Correspondance between free fermion CFT partition function and Virasoro character

Question

In these lectures on conformal field theory, the author calculates the partition function for the free fermion on a torus. By calculating the contributions from different boundary conditions, they arrive at

With this, they note

The final expression looks like the usual expression for a partition function in terms of the Virasoro characters, with each term counting the contribution from each primary field. These characters are traditionally defined as

$$\chi_{c, h>0}(\tau) = \frac{q^{h+(1-c)/24}}{\eta(\tau)}$$ $$\chi_{c, 0}(\tau) = (1-q) \frac{q^{(1-c)/24}}{\eta(\tau)}$$

where $\eta(\tau)$ is the Dedekind eta function and $q = e^{2\pi i \tau}$. However, these definitions of the Virasoro characters do not appear to match up with their definitions of $\chi_{0, 1/2, 1/16}$ as given in Eq. 5.68 - at least this not is what Mathematica gives. Are their definitions of $\chi$ entirely different? If so, what exactly goes wrong when we try to calculate the partition function using the Virasoro characters as given in the previous equation?

Answer 1

The character of a representation of the Virasoro algebra depends on the whole structure of the representation, not just of the conformal dimension $h$ of the highest-weight state. For a Verma module, the character is simply $$ \chi_{c,h}^\text{Verma}(\tau) = \frac{q^{h+\frac{1-c}{24}}}{\eta(\tau)}$$ as you wrote. However, for generic $c$ there are two highest-weight representations with $h=0$: the Verma module, and the degenerate representation, i.e. a quotient of the Verma module by a submodule generated by the singular vector. The character of the degenerate representation is then $$ \chi_{c,0}^\text{degenerate}(\tau) = (1-q)\frac{q^{\frac{1-c}{24}}}{\eta(\tau)} \neq \chi_{c,0}^\text{Verma}$$ Then, your free fermion has the central charge $c=\frac12$, and for certain rational values of $h$ there are infinitely many singular vectors in the Verma module, so there exist infinitely many highest-weight representations with the same $h$. The particular representations that you need are the maximally degenerate representations, and their characters are a bit complicated. You can find them in the textbook by Di Francesco et al. Or in my lecture notes https://arxiv.org/abs/1609.09523 , formula (A.27). The parameters are $(p,q)=(4,3)$ for $c=\frac12$, and $(r,s)=(1,1),(2,1),(1,2)$ for your three fields. (I should probably write these characters in Wikipedia one day, if nobody else does it.)