]]> 2001 American Mathematical Society Transactions of the American Mathematical Society Trans. Amer. Math. Soc. tran 354 01 20010821 April 18, 2000 February 21, 2001 August 21, 2001 363 363-385 23 S0002-9947-01-02869-0 10.1090/S0002-9947-01-02869-0 0002-9947 1088-6850 English 17B69 17B68 17B65 81R10 81T40 81T60 1991 http://www.ams.org/jourcgi/jour-getitem?pii=S0002-9947-01-02869-0 section -1 Introduction It has been known that the N 2 Neveu-Schwarz superalgebra is one of the most important algebraic objects realized in superstring theory. The N 2 superconformal field theories constructed from its discrete unitary representations of central charge c 3 are among the so-called minimal models.'' In the physics literature, there have been many conjectural connections among Calabi-Yau manifolds, Landau-Ginzburg models and these N 2 unitary minimal models. In fact, the physical construction of mirror manifolds GP used the conjectured relations G1 G2 between certain particular Calabi-Yau manifolds and certain N 2 superconformal field theories (Gepner models) constructed from unitary minimal models (see Gr for a survey). To establish these conjectures as mathematical theorems, it is necessary to construct the N 2 unitary minimal models mathematically and to study their structures in detail. In the present paper, we apply the theory of intertwining operator algebras developed by the first author in H25 , H4 and H6 and the tensor product theory for modules for a vertex operator algebra developed by Lepowsky and the first author in HL1 -- HL5 , HL6 and H1 to construct the intertwining operator algebras and vertex tensor categories for N 2 superconformal unitary minimal models. The main work in this paper is to verify that the conditions to use the general theories are satisfied for these models. The main technique used is the representation theory of the N 2 Neveu-Schwarz algebra, which has been studied by many physicists and mathematicians, especially by Eholzer and Gaberdiel EG , Feigin, Semikhatov, Sirota and Tipunin FSST FST FS , and Adamovic A2 . The present paper is organized as follows: In Section 1, we recall the notion of N 2 superconformal vertex operator superalgebra. In Section 2, we recall and prove some basic results on representations of unitary minimal N 2 superconformal vertex operator superalgebras and on representations of N 2 superconformal vertex operator superalgebras in a much more general class. Section 3 is devoted to the proof of the convergence and extension properties for products of intertwining operators for unitary minimal N 2 superconformal vertex operator superalgebras and for vertex operator superalgebras in the general class. Our main results on the intertwining operator superalgebra structure and vertex tensor category structure are given in Section 4. N 2 superconformal vertex operator superalgebras In this section we recall the notion of N 2 superconformal vertex operator algebra and basic properties of such an algebra. These algebras have been studied extensively by physicists. The following precise version of the definition is from HZ : defn An N 2 superconformal vertex operator superalgebra is a vertex operator superalgebra (V, Y, 1, ) together with odd elements , - and an even element satisfying the following axioms: Let eqnarray Y( , x) r 12 G (r)x -r-3 2 , Y( - , x) r 12 G - (r)x -r-3 2 , Y(, x) n J(n)x -n-1 . eqnarray Then V is a direct sum of eigenspaces of J(0) with integral eigenvalues which modulo 2 give the 2 grading for the vertex operator superalgebra structure, and the following N 2 Neveu-Schwarz relations hold: For m, n , r,s 12 eqnarray L(m), L(n) (m-n)L(m n) 12 (m 3-m) m n, 0 , J(m), J(n) 3m m n, 0 , L(m), J(n) -nJ m n , L(m), G (r) (2-r)G (m r), J(m), G (r) G (m r), G (r), G - (s) 2L(r s) (r-s)J(r s) 3(r 2-14) r s, 0 , G (r), G (s) 0 eqnarray where L(m) , m , are the Virasoro operators on V and c is the central charge of V . Modules and intertwining operators for an N 2 superconformal vertex operator superalgebra are modules and intertwining operators for the underlying vertex operator superalgebra. defn Note that this definition is slightly different from the one in ; in , V is not required to be a direct sum of eigenspaces of J(0) . The N 2 superconformal vertex operator superalgebra defined above is denoted by (V, Y, 1, , - , ) (without since L(-2)1 12 G (-32), G - (-12) 1 12J(-2)1) or simply by V . Note that a module W for a vertex operator superalgebra (in particular the algebra itself) has a 2 -grading called sign in addition to the -grading by weights. We shall always use W 0 and W 1 to denote the even and odd subspaces of W . If W is irreducible, there exists h such that W W 0W 1 where W 0 nh W (n) and W 1 nh 1 2 W (n) . We shall always use the notation to denote the map from the union of the even and odd subspaces of a vertex operator superalgebra or of a module for such an algebra to 2 by taking the signs of elements in the union. The notion of N 2 superconformal vertex operator superalgebra can be reformulated using odd formal variables. (In the N 1 case, this reformulation was given by Barron B1 B2 .) We need some spaces which play the role of spaces of algebraic functions and fields on certain superspaces.'' For l symbols 1, , l , consider the exterior algebra of the vector space over spanned by these symbols and denote this exterior algebra by 1, , l . For any vector space E , we also have the vector space eqnarray E 1, , l , E x 1, , x k 1, , l , E x 1, x 1 -1 , , x k, x k -1 1, , l , E x 1, , x k 1, , l , E x 1, x 1 -1 , , x k, x k -1 1, , l , E x 1, , x k 1, , l eqnarray and E((x 1, , x k)) 1, , l . If E is a 2 -graded vector space, then there are natural structures of modules over the ring x 1, , x k 1, , l on these spaces. The ring x 1, , x k 1, , l can be viewed