Correlation Functions in N=3 Superconformal Theory

0.17 MB·English

by  Dmitriy Drichel

Prepared for submission to JHEP
Correlation Functions in N=3 Superconformal Theory
1
1
0 Dmitriy Drichel,a,1 Michael Flohra
2
aInstitute for Theoretical Physics,
n
a Appelstraße 2, 30167 Hannover, Germany
J
E-mail: [e-mail],
1
[e-mail]
3

h] Abstract:Using a superspace representation of the N=3 Neveau-Schwarz super Virasoro
t algebra, we find solutions of N=3 super Ward identities. Global transformations gener-
-
p
ated by the non-abelian supercurrent require not only superfields, but also functions of
e
h Grassmann variables (in particular correlation functions and their linear combinations) to
[
be su(2) representations. As a consequence, the only admissible fields in the theory are

3
isospin singlets and doublets. We show how to compute the generic form of any N=3 n-
v
6 point function and demonstrate a construction of all su(2) representations on the space of
4
N=3 superfunctions.
3
3
6. Keywords: CFT, SCFT, Super Virasoro Algebra, N=3 Superspace, Conformal Super-
0
fields
0
1
:
v ArXiv ePrint: 1006.3346
i
X
r
a
1Corresponding author
Contents
1 Introduction 1
2 Global Transformations in N=3 Superspace 2
3 Superfields in N=3 Theory 3
4 Ward Identities and the Two-Point Functions 5
5 The n-Point Functions 10
6 Conclusion 11
1 Introduction
Superconformal extensions of the Virasoro algebra have been studied in the context of
high-energy physics [1][2], two-dimensional critical systems [3] and as a subject on its own
[4][5][6][7].
The N=1 and N=2 theories remain the most studied and well-understood super Vi-
rasoro theories. Supersymmetric extensions for different N turn out to have surprisingly
distinctive properties. Some of the features of the N=2 theories include existence of sub-
singular vectors [8] and unitarity of all rational theories [9]. Central extensions of the
superconformal algebra do not exist for N>4 [10].
In this paper, the n-point functions in the Neveau-Schwarz sector of N=3 super Vira-
soro theories, as far as they are fixed by global transformations, are studied.
We find that the superfield representation space of the non-abelian supercurrent is
restricted to the lowest two representations. This allows us to calculate arbitrary n-point
correlation functions. In fact, our results indicate that the representation content of N>3
theories with respect to the supercurrent is finite-dimensional as well, which would make
N=2 the most “interesting” super Virasoro theory with the largest representation content.
Calculation of correlation functions in supersymmetric extensions with N>2 involves
non-abelian symmetrygenerators, returningsystemsofsuperdifferentialequationscontain-
ing correlation functions of fields in different states with respect to the Cartan basis of the
supersymmetry current. The super Ward identities for N=3 lead to equations containing
correlation functionsof fieldsinarepresentation twithraisedorlowered eigenvalues |q|≤ t
of the su(2) generator TH. The situation is different in N=2, where q is an eigenvalue of a
0
U(1) supercurrent, the correlation functions are restricted by charge conservation and the
Ward identities return differential equations in only one correlation function. We will refer
to q as isospin.
– 1 –
In section 2 and 3 we fix the notation by introducing N=3 superconformaltransforma-
tions of superfunctions and superprimary fields, respectively. In section 4, we determine
the two-point correlation functions by solving super Ward identities. The general n-point
function is produced by contraction in section 5. We discuss implications of our findings
in the final part of this manuscript.
