As I understand, there are CFTs with negative central charge $c<0$. It is said that such theories break unitarity.

I think that the most famous theory with $c<0$ is ghost CFT with $c=-26$ (but I dobt sure that such theory breaks unitarity).

But there are a lot of other CFT with negative $c$, and such theories appear in the percolation problem (symplectic fermions, $c=-2$). But as I understand such theories are very very hard.

Why are $c<0$ CFTs interesting? What makes them hard?

  • $\begingroup$ In what sense is it hard? What problem are you trying to solve using such CFTs and what precise calculation turns out to be hard? $\endgroup$
    – Prahar
    Apr 24 '21 at 10:22
  • $\begingroup$ @PraharMitra for example spectr of relevant operators or critical indices $\endgroup$
    – Nikita
    Apr 24 '21 at 11:24
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    $\begingroup$ What is so special about $c<0$ that should influence the interest and difficulty of CFTs? For minimal models, whether $c<0$ or not does not make a big difference. $\endgroup$ Apr 24 '21 at 20:13
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    $\begingroup$ But why do you ask about $c<0$ rather than say $c<1$ or some other interval? Do you have a reference saying something special happens? Or a calculation showing a qualitative difference depending on the sign of $c$? If not, the answer is simply "CFTs with $c<0$ are just as interesting and as hard as CFTs with other central charges". $\endgroup$ Apr 24 '21 at 20:37
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    $\begingroup$ from a string theory perspective you can add various cft’s with all sorts of central charges and still get a consistent theory provided the total central charge vanishes. so studying the various possibilities enables one to explore the string theory landscape. eg many of these will be connected by RG flow, so one can, eg, start speaking about emergence of space and all sorts of interesting things. $\endgroup$ Apr 24 '21 at 21:43

First, allow me to explain why CFTs with negative central charge are nonunitary. The Virasoro algebra of a CFT with central charge $c$ is


Then the norm of $L_{-n}|h\rangle$ is

$$\langle h|L_nL_{-n}|h\rangle=\langle h|L_{-n}L_n+2nL_0+\frac{c}{12}n(n-1)(n+1)|h\rangle=(2nh+\frac{c}{12}n(n-1)(n+1))|h\rangle.$$

So when $n$ is large enough, this norm is negative, i.e. nonunitary.

I met these CFTs, bc-CFT, linear dilaton CFT, etc. in string theory. In string theory, people want to cancel the Weyl anomaly. And this anomaly is canceled iff the central charge of the whole CFT on the worldsheet is 0. However, the central charge of strings is positive. At this crucial time, the ghost fields, which are introduced to cancel the overcounting of string states, cure this. So from a string theory perspective, the negative central charge is not a problem at the beginning (they are not real fields!). And it is the negative central charge that makes the total central charge zero and string theory a self-consistent theory.

  • $\begingroup$ Thank you, this is well-known information. What do we know about other non-unitary CFT? $\endgroup$
    – Nikita
    May 22 '21 at 17:34
  • $\begingroup$ I heard that the Ising model near the Yang-Lee singularity is nonunitary. However, this is again a speculated model because the magnetic field is imaginary in the case of Yang-Lee singularity. I guess one can gather more intuition in condensed matter physics. But unfortunately, I do not know more. $\endgroup$
    – Youran
    May 22 '21 at 17:48

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