Extremal N=(2,2) 2D Conformal Field Theories and Constraints of Modularity

TL;DR

Modularity of the elliptic genus for ${ m N}=(2,2)$ 2D CFTs with $c=6m$ imposes strong spectral constraints and casts doubt on the existence of strictly extremal theories beyond a finite set of levels. The authors define extremal ${ m N}=(2,2)$ CFTs and prove, via polar-state counting and modular form analysis, that only a finite list of $m$ can support such theories, with nine exceptional levels identified numerically and analytically. They then develop the concept of near-extremal theories, construct near-extremal elliptic genera, and derive quantitative bounds on the spectrum, showing that a strictly extremal spectrum is incompatible with modularity at large $m$ while near-extremal spectra can be realized. The work discusses implications for pure ${ m AdS}_3$ supergravity, possible quantum corrections to the cosmic censorship bound, and potential applications to flux compactifications, while also extending the discussion to ${ m N}=4$ cases and outlining avenues for future validation. Overall, the paper provides a rigorous modular-form framework to constrain holographic spectra and to explore near-extremal holography in ${ m AdS}_3$ settings.

Abstract

We explore the constraints on the spectrum of primary fields implied by modularity of the elliptic genus of N=(2,2) 2D CFT's. We show that such constraints have nontrivial implications for the existence of "extremal" N=(2,2) conformal field theories. Applications to AdS3 supergravity and flux compactifications are addressed.