2 Global Transformations in N=3 Superspace
We consider the two-dimensional N=3 superspace with a basis Z given by a complex coor-
dinate z and three Grassmannian variables θ1, θ2 and θ3. The superconformal condition
ω′ = κ(z,θ )ω for the one-form ω = dz − dθ θ admits a set of classical generators of
i i i i
superconformaltransformations (usinginteger m, half-integer r and factors 1 in hindsight)
P 2
l = −zm z∂ + 1(m+1)θi∂ ,
m z 2 θi
gri = 12zr(cid:0)−21 zθi∂z −z∂θi + r(cid:1)+ 21 θiθj∂θj ,
(2.1)
(cid:0) (cid:0) (cid:1) (cid:1)
ti = 1zm−1 zǫ θj∂ −mθ1θ2θ3∂ ,
m 2 ijk θk θi
ψr = −21zr−21 (cid:0)θ1θ2θ3∂z + 21ǫijkθiθj∂θk (cid:1).
(cid:0) (cid:1)
The (anti-)commutation relations of the global subset of these generators are a represen-
tation of the classical N=3 superconformal algebra. The subset of global transformations
is the set of all generators that has no singularities on the bosonic coordinate of a graded
Riemann sphere [6]. This subset consists of twelve generators of the group OSp(2|3),
{l ,gi ,ti}.
±1,0 ±1 0
2
The operators l are simply an extension of the conformal N=0 group to superspace,
±1,0
gi generatesupertranslationsandgi aregeneratorsofspecialsuperconformaltransforma-
−1 1
2 2
tions. The closed set of operators ti is a manifestation of the so(3) symmetry of fermionic
0
coordinates which leaves z unaffected.
In contrast to continous rotations in Euclidian spaces, transformations generated by
so(N) operators are discrete on fermionic coordinates. The supercurrentis diagonalized by
mapping its modes ti to an su(2) basis,
m
θ+ = 2 iθ1−θ2 , θ− = 2 iθ1+θ2 , θH = iθ3,
(cid:0) (cid:1) (cid:0) (cid:1)
t+ = 2 it1 −t2 , t− = 2 it1 +t2 , tH = −2it3 ,
m m m m m m m m
(cid:0) (cid:1) (cid:0) (cid:1)
g+ = 4 g2 −ig1 , g− = 4 g2+ig1 , gH = 8ig3.
r r r r r r r r
(cid:0) (cid:1) (cid:0) (cid:1)
Because of the discrete nature of their action on nilpotent coordinates, the su(2) gen-
erators t± satisfy t± 3 = 0. As a consequence of this, the highest superfunction repre-
0 0
sentation in N=3 Grassmannian variables is the t = 1 representation. In fact, the only
(cid:0) (cid:1)
– 2 –
admissible superfunction representations are t = 0 and t = 1, which can be seen by consid-
ering coefficients in the expansion of a superfunction with respect to nilpotent varialbles,
{1,θH,θ+,θ−,θHθ+,θ+θ−,θHθ−,θHθ+θ−},
and calculating all elements through the generators of su(2),
tH = θ+∂ −θ−∂ and t± = ±1θ±∂ ∓4θH∂ .
0 θ+ θ− 0 2 θH θ∓
The terms in the expansion provide a basis for constructing t = 0 and t = 1 represen-
tations in superspace,
t = 0, q = 0 : {1,θ+θ−θH},
t = 1, q = 1 : {θ+,θ+θH},
(2.2)
t = 1, q = 0 : {θH,θ+θ−},
t = 1, q = −1 : {θ−,θ−θH}.
We can not draw any conclusions about superfield representations from this result
alone, since superfields are not guaranteed to have a superspace representation. A pri-
mary superfield written using the notation Φ (z,θ+,θ−,θH) should not be mistaken for
h,t,q
a genuine superfunction. It can not, in general, be expanded in Grassmannian variables.
However, correlation functions of superfields are ordinary functions dependent on super-
space variables, and this will allow us to draw conclusions about the representation space
of conformal superfields.
