In Sec. 8.7 of this Applied Conformal Field Theory, Ginsparg discusses the space of $1+1$ dimensional CFTs with central charge $c=1$, and the summary is in figure 14. Essentially there are two continuous families of such CFTs, one along the horizontal axis and the other along the vertical axis, together with 3 other theories labeled by T, O and I in the figure.

The theories along the horizontal axis are free compact bosons on a circle, which in the condensed matter terminology are also known as Luttinger liquids. It is known that all these theories have a common internal unitary $(U(1)_1\times U(1)_2)\rtimes Z_2$ symmetry. If we denote the compact boson by $\theta$ and its dual variable by $\phi$, the actions of the $(U(1)_1\times U(1)_2)\rtimes Z_2$ symmetry are

\begin{equation} \begin{split} &U(1)_1: \theta\rightarrow\theta+\alpha, \phi\rightarrow\phi,\\ &U(1)_2: \theta\rightarrow\theta, \phi\rightarrow\phi+\alpha,\\ &Z_2: \theta\rightarrow-\theta, \phi\rightarrow-\phi \end{split} \end{equation}

For these theories, there can be enlarged symmetry when the compactification radius takes special values. For example, in figure 14 if the compactification radius is $1/\sqrt{2}$, the theory has an $SO(4)$ symmetry.

My question is: What is the internal unitary symmetry that is in common for all theories along the vertical axis?

As I know, an example of the theory in the vertical axis is two decoupled Ising CFTs, whose internal unitary symmetry seems to be $Z_2\times Z_2\times Z_2$, where two of the $Z_2$'s are the $Z_2$ symmetry of each individual Ising CFT, and the last $Z_2$ exchanges the two copies of Ising CFTs. Do all theories along the vertical axis have a $Z_2\times Z_2\times Z_2$ symmetry? If they do, do they have an even larger internal unitary symmetry that is in common?

My other question is: What are the symmetries of the T, O and I theories?

  • $\begingroup$ Related: physics.stackexchange.com/q/27763/2451 $\endgroup$
    – Qmechanic
    Apr 30, 2023 at 14:59
  • $\begingroup$ @Qmechanic Thank you for the comments. I have to admit that I am not an expert in CFT, and I am not able to read off the symmetry of the $c=1$ CFTs from the related question. Could you be more explicit about these symmetries? A more specific question is: Is it correct that most of the $c=1$ CFTs not on the horizontal axis have no continuous internal unitary symmetry? $\endgroup$ May 1, 2023 at 12:52


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