Putting the Boot into the Swampland

Abstract

The swampland program of delineating the space of effective field theories consistent with quantum gravity appears similar to the bootstrap program of delineating the space of quantum field theories consistent with conformal symmetry. With this in mind we rewrite the effective field theory of the Large Volume Scenario in AdS space solely in terms of $R_{AdS}$, in a form suitable for holographic analysis. This rewritten EFT takes a remarkably universal (and previously unnoticed) form, which is uniquely determined in the large-volume limit up to terms suppressed by $\mathcal{O} \left( 1/\ln R_{AdS} \right)$, with no reference to any of the fluxes, brane or instanton configurations that enter the microphysics of moduli stabilisation. The putative dual 3d CFT will have two low-lying single trace scalars, an even-parity scalar $\Phi$ dual to the volume modulus with $\Delta_{\Phi} = \frac{3}{2}\left( 1 + \sqrt{19} \right) \simeq 8.038$ and an odd-parity scalar $a$ dual to the volume axion with $\Delta_a = 3$. On the AdS side the higher-point interactions are likewise uniquely determined. As the AdS theory is both subject to swampland constraints and holographically related to a CFT, we argue that holography will lead to a `bootland' --- a map between swampland constraints on the AdS side and bootstrap constraints on the CFT side. We motivate this with a discussion of swampland quantum gravity constraints on the axion decay constant in the $\mathcal{V} \to \infty$ limit and the $\langle \Phi \Phi a a \rangle$ 4-point function on the CFT side.