Verlinde Formula and Theta Function Identities

Question

The paper Fusion rules and modular transformations in 2D conformal field theory by Erik Verlinde mentions a simple case of rational conformal field theory, where the fusion algebra is just $(\mathbb{Z}_p,+)$:

$$ \phi_p \times \phi_{p'} = \phi_{p + p'}$$

Through some mysterious physics logic he obtains an identity relating various characters of the Virasoro algebra:

$$ S: \chi_p \mapsto \sum_{p'\in \mathbb{Z}_n} e^{\frac{2\pi i pp'}{N}}\chi_{p'}$$

In this particular case, the characters are just theta functions, but Verlinde doesn't specify which ones. The partition function is:

$$ Z = \mathrm{tr}[q^{L_0 - \frac{1}{24} }] = \frac{1}{\eta(q)}\sum q^{\frac{1}{2}\left(\tfrac{n}{R}+ \frac{1}{2}mR\right)^2} \tag{$\ast$}$$

The characters are given as partial traces over the different primary fields:

$$ \chi_i = \mathrm{tr}_{[\phi_i]}[q^{L_0 - \frac{1}{24} }]$$

The exercise for now is to re-write $(\ast)$ as an identity of theta functions and prove it for this very special case. How is it theta functions transform as the Discrete Fourier Transform

The trouble is I don't know how the traces for $\chi_i$ were computed. Also, the series Verlinde generates even more general than theta functions and can be defined for general Riemann surfaces, so I have been to understand them.

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I was able to find work of Arnaud Beauville where he discusses the Verlinde formula which interprets this as the dimension of a line bundle over the moduli space of curves. Still trying to understand if this may be related, but I sense this particular case should be known.

• Arnaud Beauville, Vector bundles on Riemann surfaces and Conformal Field Theory

Answer 1

For any integer lattice $L$, you can write a theta function $\theta_L$ as a generating function for lattice vectors of a given norm. That is, $$\theta_L(\tau) = \sum_{a \in L} q^{(a,a)/2}.$$ The quotient $\theta_L(\tau)/\eta(\tau)^{\textrm{rank} (L)}$ is the partition function for the lattice model vertex algebra (and the CFT by the same name), because there is a boson of momentum $a$ corresponding to each vector $a$ in the momentum lattice, and the space of states of an $n$-dimensional boson has partition function $\eta^{-n}$. This last identification is standard quantum mechanics - take the symmetric algebra on the harmonic oscillator space.

The lattice vertex algebra has modules parametrized by cosets of $L$ in the dual lattice $L^\ast$, which can be viewed as the set of vectors in $L \otimes \mathbb{R}$ satisfying $(v, \ell) \in \mathbb{Z}$ for all $\ell \in L$. These modules have partition functions given by $\chi_v = \theta_{L + v}(\tau)/\eta(\tau)^{\textrm{rank} (L)}$, where $$\theta_{L+v}(\tau) = \sum_{a \in L + v} q^{(a,a)/2}.$$ The theta functions assemble into a vector-valued modular form for the Weil representation, of weight equal to the rank of $L$ (one proof is Lemma 4.1 in Borcherds's paper Automorphic forms with singularities on Grassmannians, where you plug in $p=1, \alpha = \beta = 0$). This fact is one way to encode the discrete Fourier transform, and in general, transformation rules of theta functions can be proved using Poisson summation.

For your case, $L$ is the rank 1 lattice generated by a single vector $v$ satisfying $(v,v)=p$.

Beauville's work concerns WZW models, and they are related to this question, both by the Frenkel-Kac construction of level 1 representations from lattices, and because lattices are built from $U(1)^n$ models.