The Sine-Gordon Model in Hyperbolic Space and AdS/CFT

This post pulls out the main thread of my master’s thesis: what does the AdS/CFT correspondence look like when the bulk theory is the Sine-Gordon model, living in two-dimensional Euclidean Anti-de Sitter space?

The setting

The AdS/CFT correspondence, proposed by Maldacena, conjectures a duality between a theory in $(d+1)$-dimensional Anti-de Sitter space and a $d$-dimensional conformal field theory on its boundary. I work in the simplest non-trivial case, $\text{EAdS}_2$, in the Poincaré patch:

$$ds^2 = \frac{L^2}{z^2}\left(dz^2 + dp^2\right), \qquad R = -\frac{2}{L^2}.$$

Here $z>0$ is a radial coordinate and the conformal boundary sits at $z \to 0$. The conformal factor $L^2/z^2$ makes proper distances diverge as you approach the boundary, which is exactly where the holographic data lives.

Scalar fields and the dictionary

A free scalar of mass $m$ in the bulk is dual to a boundary operator whose scaling dimension is fixed by the Breitenlohner-Freedman analysis,

$$\Delta = \tfrac{1}{2} + \sqrt{\tfrac{1}{4} + m^2 L^2}.$$

The technical heart of the free theory is the massive bulk propagator and its boundary limit. Building it from a spectral representation and a heat-kernel decomposition gives a precise bulk-to-boundary dictionary: a recipe for translating bulk correlation functions into boundary CFT correlators.

Vertex operators

The Sine-Gordon interaction is built from normal-ordered vertex operators

$$V_\beta(\vec{x}) = {:}e^{i\beta\phi(\vec{x})}{:},$$

which are the natural observables of the theory. In the free theory their correlators follow from Wick contractions, and pushing them toward the boundary reproduces the expected CFT structure: each $V_\beta$ corresponds to a dual scalar primary.

Perturbation theory and anomalous dimensions

Turn on the interaction, and the connected two- and three-point functions of the vertex operators pick up corrections. An internal AdS loop integral produces a logarithmic IR divergence, which is a generic feature of integration over AdS. I regularize it with a cutoff $\epsilon$ a small distance from the boundary. Renormalizing then shifts the scaling dimension of the dual operator,

$$\Delta \;\longrightarrow\; \Delta_1 = \Delta + \lambda\,\Delta^{(1)} + \mathcal{O}(\lambda^2),$$

the anomalous dimension. The nice part is that the same dictionary built from the two-point function also prescribes the three-point function, which closes the analysis at this order.

The logarithmic-CFT hint

One striking feature: the corrections suggest the vertex operator splits into a primary $\mathcal{O}$ and a logarithmic multiplet $\widetilde{\mathcal{O}} = \{\hat{\mathcal{O}}, {:}\mathcal{O}^2{:}\}$, the signature of a logarithmic CFT. Whether that structure is real or just an artifact of the scheme is the question I take up in the follow-up article.

There is an interactive companion to this work in the DiffQFT project: differentiable Witten-diagram integration and a PINN for the Klein-Gordon equation on AdS.