Discreteness and Integrality in Conformal Field Theory
Justin Kaidi, Eric Perlmutter
TL;DR
The paper tackles how discreteness and integrality constrain two-dimensional CFT data beyond positivity, establishing two main mathematical anchors: (i) a theorem enforcing negative leading exponents $m_0$ for weight-0 vector-valued modular forms in integral representations, and (ii) explicit constructions of non-factorizable, non-holomorphic cusp forms that satisfy discreteness and integrality but violate positivity. These results yield bootstrap-type bounds on twist and dimension spectra for rational and irrational CFTs, and illuminate spectral determinacy: for RCFTs, the twist spectrum above the $t=c/12$ threshold is fixed by its complement, while in generic CFTs a related bound holds for dimensions above $(c-1)/12$ absent fine-tuning. The framework also connects to OPE data via torus one-point functions, showing that integral one-point blocks (notably for $h_{\, obreak O}=1$) enforce determinacy of certain OPE coefficients and place constraints on conformal manifolds and marginal deformations. Beyond 2D CFT, these modular constraints inform AdS$_3$ gravity, clarifying when gravity is dual to a unique theory versus an ensemble, and shaping understanding of black-hole thresholds and enigmatic bulk states. Collectively, the work tightly binds modular form structure to physical spectra, offering new levers to probe the space of consistent theories and their gravitational avatars.
Abstract
Various observables in compact CFTs are required to obey positivity, discreteness, and integrality. Positivity forms the crux of the conformal bootstrap, but understanding of the abstract implications of discreteness and integrality for the space of CFTs is lacking. We systematically study these constraints in two-dimensional, non-holomorphic CFTs, making use of two main mathematical results. First, we prove a theorem constraining the behavior near the cusp of integral, vector-valued modular functions. Second, we explicitly construct non-factorizable, non-holomorphic cuspidal functions satisfying discreteness and integrality, and prove the non-existence of such functions once positivity is added. Application of these results yields several bootstrap-type bounds on OPE data of both rational and irrational CFTs, including some powerful bounds for theories with conformal manifolds, as well as insights into questions of spectral determinacy. We prove that in rational CFT, the spectrum of operator twists $t\geq {c \over 12}$ is uniquely determined by its complement. Likewise, we argue that in generic CFTs, the spectrum of operator dimensions $Δ> {c-1\over 12}$ is uniquely determined by its complement, absent fine-tuning in a sense we articulate. Finally, we discuss implications for black hole physics and the (non-)uniqueness of a possible ensemble interpretation of AdS$_3$ gravity.