as the space of algebraic functions on a superspace'' consisting of elements of the form (x 1, , x k; 1, , l) where x 1, , x k are commuting coordinates and 1, , l are anticommuting coordinates. The other spaces can be viewed as spaces of suitable fields over this superspace.'' Let (V, Y, 1, , - , ) be an N 2 superconformal vertex operator superalgebra. Let eqnarray 1 1 2 ( - ), 2 1 -2 ( - - ) eqnarray and eqnarray Y( 1, x) r 12 G 1(r)x -r-3 2 , Y( 2, x) r 12 G 2(r)x -r-3 2 . eqnarray We define the vertex operator map with odd variables eqnarray Y: VV V((x)) 1, 2 uv Y(u, (x, 1, 2)) v eqnarray by eqnarray Y(u, (x, 1, 2))v Y(u, x)v 1 Y(G 1(-1 2)u, x)v 2 Y(G 2(-1 2)u, x)v 1 2Y(G 1(-1 2) G 2(-1 2)u, x)v eqnarray for u, vV . (We use the same notation Y to denote the vertex operator map and the vertex operator map with odd variables.) In particular, we have eqnarray Y(,(x, 1, 2)) Y(,x) - -1 1Y( 2,x) - -1 2 Y( 1,x)-2 -1 1 2 Y(,x). eqnarray Also, if we introduce eqnarray - 1 -1 2 2 , - 1 -1 2 2 , eqnarray then we can write eqnarray Y(,(x, 1, 2)) Y(,x) Y( ,x) -Y( -,x) 2 -Y(,x). eqnarray We have prop odd-svoa The vertex operator map with odd variables satisfies the following properties: enumerate The vacuum property : Y(1, (x, 1, 2)) 1 where 1 on the right-hand side is the identity map on V . The creation property : For any vV , Y(v, (x, 1, 2))1 V x 1, 2 , (x, 1, 2) (0, 0, 0) Y(v, (x, 1, 2))1 v. The Jacobi identity : In ( End V) x 0, x 0 -1 , x 1, x 1 -1 , x 2, x 2 -1 1, 2, 1, 2 , we have eqnarray x 0 -1 ( x 1-x 2- 1 1- 2 2 x 0 ) Y(u,(x 1, 1, 2)) Y(v,(x 2, 1, 2)) -(-1) u v x 0 -1 ( x 2-x 1 1 1 2 2 -x 0 ) Y(v,(x 2, 1, 2)) Y(u,(x 1, 1, 2)) x 2 -1 ( x 1-x 0- 1 1- 2 2 x 2 ) Y(Y(u,(x 0, 1- 1, 2- 2))v,(x 2, 1, 2)), eqnarray for u, vV which are either even or odd. The G i(-1 2) -derivative property : For any vV , i 1, 2 , Y(G i(-1 2) v, (x, 1, 2)) ( i i x )Y( v, (x, 1, 2)). The L(-1) -derivative property : For any vV , Y(L(-1) v, (x, 1, 2)) x Y(v, (x, 1, 2)). The skew-symmetry : For any u, vV which are either even or odd, eqnarray Y(u,(x, 1, 2))v (-1) u v e xL(-1) 1 G 1(-1 2) 2 G 2(-1 2) Y(v,(-x, - 1,- 2))u. 1em eqnarray enumerate prop The proof of this result is straightforward and we omit it. We can also reformulate the data and axioms for modules and intertwining operators for an N 2 superconformal vertex operator superalgebra using odd variables as in the N 1 case in B2 and HM . Here we give the details of the corresponding reformulation of the data and axioms for intertwining operators. Let W 1 , W 2 and W 3 be modules for an N 2 superconformal vertex operator superalgebra V and an intertwining operator of type W 3 W 1W 2 . We define the corresponding intertwining operator map with odd variables eqnarray : W 1W 2 W 3 x 1, 2 w (1) w (2) (w (1) , (x, 1, 2)) w (2) eqnarray by eqnarray (w (1) , (x, 1, 2)) w (2) (w (1) , x) w (2) 1(G 1(-1 2)w (1) , x) w (2) 2(G 2(-1 2)w (1) , x) w (2) 1 2(G 1(-1 2)G 2(-1 2)w (1) , x) w (2) eqnarray for u, vV . Then we have prop The intertwining operator map with odd variables satisfies the following properties: enumerate The Jacobi identity : In Hom (W 1W 2, W 3) x 0, x 1, x 2 1, 2, 1, 2 , we have eqnarray x 0 -1 ( x 1-x 2- 1 1- 2 2 x 0 ) Y(u,(x 1, 1, 2)) (w (1) ,(x 2, 1, 2)) -(-1) u w (1) x 0 -1 ( x 2-x 1 1 1 2 2 -x 0 ) (w (1) ,(x 2, 1, 2)) Y(u,(x 1, 1, 2)) x 2 -1 ( x 1-x 0- 1 1- 2 2 x 2 ) (Y(u,(x 0, 1- 1, 2- 2))w (1) , (x 2, 1, 2)), eqnarray -1pc for uV and w (1) W 1 which are either even or odd. The G i(-1 2) -derivative property : For any vV , i 1, 2 , (G i(-1 2) v, (x, 1, 2)) ( i i x )( v, (x, 1, 2)). The L(-1) -derivative property : For any vV , (L(-1) w (1) , (x, 1, 2)) x (w (1) , (x, 1, 2)). The skew-symmetry : There is a linear isomorphism : W 3 W 1 W 2 W 3 W 2 W 1 such that eqnarray ( Y )(w (1) , (x, 1, 2))w (2) (-1) w (1) w (2) e xL(-1) 1 G 1(-1 2) 2 G 2(-1 2) 4em Y (w (2) , (e -i x,- 1,- 2))w (1) eqnarray for w (1) W 1 and w (2) W 2 which are either even or odd. enumerate prop The proof of this result is similar to the proof of Proposition odd-svoa and is omitted. Unitary minimal N 2 superconformal vertex 1 operator superalgebras In this section, we recall the constructions and results on unitary minimal N 2 superconformal vertex operator superalgebras and their representations. New results needed in later sections are also proved. We then introduce in this section a class of N 2 superconformal vertex operator superalgebras and generalize most of the results for unitary minimal N 2 superconformal vertex operator superalgebras to algebras in this class. The N 2 Neveu-Schwarz Lie superalgebra is the vector space n n r 12 r r 12 - r n n equipped with the following N 2 Neveu-Schwarz relations: eqnarray L m, L n (m-n)L m n 12 (m 3-m) m n, 0 , J m, J n 3m m n, 0 , L m, J n -nJ m n , L m, G r (2-r)G m r , J m, G r G m r , G r, G - s 2L r s (r-s)J r s 3(r 2-14) r s, 0 , G r, G s 0 eqnarray for m, n , r, s 12 . For simplicity, we shall simply denote the N 2 Neveu-Schwarz Lie superalgebra by (2) in this paper. We now construct certain representations of the N 2 Neveu-Schwarz Lie superalgebra. Consider the two subalgebras eqnarray (2) n 0 n r 0 r r 0 - r n 0 n, - (2) n 0 n r 0 r r 0 - r n 0 n eqnarray of (2) . Let U() be the functor from the category of Lie superalgebras to the category of associative algebras obtained by taking the universal enveloping algebras of Lie superalgebras. For any representation of (2) , we shall use L(n) , n , G (r) , r 12 , and J(n) , n , to denote the representation images of L n , G r and J n . For any c, h, q , the Verma module M ns (2) (c, h, q) for (2) is a free U( - (2)) -module generated by 1 c, h, q such that eqnarray (2)1 c, h, q 0, L(0)1 c, h, q h1 c, h, q , C1 c, h, q c1 c, h, q , J(0)1 c, h, q q1 c, h, q . eqnarray There exists a unique maximal proper submodule J ns (2) (c, h, q) of the Verma module M ns (2) (c, h, q) . It is easy to see that when c0 , 1 c, 0, 0 , G (-3 2)1 c, 0, 0 and L(-2)1 c, 0, 0 are not in J (2) (c, 0, 0) . Let L ns (2) (c, h, q) M ns (2) (c, h, q) J ns (2) (c, h, q) and V ns (2) (c, 0, 0) M ns (2) (c, 0, 0) G (-1 2)1 c, 0, 0 , G - (-1 2)1 c, 0, 0 where G (-1 2)1 c, 0, 0 , G - (-1 2)1 c, 0, 0 is the submodule of M ns (2) (c, 0, 0) generated by G (-1 2) 1 c, 0, 0 . Then L ns (2) (c, 0, 0) and V ns (2) (c, 0, 0) have the structures of vertex operator superalgebras with the vacuum 1 c, 0, 0 , the Neveu-Schwarz elements G (-3 2)1 c, 0, 0 and the Virasoro element L(-2)1 c, 0, 0 (see ). In EG , Eholzer and Gaberdiel showed, among other things, that among vertex operator superalgebras of the form L ns (2) (c, 0, 0) , the only ones having finitely many irreducible modules are the unitary'' ones L ns (2) (c m, 0, 0) for nonnegative integers m , where c m 3m m 2 . The following result was proved by Adamovic in and A2 using the results obtained Adamovic and Milas AM , Feigin, Semikhatov and Tipunin FST and Doerrzapf : thm ad The vertex operator superalgebra L ns (2) (c, 0, 0) has only finitely many irreducible modules (up to isomorphisms) and every module for L ns (2) (c, 0, 0) is completely reducible if and only if c c m 3m m 2 where m is a nonnegative integer. A set of representatives of the equivalence classes of irreducible modules for L ns (2) (c m, 0, 0) is L ns (2) (c m, h m j,k , q m j,k ) j, k 12 , ;0j, k, j k m where 12 12, 32, 52, and eqnarray h m j, k jk-14 m 2 ,q m j, k j-k m 2 . eqnarray thm For any m0 , we call the vertex operator algebra L ns (2) (c m, 0, 0) a unitary minimal N 2 superconformal vertex operator superalgebra . prop fusion Let j i, k i 12 , i 1, 2, 3 , satisfying 0j i, k i, j i k i m and Y an intertwining operator of type equation type L (2) (c m, h m j 3, k 3 , q m j 3, k 3 ) L (2) (c m, h m j 1, k 1 , q m j 1, k 1 ) L (2) (c m, h m j 2, k 2 , q m j 2, k 2 ) . equation Then we have: enumerate 2.2-1 For any w (1) L (2) (c m, h m j 1, k 1 , q m j 1, k 1 ) and w (2) L (2) (c m, h m j 2, k 2 , q m j 2, k 2 ), (w (1) , x)w (2) x h m j 3, k 3 -h m j 1, k 1 -h m j 2, k 2 L (2) (c m, h m j 3, k 3 , q m j 3, k 3 )((x 1 2 )). 2.2-2 Let h m j 3, k 3 -h m j 1, k 1 - h m j 2, k 2 and w (i) 1 c m, h m j i, k i , q m j i, k i , i 1, 2, 3 , the lowest weight vectors in L (2) (c m, h m j i, k i , q m j i, k i ) . Then the map is uniquely determined by the maps eqnarray (w (1) ) --1 , (G (-1 2)w (1) ) --1 2 , (G - (-1 2)w (1) ) --1 2 , (G (-1 2)G - (-1 2)w (1) ) - eqnarray from the 1-dimensional subspace of W 2 spanned by w (2) to the 1-dimensional subspace of W 3 spanned by w (3) . That is, if these maps are 0 , then 0 . 2.2-3 If q m j 3, k 3 is not equal to one of the numbers q m j 1, k 1 q m j 2, k 2 , q m j 1, k 1 q m j 2, k 2 -1 and q m j 1, k 1 q m j 2, k 2 1 , then the space L (2) (c m, h m j 3, k 3 , q m j 3, k 3 ) L (2) (c m, h m j 1, k 1 , q m j 1, k 1 ) L (2) (c m, h m j 2, k 2 , q m j 2, k 2 ) of intertwining operators of type ( type ) is 0 . If q m j 3, k 3 q m j 1, k 1 q m j 2, k 2 1 , it is at most 1 -dimensional. If q m j 3, k 3 q m j 1, k 1 q m j 2, k 2 , it is at most 2 -dimensional. enumerate prop proof Conclusion 2.2-1 is clear since the three modules are irreducible. Conclusion 2.2-2 can be proved similarly to the proof of the similar statement in the N 1 case in HM . Here we give a different proof. Suppose that eqnarray (w (1) ) --1 w (2) , v1 (G (-1 2)w (1) ) --1 2 w (2) , v2 (G - (-1 2)w (1) ) --1 2 w (2) , v3 (G (-1 2)G - (-1 2)w (1) ) - w (2) v4 eqnarray are all equal to 0 but 0 . Using the associator formula (obtained by taking residue in x 1 in the Jacobi identity defining intertwining operators) eqnarray (Y(u, x 0)w, x 2) Y(u, x 0 x 2)(w, x 2) .9em -(-1) u w x 1 x 0 -1 ( x 2-x 1 -x 0 ) (w, x 2)Y(u, x 1) eqnarray repeatedly, we see that 0 if (w (1) , x) 0 . Thus (w (1) , x)0 . Using the commutator formula (obtained by taking residue in x 0 in the Jacobi identity defining intertwining operators) eqnarray Y(u, x 1)(w, x 2)-(-1) u w (w, x 2)Y(w, x 1) x 0 x 2 -1 ( x 1-x 0 x 2 ) (Y(u, x 0)w, x 2) eqnarray together with the N 2 Neveu-Schwarz algebra relations, the L(-1) -derivative property and the definition of the lowest weight vector w (1) repeatedly, we see that (w (1) , x) 0 if ( v1 )--( v4 ) are all equal to 0 . Thus these four vectors cannot be all 0 . We have a contradiction. We prove Conclusion 2.2-3 now. By Conclusion 2.2-2 , we need only estimate the number of nonzero vectors in the set of four vectors ( v1 )--( v4 ). We need the following: lemma u1c The following equalities hold: equation q m j 3, k 3 (w (1) ) --1 w (2) (q m j 1, k 1 q m j 2, k 2 ) (w (1) ) --1 w (2) , u1c1 equation eqnarray q m j 3, k 3 (G (-1 2)w (1) ) --1 2 w (2) u1c2 (q m j 1, k 1 q m j 2, k 2 1) (G (-1 2)w (1) ) --1 2 w (2) , eqnarray eqnarray q m j 3, k 3 (G (-1 2)w (1) ) --1 2 w (2) u1c3 (q m j 1, k 1 q m j 2, k 2 -1) (G (-1 2)w (1) ) --1 2 w (2) , eqnarray eqnarray u1c4 q m j 3, k 3 (G (-1 2)G -(-1 2) w (1) ) - w (2) (q m j 1, k 1 q m j 2, k 2 ) (G (-1 2)G -(-1 2)w (1) ) - w (2) (2h m j 1, k 1 q m j 1, k 1 ) (w (1) ) --1 w (2) . eqnarray lemma proof A straightforward calculation gives J(0)(w (1) ) --1 w (2) (q m j 1, k 1 q m j 2, k 2 ) (w (1) ) --1 w (2) , eqnarray J(0)(G (-1 2)w (1) ) --1 2 w (2) (q m j 1, k 1 q m j 2, k 2 1) (G (-1 2)w (1) ) --1 2 w (2) , eqnarray eqnarray J(0)(G (-1 2)w (1) ) --1 2 w (2) (q m j 1, k 1 q m j 2, k 2 -1) (G (-1 2)w (1) ) --1 2 w (2) , eqnarray eqnarray J(0)(G (-1 2)G -(-1 2) w (1) ) - w (2) (q m j 1, k 1 q m j 2, k 2 ) (G (-1 2)G -(-1 2)w (1) ) - w (2) (2h m j 1, k 1 q m j 1, k 1 ) (w (1) ) --1 w (2) . eqnarray But on the other hand, note that ( v1 )--( v4 ) are all (zero or nonzero) constant multiples of w (3) and thus all have