The paper [Fusion rules and modular transformations in 2D conformal field theory](http://www.sciencedirect.com/science/article/pii/0550321388906037) 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](https://en.wikipedia.org/wiki/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](http://ncatlab.org/nlab/show/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](http://math1.unice.fr/~beauvill/pubs/Kaciveli.pdf)



Verlinde Formula and Theta Function Identities
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](http://arxiv.org/abs/alg-geom/9609022), 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.


