Bounds on Operator Dimensions in 2D Conformal Field Theories

Abstract

We extend the work of Hellerman (arxiv:0902.2790) to derive an upper bound on the conformal dimension ${\Delta }_2$ 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 ${\Delta }_1$: ${\Delta }_2\le c_{tot}/12+O(1)$. We find a similar bound on the conformal dimension ${\Delta }_3$, and present a method for deriving bounds on ${\Delta }_n$ for any $n$, under slightly modified assumptions. For asymptotically large $c_{tot}$ and fixed $n$, we show that ${\Delta }_n\le \frac{c_{tot}}{12}+O(1)$. We conclude with a brief discussion of the gravitational implications of these results.

Corrected typos; revised arguments (adding detail) for clarity, results unchanged

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• Three-Dimensional Gravity Revisited
• The $(2,0)$ superconformal bootstrap
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