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We consider the Y-systems satisfied by the A1(1) , A2(1) , A2(2) vertex and loop models at roots of unity with twisted boundary conditions on the cylinder. The vertex models are the 6-, 15- and Izergin–Korepin 19-vertex models respectively. The corresponding loop models are the dense, fully packed and dilute Temperley–Lieb loop models respectively. For all three models, our focus is on roots of unity values of e iλ with the crossing parameter λ corresponding to the principal and dual series of these models. Converting the known functional equations to nonlinear integral equations in the form of thermodynamic Bethe ansatz equations, we solve the Y-systems for the finite-size 1N corrections to the groundstate eigenvalue following the methods of Klümper and Pearce. The resulting expressions for c − 24Δ, where c is the central charge and Δ is the conformal weight associated with the groundstate, are simplified using various dilogarithm identities. Our analytic results are in agreement with previous results obtained by different methods.

Groundstate finite-size corrections and dilogarithm identities for the twisted A1(1) , A2(1) and A2(2) models