Twisted Verlinde Formula for Vertex Operator Algebras: The Prime Case

For a rational and \(C_2\)-cofinite vertex operator algebra V with an automorphism group G of prime order, the fusion rules for twisted V-modules are studied, a twisted Verlinde formula which relates fusion rules for g-twisted modules to the S-matrix in the orbifold theory is established. As an application of the twisted Verlinde formula, a twisted analogue of the Kac-Walton formula is proved, which gives fusion rules between twisted modules of affine vertex operator algebras at positive integer levels in terms of Clebsch–Gordan coefficients associated to the corresponding finite dimensional simple Lie algebras.