Table Of Content

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: [email protected], 1 [email protected] 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 superconformaltransformations 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 coordinate 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 representation 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) generators 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 considering 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 representations 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 primary 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 superspace 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 superspace 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 superdifferences 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 superspace 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 –