C1-Cofiniteness and Fusion Products for Vertex Operator Algebras
Let V be a vertex operator algebra. We prove that if U and W are C 1-cofinite \(\mathbb{N}\) -gradable V-modules, then a fusion product U⊠W is also a C 1-cofinite \(\mathbb{N}\) -gradable V-module, where the fusion product is defined by (logarithmic) inte
- PDF / 278,316 Bytes
- 9 Pages / 441 x 666 pts Page_size
- 10 Downloads / 204 Views
Abstract Let V be a vertex operator algebra. We prove that if U and W are C1 cofinite N-gradable V -modules, then a fusion product U W is also a C1 -cofinite N-gradable V -module, where the fusion product is defined by (logarithmic) intertwining operators.
1 Introduction The tensor product theory is a powerful tool in the theory of representations. Unfortunately, in the theory of vertex operator algebras (shortly VOA), a tensor product (we call “fusion product”) for some modules may not exist in the category of modules of vertex operator algebras. In order to avoid such an ambiguity, we will introduce a new approach to treat fusion products. Let us explain it briefly. The details are given in Sect. 3. Let V = ∞ n=K Vn be a vertex operator algebra (shortly VOA) and modN (V ) denote the set of N-gradable (weak) V -modules, where a (weak) V module W is called N-gradable if W = ∞ m=0 W(m) such that vk w ∈ W(m+wt(v)−k−1) for any homogeneous element v ∈ Vwt(v) , k ∈ Z and w ∈ W(m) . It is well-known that d −1 g(V ) := V ⊗C C x, x V ⊗C C x, x −1 L(−1) ⊗ 1 − 1 ⊗ dx has a Lie algebra structure and all (weak) V -modules are g(V )-modules (see [1]). For U, W ∈ modN (V ), we introduce a g(V )-module U W (or its isomorphism class) as a projective limit of a direct set of V -modules (by viewing them as g(V )modules). So, a g(V )-module U W always exists. The key point is that a fusion product for U, W ∈ modN (V ) exists if and only if U W is a V -module (and so it is a fusion product). The main purpose of this paper is to explain the fusion products by emphasizing the importance of C1 -cofiniteness. The importance of the C1 -cofiniteness conditions on modules was firstly noticed by Huang in [2], where he has proved that M. Miyamoto Institute of Mathematics, University of Tsukuba, Tsukuba 305, Japan C. Bai et al. (eds.), Conformal Field Theories and Tensor Categories, Mathematical Lectures from Peking University, DOI 10.1007/978-3-642-39383-9_7, © Springer-Verlag Berlin Heidelberg 2014
271
272
M. Miyamoto
intertwining operators of a C1 -cofinite N-gradable module from a C1 -cofinite Ngradable module to an N-gradable module satisfy a differential equation. He has also shown the associativity of intertwining operators among C1 -cofinite N-gradable modules by using the space of solutions of this differential equation. We will prove Key Theorem by using his idea. In order to follow his arguments, we give a slightly different definition of Cm -cofiniteness for modules. Definition 1 Set m = 1, 2, . . . . A V -module U is said to be “Cm -cofinite as a V module” if Cm (U ) := SpanC {v−m u | u ∈ U, v ∈ V , wt(v) > 1 − m} has a finite codimension in U . This is slightly different from the old one. For example, any VOA V is always C1 -cofinite as a V -module in our definition. Since (L(−1)v)−m = mv−m−1 and wt(L(−1)v) = wt(v) + 1, Cm -cofiniteness implies Cm−1 -cofiniteness for m = 2, 3, . . . . We will prove the following theorem. Key Theorem Let V be a VOA. For each m = 1, 2, . . . and Cm -cofinite N-gradable
Data Loading...
