  We study conjectures of Ben-Zvi--Sakellaridis--Venkatesh that categorify the
relationship between automorphic periods and $L$-functions in the context of
the Geometric Langlands equivalence. We provide evidence for these conjectures
in some low-rank examples, by using derived Fourier analysis and the theory of
chiral algebras to categorify the Rankin-Selberg unfolding method.
