If by 'Ising CFT' I mean the conformal field theory describing the critical quantum Ising chain $ H = \sum_n \left( \sigma^z_n - \sigma^x_n \sigma^x_{n+1} \right)$ and by 'Majorana CFT' I mean the conformal field theory describing its Jordan-Wigner transform (or for field theory enthusiasts, $S \propto \int \mathrm d^2 x \; \left( \chi \tilde \partial \chi + \tilde \chi\partial \tilde\chi \right)$ ), is it true that despite both of these being CFTs with $c= \frac{1}{2}$ they are in fact different CFTs?

Clearly anything that can be calculated for one can be calculated in the other language (since they map to each other), however it seems their physics is quite different (in the same way that the symmetry breaking quantum Ising chain maps to the topological Kitaev chain). In particular, the Ising CFT has the three primaries $1,\epsilon,\sigma$ with respective conformal dimensions $(0,0),\left( \frac{1}{2}, \frac{1}{2} \right)$ and $\left( \frac{1}{16}, \frac{1}{16} \right)$. On the other hand the Majorana CFT has the three primaries $1,\chi,\tilde \chi$ with respective conformal dimensions $(0,0),\left( \frac{1}{2}, 0 \right)$ and $\left( 0, \frac{1}{2} \right)$. It is true that I can write the primaries of one as non-local operators in the other (e.g. the $\sigma$ of the Ising CFT can be written as a stringy object in the fermionic language), but since primaries are by definition local objects, I do not call those non-local objects primaries, correct?

I do not want to make this an issue of semantics, but rather of physics. I would like to get confirmation (or refutation) of the physical difference between these two CFTs. In particular I am wondering to what extent I should (not) consider the $\sigma$ operator a primary in the Majorana CFT. Two possible physical criteria come to mind:

  1. If I do finite-size scaling of the critical Ising chain, by looking at the energy spectrum I can e.g. extract the $\frac{1}{16} + \frac{1}{16} = \frac{1}{8}$ scaling dimension. It is my understanding that I a finite-size scaling of the critical Majorana chain would not give that scaling dimension. This would be an objective criterion for saying the primaries of both CFTs are distinct.
  2. If I would for example look at something like $\textrm{tr} q^{L_0}$ for the Majorana CFT, would there be a $\frac{1}{16}$ contribution? It is my understanding that if I look at the partition function (a related but slightly different object) of the Majorana CFT, then depending on the boundary conditions of the fermions, I do (not) get that contribution. In particular if I take periodic boundary conditions in space and time for my Majorana, then modular invariance does not imply the presence of that $\frac{1}{16}$ conformal dimension. The part I am not sure about: are those boundary conditions the ones that are natural if I am purely living on the fermionic side? (On first eye it would seem so, but my reading of Di Francesco et al. [for those curious: section 10.3, p346] seems to imply that antiperiodic boundary conditions in time are natural for fermions due to time-ordering, then again they do not say it in those words, so my reading might very well be off!)

I would say that while the Ising CFT is a CFT of the usual sort, the Majorana CFT is a more refined object which can be examined in any spin structure on the spacetime surface. The two are related by bosonization. That is, the Ising CFT is obtained from the Majorana CFT by summing over all possible spin structures, weighted by the Arf invariant. This means that different states (equivalently vertex operators) of the Ising CFT states may be in different spin structure sectors (antiperiodic or periodic) of the Majorana CFT. I believe this relationship between partition functions can be found in the big yellow book. Anton Kapustin and I also wrote a bit about this in this paper(pdf) starting on page 4.

Update: I wrote about the bosonization of the free Majorana and Dirac fermions in 1+1D in great detail in a recent paper, which you can read on the arxiv.

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    $\begingroup$ Sorry for bumping an old question, but it's not obvious to me how the Ising and Majorana CFTs are related by the JW transformation. Naively, taking the Ising model in its discrete form and performing a JW transformation, we obtain a Majorana theory that's periodic /only/ when the fermion parity is even, and antiperiodic /only/ when the fermion parity is odd. But somehow when we go to the continuum CFTs, this distinction between even/odd parity is lost and we just sum over all different spin structures? I'm having difficulty understanding rigorously how that happens. $\endgroup$ – Henry Shackleton Jul 31 '19 at 21:07
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    $\begingroup$ Hi @HenryShackleton you can look here arxiv.org/abs/1701.08264 at section 2.1 to see how the sectors are supposed to match. You should be able to derive this from JW. $\endgroup$ – Ryan Thorngren Aug 1 '19 at 11:00
  • $\begingroup$ Thanks - I think I'm still having trouble seeing the correspondence between the two models. If we sum over all different spin structures in the fermion theory, then according to this paper, we're summing over both the $\mathbb{Z}_2$-twisted and $\mathbb{Z}_2$-untwisted sector of the bosonic theory? Does this mean that the Ising CFT includes both periodic and anti-periodic boundary conditions on the quantum Ising chain? $\endgroup$ – Henry Shackleton Aug 1 '19 at 15:09
  • $\begingroup$ Additionally, it says that the bosonic $\mathbb{Z}_2$-untwisted sector corresponds to the NS and R sectors with even fermion parity, which is not what the JW transformation does in the quantum Ising chain - the coupling between the last and the first fermion after the JW transformation is $\exp(i \pi \sum_i c_i^\dagger c_i)$, which imposes periodic boundary conditions if fermion parity is even, and anti-periodic boundary conditions if fermion parity is odd. Using the notation from the paper, I would have expected that $\mathcal{B}_0 = \mathcal{F}_R^+ \oplus \mathcal{F}_{NS}^-$. $\endgroup$ – Henry Shackleton Aug 1 '19 at 15:12
  • $\begingroup$ @HenryShackleton I think maybe what's confusing is that what looks periodic in the spin operators is actually antiperiodic for the fields. Indeed, recall that even away from any boundaries, around a small circle bounding a disc a fermion field has to have antiperiodic BC. The fermion parity of the ground state of a periodic spin-structure Kitaev wire should be odd, for example. $\endgroup$ – Ryan Thorngren Aug 1 '19 at 18:46

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