The A-D-E classification of minimal and \(A_ 1^{(1)}\) conformal invariant theories
The Hilbert space H of a conformal field theory is a representation of two commuting copies of the Virasoro algebra. Write \(\{L_ n\}\), \(\{\bar L_ n\}\) \((n\in {\mathbb{Z}})\) for the generators of the two Virasoro algebras. The partition function of the theory is \[ Z(\tau)=tr_ H q^{L_ 0-c/24} \bar q^{-\bar L_ 0-c/24},\text{ where } q=e^{2\pi i\tau}. \] If the central charge \(c\in (0,1)\), the multiplicities of the irreducible components of H are finite; then Z is well-defined, is real- analytic on the upper half-plane, and is expressible as a sesquilinear form in the characters of the irreducible highest weight representations of the Virasoro algebra: \[ (*)\;Z(\tau)=\sum_{(h,\bar h)}N_{h\bar h} \chi_ h(q)\chi_{\bar h}(\bar q), \] where \((h,\bar h)\) runs over distinct pairs of highest weights for the central charge c and the multiplicities \(N_{h\bar h}\) are non-negative integers. It is an axiom of such theories that Z should be an \(SL_ 2({\mathbb{Z}})\)- invariant function in the upper half-plane. One can therefore ``classify'' such theories by determining all expressions (*), with \(N_{h\bar h}\in {\mathbb{Z}}_+\), which are modular invariant. This problem is solved in the present paper. The surprising result is that there is a one-to-one correspondence between such theories and the simple complex Lie algebras of types A, D or E (omitting a few low rank cases). The precise role of these algebras in conformal field theory seems not to be understood at present, but data associated to the algebras appear in the corresponding theory. For example, the central charge of the theory is simply related to the Coxeter number of the corresponding algebra. The authors also consider the problem of finding all modular invariant expressions (*), with \(N_{h\bar h}\in {\mathbb{Z}}_+\), but with the \(\chi\) 's being characters of the irreducible highest weight representations of the affine algebra of \(s\ell_ 2\). Again an A-D-E classification is obtained.
simple complex Lie algebras of types A, D or E
scientific article; zbMATH DE number 4042123
scientific article