Redundancy Channels in the Conformal Bootstrap

Abstract

A method for obstructing symmetry enhancement in numerical conformal bootstrap calculations is proposed. Symmetry enhancement refers to situations where bootstrap studies initialised with a certain symmetry end up allowing theories with higher symmetry. In such cases, it is shown that redundant operators in the less symmetric theory can descend from primary scaling operators of the more symmetric one, motivating the imposition of spectral gaps that are justified in the former but not the latter. The same mechanism can also be used to differentiate between decoupled and fully coupled theories which otherwise have the same global symmetry. A systematic understanding of this mechanism is developed and applied to distinguish the cubic from the $O(3)$ model in three dimensions, where a strip of disallowed parameter space, referred to as the cubic redundancy channel, emerges once a gap associated with a redundant operator of the cubic theory is imposed. The channel corresponds precisely to the region of parameter space where the assumed cubic symmetry would be enhanced to $O(3)$.

Figures (10)

• Figure 1: Depending on the value of $N$, the IR stable fixed point is either the $O(N)$ model, denoted by H, or the hypercubic theory, denoted by C. The free theory is denoted by G and decoupled Ising models by I. The region between the hatched lines is the basin of attraction of the IR stable fixed point.
• Figure 2: Channel in the $\Delta_Z$-$\Delta_X$ plane obstructing symmetry enhancement, calculated at $\Lambda=11$. The numerical parameters used are Set A and Set 1 of Appendix \ref{['AppendixParameters']}. The gap assumptions on the spectrum are $\Delta_S \geqslant 1.5$, $\Delta_{X^\prime}\geqslant2.8$, $\Delta_{Z^\prime}\geqslant2.8$, $\Delta_{\overline{X\space X}_{\mu}}\geqslant 3.0$ and $\Delta_{T_{\mu\nu}^\prime} \geqslant 3.5$. Names and constructions of representations are given in Bednyakov:2023lfj. Primes denote a subleading operator in a given sector. A twist gap of $\delta =10^{-6}$ is imposed on all operators not mentioned. We also impose that the ratio of OPE coefficients $\frac{\lambda_{XXT_{\mu \nu}}}{\lambda_{ZZT_{\mu\nu}}}=\frac{\Delta_X}{\Delta_Z}$ is fixed. All assumptions are comfortably in agreement with the calculations of Bednyakov:2023lfjHenriksson:2025hwiHenriksson:2025vyi.
• Figure 3: Channel in the $\Delta_Z$-$\Delta_X$ plane obstructing symmetry enhancement, calculated at $\Lambda=19$. The numerical parameters used are Set B and Set 1 of Appendix \ref{['AppendixParameters']}. The gaps on the spectrum and other assumptions are as in Fig. \ref{['fig:ZXLambda11']}.
• Figure 4: Channel in the $\Delta_Z$-$\Delta_X$ plane obstructing symmetry enhancement, calculated at $\Lambda=27$. The numerical parameters used are Set B and Set 1 of Appendix \ref{['AppendixParameters']}. The gaps on the spectrum and other assumptions are as in Fig. \ref{['fig:ZXLambda11']}.
• Figure 5: Island in the $\Delta_\phi$-$\Delta_Z$ plane at $\Lambda = 19$. The island contains both the $C_3$ and $O(3)$ CFTs. This will be remedied when we study the full $\phi$-$X$-$Z$ correlator system. The numerical parameters used are Set B and Set 1 of Appendix \ref{['AppendixParameters']}. The gap assumptions on the spectrum are $\Delta_S \geqslant 1.5$, $\Delta_{Z^\prime}\geqslant2.8$, $\Delta_{\phi^\prime}\geqslant 1.5$, $\Delta_{Z_3}\geqslant 1.5$, $\Delta_{XV}\geqslant 1.5$ and $\Delta_{T_{\mu\nu}^\prime} \geqslant 4.0$. Names and constructions of representations are given in Bednyakov:2023lfj. Primes denote a subleading operator in a given sector. A twist gap of $\delta =10^{-6}$ is imposed on all operators not mentioned, except for the leading $B$ spin-one operator (the would be $O(3)$ current). We also impose that the ratio of OPE coefficients $\frac{\lambda_{\phi \phi T_{\mu \nu}}}{\lambda_{ZZT_{\mu\nu}}}=\frac{\Delta_\phi}{\Delta_Z}$ is fixed. All assumptions are comfortably in agreement with the calculations of Bednyakov:2023lfjHenriksson:2025hwiHenriksson:2025vyi. The blue circle represents the central value of the $O(3)$ bootstrap determination Chester:2020iyt, $(\Delta_\phi, \Delta_t) =(0.518936(67), 1.20954(32))$. The $O(3)$ theory lies deep in the allowed region, and without our analysis on redundant operators would be particularly hard to exclude. The yellow square gives the location of the $C_3$ theory using the central values of the results $\Delta_\phi=0.51891(7)$ from Hasenbusch:2022zur and $\Delta_Z=1.1988(24)$ from Rong:2023owx.