Fusion rules for the vertex operator algebra VL2A4

Abstract

The fusion rules for vertex operator algebra V A4 L2 are determined.

A Characterization of the Vertex Operator Algebra $$V _{L_{2}}^{A_{4}}$$

“…We may identify the Virasoro vertex operator subalgebra L(1, 0) both in V 1 and V 2 . Let φ : L(1, 0) ⊕ V (1,9) ⊕ V (1,16) → L(1, 0) ⊕ V (2,9) ⊕ V (2,16) be an L(1, 0)-module isomorphism such that φω = ω, φx 1 = x 2 , φy 1 = y 2 . (1,9) ⊕ V (1,16) .…”

“…Let φ : L(1, 0) ⊕ V (1,9) ⊕ V (1,16) → L(1, 0) ⊕ V (2,9) ⊕ V (2,16) be an L(1, 0)-module isomorphism such that φω = ω, φx 1 = x 2 , φy 1 = y 2 . (1,9) ⊕ V (1,16) . (1,9) be the intertwining operator of type L(1, 0) V (1,9) V (1,9) , and I 0 (φu, z)φv = Q 0 • Y (φu, z)φv for u, v ∈ V 1 be the intertwining operator of type L(1, 0) V (2,9) V (2,9) , where P 0 , Q 0 are the projections of V 1 and V 2 to L(1, 0) respectively.…”

“…for u, v ∈ V (1,9) . Similarly, let (1,16) be the intertwining operator of type L(1, 0) V (1,16) V (1,16) ,…”

“…for u ∈ V (1,16) , v ∈ V (1,9) be the intertwining operator of type V (1,9) V (1,16) V (1,9) , and I 2 (φu, z)φv = Q 2 • Y (φu, z)φv for u ∈ V (1,16) , v ∈ V (1,9) be the intertwining operator of type V (2,9) V (2,16) V (2,9) , where P 2 , Q 2 are the projections of V 1 and V 2 to V (1,9) and V (2,9) respectively. Then we have the following lemma.…”

“…This allows us to use the fusion rules for the irreducible modules for the Virasoro vertex operator algebra L(1, 0) obtained in [12] and [35] to understand the structure of both vertex operator algebras in terms of generators. The other one is that V A 4 L 2 is generated by a weight 9 primary vector x 1 and has a spanning set in terms Virasoro algebra, the component operators of x 1 and the 1 Supported bf NSF grants 2 Supported by China NSF grants 10931006,11371245, the RFDP of China(20100073110052), and the Innovation Program of Shanghai Municipal Education Commission (11ZZ18) component operators of y 1 which is a primary vector of weight 16. From the q-characters we know that V also has primary vectors x 2 , y 2 of weights 9, 16, respectively.…”