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$\mathcal{N}=2$ central charge bounds from $2d$ chiral algebras

Madalena Lemos, Pedro Liendo

TL;DR

This work leverages the 2d chiral algebra (Schur sector) of 4d ${\mathcal N}=2$ SCFTs to derive a new analytic bound on the central charges relating $c_{4d}$ and the flavor central charge $k_{4d}$, expressed through Lie-algebra data. By analyzing mixed correlators of stress-tensor and flavor currents and mapping 4d OPE data to 2d conformal data, the authors establish a universal inequality $k_{4d}(-180 c_{4d}^2+66 c_{4d}+3\dim_G)+60 c_{4d}^2 h^{\vee}-22 c_{4d} h^{\vee} \le 0$, which, together with existing bounds, sharply narrows the allowed ${\mathcal N}=2$ theory space. The study shows that certain theories, including canonical rank-one Kodaira models, lie at the intersection where old and new bounds fix $c_{4d}$ given $k_{4d}$, and interprets saturations in terms of a Sugawara construction in the 2d chiral algebra with a null state at level $h=4$. These results inform both the analytic bootstrap and numerical bootstrap programs for ${\mathcal N}=2$ theories and highlight deep links between 2d chiral algebras and 4d SCFT dynamics.

Abstract

We study protected correlation functions in $\mathcal{N} = 2$ SCFT whose description is captured by a two-dimensional chiral algebra. Our analysis implies a new analytic bound for the $c$-anomaly as a function of the flavor central charge $k$, valid for any theory with a flavor symmetry $G$. Combining our result with older bounds in the literature puts strong constraints on the parameter space of $\mathcal{N}=2$ theories. In particular, it singles out a special set of models whose value of $c$ is uniquely fixed once $k$ is given. This set includes the canonical rank one $\mathcal{N}=2$ SCFTs given by Kodaira's classification.

$\mathcal{N}=2$ central charge bounds from $2d$ chiral algebras

TL;DR

This work leverages the 2d chiral algebra (Schur sector) of 4d SCFTs to derive a new analytic bound on the central charges relating and the flavor central charge , expressed through Lie-algebra data. By analyzing mixed correlators of stress-tensor and flavor currents and mapping 4d OPE data to 2d conformal data, the authors establish a universal inequality , which, together with existing bounds, sharply narrows the allowed theory space. The study shows that certain theories, including canonical rank-one Kodaira models, lie at the intersection where old and new bounds fix given , and interprets saturations in terms of a Sugawara construction in the 2d chiral algebra with a null state at level . These results inform both the analytic bootstrap and numerical bootstrap programs for theories and highlight deep links between 2d chiral algebras and 4d SCFT dynamics.

Abstract

We study protected correlation functions in SCFT whose description is captured by a two-dimensional chiral algebra. Our analysis implies a new analytic bound for the -anomaly as a function of the flavor central charge , valid for any theory with a flavor symmetry . Combining our result with older bounds in the literature puts strong constraints on the parameter space of theories. In particular, it singles out a special set of models whose value of is uniquely fixed once is given. This set includes the canonical rank one SCFTs given by Kodaira's classification.

Paper Structure

This paper contains 6 sections, 17 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction and summary
  2. A new analytic bound
  3. Schur correlators in 4d
  4. An orthonormal basis:
  5. Twisted correlators in 2d
  6. Consequences for the N=2 landscape

Figures (2)

  • Figure 1: Constraints on the $(c_{4d},k_{4d})$ plane for $\mathfrak{su}(2)$ flavor symmetries arising as a combination of the analytic bound \ref{['eq:newbound']} (blue), the analytic bounds of Beem:2013sza (red) and the numerical bound of Beem:2014zpa (gray). We have marked the position of some known ${\mathcal{N}}=2$ theories which contain an $\mathfrak{su}(2)$ flavor symmetry, namely the free hyper multiplet, ${\mathcal{N}}=4$$SU(N)$ SYM, and the theories obtained by $N$$D3$-branes probing an F-theory singularity of type $H_1$ (starting with the rank $N=1$ of Tab. \ref{['Tab:rank1theories']}), and $H_0$ (for $N \geqslant2$ when it has an $\mathfrak{su}(2)$ flavor symmetry) Aharony:2007dj.
  • Figure 2: Constraints on the $(c_{4d},k_{4d})$ plane for two different flavor symmetries arising as a combination of the analytic bound \ref{['eq:newbound']} (blue) and the analytic bounds of Beem:2013sza (red). For the case of $\mathfrak{e}_6$ the numerical bound of Beem:2014zpa is also shown in gray. We also marked the position of several theories with at least $\mathfrak{e}_6$ and $\mathfrak{su}(4)$ flavor symmetry, namely the rank $N$ version of some of the theories given in Tab. \ref{['Tab:rank1theories']}Aharony:2007dj, the $T_{N\geqslant4}$ family of theories of Gaiotto:2009we which have an $\mathfrak{su}(N)^3$ flavor symmetry, and the $R_{0,N\geqslant3}$ of Chacaltana:2010ks which have an $\mathfrak{su}(2) \times \mathfrak{su}(2N)$ flavor symmetry (for $N=3$ the flavor symmetry enhances and it corresponds to the rank one $\mathfrak{e}_6$).