U(1) charge q m j 3, k 3 . So we have ( u1c1 )--( u1c4 ). proof We prove Conclusion 2.2-3 using this lemma now. If q m j 3, k 3 is not equal to one of the numbers q m j 1, k 1 q m j 2, k 2 q m j 1, k 1 q m j 2, k 2 -1 and q m j 1, k 1 q m j 2, k 2 1 , then from ( u1c1 )--( u1c4 ), we conclude that ( v1 )--( v4 ) are all equal to 0 . Thus the space of intertwining operators is 0 . If q m j 3, k 3 q m j 1, k 1 q m j 2, k 2 1 , then by ( u1c1 ), ( u1c3 ) and ( u1c4 ), ( v1 ), ( v3 ) and ( v4 ) must be 0 . Thus there is at most one nonzero vector ( v2 ). So the dimension is at most 1 . Similarly in the case of q m j 3, k 3 q m j 1, k 1 q m j 2, k 2 -1 , we can show that (w (1) ) --1 w (2) , ( v1 ), ( v2 ) and ( v4 ) must be 0 and thus the dimension is at most 1 . If q m j 3, k 3 q m j 1, k 1 q m j 2, k 2 , then by ( u1c2 ) and ( u1c3 ), ( v2 ) and ( v3 ) must both be 0 . Thus we have at most two nonzero vectors ( v1 ) and ( v4 ) and the dimension is at most 2 . proof rema In the proofs above we do not use the particular properties, except the irreducibility, of L ns (2) (c m, h m j i, k i , q m j i, k i ) , i 1, 2, 3 . Thus the conclusions of Proposition fusion remain true if we replace c m by an arbitrary c and L ns (2) (c m,h m j i,k i ,q m j i,k i ) , i 1,2,3 , by L ns (2) (c,0,0) -modules L ns (2) (c,h i,q i) , i 1,2,3 , if L ns (2) (c,h i,q i) are irreducible. rema defn An irreducible module for L (2) (c, 0, 0) is said to be chiral (anti-chiral) if G (-1 2)w 0, (G -(-1 2)w 0) where w is a nonzero lowest weight vector of the module. defn Note that in the case c c m we have only finitely many chiral (anti-chiral) modules. cor fusion-ch Assume that L (2) (c m, h m j 1, k 1 , q m j 1, k 1 ) is chiral or anti-chiral. Then the dimension of the space L (2) (c m, h m j 3, k 3 ) L (2) (c m, h m j 1, k 1 )L (2) (c m, h m j 2, k 2 ) is at most 1 . cor proof Assume that L (2) (c m, h m j 1, k 1 , q m j 1, k 1 ) is chiral. Then (G (-1 2)w (1) ) --1 2 w 2 0. We claim that in this case equation g g-1 (G (-1 2)G - (-1 2)w (1) ) - w (2) 2(w (1) ) --1 w (2) . equation In fact, the commutator formula for G (-1 2) and G - (-1 2) gives G (-1 2)G - (-1 2) -G - (-1 2)G (-1 2) 2L(-1). Thus eqnarray g g-2 (G (-1 2)G - (-1 2)w (1) ) - w 2 -(G - (-1 2)G (-1 2) w (1) ) - w 2 2(L(-1)w (1) ) - w 2 2(L(-1)w (1) ) - w 2. eqnarray From the L(-1) -derivative property for intertwining operators, we obtain (L(-1)w (1) ) - (w (1) ) --1 . Thus the right-hand side of ( g g-2 ) becomes 2(w (1) ) --1 w (2) , proving ( g g-1 ). On the other hand, from ( u1c1 ) and ( u1c3 ), we see that the vectors (w (1) ) --1 w 2 and (G - (-1 2)w (1) ) --1 2 w 2 cannot be nonzero at the same time. Thus in this case, the dimension of the space spanned by the four vectors ( v1 )--( v4 ) is at most 1 . Equivalently, the corollary is proved. proof rema Note that the operator J(0) plays an essential role in the proofs of Conclusion 2.2-3 in Proposition fusion and Corollary fusion-ch . rema rema After the first version of the present paper was finished, we received a preprint A3 from Adamovic in which the fusion rules for L (2) (c m, 0, 0) are calculated explicitly and a stronger complete reducibility theorem is proved. But for the purpose of the present paper, we shall not need these stronger results. rema Combining Theorem ad and the second or third conclusion of Proposition fusion , we obtain cor rat The unitary minimal N 2 superconformal vertex operator superalgebras are rational in the sense of HL1 , that is, the following three conditions are satisfied: enumerate Every module for such an algebra is completely reducible. There are only finitely many inequivalent irreducible modules for such an algebra. The fusion rules among any three (irreducible) modules are finite. enumerate cor We also have prop Any finitely-generated lower truncated generalized module W for L (2) (c m,0, 0) is an ordinary module. prop proof The proof is the same as the corresponding result in HM . We repeat it here since it is simple. Suppose that W is generated by a single vector w W . Then by the Poincare-Birkhoff-Witt theorem and the lower truncation condition, every homogeneous subspace of U((2))w is finite-dimensional, proving the result. proof Let m i , i 1, , n , be positive integers and let V L ns (2) (c m 1 , 0, 0)L ns (2) (c m n , 0, 0). From the trivial generalizations of the results proved in FHL and DMZ to vertex operator superalgebras, V is a rational N 2 superconformal vertex operator superalgebra, a set of representatives of equivalence classes of irreducible modules for V can be listed explicitly and the fusion rules for V can be calculated easily. We introduce a class of N 2 superconformal vertex operator superalgebras: defn Let m i , i 1, , n , be positive integers. An N 2 superconformal vertex operator superalgebra V is said to be in the class m 1; ; m n if V has a vertex operator subalgebra isomorphic to L (2) (c m 1 , 0, 0)L (2) (c m n , 0, 0) . defn prop 2-7 Let V be an N 2 superconformal vertex operator superalgebra in the class m 1; ; m n . Then any finitely-generated lower truncated generalized V -module W is an ordinary module. prop proof The proof is similar to the proofs of Proposition 3.7 in H2 and Proposition 2.7 in HM . Here we only point out the main difference. As in H2 and HM , we discuss only the case n 2 . Similar to the proofs of Proposition 3.7 in H2 and Proposition 2.7 in HM , using the Jacobi identity, the N 2 Neveu-Schwarz relations, in particular, the formulas G (-1 2), G - (-1 2) 2L(-1) , (G (-1 2)) 2 (G - (-1 2)) 2 0 , and Theorem 4.7.4 of FHL , we can reduce our proof in the case of n 2 to the finite-dimensionality of the space spanned by the elements of the form eqnarray generators A (L(-1) l 1 J(-1) m 1 G (-1 2) k 1 G - (-1 2) k 2 u (j) (1) ) j 1 Bw (t) (1) C (L(-1) l 2 J(-1) m 2 G (-1 2) k 3 G - (-1 2) k 4 u (j) (2) ) j 2 Dw (t) (2) , eqnarray l 1, l 2 , m 1, m 2, k 1, k 2, k 3, k 4 0, 1 , j 1, j 2 , t 1, , c , j 1, , d , where A (or C ) are products of operators of the forms L(-a) 1 (or L(-a) 2 ), J(-a) 1 (or J(-a) 2 ), a , and G (-b) 1 (or G (-b) 2 ), b 12 , B (or D ) are products of operators of the forms L(a) 1 (or L(a) 2 ), J(a) 1 (or J(a) 2 ), a , and G (b) 1 (or G (b) 2 ), b 12 , where u (i) (j) , j 1, , d , i 1, 2 , are homogeneous elements of V such that the L (2) (c m i , 0, 0) -submodules generated by them are isomorphic to L(c m i , h m i r j,s j , q m i r j,s j ) for some r j,s j 12 satisfying 0r j,s j, r j s j m i with the images of u (i) (j) , j 1, , d , i 1, 2 , as the lowest weight vectors and such that V is isomorphic to the direct sum of these submodules, and where w (t) (i) , t 1, , c , i 1, 2 , are homogeneous elements of some irreducible L (2) (c m i , 0, 0) -modules. Because of the L(-1) -property, we may assume that l 1 l 2 0 . Because W is lower truncated, there are only finitely many elements of the form ( generators ) of the fixed weight. Thus W is a V -module. proof The convergence and extension property for products of intertwining operators In this section, we study products of intertwining operators for the unitary minimal N 2 superconformal vertex operator superalgebras. The main result is the following: thm cep2 Let m be a positive integer. Then intertwining operators for the vertex operator superalgebra L(c m,0,0) satisfy the convergence and extension property for products of intertwining operators introduced in H1 . thm An immediate consequence is a similar result for the vertex operator algebras in the class m 1; ; m n . See Theorem cep1 . Instead of proving Theorem cep2 by deriving differential equations with regular singularities satisfied by the matrix elements of products of intertwining operators of lowest weight vectors, as in H1 , HL55 and HM , we use the so-called anti-Kazama-Suzuki mapping FST , which reduces the problem to the study of intertwining operators for a vertex operator algebra constructed from an 2 integrable lowest weight representation. First we give some auxiliary constructions and discuss some results obtained using the so-called anti-Kazama-Suzuki mapping'' (introduced in FST ). Fix a positive integer m . As before, we let c m 3m m 2 and denote the unitary minimal N 2 superconformal vertex operator superalgebra by L (2) (c m,0,0) . Let L be a rank one lattice generated by with the bilinear form , given by , -1, and let l L . As in FLM , we have a vertex superalgebra V L S( -) L . Note that V L is super since L is odd and V L does not satisfy the grading-restriction conditions for the grading obtained from the usual Virasoro elements for lattice vertex algebras since the bilinear form is not positive definite. Let V L - (-1) 2 2 (-2) 2. It can be verified easily that the component operators of the vertex operator associated to V L satisfy the Virasoro relations with central charge 4 . In particular, the component operator for the -2 nd power of x gives a 2 -grading for V L . With this grading, V L is a 2 -graded vertex superalgebra. However, it is easy to see that this grading is not truncated from below. Thus V L with V L as its Virasoro element fails to be a vertex operator superalgebra. We also need the following construction of the so-called Liouville scalar model:'' Let be the Lie algebra with a basis a(n) , n , and d , satisfying the bracket relations eqnarray a(m),a(n) m m n,0 d, a(m), d 0 eqnarray for m, n . Let M(1,s) be the corresponding irreducible highest weight module with central charge 1 and highest weight s . It is well-known that M(1,0) S( -) has a vertex operator algebra structure with the Virasoro element M(1, 0) a(-1) 2 2 . Consider the vertex algebra structure on M(1,0) together with a different Virasoro element M(1,0) a(-1) 2 2 ia(-2) 2. A straightforward calculation shows that the vertex algebra structure on M(1,0) together with the Virasoro element M(1,0) is a vertex operator algebra with the central charge 4 (see, for example, and FF ). We shall denote this vertex operator algebra by V Liou (the vertex operator algebra associated to the Liouville scalar model). In addition, every M(1,s) , s , is an irreducible V Liou -module and any V Liou -module on which a(0) acts semisimply is completely reducible. We will work only with such V Liou -modules, which are enough for our purposes. The anti-Kazama-Suzuki mapping gives us a structure of an sl 2 -module on L ns (2) (c m,h m j, k ,q m j, k ) V L for j, k 12 , 0 j, k, j k m . Consider the vectors (as in FST and ) eqnarray G (-3 2)1 c m, 0, 0 e - , m 2 2 G - (-3 2)1 c m, 0, 0 e , -m1 c m, 0, 0 (-1) (m 2)J(-1)1 c m, 0, 0 e 0 eqnarray in L ns (2) (c m, 0, 0) V L . Then the vertex operators Y(,x) , Y(,x) and Y(,x) for the L ns (2) (c m,0,0) V L -module L ns (2) (c m,h m j, k ,q m j, k ) V L give a representation of sl 2 of level m on L ns (2) (c m,h m j, k ,q m j, k ) V L . The main observation in FST is that L ns (2) (c m,h m j, k ,q m j, k ) V L is completely reducible as an sl 2 -module. In the special case h m j, k q m j, k 0 , we obtain a vertex subalgebra of L ns (2) (c m,0,0) V L which is isomorphic as a vertex algebra to the underlying vertex algebra of the vertex operator algebra L sl (2) (m, 0) on the integrable highest-weight sl (2) -module of level m and highest weight 0 (as in ). The Virasoro element for L sl (2) (m, 0) is given by the Sugawara-Segal construction, and if we identify this vertex subalgebra with L sl (2) (m, 0) , then eqnarray L sl (2) (m,0) L ns (2) (c m,0,0) e 0 m 2 4J(-1) 21 c m, 0, 0 e 0 - m 2 2J(-1)1 c m, 0, 0 (-1) 41 c m, 0, 0 (-1) 2, eqnarray where L sl (2) (m,0) and L ns (2) (c m,0,0) are the Virasoro elements for the vertex