Bounds on Operator Dimensions in 2D Conformal Field Theories

TL;DR

The paper extends Hellerman's modular-invariance-based bound on the lowest nontrivial primary in unitary 2D CFTs to the next two primaries, demonstrating that $\\Delta_2$ and $\\Delta_3$ obey bounds of the form $\\Delta_n \\le rac{c_{tot}}{12} + O(1)$ in the appropriate regime. By formulating and analyzing differential constraints with polynomials $f_p(z)$ and $b_p(z)$, it provides explicit expressions for bounds such as $\\Delta_1 \\le rac{c_{tot}}{12} + 0.4736...$, $\\Delta_2 \\le rac{c_{tot}}{12} + 0.5338...$, and $\\Delta_3 \\le rac{c_{tot}}{12} + 0.8795...$ in the large-$c_{tot}$ limit, and develops a general method for $\\Delta_n$ subject to a $c_{tot}$ threshold that grows with $n$ (via a Lambert $W$-function-based bound). The results imply an exponential growth in the number of primaries with $\\Delta \\lesssim rac{c_{tot}}{12}$ and have gravitational interpretations: within AdS$_3$/CFT$_2$, the bulk spectrum must contain many states with rest energies bounded by $M_n^+$, and in the flat limit this is tied to a bound near the BTZ black hole mass. Overall, the work extends a key modular-invariance constraint to higher-lying primaries and clarifies the conditions under which tight $rac{c_{tot}}{12}$-type bounds persist at large central charge.

Abstract

We extend the work of Hellerman (arxiv:0902.2790) to derive an upper bound on the conformal dimension $Δ_2$ of the next-to-lowest nontrival primary operator in unitary two-dimensional conformal field theories without chiral primary operators. The bound we find is of the same form as found for $Δ_1$: $Δ_2 \leq c_{tot}/12 + O(1)$. We find a similar bound on the conformal dimension $Δ_3$, and present a method for deriving bounds on $Δ_n$ for any $n$, under slightly modified assumptions. For asymptotically large $c_{tot}$ and fixed $n$, we show that $Δ_n \leq \frac{c_{tot}}{12}+O(1)$. We conclude with a brief discussion of the gravitational implications of these results.