,
(D.8)
then from p−1 = p† = pt ,
pcp−1 =
0 1 −1 0
!
=j,
(D.9)
we note that eq.(D.7a) is equivalent to the sp(2, R) condition m ˜∗ = m ˜,
jm ˜ +m ˜ tj = 0 .
(D.10)
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