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We consider the $Y$-systems satisfied by the $A_1^{(1)}$, $A_2^{(1)}$,$A_2^{(2)}$ vertex and loop models at roots of unity with twisted boundaryconditions on the cylinder. The vertex models are the 6-, 15- andIzergin-Korepin 19-vertex models respectively. The corresponding loop modelsare the dense, fully packed and dilute Temperley-Lieb loop models respectively.For all three models, our focus is on roots of unity values of $e^{i\lambda}$with the crossing parameter $\lambda$ corresponding to the principal and dualseries of these models. Converting the known functional equations to nonlinearintegral equations in the form of Thermodynamic Bethe Ansatz (TBA) equations,we solve the $Y$-systems for the finite-size $\frac 1N$ corrections to thegroundstate eigenvalue following the methods of Kl\"umper and Pearce. Theresulting expressions for $c-24\Delta$, where $c$ is the central charge and$\Delta$ is the conformal weight associated with the groundstate, aresimplified using various dilogarithm identities. Our analytic results are inagreement with previous results obtained by different methods and are new forthe dual series of the $A_2^{(1)}$ model.

Groundstate finite-size corrections and dilogarithm identities for the twisted $A_1^{(1)}$, $A_2^{(1)}$ and $A_2^{(2)}$ models