Multiplet Recombination and the CFT Distance Conjecture

Abstract

Motivated by quantum gravity and the CFT Distance Conjecture, we study infinite-distance limits in four-dimensional ${\cal N}=2$ superconformal field theories with higher-dimensional conformal manifolds and their AdS duals. We focus on partial decoupling limits where a gauge sector becomes weakly coupled while an interacting sector persists. We analyse the structure of towers of states emerging in these limits. The weakly coupled sector contributes, among others, the massless higher-spin tower predicted by the CFT Distance Conjecture exhibiting polynomial degeneracy. The key novelty is the appearance of a protected BPS tower in the interacting sector, characterised by exponential degeneracy and masses at the AdS scale. This structure follows from multiplet recombination in the ${\cal N}=2$ superconformal algebra: As unprotected long multiplets hit the unitarity bound at weak coupling, they recombine into protected short multiplets. We verify this picture through an explicit one-loop computation in the simplest two-node quiver gauge theory with a two-dimensional conformal manifold.

Figures (12)

• Figure 1: The CFT Distance Conjecture in AdS space visualised in the case of a single decoupling gauge node. At large $N$, the initial scenario is completely analogous to the situation in $\mathcal{N}=4$ SYM, while the spectrum decomposes into five separate sectors in the limit $\check{\lambda} \to 0$. The precise mass of the HS tower in relation to $\check{M}_s$ is discussed in the main text.
• Figure 2: The $\mathcal{N}=2$ quivers of the two-node necklace theory and of SCQCD + decoupled vector multiplet.
• Figure 3: The Swampland Distance Conjecture in flat space visualised. Tower 1 sits at the string scale, 2 at the KK scale. The scale separation between them is determined by the volume of the internal manifold $\mathcal{V}(\mathcal{M})$, as prescribed by equation \ref{['eq: scales flat']}. In this cartoon, we are already considering $\mathcal{V}(\mathcal{M}) \sim \mathcal{O}(1)$ and fixed, such that the string and the KK scales are not separated and are "locked" with each other. In the $g_s \to 0$ limit, both towers become asymptotically massless.
• Figure 4: The CFT Distance Conjecture in AdS space visualised in the case of a one-dimensional conformal manifold on the boundary, with $\mathcal{N}=4$ SYM being employed as an illustrative example. At large $N$, the AdS and the Planck scales are separated as prescribed by equation \ref{['lpllAdS']}. Tower 1 becomes light with the rate given by \ref{['effective-mass']}; tower 2 is a tower of BPS states and sits at the KK scale. In this case, the separation between 1 and 2 is governed by $\lambda$ (see eq. \ref{['lslAds']}), and both tower scales are not locked together. In the infinite-distance limit, the AdS scale does not renormalise (as indicated by the dashed horizontal line), while the string tower goes asymptotically to zero, generating a tower of massless states (dual to HS currents on the gauge theory side). The tower 3 represents all the new BPS states that belonged to the tower 1 and got "stuck" at the AdS scale.
• Figure 5: We illustrate the generation of extra BPS states from the long multiplet $\mathcal{A}^{\Delta}_{0,0(j,\bar{\jmath})}$. We start by considering the long multiplet at a finite distance point on the conformal manifold, i.e. away from the cusp. When we move towards the cusp, $\check{g} \ll 1$, and the conformal dimension becomes perturbative in the coupling $\check{g}$. The infinite-distance limit coincides with reaching the unitarity bound, and the long multiplet breaks down as shown in equation \ref{['eq: longHS']}. The same reasoning applies to other long multiplets reaching the unitarity bound in the same limit.