3 Superfields in N=3 Theory
An immediate consequence of the discussion in the previous section is that OSp(2|3) is
the largest subgroup of transformations that annihilates the conformal N=3 vacuum. To
obtain aquantumalgebra of infinitesimaltransformations of superconformalquantumfield
theory, thealgebraofclassical generators iscentrally extended. Consistency withthesuper
– 3 –
Jacobi identity leads to the complete quantum N=3 Neveau-Schwarz algebra
[L ,L ] = (m−n)L + km m2−1 δ , [L ,T±,H] = −nT±,H,
m n m+n 4 m+n,0 m n m+n
{GH,GH} = −32L −16k r2−(cid:0)1 δ (cid:1), [L ,G±,H] = m −r G±,H,
r s r+s 4 r+s,0 m r 2 r+m
(cid:0) (cid:1) (cid:0) (cid:1)
{G+,G−} = 16L +8k r2− 1 δ [L ,ψ ] = − m +s ψ ,
r s r+s 4 r+s,0 m s 2 m+s
(cid:0) (cid:1) (cid:0) (cid:1)
+8(r−s)TH , [T∓,G±] = −GH ±8mψ ,
r+s m r r+m r+m
{G±,GH} = 8(r−s)T± , [T±,GH] = −2G± ,
r s r+s m r m+r
[T+,T−] = 2TH +2kmδ , [TH,G±] = ±G± ,
m n m+n m+n,0 m r r+m
[TH,TH] = kmδ , [TH,GH] = −2TH ,
n m m+n,0 m r r+s
[TH,T±] = ±T± , {ψ ,G±} = ∓T± ,
m n m+n s r r+s
[ψ ,ψ ] = −kδ , {ψ ,GH} = −2TH ,
r s 4 r+s,0 s r r+s
with all other (anti-)commutator relations vanishing.
The Cartan subalgebra is two-dimensional. A highest-weight state is simultaneously
diagonalizable in its conformal weight and isospin,
L |h,qi = h|h,qi, TH|h,qi = q|h,qi. (3.1)
0 0
We further expect the highest-weight states to satisfy
L |h,qi = T±,H|h,qi = G±,H|h,qi = ψ |h,qi = 0, n, r > 0.
n n ±r ±r
Theactionofgenerators oftransformationscanbegiven byintroducingalgebraicsu(2)
operators J±, JH which account for the transformation of index q carried by a primary
field
[JH,J±] = ±J±, [J+,J−] = 2JH. (3.2)
Theseoperatorsaredefinedtohavefollowingrelationswiththeunderlyingso(3)generators
Ji
J+ = 2(iJ1 −J2), J− = 2(iJ1 +J2), JH = −2iJ3. (3.3)
We choose the convenient normalization
JHΦ (Z) = qΦ (Z),
h,j,q h,t,q
(3.4)
J±Φ (Z) = Φ (Z).
h,j,q h,t,q±1
Thehighest-weight conditions on representation spaces of L and J have been treated
0 0
on the same footing in the N=2 theory, where q is a U(1)-isospin [13]. In contrast to N=2,
– 4 –
the isospin representation in N=3 theory is finite-dimensional, since q can take only integer
or half-integer values dueto (3.2). For every highest-weight state |h,t,qi, there is a lowest-
weight state |h,t,−qi. We therefore reserve the term “primary” for fields that satisfy (3.1)
without imposing conditions T±|h,±qi = 0. Then the infinitesimal transformations of
0
primary fields read
[L ,Φ(Z)] = L Φ(Z) = ∂ Φ(Z), (3.5)
−1 −1 z
1
[L ,Φ(Z)] = L Φ(Z) = h+z∂ + θ+∂ +θ−∂ +θH∂ Φ(Z), (3.6)
0 0 z θ+ θ− θH
2
(cid:16) (cid:17)
[L ,Φ(Z)] = L Φ(Z) = 2hz +z z∂ (cid:0)+θ+∂ +θ−∂ +θH∂(cid:1)
