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Generalized bootstrap equations for N=4 SCFT

Luis F. Alday, Agnese Bissi

TL;DR

This work extends the conformal bootstrap to four-point functions of identical half-BPS operators in ${\cal N}=4$ SCFT with weight ${p}$, deriving ${p(p-1)}/2$ coupled bootstrap equations that respect the ${SU(4)}$ R-symmetry. The authors decompose intermediate states into SU(4) reps ${[n-m,2m,n-m]}$, introduce left-hand side blocks ${F_{\Delta,\ell}^{(p)}}$ and ${H_{\Delta,\ell}^{(p)}}$ and account for short multiplets via ${F_{(p)}^{[nm]}}$, with the central charge $c$ and color factors encoding model dependence. Using a derivative-based numerical functional method with ${\Lambda=11}$ on ${SU(N)}$ theories (with ${N\ge3}$), they obtain non-perturbative upper bounds on leading-twist unprotected operators in representations ${[1,0,1]}$ and ${[0,2,0]}$, showing tighter constraints than the singlet case and revealing explicit large-${N}$ behavior ($\Delta^{[10]}\lesssim7.54$, $\Delta^{[11]}\lesssim6.58$) that align with double-trace expectations. The results illustrate how additional OPE data, beyond the central charge, constrain the spectrum and suggest pathways to sharpen bounds and extend to higher weights ${p}$, other representations, and analytic insights at large spin or through S-duality considerations.

Abstract

We study the consistency of four-point functions of half-BPS chiral primary operators of weight p in four-dimensional N=4 superconformal field theories. The resulting conformal bootstrap equations impose non-trivial bounds for the scaling dimension of unprotected local operators transforming in various representations of the R-symmetry group. These bounds generalize recent bounds for operators in the singlet representation, arising from consistency of the four-point function of the stress-energy tensor multiplet.

Generalized bootstrap equations for N=4 SCFT

TL;DR

This work extends the conformal bootstrap to four-point functions of identical half-BPS operators in SCFT with weight , deriving coupled bootstrap equations that respect the R-symmetry. The authors decompose intermediate states into SU(4) reps , introduce left-hand side blocks and and account for short multiplets via , with the central charge and color factors encoding model dependence. Using a derivative-based numerical functional method with on theories (with ), they obtain non-perturbative upper bounds on leading-twist unprotected operators in representations and , showing tighter constraints than the singlet case and revealing explicit large- behavior (, ) that align with double-trace expectations. The results illustrate how additional OPE data, beyond the central charge, constrain the spectrum and suggest pathways to sharpen bounds and extend to higher weights , other representations, and analytic insights at large spin or through S-duality considerations.

Abstract

We study the consistency of four-point functions of half-BPS chiral primary operators of weight p in four-dimensional N=4 superconformal field theories. The resulting conformal bootstrap equations impose non-trivial bounds for the scaling dimension of unprotected local operators transforming in various representations of the R-symmetry group. These bounds generalize recent bounds for operators in the singlet representation, arising from consistency of the four-point function of the stress-energy tensor multiplet.

Paper Structure

This paper contains 8 sections, 62 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Generalized bootstrap equations
  3. Numerical bounds
  4. Setup
  5. Results
  6. Discussion
  7. Expressions for $F_3^{[nm]}(z,\bar{z})$
  8. Color factors

Figures (3)

  • Figure 1: Bounds for the scaling dimension of the leading twist unprotected superconformal primary in the $[1,0,1]$ representation of the R-symmetry group, with $\ell=1$. The given bounds correspond to $\Lambda=11$, while the upper curve shows the results for $\Lambda=9$.
  • Figure 2: Bounds for the scaling dimension of the leading twist unprotected superconformal primary in the $[0,2,0]$ representation of the R-symmetry group, with $\ell=0$. The stronger bounds are for $\Lambda=11$ while the upper curve corresponds to $\Lambda=9$.
  • Figure 3: Topologies corresponding to color factors $a_1$ (left), $a_2$ (middle) and $a_3$ (right).