Discreteness and integrality in Conformal Field Theory

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 $$ t\ge \frac{c}{12} $$ t ≥ c 12 is uniquely determined by its complement. Likewise, we argue that in generic CFTs, the spectrum of operator dimensions $$ \Delta >\frac{c-1}{12} $$ Δ > c − 1 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 AdS3 gravity.

Harmonic analysis of 2d CFT partition functions

“…One rigorous consequence of this structure that emerges rather easily, as shown in section 4.4, is a new result on spectral determinacy, i.e. the minimal content necessary to fully determine the spectrum of a 2d CFT [31,32]. We show that the entire primary spectrum of a 2d CFT is uniquely fixed by the light spectrum, the scalar spectrum, and the spectrum of any single nonzero integer spin (see figure 1).…”

“…arguments emphasizing the strong constraints implied by compactness were given in [32]. As we discuss in the next subsection, this impossibility is a reasonable expectation because cusp forms are, in a precise sense we will articulate in the next subsection, chaotic.…”

“…12 is fully determined by its complement, this time due to properties of vector-valued modular forms [32]. Note that spectral determinacy in ∆ also applies to rational CFTs: performing the algorithm with respect to the full partition function Z(τ ) rather than to Z p (τ ), whose form is obfuscated by non-trivial null states, yields essentially the same result, marginally weaker due to the lack of distinction between primaries and descendants.…”

Universality in asymptotic bounds and its saturation in 2D CFT

“…Our results naturally fall into the scope of conformal bootstrap program [25,[30][31][32][33][34][35][36][37][38][39][40][41][42][43][44]. We hope the techniques/functions used here would be useful in the broader context of this program, especially for the studies related to extremal functionals and dispersive sum rules [45][46][47][48][49] and its connection to analyticity in replica correlator [50].…”

A CFT distance conjecture

“…[52,53]), but it remains an outstanding problem for the conformal bootstrap to develop robust methods for solving CFTs with conformal manifolds. (See [52][53][54][55][56][57][58] for some initial approaches.) Of course, this is not just a bespoke bootstrap problem: the reasons to study conformal manifolds are manifold, with a long and rich history.…”