Draft:The c-d conjecture

In an arXiv preprint, José Ignacio Latorre and Germán Sierra made the following conjecture about the upper bound of the central charge for one-dimensional quantum critical lattice Hamiltonians with nearest-neighbor interactions:


 * If the local Hilbert space dimension of the lattice model is $$d$$, the maximal central charge that the model can reach is $$c_{\mathrm{max}} = d-1$$.

Examples
The currently known examples are consistent with this conjecture.

The upper bound is saturated for the SU($$n$$) Uimin-Lai-Sutherland model, whose low-energy effective theory is the SU($$n$$) level 1 Wess-Zumino-Witten model. The local Hilbert space dimension of the lattice model is $$d = n$$, and the SU($$n$$) level 1 Wess-Zumino-Witten model has central charge $$c = n-1$$.