1 1 z θ+ θ− θH
(cid:16)+ 1θ+θ−(cid:0)JH + 1θ+θHJ−− 1θ−θHJ+ Φ(cid:1) (Z), (3.7)
8 4 4
[GH ,Φ(Z)] = GH Φ(Z) = −4 θH∂ +∂ Φ(Z), (cid:17) (3.8)
−1 −1 z θH
2 2
[GH,Φ(Z)] = GHΦ(Z) = −(cid:0)8hθH −4θH(cid:1)z∂ −4z∂ −4θHθ−∂
1 1 z θH θ−
2 2
(cid:16)
−4θHθ+∂ +θ−J+−θ+J− Φ(Z), (3.9)
θ+
[G± ,Φ(Z)] = G± Φ(Z) = ± θ±∂ +8∂ Φ(Z), (cid:17) (3.10)
−1 −1 z θ∓
2 2
[G±,Φ(Z)] = G±Φ(Z) = ±(cid:0) 2hθ±±θ±z(cid:1)∂ ±8z∂ +θ+θ−∂
1 1 z θ∓ θ∓
2 2
(cid:16)
+2θHJ± +θ±JH Φ(Z), (3.11)
[TH,Φ(Z)] = THΦ(Z) = θ−∂ −θ+∂ +J(cid:17)H Φ(Z), (3.12)
0 0 θ− θ+
1
[T±,Φ(Z)] = T±Φ(Z) = (cid:0) ∓ θ±∂ ±4θH∂ (cid:1)+J± Φ(Z). (3.13)
0 0 2 θH θ∓
(cid:16) (cid:17)
These transformations are a representation of the superconformal algebra in N=3 super-
space and can be used in the form of Ward identities to impose conditions on n-point
superfunctions. Due to nilpotency of θ±,H and the su(2) nature of isospin, the space of
solutions turns out to be surprisingly small.
4 Ward Identities and the Two-Point Functions
We will use upper indices in parenthesis on superdifferential operators in (2.1) and (3.5)-
(3.13)forindicatingsuperspacepositions{z ,θ+,θ−,θH}onwhichtheoperator acts. Then
i i i i
the super Ward identities can be written as sums of one of the differential operators at
different superpoints acting on an n-point function F ,
n
n
L(i) F (Z ,...,Z ) = L(1,...,n)F (Z ,...,Z ) = 0,
±1,0 n 1 n ±1,0 n 1 n
i
X
n
G±,H(i)F (Z ,...,Z ) = G±,H(1,...,n)F (Z ,...,Z ) = 0,
±1 n 1 n ±1 n 1 n
i 2 2
X
n
T±,H(i)F (Z ,...,Z ) = T±,H(1,..,n)F (Z ,...,Z )= 0. (4.1)
0 n 1 n 0 n 1 n
i
X
– 5 –
The Ward identity from the generator of translations L imposes dependence on differ-
−1
ences z −z . Supertranslation operators G±,H further impose dependence on superdiffer-
i j −1
ences 2
1
Z = z −z + θ−θ++θ+θ− +θHθH, (4.2)
ij i j 8 i j i j i j
θ±,H = θ±,H −θ±,H(cid:16). (cid:17) (4.3)
ij i j
Writing down other Ward identities explicitly and transforming them to new coordi-
natescanbecometediousforhighvaluesofn. Fortunately, thisturnsouttobeunnecessary
by using the product ansatz
hΦ (Z )...Φ (Z )i= Υ (Z ,...,Z )Ω (Z ,...Z ). (4.4)
h1,t1,q1 1 hn,tn,qn n h1,...,hn 12 n−1,n t1,q1;...;tnqn 1 n
While Υ is a function of (4.2) and satisfies the action of the conformal group (3.5)-
h1,...,hn
(3.7), Ω depends on both (4.2) and (4.3). For correlation functions of n fields,
t1,q1;...;tnqn
the parameterization of n(n−1) ×4 superdifferences is chosen such that i < j. Using (3.7),
2
the solution to Υ can be found to be
h1,...,hn
n= 2 : Z−2h1δ
12 h1,h2
Υ (Z ,...,Z ) =  (4.5)
h1,...,hn 12 n−1,n n> 2 : F (x1,...,xn−3) Zi−j∆ij
i<j
Y

∆ =  h , ∆ = 2h .
