A Bootstrap Study of Confinement in AdS

Abstract

Yang-Mills theory in AdS$_{4}$ with Dirichlet boundary conditions is expected to undergo a transition as the AdS radius varies, since the boundary data is incompatible with confinement in flat space. Various mechanisms have been proposed for the disappearance of the Dirichlet boundary condition. From the boundary viewpoint, the associated $3d$ CFT is a deformation of a generalised free theory of non-Abelian conserved currents, with the deformation governed by the bulk gauge coupling. We test these scenarios by deriving non-perturbative constraints from the numerical conformal bootstrap of the four-point function of non-Abelian conserved currents. We rule out the scenario in which the boundary current decouples. Bounds on the lightest scalar operators disfavour a bulk Higgs mechanism and instead point to a transition driven by a scalar singlet becoming marginal. We also obtain bounds on other scalar operators and on the current central charge, and we refine character-based techniques incorporating parity and charge-conjugation symmetry to determine the operator spectrum of the $3d$ Generalised Free Vector theory. These results may be of independent interest beyond Yang-Mills theory in AdS.

Figures (17)

• Figure 1: Sketch of the boundary conditions available in terms of real boundary CFTs as the coupling $g\textormath{\textsubscript{YM}}{_{\mathrm{YM}}}$ grows towards flat space. Here we illustrate the simplest possible scenario, namely merging with transition to Neumann. At some critical value, the Dirichlet boundary condition is assumed to merge with a hitherto unknown boundary condition dubbed $\mathrm{D}^{\star}$ and annihilate into the complex plane. Thus, sufficiently close to flat space, one naively expects only the Neumann boundary condition to remain. Note that the vertical axis does not represent a definite quantity; it is merely used to juxtapose the different options.
• Figure 2: Two sketches of possible Higgsing scenarios. In both panels, at $(L\Lambda)\textormath{\textsubscript{H}}{_{\mathrm{H}}}< (L \Lambda)\textormath{\textsubscript{M}}{_{\mathrm{M}}}$ a charged operator under $G\textormath{\textsubscript{YM}}{_{\mathrm{YM}}}$ becomes relevant, guaranteeing the existence of a boundary condition with smaller global symmetry. This new boundary condition may itself contain other charged operators which eventually also become relevant, leading to a cascade of symmetry breaking until no symmetry is left. In panel (a), the Neumann bc extrapolates to flat space. However, since H emerges before the merging transition, the latter has one more possible endpoint. In panel (b), a completely Higgsed boundary condition H$'$, rather than N, extrapolates to flat space.
• Figure 3: Lower bound on $C_{J}$ in the case of $SU(2)$ as a function of $\alpha_{JJJ}$. The shaded area is allowed, darker colour corresponding to higher $\Lambda$, we have $\Lambda = 11,19$ and $23$ respectively. The dashed vertical lines and the blue dots at $\alpha_{JJJ} = 1/2, 1$ denote the free boson and free fermion theory, respectively. We see that $C_J=0$ is excluded for any choice of $\alpha_{JJJ}$.
• Figure 4: Lower bound on $C_{J}$ in the case of $SU(3)$ as a function of $\alpha_{JJJ}$. The shaded area is allowed, the darker area corresponding to $\Lambda=19$ (and the rest to $\Lambda=11$). The dashed vertical lines and the blue dots at $\alpha_{JJJ} = 1/2, 1$ denote the free boson and free fermion theory, respectively. As in the $SU(2)$ case, $C_J=0$ is excluded for any choice of $\alpha_{JJJ}$.
• Figure 5: Lower bound on $C_{J}$ in the case of $SU(2)$ as a function of a compact parametrization of the OPE ratio $\chi\in[-1,1]$. The shaded area is allowed and corresponds to the $\Lambda=19$ computation of Figure \ref{['fig:SU2_CJ-bound_pd-11-19-23']}. The vertical axis is on a logarithmic scale.