: An analogue of the classical Frobenius-Schur indicator is introduced in order to distinguish between real and pseudo-real self-conjugate primary fields, and an explicit expression for this quantity is derived from the trace of the braiding operator.

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Explicit formulae describing the genus one characters and modular transformation properties of permutation orbifolds of arbitrary Rational Conformal Field Theories are presented, and their relation to the theory of covering surfaces is investigated.

A general theory of vector-valued modular functions, holomorphic in the upper half-plane, is presented for finite-dimensional representations of the modular group. This also provides a description of vector-valued modular forms of arbitrary half-integer weight. It is shown that the space of these modular functions is spanned, as a module over the polynomials in J, by the columns of a matrix that satisfies an abstract hypergeometric equation, providing a simple solution of the Riemann-Hilbert problem for representations of the modular group. Restrictions on the coefficients of this differential equation implied by analyticity are discussed, and an inversion formula is presented that allows the determination of an arbitrary vector-valued modular function from its singular behavior. Questions of rationality and positivity of expansion coefficients are addressed. Closed expressions for the number of vector-valued modular forms of half-integer weight are given, and the general theory is illustrated on simple examples.

A general theory of permutation orbifolds is developed for arbitrary twist groups. Explicit expressions for the number of primaries, the partition function, the genus one characters, the matrix elements of modular transformations and for fusion rule coefficients are presented, together with the relevant mathematical concepts, such as Lambda-matrices and twisted dimensions. The arithmetic restrictions implied by the theory for the allowed modular representations in CFT are discussed. The simplest nonabelian example with twist group S_3 is described to illustrate the general theory.Comment: 10 page