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Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$ and let $\hat{\mathfrak{g}} := (\mathfrak{g}[t^{\pm 1}] \oplus \mathbb{C} c) \rtimes \mathbb{C} D$ be the corresponding untwisted affine Kac-Moody algebra. Let $\mathfrak{h}$ denote a fixed Cartan subalgebra of $\mathfrak{g}$, and let $Q$ denote the root lattice of $\mathfrak{g}$. Denote the set Chevalley-Serre generators of $\hat{\mathfrak{g}}$ by $\{x_i^{\pm}, h_i\}_{i = 0}^{\dim \mathfrak{h}}$.

Set: $$V_Q := Sym(t^{-1}\mathfrak{h}[t^{-1}]) \otimes_{\mathbb{C}} \mathbb{C}Q$$ where $\mathbb{C}Q$ is the group algebra of $Q$.

Also, for each $x \in \mathfrak{g}$, let us write $x^{(m)} := x \otimes t^m$ and $x(z) := \sum_{m \in \mathbb{Z}} x^{(m)} z^{-m - 1}$.

In Kac's book "Infinite-dimensional Lie algebras", there is Theorem 14.8, which states that the basic representation of $\hat{\mathfrak{g}}$ (i.e. the unique simple $\hat{\mathfrak{g}}$-module with the first fundamental weight as its highest weight) can be specified by a map assigning the fields $x_i^{\pm}(z), h_i(z) \in \hat{\mathfrak{g}}[\![z^{\pm 1}]\!]$ (i \not = 0) to certain power series in $\operatorname{End}(V_Q)[\![z^{\pm 1}]\!]$ (so-called "vertex operators"). My understanding is that this is the same as the Frenkel-Kac construction of the basic representation of $\hat{\mathfrak{g}}$ via vertex operators. My questions are then:

  1. Why is $V_Q$ instead of any other representation the right one for establishing the Frenkel-Kac construction ? My impression is that, since $V_Q$ is the direct sum over all $\lambda \in Q$ of the vacuum modules of highest-weights $\lambda$ of the infinite-dimensional Heisenberg algebra associated to $Q$ (the modules denoted by $\pi_{\lambda}$ in "Vertex algebras and algebraic curves" by Frenkel and Ben-Zvi), and since we know how to realise these modules in terms of vertex operators, we would try to induce another vertex realisation using $V_Q$.

  2. What is the motivation for the construction of the vertex operators that the fields $x_i^{\pm}(z), h_i(z)$ get sent to (as in Theorem 14.8 of Kac's book) ? I have seen that they are modelled after vertex operators in bosonic string theory somehow, but I do not know enough physics to be able to understand the original construction on my own. Is there a representation-theoretic explanation ?

Thanks a lot!

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