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Intertwining Operator Superalgebras and Vertex Tensor Categories for Superconformal Algebras, Ii

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Intertwining Operator Superalgebras and Vertex Tensor Categories for Superconformal Algebras, Ii TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 354, Number 1, Pages 363{385 S 0002-9947(01)02869-0 Article electronically published on August 21, 2001 INTERTWINING OPERATOR SUPERALGEBRAS AND VERTEX TENSOR CATEGORIES FOR SUPERCONFORMAL ALGEBRAS, II YI-ZHI HUANG AND ANTUN MILAS Abstract. We construct the intertwining operator superalgebras and vertex tensor categories for the N = 2 superconformal unitary minimal models and other related models. 0. 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 =2su- perconformal field theories constructed from its discrete unitary representations of central charge c<3 are among the so-called \minimal models." In the physics liter- ature, 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 [Ge1] [Ge2] between certain particular Calabi-Yau manifolds and certain N =2super- conformal 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 [H3], [H5] and [H6] and the tensor product theory for modules for a vertex operator algebra developed by Lepowsky and the first author in [HL1]{[HL6], [HL8] 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 Adamovi´c [A1] [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 Received by the editors April 18, 2000 and, in revised form, February 21, 2001. 1991 Mathematics Subject Classification. Primary 17B69, 17B68; Secondary 17B65, 81R10, 81T40, 81T60. Key words and phrases. N = 2 superconformal algebras, intertwining operator superalgebras, vertex tensor categories. The research of Y.-Z. H. is supported in part by NSF grants DMS-9622961 and DMS-0070800. The research of A. M. is supported in part by NSF grants. c 2001 American Mathematical Society 363 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 364 YI-ZHI HUANG AND ANTUN MILAS prove some basic results on representations of unitary minimal N = 2 superconfor- mal 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. 1. N =2superconformal 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]: Definition 1.1. An N =2superconformal vertex operator superalgebra is a vertex operator superalgebra (V;Y;1;!) together with odd elements τ +, τ − andaneven element µ satisfying the following axioms: Let X Y (τ +;x)= G+(r)x−r−3=2; r2Z+ 1 X 2 Y (τ −;x)= G−(r)x−r−3=2; r2Z+ 1 X 2 Y (µ, x)= J(n)x−n−1: n2Z Then V is a direct sum of eigenspaces of J(0) with integral eigenvalues which modulo 2Z give the Z2 grading for the vertex operator superalgebra structure, and 2 Z 2 Z 1 the following N = 2 Neveu-Schwarz relations hold: For m; n , r; s + 2 c [L(m);L(n)] = (m − n)L(m + n)+ (m3 − m)δ ; 12 m+n;0 c [J(m);J(n)] = mδ ; 3 m+n;0 − [L(m);J(n)] = nJm+n; m [L(m);G(r)] = − r G(m + r); 2 [J(m);G(r)] = G(m + r); c 1 [G+(r);G−(s)] = 2L(r + s)+(r − s)J(r + s)+ (r2 − )δ ; 3 4 r+s;0 [G(r);G(s)] = 0 where L(m), m 2 Z, are the Virasoro operators on V and c is the central charge of V . Modules and intertwining operators for an N = 2 superconformal vertex opera- tor superalgebra are modules and intertwining operators for the underlying vertex operator superalgebra. Note that this definition is slightly different from the one in [A1]; in [A1], V is not required to be a direct sum of eigenspaces of J(0). