Conformal defects

Question

 I have a few questions on conformal defects in 2d or (1+1)-d CFTs. Relevant references are [1],[2],[3].

To summarize: a CFT can host conformal defects. These can be thought of as cutting the space-time surface along the defect line and rejoining the two sides by an appropriate boundary condition. The boundary condition is a prescription on how the bulk fields behave when they cross the cut. Demanding preservation of conformal symmetry implies that the conformal defects also fall into representations of the Virasoro algebra, so, they can be labelled by the primary fields in the CFT.

For example, the $c = 1/2$ Ising CFT has 3 defects: $D_I, D_{\sigma}, D_{\epsilon}$.

This can be realized in a microscopic model: if you take the Ising model on spins on a ring,

$$H = -\sum_{i=1}^{N} ( \sigma^z_i \sigma^z_{i+1} + \sigma_i^x)$$

with periodic boundary conditions, $\sigma_{N+1} = \sigma_1$, then in the continuum limit this becomes the $c=1/2$ CFT with the insertion of the trivial defect $D_I$, which gives a partition function (on a torus) consisting of diagonal pairings of holomorphic and antiholomorphic representations of the Virasoro algebra; we get the primary fields $(0,0), (1/2,1/2), (1/16, 1/16)$ which correspond to the identity $I$, the energy density $\epsilon$ and spin $\sigma$ primary operators respectively. This partition function is modular invariant on the torus.

If instead we choose antiperiodic boundary conditions $\sigma_{N+1} = -\sigma_1$, then we get a twisted partition function with the combinations $(1/16,1/16), (1/2,0), (0,1/2)$ which correspond to the disorder $\mu$ and the fermions $\psi, \bar{\psi}$ operators. This is the CFT with a $D_\epsilon$ defect. This partition function is modular invariant modulo a symmetry twist.

It turns out that there is one more possible way of twisting the partition function, you put in a duality (Kramers-Wannier) defect, which gives $(1/16,0),(1/16,1/2)$ and $(0,1/16),(1/2,1/16)$ primary fields. This partition function is modular invariant modulo this duality twist and is the CFT with a $D_\sigma$ defect. You can write down a duality-twisted lattice Ising model on spins which has in the continuum limit has these primary fields.

In this way, by inserting defects, we can recover all possible combinations of the $m=3$ Kac table.

My questions:

1) Should we think of the defected theories, namely the antiperiodic and duality-twisted theories as $c=1/2$ CFTs? That is, do we say that they fall under the same universality class as the 2D Ising CFT? Or are they strictly not CFTs in the usual sense because they are really 'defective'? Does it even make sense to assign a central charge $c$ to them?

2) There are many ways of pairing the holomorphic and antiholomorphic representations of the Virasoro algebra. But this pairing has to be consistent; modular invariance provides one such constraint. It turns out that the diagonal pairing (e.g. $(0,0), (1/2,1/2), (1/16,16/)$) in any CFT is always modular invariant (there might be off diagonal ones), and if the model has additional symmetries one can get extra pairings up to the twist due to the symmetries. (Hence this gives the additional pairings $(1/16,1/16),(1/2,0),(0,1/2)$ and $(1/16,0),(1/16,1/2), (0,1/16), (1/2,1/16)$ in the Ising CFT). All these are given by inserting the possible defects in the theory. Thus it appears to me that these should be all the pairings I can get (following the papers I referenced above); however if I consider a bunch of $2N$ Majorana fermions on a ring,

$$H = -\sum_{i=1}^{2N} i \chi_{i} \chi_{i+1}$$

with antiperiodic boundary conditions $\chi_{2N} = -\chi_{1}$, then I will see the primaries $(0,0), (1/2,1/2), (1/2,0),(0,1/2)$. But this is not any of the partition functions twisted by the local symmetry or a duality symmetry, and is not due to the insertion of any defects, so why is this combination allowed? (Perhaps I am misunderstanding the importance of modular invariance.)

Put it in another way, would you say that this model of $2N$ Majorana fermions in the continuum limit is described by a CFT? If so, what CFT? 

Edit after thinking about how to rephrase qn 2):

One way to define a CFT non-perturbatively [4],[5] is in specifying the list of operators in a theory and all its 3 point OPE coefficients: e.g. $\{ (0,0), (\frac{1}{2},0), (0,\frac{1}{2}), (\frac{1}{16},\frac{1}{16}) , \cdots, (2,0), (0,2), \cdots \}$ and all the $c_{ijk}$s. We can restrict to just the primary fields.

Now if I say $c = 1/2$, then automatically I get a finite set of conformal weights, $(0, \frac{1}{2}, \frac{1}{16})$, representing both the possible holomorphic and antiholomorphic representations of the Virasoro algebra. So all the possible primaries on the plane will be the set $(0, \frac{1}{2}, \frac{1}{16})$ tensor producted with $(0, \frac{1}{2}, \frac{1}{16})$.

Now, is there a constraint that restricts which pairing can happen? I thought that the restriction was modular invariance modulo insertion of conformal defects - which gives only 3 possible types of pairings: $a) ( (0,0), (1/2,1/2), (1/16, 1/16))$; $b) ((1/16,1/16),(1/2,0),(0,1/2))$; $c) ((1/16,0),(1/16,1/2),(1/2,1/16),(0,1/16))$ but the example of a free fermion system (with either NS or Ramond sectors) gives more pairings.

So, firstly would I say $a), b), c)$ are not the same CFTs (from my non-perturbative definition of CFTs), and secondly what pairings can happen?

Comments

The mapping from the Majorana fermions on a ring to a spin chain is subtle: if you perform a Jordan-Wigner transformation, the boundary conditions of the Ising model you get depends on the $Z_2$ global fermion parity of the Majorana fermions. This should explain the discrepancy you mentioned.

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Yes of course. I know how to relate the spin model charge sectors to the fermion model charge sectors. Everything is consistent and works out if you do a JW transformation. But I'm asking more from the point of view of defects - how can I understand the fermion model in terms of the usual CFT language with insertion of defects $D_I, D_\epsilon, D_\sigma$? Insertion of the defects don't give me the spectrum I see; put it another way, I expect only three kinds of partition functions, yet this combination is not in any of the three (maybe my expectation is just misfounded). Also, would you say the fermion model in the continuum limit is in the "universality class" of the 2D Ising CFT?

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The fermion model includes both $D_I$ and $D_\epsilon$ depending on global $Z_2$ fermion parity. This is the answer to your question.

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That is true, thank you. My main question (see edited post), I guess, is actually: is there any constraint that restricts what kind of operator content appears in a theory? A side qn whose answer is perhaps just definitions/semantics: would you say that the free fermion with NS is the 'same' CFT as the free fermion with Ramond boundary conditions? After all, they have different operator content.

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The constraint on the operator content is modular invariance on torus, and as you already discussed in the question this basically determines how the holomorphic/antiholomorphic sectors are glued. For fermions there is an additional complication of spin structure and global fermion parity. But after all, boundary conditions are only visible on a finite system (for example, if you look at the energy levels on a finite size system), in the thermodynamic limit correlation functions do not care about the boundary condition, so as far as local properties are concerned they are the same CFT.

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I see. But the notion of locality differs between the spin model and the fermionic model, for example the spin order parameter is, I believe, a non-local field in terms of the fermions - thus, the correlation functions of local observables might be different..? What I mean is if I'm only allowed to measure local fields I might be able to distinguish between a spin and a fermion system, even in the thermodynamic limit. (what I'm saying might be bs, though)