Conformal partition functions of critical percolation from $D_3$ Thermodynamic Bethe Ansatz equations

Using the planar Temperley-Lieb algebra, critical bond percolation on the square lattice is incorporated as ${\cal LM}(2,3)$ in the family of Yang-Baxter integrable logarithmic minimal models ${\cal LM}(p,p')$. We consider this model in the presence of boundaries and with periodic boundary conditions. Inspired by Kuniba, Sakai and Suzuki, we rewrite the recently obtained infinite $Y$-system of functional equations. We obtain nonlinear integral equations in the form of a closed finite set of TBA equations described by a $D_3$ Dynkin diagram. Following the methods of Kl\"umper and Pearce, we solve the TBA equations for the conformal finite-size corrections. For the ground states of the standard modules on the strip, these agree with the known central charge $c=0$ and conformal weights $\Delta_{1,s}$ for $s\in {\Bbb Z_{\ge 1}}$ with $\Delta_{r,s}= \big((3r-2s)^2-1\big)/24$. For the periodic case, the finite-size corrections agree with the conformal weights $\Delta_{0,s}$, $\Delta_{1,s}$ with $s\in\frac12\Bbb Z_{\ge0}$. These are obtained analytically using Rogers dilogarithm identities. We incorporate all finite excitations by formulating empirical selection rules for the patterns of zeros of all the eigenvalues of the standard modules. We thus obtain the conformal partition functions on the cylinder and the modular invariant partition function (MIPF) on the torus. By applying $q$-binomial identities, it is shown that our finitized characters on the strip agree with those of Pearce, Rasmussen and Zuber. On the torus, the MIPF is a non-diagonal sesquilinear form in affine $u(1)$ characters given by the $u(1)$ partition function $Z_{2,3}(q)=Z_{2,3}^{\rm{Circ}}(q)$. This is compatible with the general conjecture of Pearce and Rasmussen, namely $Z_{p,p'}(q)=Z^{\rm{Proj}}_{p,p'}(q)+n_{p,p'}Z^{\rm{Min}}_{p,p'}(q)$ with $n_{p,p'}\in {\Bbb Z}$, and the lattice derivation fixes $n_{2,3}=-1$.