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use INTERTWINING OPERATOR SUPERALGEBRAS 365 The N = 2 superconformal vertex operator superalgebra defined above is de- noted by (V;Y;1,τ+,τ−,µ)(without! since 1 3 1 1 ! = L(−2)1 = [G+(− );G−(− )]1 + J(−2)1) 2 2 2 2 or simply by V . Note that a module W for a vertex operator superalgebra (in particular the algebra itself) has a Z2-grading called sign in addition to the C- grading by weights. We shall always use W 0 and W 1 to denote the even and odd 0 1 subspaces of WL.IfW is irreducible, thereL exists h 2 C such that W = W ⊕ W 0 1 where W = n2h+Z W(n) and W = n2h+Z+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 Z2 by taking the signs of elements in the union. The notion of N = 2 superconformal vertex operator superalgebra can be refor- mulatedusingoddformalvariables.(IntheN = 1 case, this reformulation was given by Barron [Ba1] [Ba2].) 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 C spanned by these symbols and denote this exterior algebra by C['1;:::;'l]. For any vector space E, we also have the vector space E['1;:::;'l]; E[x1;:::;xk]['1;:::;'l]; −1 −1 E[x1;x1 ;:::;xk;xk ]['1;:::;'l]; E[[x1;:::;xk]]['1;:::;'l]; −1 −1 E[[x1;x1 ;:::;xk;xk ]]['1;:::;'l]; Efx1;:::;xkg['1;:::;'l] and E((x1;:::;xk))['1;:::;'l]: If E is a Z2-graded vector space, then there are natural structures of modules over the ring C[x1;:::;xk]['1;:::;'l] on these spaces. The ring C[x1;:::;xk]['1;:::;'l] can be viewed as the space of algebraic functions on a \superspace" consisting of elements of the form (x1;:::;xk; '1;:::;'l)wherex1;:::;xk 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,τ+,τ−,µ)beanN = 2 superconformal vertex operator superalge- bra. Let 1 + − τ1 = p (τ + τ ); 2 1 + − τ2 = p (τ − τ ) −2 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 366 YI-ZHI HUANG AND ANTUN MILAS and X −r−3=2 Y (τ1;x)= G1(r)x ; r2Z+ 1 X 2 −r−3=2 Y (τ2;x)= G2(r)x : 2Z 1 r + 2 We define the vertex operator map with odd variables Y : V ⊗ V ! V ((x))['1;'2] u ⊗ v 7! Y (u; (x; '1;'2))v by Y (u; (x; '1;'2))v = Y (u; x)v + '1Y (G1(−1=2)u; x)v +'2Y (G2(−1=2)u; x)v +'1'2Y (G1(−1=2)G2(−1=2)u; x)v for u; v 2 V . (We use the same notation Y to denote the vertex operator map and the vertex operator map with odd variables.) In particular, we have p Y (µ, (x; ' ;' )) = Y (µ, x) − −1' Y (τ ;x) 1 2 p 1 p2 − −1'2Y (τ1;x) − 2 −1'1'2Y (!;x): Also, if we introduce p −' + −1' '+ = 1 p 2 ; p 2 ' + −1' '− = 1 p 2 ; 2 then we can write + + Y (µ, (x; '1;'2)) = Y (µ, x)+' Y (τ ;x) +'−Y (τ −;x)+2'+'−Y (!;x): We have Proposition 1.2. The vertex operator map with odd variables satisfies the follow- ing properties: 1. The vacuum property: Y (1; (x; '1;'2)) = 1 where 1 on the right-hand side is the identity map on V . 2. The creation property: For any v 2 V , Y (v; (x; '1;'2))1 2 V [[x]]['1;'2]; lim Y (v; (x; '1;'2))1 = v: (x;'1;'2)7!(0;0;0) 3. The Jacobi identity: In −1 −1 −1 (End V )[[x0;x0 ;x1;x1 ;x2;x2 ]]['1;'2; 1; 2]; License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use INTERTWINING OPERATOR SUPERALGEBRAS 367 we have x − x − ' − ' x−1δ 1 2 1 1 2 2 Y (u; (x ;' ;' ))Y (v; (x ; ; )) 0 x 1 1 2 2 1 2 0 − − − jujjvj −1 x2 x1 + '1 1 + '2 2 ( 1) x0 δ −x0 · Y (v; (x2; 1; 2))Y (u; (x1;'1;'2)) − − − −1 x1 x0 '1 1 '2 2 = x2 δ x2 ·Y (Y (u; (x0;'1 − 1;'2 − 2))v; (x2; 1; 2)); for u; v 2 V which are either even or odd.
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