The 3d Stress-Tensor Bootstrap

Abstract

We study the conformal bootstrap for 4-point functions of stress tensors in parity-preserving 3d CFTs. To set up the bootstrap equations, we analyze the constraints of conformal symmetry, permutation symmetry, and conservation on the stress-tensor 4-point function and identify a non-redundant set of crossing equations. Studying these equations numerically using semidefinite optimization, we compute bounds on the central charge as a function of the independent coefficient in the stress-tensor 3-point function. With no additional assumptions, these bounds numerically reproduce the conformal collider bounds and give a general lower bound on the central charge. We also study the effect of gaps in the scalar, spin-2, and spin-4 spectra on the central charge bound. We find general upper bounds on these gaps as well as tighter restrictions on the stress-tensor 3-point function coefficients for theories with moderate gaps. When the gap for the leading scalar or spin-2 operator is sufficiently large to exclude large N theories, we also obtain upper bounds on the central charge, thus finding compact allowed regions. Finally, assuming the known low-lying spectrum and central charge of the critical 3d Ising model, we determine its stress-tensor 3-point function and derive a bound on its leading parity-odd scalar.

Figures (17)

• Figure 1: Parameters of scalar conformal blocks for the even-even (blue dots) and odd-odd (red squares) cases.
• Figure 2: A series of lower bounds on $C_T$ as a function of $\theta$, valid in any unitary parity-preserving 3d CFT. The shaded region is allowed.
• Figure 3: A lower bound on $C_T$ as a function of $\theta$ in 3d CFTs with no relevant parity-even scalars.
• Figure 4: Bounds on $(\theta,C_T)$ with varying gaps in the parity-even scalar sector. When $\Delta_{\text{even}}^{\min}=4.0,\dots,6.0$, we have a series of lower bounds on $C_T$ as a function of $\theta$. When $\Delta_{\text{even}}^{\min}>6.0$, we have closed islands which eventually shrink to zero size.
• Figure 5: Bounds on $(\theta,C_T)$ with varying gaps in the parity-odd scalar sector. When the value of the gap $\Delta_{\textrm{odd}}^{\min}>7$, it becomes possible to find both upper and lower bounds on $C_T$ as.