operator (super)algebras L sl (2) (m,0) and L ns (2) (c m,0,0) , respectively. We see that under the isomorphism from this vertex subalgebra of L ns (2) (c m,0,0) V L to L sl (2) (m, 0) , the Virasoro element in L sl (2) (m, 0) is not the image of the Virasoro element of L ns (2) (c m,0,0) V L . To get the correct Virasoro element (as in FST ), we consider the vertex subalgebra of L ns (2) (c m,0,0) V L generated by the element m 2 2 (J(-1)1 c m, 0, 0 e 0- 1 c m, 0, 0 (-1)). It is straightforward to verify that this vertex subalgebra is actually isomorphic to V Liou . Straightforward calculations also show that Y(,x) commutes with sl 2 generators. So L sl (2) (m,0) V Liou is isomorphic to a vertex subalgebra of L ns (2) (c m,0,0) V L as well and we shall, for convenience, identify L sl (2) (m,0) V Liou with this vertex subalgebra. It is easy to see that the Virasoro element of L sl (2) (m,0) V Liou is identified with the Virasoro element of L ns (2) (c m,0,0)V L . Thus L sl (2) (m,0) V Liou is a vertex operator subalgebra. Now we use a key result in FST and FSST (in the case of unitary modules), slightly reformulated in the language of vertex operator algebras: thm fst As a generalized L sl (2) (m,0) V Liou -module, L ns (2) (c m,h,q) V L decomposes as equation last k 0,1,,m s I s L sl (2) (m,k) M(1,s). equation where s runs through a certain infinite index set I s . thm proof The main result from FSST , applied to the case of irreducible admissible modules (cf. AM ), yields a decomposition similar to the above decomposition but with twisted admissible modules for sl (2) also appearing in ( last ). In the case of unitary modules (which are admissible modules), the situation is simpler because only ordinary unitary modules appear (see also Remark 4.7 in FS ). So the theorem holds. proof rema The proof of the theorem above illustrates why the non-unitary minimal models for the superalgebra ns (2) are harder to study than the unitary ones (see also ). rema We know that L sl (2) (m,0) is rational (see FZ ). Although V Liou is not rational, any module on which a(0) acts semisimply is completely reducible, as we mentioned above. Any irreducible module for L sl (2) (m,0) is isomorphic to L sl (2) (m,i) for some i 1, , m and any irreducible module for V Liou is isomorphic to M(1,s) for some s . Thus by the result in FHL on modules for a tensor product of vertex operator algebras, any irreducible module for L sl (2) (m,0)V Liou is isomorphic to L sl (2) (m,i) M(1,s) for some i 0,...,m , s C . Proposition 2.7 in DMZ and its proof can be generalized trivially to the case where one of the vertex operator algebra is an irrational vertex operator algebra like V Liou , such that in particular, any L sl (2) (m,0) V Liou -module with 1 L sl (2) (m,0) a(0) ( 1 L sl (2) (m,0) being the vacuum vector of L sl (2) (m,0) ) acting semi-simply is completely reducible. Now suppose that M is such an L sl (2) (m,0) V Liou -module. Then it follows that M is completely reducible. So it has a decomposition B M where B is an index set. Note that here the sum might be infinite (comparing with the rational case where this sum is always finite). Since any irreducible L sl (2) (m,0) V Liou -module is isomorphic to L sl (2) (m,i) M(1,s) for some i 0,...,m , s C , M for any B is isomorphic to such a module.We need the following: lemma facint Let be an intertwining operator of type L sl (2) (m,i 3) M(1,s 3) L sl (2) (m,i 1) M(1,s 1) L sl (2) (m,i 2) M(1,s 2) . Then ' '' where ' and '' are intertwining operators of types L sl (2) (m,i 3) L sl (2) (m,i 1) L sl (2) (m,i 2) and M(1,s 3) M(1,s 1) M(1,s 2) , respectively. In particular, all fusion rules for irreducible modules for L sl (2) (m,0) V Liou are 0 or 1 . lemma proof It is enough to show that there is a linear injective map from L sl (2) (m,i 3) M(1,s 3) (L sl (2) (m,i 1) M(1,s 1)) (L sl (2) (m,i 2) M(1,s 2)) , the space of intertwining operators of type L sl (2) (m,i 3) M(1,s 3) L sl (2) (m,i 1) M(1,s 1) L sl (2) (m,i 2) M(1,s 2) , to L sl (2) (m,i 3) L sl (2) (m,i 1) L sl (2) (m,i 2) M(1,s 3) M(1,s 1) M(1,s 2) , where L sl (2) (m,i 3) L sl (2) (m,i 1) L sl (2) (m,i 2) and M(1,s 3) M(1,s 1) M(1,s 2) are the space of intertwining operators of type L sl (2) (m,i 3) L sl (2) (m,i 1) L sl (2) (m,i 2) and M(1,s 3) M(1,s 1) M(1,s 2) , respectively. But this follows from Proposition 2.10 in DMZ which in turn is a consequence of a result in FHL on irreducible modules for a tensor product vertex operator algebra and a result in FZ giving an isomorphism between a space of intertwining operators and a certain vector space. Since the fusion rules for irreducible L sl (2) (m,0) -modules are 0 and 1 (see FZ ) and the same is true for irreducible V Liou -modules, the lemma is proved. proof rema Note that this result should be very useful in calculating the fusion algebra for L ns (2) (c m,0,0) since every such intertwining operator for L ns (2) (c m,0,0) factors as a tensor product of an intertwining operator for vertex operator algebra L sl (2) (m,0) and an intertwining operator for the vertex operator algebra associated to the Heisenberg algebra. In the present paper, the exact values of the fusion rules are not what we are interested in and thus we shall not calculate them here. (After the first version of the present paper was finished, we received a preprint A3 from Adamovic in which the fusion rules for L ns (2) (c m,0,0) are calculated explicitly.) rema Using Lemmas facint , we obtain prop propfinite For fixed i 1, i 2 1, , m and s 1, s 2 , if s 3s 1 s 2 , the space (L sl (2) (m,i 1) M(1,s 1))(L sl (2) (m,i 2) M(1,s 2)) L sl (2) (m,i 3) M(1,s 3) is 0 . In particular, there are only finitely many pairs (i 3, s 3) 1, ,m such that the space (L sl (2) (m,i 1) M(1,s 1))(L sl (2) (m,i 2) M(1,s 2)) L sl (2) (m,i 3) M(1,s 3) are