i,j  i ij i
i,j;i<j i j;i<j
X X X
This solution is the same as in N=0,1,2 theories, up to a redefinition of Z
ij
Z = z −z for N=0,
ij i j
Z = z −z −θ θ for N=1,
ij i j i j
Z = z −z −θ+θ−−θ−θ+ for N=2.
ij i j i j i j
The function Υ (Z ,...,Z ) is expected to remain valid in the ansatz (4.4) for
h1,...,hn 12 n−1,n
N=4 super algebra generated by OSp(2|4) for supertranslationally invariant superdiffer-
ences Z . The function F(x ,...,x ) depends on n −3 independent cross-ratios. Ad-
ij 1 n−3
ditional constraints on the function F can be obtained from singular vectors in Verma
modules. In case of logarithmic dependence [14][15] the representations in the correlation
function are indecomposable with respect to L . One can compute correlation functions of
0
indecomposable representations from (4.5) using the “derivation trick” [12][16].
In N=1 theory, the function Ω is given by 1 for the two-point function and by increas-
ingly longer expressions with growing n [11].
– 6 –
In the case of the N=2 n-point function, Ω is given by the simple, closed form [13]
n θ+θ−
ij ij
Ω = exp Aij Zij δPni=1qi,0
i<j
X
 
n
A = −q , A = −A ,
ij i ij ji
j=1,i6=j
X
where q is the eigenvalue to the U(1)-isospin operator.
i
In N=3 theory, the form of Ω depends on the value of the isospin. In the following, we
will derive Ω .
t1,q1;...;tnqn
The Ward identities arising from the action of T±,H on Ω can be rewritten
0 t1,q1;...;tnqn
using classical operators t±,H (2.1). The n-point function, up to normalization, is an
0
±H(1...n)
eigenstate of su(2) operators t
0
n
tH(1...n)Ω = θ+∂ −θ−∂ Ω = q Ω
0 q1,...,qn  ij θi+j ij θi−j  q1,...,qn i! t1,q1;...;tn,qn
Xi<j (cid:16) (cid:17) Xi=1
 
t±(1...n)Ω = ±1θ±∂ ∓4θH∂ Ω (4.6)
0 q1,...,qn  2 ij θiHj ij θi∓j  q1,...,qn
i<j (cid:18) (cid:19)
X
 
= Ω +...+Ω .
t1,q1±1;...;tn,qn t1,q1;...;tn,qn±1
Considering the special case of the two-point function, we see that the possible super-
space components of the two-point function are just (2.2) with an implied position index
on all Grassmann variables. It will be seen that L does not allow terms with an uneven
1
number of Grassmann variables. We immediately conclude
Ω = 1.
0,0;0,0
For q = t , q =t , t +t 6= 0, (4.6) returns t +t = 1. Thereare two solutions, t =
1 1 2 2 1 2 1 2 1
t = 1 andt = 1, t = 0. Thelatter ansatz does notsatisfy theremainingWard identities.
2 2 1 2
We are left with the result that the t = 1 superfunction representation of OSp(2|3) is just
Ω . All representations with t 6= 0, t 6= 1 decouple from the theory since they do
1,1;1,1 2
2 2 2 2
not correlate with any other field, including themselves. This result is consistent with [4],
where states h = 0, q = 0 and h = 1, q = 1 were identified in the c = 3 theory.
4 2 2
Since there are only t = 0 and t = 1 primary fields in the N=3 theory, we use the
2
short-hand notation
Ω ≡ Ω , Ω ≡ Ω , Ω ≡ Ω .
0,0;0,0 00 1,1;1,1 ++ 1,−1;1,−1 −−
2 2 2 2 2 2 2 2
Ω ≡ Ω Ω ≡ Ω .
