Modular invariance and thermal effective field theory in CFT

We use thermal effective field theory to derive that the coefficient of the first subleading piece of the thermal free energy, $c_1$, is equal to the coefficient of the subleading piece of the Casimir energy on $S^1 \times S^{d-2}$ for $d \geq 4$. We conjecture that this coefficient obeys a sign constraint $c_1 \geq 0$ in CFT and collect some evidence for this bound. We discuss various applications of the thermal effective field theory, including placing the CFT on different spatial backgrounds and turning on chemical potentials for $U(1)$ charge and angular momentum. Along the way, we derive the high-temperature partition function on a sphere with arbitrary angular velocities using only time dilation and length contraction.