# Different views on Highest weight irreducible modules of the Virasoro algebra

Every highest weight irreducible representation of the Virasoro algebra can be labelled uniquely by a pair $$(c,h)$$ of complex numbers [1]. This module can be written as quotient of the unique (up to iso) Verma module with highest weight $$(c,h)$$.

Since a coset model $$G/H$$ is a highest weight irreducible Vir-module, then it can be uniquely labelled by such a pair $$(c,h)$$.

On the other hand, a coset model $$G/H$$ can also be labelled (non-uniquely) by $$(\Lambda,\lambda)$$, where $$\Lambda$$ and $$\lambda$$ are highest weights of $$\mathfrak g$$ and $$\mathfrak h$$, respectively [2].

The Question:
Is there a way to write $$h$$ and $$c$$ in terms of $$\Lambda$$ and $$\lambda$$? What about a way to write $$\Lambda$$ and $$\lambda$$ in terms of $$h$$ and $$c$$?
Finally, I know that $$h$$ can be written in terms of the central charge $$c$$ for the case $$c<1$$ [3]. Is it possible to write $$h$$ in terms of $$c$$ for when $$c \ge 1$$?
(I am particularly interested in the coset models $$\frac{\widehat{S U}(n+1)_{k} \times \widehat{S O}(2 n)_{1}}{\widehat{S U}(n)_{k+1} \times \widehat{U}(1)_{n(n+1)(k+n+1)}}$$, and even more specifically in the case $$n=2$$).

[1] Kac, V. G., Raina, A. K., and Rozhkovskaya, N. (2013). Bombay lectures on highest weight representations of infinite dimensional Lie algebras, volume 29. World scientific.

[2] Nozaki, M. (2002). Comments on d-branes in kazama-suzuki models and landau-ginzburg theories. Journal of High Energy Physics, 2002(03):027.

[3] Goddard, P., Kent, A., and Olive, D. (1986). Unitary representations of the Virasoro and super-Virasoro algebras. Communica- tions In Mathematical Physics, 103(1):105–119.

• -1 for the confused writing. Please distinguish more clearly models and modules, with the central charge or the level of a model on the one hand, and the conformal dimension or the highest weight of a module on the other hand. Aug 26, 2019 at 18:22
• @SylvainRibault I honestly do not know what you mean. Models here are "coset models", a construction due to GKO of Virasoro representations. They have a certain central charge and highest weight. Do you know what a coset model is? (In conformal field theory)
– Soap
Aug 26, 2019 at 20:03