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Rationality in conformal field theory. (English) Zbl 0647.17012

The Hilbert space of a conformal field theory (CFT) is a positive-energy representation of the direct sum of two commuting copies of the Virasoro algebra, and hence decomposes \[ (*)\quad H=\oplus_{h,\bar h\geq 0}V(h,c)\oplus \bar V(\bar h,c) \] where V(h,c) is the irreducible representation of the Virasoro algebra with highest weight h and central charge c, the multiplicity \(N_{h\bar h}\in Z_+\) and the bar refers to the second copy of the Virasoro. The partition function \(Z=\sum_{h,\bar h\geq 0}N_{h\bar h}\chi (h,c){\bar \chi}(\bar h,c)\) where \(\chi\) (h,c) is the character of V(h,c), is holomorphic in the unit disc. The CFT is said to be rational if the matrix \((N_{h\bar h})\) is of finite rank, and modular invariant if Z is modular invariant. CFT’s which are modular invariant and rational include the unitary discrete series and Wess- Zumino-Witten models, among others.
In this paper, it is proved that, in any rational, modular invariant CFT, the central charge c and all the highest weights h, \(\bar h\) which occur are rational numbers.
Reviewer: A.N.Pressley

MSC:

17B65 Infinite-dimensional Lie (super)algebras
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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