I've been following Ginsparg's notes (http://arxiv.org/abs/hep-th/9108028) and Francesco et al's big yellow book on modular invariance of CFT on a torus, particularly on the $c=1/2$ (free fermion) case.

I thought I understood the derivation of the modular-invariant partition function, but then I realized I did not. For example, in Ginsparg's notes, the computation of the partition function relies on the mode expansion of the free fermion field (let's say antiperiodic case) $$\psi(z)=\sum_{n\in\mathbb{Z}} b_n z^{-n-1/2},$$ where $[L_0, b_{-n}]=nb_n$, and then the partition function is just summing over the $b_{-n}|0\rangle$ states. Taking this, I can follow the rest of his derivation.

However, this mode expansion is for a plane (which can be transformed into a cylinder, but not into a torus) -- it is only anti-periodic (periodic if you drop the $1/2$) in one direction (here angular direction). The above expansion can never satisfy the periodic or anti-periodic boundary condition in the other direction, which in $z$ plane is the radial direction.

So to be specific, when Mr Ginsparg was calculating the Virasoro character for (A,A) sector (note the second A), say in Eq. (7.13 a), what are his eigenbases for $L_0$?

  • $\begingroup$ Is there a precise equation you don't understand (eg in the yellow book) ? $\endgroup$
    – Antoine
    Mar 31, 2017 at 13:47
  • $\begingroup$ @user40085 Eq. (7.13 a) in Ginsparg's notes, or the counterpart in the big yellow book. $\endgroup$ Mar 31, 2017 at 15:23

1 Answer 1


First of all, it is not correct to say that the plane can not be transformed into a torus. This is precisely what the trace in equation (7.13a) does. Let me be more precise. When you compute the partition function for the free fermion on the torus, you need to specify two kinds of periodicity.

  • The first one is the "spatial" periodicity, if you think of the CFT as a worldsheet CFT in string theory. It is the periodicity along the angular direction, if you think on the plane. Depending on this, $L_0$ takes two slightly different forms:

    1. If the fermion is periodic along the angular direction on the plane (this is the Ramond condition), then $$L_0^R = \frac{1}{24} + \sum\limits_{m=1}^{\infty} m b_{-m} b_m$$
    2. If the fermion is anti-periodic along the angular direction on the plane (this is the Neveu-Schwarz condition), then (here the sum runs over half-integers $r$) $$L_0^{NS} = -\frac{1}{48} + \sum\limits_{r=1/2}^{\infty} r b_{-r} b_r$$
  • The second periodicity that you have to choose is the "temporal" periodicity (in the worldsheet point of view), or radial direction on the plane. This is imposed by the way you take the trace, denoted by $\mathrm{tr}_P$ or $\mathrm{tr}_A$ in Ginsparg's notes, and it boils down to adding a $(-1)^F$ for periodic conditions, and nothing for anti-periodic conditions.

Now you can compute, say, the partition function on the torus with anti-periodic conditions in the two directions, as in (7.13a). So you need no $(-1)^F$, and you have to use $L_0^{NS}$. This gives $$\mathrm{tr} \, q^{L_0^{NS}} = q^{-1/48} \prod\limits_{r=1/2}^{\infty} \sum\limits_{N_r \in \{0,1\}} q^{rN_r} = q^{-1/48} \prod\limits_{r=1/2}^{\infty} (1+q^r) = \frac{\theta_3^4}{\eta} \, . $$

  • $\begingroup$ Thanks, I agree. My initial puzzle is that the mode expansion of $\psi$, which is also the definition of $b_m$'s, cannot satisfy the boundary condition in the temporal direction. Now I realize this is a mode expansion for an infinite cylinder, and the antiperiodic/periodic condition along temporal direction is realized by taking the trace (or trace with (-)^F) on a finite chunk of that cylinder. In short, the idea is not to directly find the eigenmodes of the fermion field kernel that is (anti)-periodic in both directions. I do think it is possible to do it that way though. $\endgroup$ Mar 31, 2017 at 16:37

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