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What we have not been able to show is that the Hamiltonian has only real eigenvalues because we do not know of a real structure for |q| = 1 and |t|= 1, which is compatible with the coalgebra structure; it exists for |q| = 1 and t real, but this is not the case of interest.
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From Kogan's approach (Ref. 6) one might infer that it is essential to work in the symmetric gauge; but on the torus one must stay in the unsymmetric gauge.
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For nonzero values of the phases μ. and v, the representations generally belong to the cyclic series. However, there is one special case for which, again, a highest weight representation is obtained. Namely, if the ground state wave functions are required to be modular invariant, one can show that this requirement is met if the phases take the values μ= - v=1/2; hence, there are no spin structures. It is straightforward to prove that the resulting representation is equivalent to (29).
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40
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85033902179
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note
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m in the notation of Sec. III. Also note the crucial role of the choice ∈= 1 in (12).
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