A genus-2 crossing equation in $d\geq 2$

We explore a "genus-2" crossing equation obeyed by CFTs in general dimensions $d\geq 2$. This crossing equation relates two different decompositions of the "genus-2 partition function" -- namely the partition function on the connected sum $M_2=(S^1\times S^{d-1})\sharp (S^1\times S^{d-1})$. The "sunrise" channel decomposition expresses $M_2$ as a pair of three-punctured spheres glued together with cylinders, while the "dumbbell" channel decomposition expresses $M_2$ as a gluing of two one-point functions on $S^1\times S^{d-1}$. We introduce coordinates to describe each channel, and write down Casimir equations obeyed by the corresponding blocks. We also explain why equality between the two channels guarantees mapping class group invariance of the genus-2 partition function in 3d CFTs. As an application of the genus-2 crossing equation, we derive a novel relation between asymptotics of "heavy-heavy-heavier" OPE coefficients and squares of thermal one-point coefficients in 3d CFTs. Along the way, we demonstrate how expectation values of conformal generators can help locate saddle points in large quantum number limits.