A newN= 6 superconformal algebra

Classification of Finite Irreducible Modules over the Lie Conformal Superalgebra CK 6

“…After the reduction of this linear system, we obtained at the end of the file "m2-macaulay" or in file "m2-ecuations.pdf" a simplified list of 192 equations (see Appendix A for the details of this reduction). The 15th and 16th equations of this list are the following 0 = −i * F 1,2 * v 3,4,5,6 + E * v 3,4,5,6 − 5 * v 3,4,5,6 + i * v 1,2,3,4,5,6 (4.184) 0 = E * v 1,2,3,4,5,6 − 3 * v 1,2,3,4,5,6 (4.185) and at the end of this list we have the conditions Therefore, if m 2 = ∂ |I|=6 ξ I ⊗ v I + |I|=4 ξ I ⊗ v I is a singular vector in Ind(F µ ), using equations (4.186), we prove that v 1,2,3,4,5,6 ∈ F µ is annihilated by the Borel subalgebra of so(6) (see (4.4)), and using that F µ is irreducible, we get that v 1,2,3,4,5,6 is a highest weight vector. Now, we shall compute the corresponding weight µ.…”

“…All degenerate E(1, 6)-modules Ind(F ) can be represented by the diagram below (very similar to that for E (5,10) in [10]), with the point (4,0) excluded, where the nodes represent the highest weights of the modules Ind(F ), and arrows represent the morphisms between these modules. Here λ 2 , λ 1 , λ 3 are the fundamental weights of so 6 = A 3 (where λ 1 is attached to the middle node of the Dynkin diagram).…”

“…The definition of Q is the following: (2,3,4,5,6) => z * u 1, v (1,3,4,5,6) => −u 1, v (1,2,4,5,6) => z * u 3, v (1, 2, 3, 5, 6) => −u 3, v (1, 2, 3, 4, 6) => z * u 5, v (1,2,3,4,5) => −u 5, v (1,2,3,4,5,6) => u (1,2,3,4,5,6)}, Q = map(P, R, Qvari); * Input 23-27: We define a new 'wvari', which is the list of variables that will appear in the equations. More precisely, wvari=list{u (I), h k * u (I), e (i, j) * u (I), em (i, j) * u (I), me (i, j) * u (I), mem (i, j) * u (I), E0 * u (I)}, where u (I) is restricted in this case to the set {u 1, u 3, u 5, u (1,3,5)}.…”

“…* Input 22: We define a map Q : R → P , that change of variables (A.16) and change the basis in so(6) using the notation (A.9). The definition of Q is the following: Qvari = {x i => t i, F (1, 2) => −z * h 1, F (3, 4) => −z * h 2, F (5, 6) => −z * h 3, F (2 * i − 1, 2 * j − 1) => (e (i, j) + em (i, j) + me (i, j) + mem (i, j))/4), F (2 * i, 2 * j) => (e (i, j) − em (i, j) + me (i, j) − mem (i, j))/4), F (2 * i − 1, 2 * j) => −z * (e (i, j) − em (i, j) − me (i, j) + mem (i, j))/4), F (2 * i, 2 * j − 1) => −z * (−e (i, j) − em (i, j) + me (i, j) + mem (i, j))/4), E => E0, v i => u i, v (i, j) => u (i, j), v (i, j, k) => u (i, j, k), v (i, j, k, l) => u (i, j, k, l), v (2,3,4,5,6) => z * (omega 1 − domega 1), v (1,2,4,5,6) => z * (omega 2 − domega 2), v (1,2,3,4,6) => z * (omega 3 − domega 3), v (1,3,4,5,6) => (omega 1 + domega 1), v (1,2,3,5,6) => (omega 2 + domega 2), v (1,2,3,4,5) => (omega 3 + domega 3), v (1,2,3,4,5,6) => u (1,2,3,4,5,…”

Singular Dimensions of the¶

“…Since then the study of superconformal algebras has made much progress on both mathematics and physics. On the mathematical side Kac and van de Leuer 23 and Cheng and Kac 6 have classified all possible superconformal algebras and Kac recently has proved that their classification is complete (see footnote in Ref. 22).…”