Unitary Representations in Conformal Field Theory

Question

So I am currently studying conformal field theory from the perspective of the representation theory of Lie algebras. I am trying to understand exactly why we care about unitarizable Verma modules. For reference, I am reading "Affine Lie Algebras and Quantum Groups" and "Symmetries, Lie Algebras, and Representations". By unitarizable, I mean a module that admits a Hermitian inner product such that $(E_{\pm}^i)^\dagger=E_{\mp^i}$ and $(H^i)^\dagger=H^i$ defines an adjoint on the Chevalley-Serre basis.

From what I understand, a unitary representation $V$ of a Lie group $G$ is a representation that admits an inner product satisfying $$(Uv,Uw)=(v,w)$$ for all $v,w \in V$ and $U \in G$. This implies that $U$ is unitary. Now, if $U$ is in some connected component of the identity of $G$, we can write it in terms of elements of the corresponding Lie algbera $\mathfrak{g}$ using the exponential map. Given a basis $\{J^a\}$ for $\mathfrak{g}$, we have $$U=\text{exp}\left(\sum_{a} \alpha_a J^a \right) \ \ \ \ \ U^{-1}=\text{exp}\left(-\sum_{a} \alpha_a J^a \right).$$ Since $U$ is unitary, we can conclude that $(J^a)^\dagger=-J^a$. Extending this to complex Lie algebras gives us the required adjoint on the Chevalley-Serre basis.

Now, to my questions. First, why do we care about unitary representations of Lie groups? Second, what exactly is more fundamental here; the representation of the Lie group or the representation of the Lie algebra? I have an issue with saying that these are equivalent. For example, Lie's third theorem only applies to finite-dimensional Lie algebras (not the Virasoro algebra of affine Lie algberas, which are very fundamental in CFT). From what I understand, we are also interested in infinite-dimensional (although irreducible) representations of Lie algebras.

Answer 1

We care about unitary representations of Lie algebras/groups because physical symmetries corresponding to unitary operators by Wigner's theorem. So if we know that a quantum theory should have a symmetry group $G$, then we know its space of states must furnish a unitary (or rather, projective, see this Q&A of mine for the mathematical details) representation of $G$.

In particular, if the algebra includes operators that physically are related to time evolution, such as the Virasoro algebra's $L_0$ that is essentially the Hamiltonian driving time evolution, then non-unitarity of the representation means a failure of the physical probabilities to always add up to 1, i.e. the theory is inconsistent.

Since we're looking at projective representations, it is "more fundamental" to care about representations of the Lie algebra, since this already covers non-linear representations induced from covering groups of the original group. Additionally, the Virasoro algebra has no unique associated group, although there are various candidates that are each slightly unsatisfactory, see e.g. Schottenloher's book on CFT for a discussion of this.