not 0 . prop proof Let (L sl (2) (m,i 1) M(1,s 1))(L sl (2) (m,i 2) M(1,s 2)) L sl (2) (m,i 3) M(1,s 3) . By Lemma facint , ''' where ' and '' are intertwining operators of types L sl (2) (m,i 3) L sl (2) (m,i 1) L sl (2) (m,i 2) and M(1,s 3) M(1,s 1) M(1,s 2) , respectively. It is clear that if s 3s 1 s 2 , '' 0 , proving the result. proof proof Proof of Theorem cep2 Let W 1, , W 5 be irreducible L ns (2) (c m,0,0) -modules and 1 and 2 intertwining operators of types W 4 W 1 W 5 and W 5 W 2 W 3 , respectively. Consider the (formal) matrix coefficients equation matrix1 w' (4) , 1(w (1) ,x 1) 2(w (2) ,x 2) w (3) , equation where w (l) W l , l 1,2,3 and w' (4) W' 4 . We shall identify W l , l 1, 2, 3 , with W le 0 in W l V L and W' 4 with W' 4e 0 in W' 4 V L . In particular, we use the same notations w (l) , l 1,2,3 , to denote w (l) e 0 , and w' (4) to denote w' (4) e 0 . We extend intertwining operators 1 and 2 uniquely to intertwining operators (denoted by the same notations 1 and 2 ) of types W 4 V L W 1V L W 5V L , and W 5 V L W 2V L W 3V L , respectively. By Theorem fst , W l V L , l 1, 2, 3 , and W' 4V L are generalized modules for L sl (2) (m,0) V Liou and are completely reducible. So w (l) k 1 p l w (l) (k) , l 1, 2, 3 , and w' (4) k 1 p 4 w (4) (k) where w (l) (k) , k 1, , p i , l 1, 2, 3, 4 , are elements of direct summands M l (k) (irreducible L sl (2) (m,0) V Liou -modules) in W i for i 1, 2, 3 or W' 4 for i 4 . Thus ( matrix1 ) is equal to equation matrix2 k 1 1 p 1 k 2 1 p 2 k 3 1 p 3 k 4 1 p 4 w (4) (k 4) , k 4 k 1W 5 (w (1) (k 1) , x 1) W 5 k 2k 3 (w (2) (k 2) , x 2) w (3) (k 3) , equation where k 4 k 1W 5 , W 5 k 2k 3 are intertwining operators of types M 4 (k 4) M 1 (k 1) W 5 , W 5 M 2 (k 2) M 3 (k 3) , respectively. By Theorem fst , W 5 is a completely reducible generalized module for L sl (2) (m,0)V Liou . By Proposition propfinite , ( matrix2 ) is equal to equation matrix3 k 1 1 p 1 k 2 1 p 2 k 3 1 p 3 k 4 1 p 4 k 5 1 p 5 w (4) (k 4) , k 4 k 1k 5 (w (1) (k 1) , x 1) k 5 k 2k 3 (w (2) (k 2) , x 2) w (3) (k 3) , equation where k 4 k 1k 5 , k 5 k 2k 3 are intertwining operators of types M 4 (k 4) M 1 (k 1) M 5 (k 5) , M 5 (k 5) M 2 (k 2) M 3 (k 3) , respectively, and M 5 (k 5) , k 5 1, , p 5 , are irreducible L sl (2) (m,0) V Liou -sub -mod -ules of W 5 as L sl (2) (m,0) V Liou -modules. By Proposition facint , k 4 k 1k 5 ( k 4 k 1k 5 )' ( k 4 k 1k 5 )'' and k 5 k 2k 3 ( k 5 k 2k 3 )' ( k 5 k 2k 3 )'', where ( k 4 k 1k 5 )' and ( k 5 k 2k 3 )' are intertwining operators for the vertex operator algebras L sl (2) (m, 0) and ( k 4 k 1k 5 )'' and ( k 5 k 2k 3 )'' are intertwining operators for the vertex operator algebras V Liou . Thus ( matrix3 ) is equal to a finite sum of series of the form equation matrix (4) , 1( (1) , x 1) 2( (2) , x 2) (3) (4) , 1( (1) , x 1) 2( (2) , x 2) , (3) , equation where 1 and 2 are intertwining operators among irreducible modules for L sl (2) (m,0) and 1 and 2 are intertwining operators among irreducible modules for V Liou . In HL55 , it was proved that intertwining operators for the vertex operator algebra L sl (2) (m,0) satisfy the convergence and extension property for products using the Knizhnik-Zamolodchikov equations. The convergence and extension property for products of intertwining operators for the vertex operator algebra V Liou can be proved trivially by a straightforward calculation. Using the convergence and extension properties for products of intertwining operators for these vertex operator algebras and using the fact proved above that ( matrix1 ) is a finite sum of series of the form ( matrix ), we conclude that ( matrix1 ) is convergent when we substitute e n i z i for x i n with z 1, z 2 satisfying z 1 z 2 0 and it can be analytically extended to an analytic function in the region z 2 z 1-z 2 0 of the form equation extint i 1 j z 2 r i (z 1-z 2) s i f i ( z 1-z 2 z 2 ). equation We still need to prove the following: There exists N (which does not depend on w (1) and w (2) ) such that equation ineq wt (w (1) ) wt (w (2) ) s i N, equation for i 1,...,j . The existence of N follows (as in the cases in H2 and HM , respectively) from an induction argument for the N 2 superconformal algebra. Since any L (2) (c m,0,0) -module is completely reducible and since there are only finitely many irreducible L (2) (c m,0,0) -modules, we need only prove the existence in the case where W 1 and W 2 are irreducible. When w (1) and w (2) are lowest weight vectors, ( matrix1 ) is absolutely convergent in the region z 1 z 2 0 and can be analytically extended to an analytic function in in the region z 2 z 1-z 2 0 of the form ( extint ). We choose an N such that for these lowest weight vectors, ( ineq ) holds. For general w (1) and w (2) , we use induction instead of the proof above to show that ( matrix1 ) converges absolutely in the region z 1 z 2 0 and can be analytically extended to an analytic function in the region z 2 z 1-z 2 0 of the form ( extint ). In addition, the induction also shows that ( ineq ) holds for the N we choose. proof An immediate consequence of Theorem cep2 is the following: thm cep1 Let m i , i 1, , n , be positive integers and V an N 2 superconformal vertex operator superalgebra in the class m 1;, m n . Then intertwining operators for V satisfy the convergence and extension property for products of intertwining operators introduced in H1 . thm We omit the proof since it is the same as the corresponding result in H2 and HL55 . Intertwining operator superalgebras and vertex tensor categories for N 2 unitary minimal models Let m i , i 1, , n , be n nonnegative integers and V a vertex operator superalgebra in the class m 1; ; m n . Using Corollary rat , Proposition 2-7 and Theorem cep1 above, and Theorems 3.1 and 3.2 in H2 , which are proved in H2 using results in