Ali Nassar, Mark A. Walton
Gukov and Vafa gave a simple, geometric criterion for rationality of a conformal $\sigma$-model with a two-dimensional torus as a target space. The modular parameter $\tau$ and K\"{a}hler parameter $\rho$ must take special values in an imaginary quadratic number field so that the torus and its mirror possesses the property of complex multiplication. On the other hand, Gannon has classified the modular invariant partition functions of the algebras $U_{m,K}$: Abeliam current algebras of the kind realized on a torus, but with a matrix-valued level $K$. We investigate the relation between the Gukov-Vafa geometric characterization of rationality and the algebraic results of Gannon. The geometric interpretation of the $U_{m,K}$ will be given for $m=2$. The Gauss product is used to give a geometric interpretation of $U_{2,K}$ in terms of rational points in the Narain moduli space which correspond to complex multiplication tori. We also show that rational points on the Grassmannian of self-dual lattices correspond to complex multiplication tori. This is another way to show that the set of rational conformal field theories is dense in the Narain moduli space.
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http://arxiv.org/abs/1211.2728
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