HET Brown Bag | 4d modularity

Modular invariance plays an important role in the study of two-dimensional CFTs. Most famously, it exhibits the universality of CFT spectra at high energy, but there are numerous other applications, including in the context of holography. In recent years, a combination of exact results in supersymmetric CFTs and developments in AdS/CFT have sparked renewed interest in possible generalizations of modularity to CFTs in dimensions greater than two. We briefly survey these developments for both supersymmetric and non-supersymmetric CFTs and note that a satisfactory geometric understanding is lacking. We aim to improve on this situation in the context of the 4d N=1 superconformal index, using the free chiral multiplet as our main example. We argue that a factorization of the BPS Hilbert space allows a KK reduction on the base of the Hopf fibration to a two-dimensional torus, comprised of the Hopf fiber and the temporal circle. This provides a 2d description, in terms of two infinite KK towers, of the 4d BPS Hilbert space. We argue, and prove, that this implies the unconventional modular property of the superconformal index, referred to before as “modular factorization”. We comment on the generalized notion of modularity, on generalizations to more interesting 4d SCFTs and other dimensions. If time permits, we discuss an SL(2,Z) family of 3d limits of the 4d index, realizing lens space partition functions of the dimensionally reduced theory.