1,1;1,−1 +− 1,−1;1,1 −+
2 2 2 2 2 2 2 2
Toobtain furtherconstraints, oneneedstousetheremainingfiveWard identities. The
action of L on the two-point function is
0
1
L(012)F2 = h1+h2+Z12∂Z12 + 2 θ1+2∂θ1+2 +θ1−2∂θ1−2 +θ1H2∂θ1H2 F2 = 0. (4.7)
(cid:18) (cid:19)
(cid:16) (cid:17)
– 7 –
Besides imposing similar conformal conditions as in the N=0 theory, it scales all terms
containing nilpotent variables to the right dimension. Note that Ω is dimensionless.
The map from superspace points θ , z to superdifferences θ±,H, Z is a projection.
i i ij ij
Because of that, we have to introduce new combinations of superspace coordinates ξ±,H,
ij
W (“supersums”) to express down Ward identities for L and G±,H,
ij 1 1
2
ξ±,H = θ±,H +θ±,H,
ij i j
1
W = z +z + θ−θ++θ+θ− +θHθH.
ij i j 8 i j i j i j
(cid:16) (cid:17)
The remaining four Ward identities are the most difficult to compute and understand intu-
itively. They arise from generators of special and special superconformal transformations
which mix bosonic and fermionic coordinates in a non-trivial way. The most computation-
ally difficult terms are of the form Z∂ . The desired Ward identities are
θ
L(1)+L(2) F = h W +Z − 1 θ−ξ+ +θ+ξ− −θHξH +h (W −Z )
1 1 2 1 12 12 8 12 12 12 12 12 12 2 12 12
(cid:16) (cid:17) (cid:16)+ 1(cid:0) ξ+ +θ+ ξH(cid:0)+θH J−+ 1(cid:1) ξ+ −θ+(cid:1) ξH −θH J−
16 12 12 12 12 1 16 12 12 12 12 2
− 1 (cid:0)ξ− +θ−(cid:1)(cid:0)ξH +θH(cid:1)J+− 1 (cid:0)ξ− −θ−(cid:1)(cid:0)ξH −θH(cid:1)J+
16 12 12 12 12 1 16 12 12 12 12 2
+ 1 (cid:0)ξ+ +θ+(cid:1)(cid:0)ξ− +θ−(cid:1)JH + 1 (cid:0)ξ+ −θ+(cid:1)(cid:0)ξ− −θ−(cid:1)JH
32 12 12 12 12 1 32 12 12 12 12 2
+ W(cid:0)12Z12− 116(cid:1)Z(cid:0)12 θ1−2ξ1+2(cid:1)+θ1+2ξ1−2 (cid:0)− 12Z12θ1H2(cid:1)ξ(cid:0)1H2 ∂Z12 (cid:1) (4.8)
+(cid:0)1 Z ξ+ +W θ+(cid:0) + 1 θ+ξ+ ξ−(cid:1)+θ− (cid:1)
2 12 12 12 12 16 12 12 12 12
− 1(cid:0)θHξH ξ+ +θ+ ∂ + 1 Z(cid:0) ξ− +W(cid:1) θ−
2 12 12 12 12 θ1+2 2 12 12 12 12
+ 1 θ−ξ−(cid:0) ξ+ +θ+(cid:1)(cid:1)− 1θHξH(cid:0) ξ− +θ− ∂
16 12 12 12 12 2 12 12 12 12 θ−
12
+ 12 Z12ξ1H2(cid:0)+W12θ1H2(cid:1)− 116 θ1−2ξ1+(cid:0)2+θ1+2ξ1−2(cid:1)(cid:1)ξ1H2+θ1H2 ∂θ1H2 F2 =0,
(cid:0) (cid:0) (cid:1)(cid:0) (cid:1)(cid:1) (cid:17)
GH(1)+GH(2) F = −4 (h +h )ξH +(h −h )θH
1 1 2 1 2 12 1 2 12
2 2
(cid:16) (cid:17) (cid:16)− 1 ξ(cid:0)+ +θ+ J−− 1 ξ+ −θ+ J(cid:1)−
2 12 12 1 2 12 12 2
