Invariant skew-symmetric differential operators on the line and Verma modules over the Virasoro algebra

Conformal boundary states for free bosons and fermions

“…. ; in this case, the Virasoro representation has a single null-vector at level 2j + 1 [9]. † The corresponding characters are then given by …”

“…In order to describe the set of Ishibashi states for a general radius of the above form, it is convenient to look at the self-dual case M = N = 1 first (see also [10,6,11]). In this case, the momenta are of the form 9) and the corresponding Virasoro highest weights are given by provided that both |m| and |n| are less or equal than j and that j−m (and j−n) is integer. Conversely, if (m, n) satisfy these constraints, the representation H Vir j ⊗ H Vir j appears precisely once in the corresponding U(1) × U(1) representation.…”

Short Distance Expansion from the Dual Representation of Infinite Dimensional Lie Algebras

“…We note that just as in the case of Virasoro theories and some other conformal field theory based operators, the central extension can be further restricted by demanding unitarity. Choices of the central extension can be determined through the Kac determinant [18,13,11]. This viewpoint has been applied to the model-independent N -extended supersymmetric GR Virasoro algebra [15] to obtain for the first time its representation in terms of short distance expansions.…”

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“…This sum in this result is obviously the difference of two Jacobi theta functions and thus we see that all the RSOS models lead to theta functions. But most remarkably the exact same expression (10) was discovered at the same time to arise in the expression of the characters [33]- [34] of the minimal models M(p, p ′ ) conformal field theory [35] and these models were soon thereafter obtained as cosets [36] of the affine Lie algebra A…”