Two appearances of the Jacobi Triple Product identity

So far, I've seen two ways of deriving the Jacobi Triple Product (JTP) formula $$\sum_{m \in \mathbb{Z}} (-u)^m q^{\frac{m(m-1)}{2}}= \prod_{j=0}^{\infty} (1-uq^{j})(1-q^{j+1})(1-u^{-1} q^{j+1})$$ using infinite-dimensional representation theory. On the one hand, it is the Weyl-Kac denominator formula for $\widehat{\mathfrak{sl}_2}$. On the other hand, it is the character $$\text{Tr}(q^{L_0} u^{\alpha_0})$$ computed in two ways using the boson-fermion correspondence. Here $$\alpha_0 = \sum_{i \in \mathbb{Z}'} : \psi_i \psi_i^* :$$ is the zero mode of the Heisenberg algebra which computes the charge of the Dirac sea of free fermions, and $$L_0 = \sum_{i \in \mathbb{Z}'} i : \psi_i \psi_i^* :$$ is the energy operator for a particular representation of the Virasoro algebra on this semi-infinite wedge space.

• Is there a structural relationship between these two appearances of JTP?
• I suspect the answer will turn out to be yes, but while waiting for an expert to confirm this you might want to add a reference for the second approach. (And maybe read all the papers having Victor Kac or Jim Lepowsky as an author.) – Jim Humphreys Feb 26 '13 at 22:45
• The second approach is essentially the proof given in en.wikipedia.org/wiki/Jacobi_triple_product – Alexander Moll Feb 26 '13 at 22:57
• Also, I'm hoping to use this observation as an opportunity to start investigating such papers - I'm just not sure where to begin! – Alexander Moll Feb 26 '13 at 23:05
• I've never quite penetrated the physics language, but there is an intriguing book (with a Japanese translator) Infinite-Dimensional Lie Algebras by M. Wakimoto, in an AMS translation series (2001). The language tends to be quirky and informal. The Jacobi identity arises from affine Lie algebra representations in Chapter 3; then in Chapter 6 some connections are worked out with the Virasoro algebra. One quote, "In the mid-1980's, The Virasoro algebra fell in passionate love with affine Lie algebras." But I don't see an explicit connection made with Jacobi identity there. – Jim Humphreys Feb 27 '13 at 18:48