3d and 5d Gauge Theory Partition Functions as q-deformed CFT Correlators

3d partition functions on the squashed three-sphere and on the twisted product have been shown to factorize into sums of squares of solid tori partition functions, the so-called holomorphic blocks. The same set of holomorphic blocks realizes and partition functions but the two cases involve different inner products, the S-pairing and the id-pairing, respectively. We define a class of q-deformed CFT correlators where chiral blocks are controlled by a deformation of Virasoro symmetry and are paired by S-pairing and id-pairing, respectively. Applying the bootstrap approach to a class of degenerate correlators we are able to derive three-point functions. We show that degenerate correlators can be mapped to 3d partition functions while the crossing symmetry of correlators corresponds to the flop symmetry of 3d gauge theories. We explore how non-degenerate q-deformed correlators are related to 5d partition functions. We argue that id-pairing correlators are associated with the superconformal index on while S-pairing three-point function factors capture the one-loop part of S ⁵ partition functions. This is consistent with the interpretation of and gauge theories as codimension two defect theories inside and S ⁵, respectively.