  We analyze a continuous spin Gaussian model on a toroidal triangular lattice
with periods $L_0$ and $L_1$ where the spins carry a representation of the
fundamental group of the torus labeled by phases $u_0$ and $u_1$. We find the
{\it exact finite size and lattice corrections}, to the partition function $Z$,
for arbitrary mass $m$ and phases $u_i$. Summing $Z^{-1/2}$ over phases gives
the corresponding result for the Ising model. The limits $m\rightarrow0$ and
$u_i\rightarrow0$ do not commute. With $m=0$ the model exhibits a {\it vortex
critical phase} when at least one of the $u_i$ is non-zero. In the continuum or
scaling limit, for arbitrary $m$, the finite size corrections to $-\ln Z$ are
{\it modular invariant} and for the critical phase are given by elliptic theta
functions. In the cylinder limit $L_1\rightarrow\infty$ the ``cylinder charge''
$c(u_0,m^2L_0^2)$ is a non-monotonic function of $m$ that ranges from
$2(1+6u_0(u_0-1))$ for $m=0$ to zero for $m\rightarrow\infty$.