HL1 -- HL5 and H1 , we obtain the following: thm associativity for intertwining operators enumerate For any V -modules W 0 , W 1 , W 2 , W 3 and W 4 , any intertwining operators 1 and 2 of types W 0 W 1W 4 and W 4 W 2W 3 , respectively, and any choice of z 1 and z 2 , w' (0) , 1(w (1) , x 1) 2(w (2) , x 2)w (3) x n 1 e nz 1 , ; x n 2 e nz 2 , ; n is absolutely convergent when z 1 z 2 0 for w' (0) W' 0 , w (1) W 1 , w (2) W 2 and w (3) W 3 . For any modules W 0 , W 1 , W 2 , W 3 , and W 5 and any intertwining operators 3 and 4 of types W 5 W 1W 2 and W 0 W 5W 3 , respectively, and any choice of z 2 and (z 1-z 2) , w' (0) , 4( 3(w (1) , x 0)w (2) , x 2)w (3) x n 0 e n (z 1-z 2) , ; x n 2 e nz 2 , ; n is absolutely convergent when z 2 z 1-z 2 0 for w' (0) W' 0 , w (1) W 1 , w (2) W 2 and w (3) W 3 . For any V -modules W 0 , W 1 , W 2 , W 3 and W 4 , any intertwining operators 1 and 2 of types W 0 W 1W 4 and W 4 W 2W 3 , respectively, there exist a module W 5 and intertwining operators 3 and 4 of types W 5 W 1W 2 and W 0 W 5W 3 , respectively, such that for any z 1, z 2 satisfying z 1 z 2 z 1-z 2 0 and for any w' (0) W' 0 , w (1) W 1 , w (2) W 2 and w (3) W 3 , eqnarray 1-1 w' (0) , 1(w (1) , x 1) 2(w (2) , x 2)w (3) x 1 n e nz 1 , x 2 n e nz 2 , n w' (0) , 4( 3(w (1) , x 0)w (2) , x 2)w (3) x 0 n e n(z 1-z 2) , x 2 n e nz 2 , n , eqnarray where z 1 z 1 iz 1 , z 2 z 2 iz 2 and (z 1-z 2) z 1-z 2 i(z 1-z 2) are the values of the logarithms of z 1 , z 2 and z 1-z 2 such that 0z 1, z 2, (z 1-z 2)2 . For any modules W 0 , W 1 , W 2 , W 3 , and W 5 , any intertwining operators 3 and 4 of types W 5 W 1W 2 and W 0 W 5W 3 , respectively, there exist a module W 4 and intertwining operators 1 and 2 of types W 0 W 1W 4 and W 4 W 2W 3 , respectively, such that for any z 1, z 2 satisfying z 1 z 2 z 1-z 2 0 and for any w' (0) W' 0 , w (1) W 1 , w (2) W 2 and w (3) W 3 , the equality ( 1-1 ) holds. enumerate thm thm commutativity for intertwining operators For any V -modules W 0 , W 1 , W 2 , W 3 and W 4 and any intertwining operators 1 and 2 of types W 0 W 1W 4 and W 4 W 2W 3 , respectively, there exist a module W 5 and intertwining operators 3 and 4 of types W 0 W 2W 5 and W 5 W 1W 3 , respectively, such that for any homogeneous w' (0) W' 0 , w (1) W 1 , w (2) W 2 and w (3) W 3 , the multivalued analytic function w' (0) , 1(w (1) , x 1) 2(w (2) , x 2)w (3) x 1 z 1, ; x 2 z 2 of z 1 and z 2 in the region z 1 z 2 0 and the multivalued analytic function (-1) w (1) w (2) w' (0) , 3(w (2) , x 2) 4(w (1) , x 1)w (3) x 1 z 1, ; x 2 z 2 of z 1 and z 2 in the region z 2 z 1 0 are analytic extensions of each other. thm The notions of intertwining operator algebra in H25 (see also H4 and H6 ) and N 1 superconformal intertwining operator superalgebra in HM can be generalized easily to the following notion: defn An N 2 superconformal intertwining operator superalgebra is an intertwining operator superalgebra W together with three elements , - and such that (W e, Y, 1, , - , ) is an N 2 superconformal vertex operator algebra. defn Then we have thm Assume in addition that V is rational. Let a i i 1 m be the set of all equivalence classes of irreducible V -modules. Let W a 1 , , W a m be representatives of a 1, , a m , respectively. Let W i 1 mW a i , and let a 1a 2 a 3 , for a 1, a 2, a 3 , be the space of intertwining operators of type W a 3 W a 1 W a 2 . Then (W, , a 1a 2 a 3 , 1, , - , ) (where 1 , , - and are the distinguished elements of V ) is an N 2 superconformal intertwining operator superalgebra. thm In particular, we have thm For any nonnegative integer m , the direct sum j, k 12 , ;0j, k, j k m L ns (2) (c m, h m j,k , q m j,k ) together with the finite set j, k 12 ; ; ;0j, k, j k m , the spaces of intertwining operators of type L ns (2) (c m, h m j 3,k 3 , q m j 3,k 3 ) L ns (2) (c m, h m j 1,k 1 , q m j 1,k 1 ) L ns (2) (c m, h m j 2,k 2 , q m j 2,k 2 ) for j i, k i 12 , ;0j i, k i, j i k i m , i 1, 2, 3 , and the vacuum and the Neveu-Schwarz elements of L (2) (c m, 0, 0) is an N 2 superconformal intertwining operator superalgebra. thm Recall the sphere partial operad K K(j) j , the vertex partial operads c c(j) j of central charge c constructed in H3 and the definition of vertex tensor category in HL4 and HL6 . For any c and j , c(j) is a trivial holomorphic line bundle over K(j) and we have a canonical holomorphic section j . Given a vertex tensor category, we have, among other things, a tensor product bifunctor for each c(2) . In particular, 2(P(z)) c(2) and thus there is a tensor product bifunctor 2(P(z)) . Note that P(z) constructed in HL5 can be generalized without any difficulty to categories of modules for vertex operator superalgebras. By Proposition 2-7 and Theorem cep1 , and Theorem 3.2 and Corollary 3.3 in H2 , we obtain thm vtc Let c be the central charge of V . Then the category of V -modules has a natural structure of vertex tensor category of central charge c such that for each z , the tensor product bifunctor 2(P(z)) associated with 2(P(z)) c(2) is equal to the generalization to the category of V -modules of P(z) constructed in HL5 . thm Combining Theorem vtc with Theorem 4.4 in HL4 (see HL6 for the proof), we obtain cor The category of V -modules has a natural structure of braided tensor category such that the tensor product bifunctor is P(1) . In particular, the category of L (2) (c m 1 , 0, 0)L (2) (c m n , 0, 0) -modules has a natural structure of braided tensor category. cor In particular, the special case V L (2) (c m, 0, 0) gives thm For any nonnegative integer m , the category of modules for the N 2 Neveu-Schwarz Lie superalgebra isomorphic to finite direct sums of L (2) (c m, h m j,k , q m j,k ),j, k 12 , 0j, k, j k m, has a natural structure of braided tensor category such that the tensor product bifunctor is P(1) . thm