Q:

Correspondance between free fermion CFT partition function and Virasoro character

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? 

A:

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.) 