3-State Potts Model and Automorphisms of Vertex Operator Algebras of Order 3

We study the fusion rules of a vertex operator algebra W(0)+R, which is a VOA over the real number field R and has a positive definite invariant bilinear form, and such that its complexification CW(0)+R is a direct sum of the 3-state Potts model L(45,0) and its module L(45,3). As an application, we define an automorphism of order 3 of a VOA VR over R if VR contains W(0)+R as a sub VOA.