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Nonexistence of modular fusion algebras whose kernels are certain noncongruence subgroups. (English) Zbl 1048.11033

For the classification of conformal field theory models their associated fusion algebras are of importance. A fusion algebra is an associative and commutative finite-dimensional algebra over \(\mathbb C\). A fusion algebra \(A\) is called a modular fusion algebra (MFA), and denoted by \((A,\rho)\), if it carries a representation \(\rho\) of the modular group \(\text{SL}(2,\mathbb Z)\) of certain type. A representation \(\rho:\text{SL}(2,\mathbb Z)\to \text{GL}(N,\mathbb C)\) is called admissible if there exists a fusion algebra structure \(A\) on \(\mathbb C^N\) such that \((A,\rho)\) is a MFA. It is conjectured that for a MFA the kernel of \(\rho\) is always a congruence subgroup. Such MFA are called strongly modular fusion algebras. W. Eholzer [Commun. Math. Phys. 172, 623–659 (1995; Zbl 0884.17018)] classified for lower dimensions all strongly MFA.
In the article under review the author considers representations of the quotient \(\Gamma/G\) with \(\Gamma=\text{PSL}(2,\mathbb Z)\) and \(G\) a normal subgroup containing the transformation \(\tau\mapsto \tau+6\). Such groups \(G\) are “not so far away” from congruence subgroups, but there exists infinitely many of these groups which are not congruence subgroups. It was the starting point for the author, that they could supply counterexamples for the conjecture. But from the classification results obtained in this article, he concludes that there does not exist an irreducible representation of \(\Gamma/G\) of degree \(\neq 1\) which is admissible. Hence, no counterexample is produced.
In the context of the conjecture, it should be pointed out that recently P. Bantay proved that for rational conformal field theories the kernel is always a congruence subgroup [Commun. Math. Phys. 233, 423–438 (2003; Zbl 1035.81055)]

MSC:

11F06 Structure of modular groups and generalizations; arithmetic groups
05E99 Algebraic combinatorics
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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