+ 1(cid:0)ξ− +θ−(cid:1)J++ 1(cid:0)ξ− −θ−(cid:1)J+ (4.9)
2 12 12 1 2 12 12 2
−4Z(cid:0)12ξ1H2∂Z12(cid:1)−2 ξ1H2θ(cid:0)1−2+θ1H2ξ1−2(cid:1) ∂θ1−2 −2 ξ1H2θ1+2+θ1H2ξ1+2 ∂θ1+2
−4 Z12− 116 θ1−2ξ(cid:0)1+2+θ1+2ξ1−2 − 12(cid:1)θ1H2ξ1H2 ∂θ(cid:0)1H2 F2 =0, (cid:1)
(cid:0) (cid:0) (cid:1) (cid:1) (cid:17)
– 8 –
G±(1)+G±(2) F = ±h ξ± +θ± ±h ξ± −θ±
1 1 2 1 12 12 2 12 12
2 2
(cid:16) (cid:17) (cid:16)+ ξH(cid:0)+θH J±(cid:1)+ ξH(cid:0)−θH J±(cid:1)
12 12 1 12 12 2
+(cid:0)21 ξ1±2+θ1±(cid:1)2 J1H +(cid:0) 12 ξ1±2−(cid:1)θ1±2 J2H ±ξ1±2Z12∂Z12 (4.10)
+ ±(cid:0)8Z ∓θ∓(cid:1)ξ± −4θ(cid:0)HξH ∂ (cid:1)
12 12 12 12 12 θ∓
12
±(cid:0)21 θ1±2ξ1H2+ξ1±2θ1H2 ∂θ1H2 F2(cid:1)=0.
Since G±,H is generated by L and G(cid:0)±,H, we do no(cid:1)t hav(cid:17)e to account for their action if the
1 1 −1
2 2
remaining Ward identities are satisfied.1
It is now easy to determine all two-point functions. Using (4.8) we find that the terms
withanunevennumberofGrassmannvariablesarenotcontainedinthetwo-pointfunction.
We are left with
θ+θH θ+θ− θ−θH
Ω ∼ 12 12, Ω +Ω ∼ 12 12 and Ω ∼ 12 12.
++ Z +− −+ Z −− Z
12 12 12
Using equations (3.9), (3.11) we obtain conditions for mixed correlation functions
θ−θ+
Ω +Ω = 12 12 and Ω −Ω = 8.
−+ +− 2Z −+ +−
12
Then the complete set of two-point correlation functions, up to a constant, is given by
θ+θH θ−θ+
Ω = ij 12, Ω = 12 12 +4,
++ Z −+ 4Z
Ω = 1, 12 12 (4.11)
00 θ−θH θ−θ+
Ω = 12 12, Ω = 12 12 −4.
−− Z +− 4Z
12 12
The solutions Ω and Ω have been found in [6]. Determining the normalization, we
00 ++
find that the j = 1 superfunction eigenstates
tH(12)Ω = qΩ ,
1,q 1,q
(4.12)
t±(12)Ω = Ω
1,q 1,q±1
1WeshouldnotethattherealsoexistsaquadraticCasimir operatorTC ofthetheorycontainingdouble
0
derivatives with respect to nilpotent variables. This operator turns out to be of no particular use for
determining correlation functions. Weshow howit worksfor correlation functions ofprimary fieldsforthe
simple case T±|h,±qi=0 (equivalently,q=±t). We compute
0
2
TC =T∓T±+ TH ±TH
0 0 0 0 0
(cid:16) (cid:17)
through the two-point function. Since the first term disappears and the second one returns an already
known identity,we obtain the superdifferential equation
TH(1) 2+2TH(1)TH(2)+ TH(2) 2 F =
(cid:18)(cid:16) 0 (cid:17) 0 0 (cid:16) 0 (cid:17) (cid:19) 2
(cid:16)−2θ1+2θ1−2∂θ1−2∂θ1+2 +(1−2(q1+q2))θ1+2∂θ1+2 +(1+2(q1+q2))θ1−2∂θ1−2 +(q1+q2)2(cid:17)F2 =0.
– 9 –