# Irrational Conformal Field Theory v.s. Non-Unitary Conformal Field Theory?

Unitary conformal field theories (CFTs) with irrational (or including the special case of rational) central charge is called irrational conformal field theory (ICFT).

Irrational conformal field theory (ICFT) is said to include all conformal field theories, and, in particular, ICFT includes rational conformal field theory (RCFT).

ICFT $$\supset$$ RCFT.

RCFT is only a tiny subset inside ICFT.

However, do we have examples of non-unitary Conformal Field Theory (NUCFT)? If so, what are the relations between:

• unitary RCFT
• unitary ICFT
• non-unitary RCFT
• non-unitary ICFT

Are there good quantum mechanical lattice realizations of these CFTs?

What are the relations of their sets?

• unitarity at $c<1$ implies rationality, no? – AccidentalFourierTransform Sep 5 '19 at 1:06
• Rational theories are, generally speaking, theories which have finite number of primaries of their respective chiral algebras (in the simplest case of Virasoro algebra, but one can have Kac-Moody algebras, etc.), although there may be people with stronger and/or more abstract opinions on this term. Also, there exist non-unitary theories, which can be rational or non-rational. I don't know whether anything useful can be said beyond these two facts. – Peter Kravchuk Sep 5 '19 at 2:01
• Also, your definitions are not consistent. You define ICFT to have irrational central charge, and then you say that RCFT is a special case of ICFT. It can't be since RCFT's have rational central charges. – Peter Kravchuk Sep 5 '19 at 2:03
• Minimal models and Liouville theory provide examples of rational and non-rational CFTs, which can be unitary or not depending on the central charge. See Wikipedia. – Sylvain Ribault Sep 5 '19 at 14:24