Machine Learning the Conformal Manifold of Holographic CFT$_{2}$s

TL;DR

The paper tackles identifying conformal manifolds in holographic CFT$_{2}$s by exploring flat directions of the scalar potential in a gauged supergravity truncation of AdS$_3$ backgrounds. It develops a hybrid workflow that combines gradient-descent sampling of the conformal manifold with a novel Annealed Sequential Monte Carlo–based symbolic regression to extract analytic polynomial constraints, providing an explicit 3D manifold within a 5D truncation and eight annihilator polynomials. The authors derive analytic relations that parametrize the solutions, compute the Zamolodchikov metric, and analyze the spectrum for stability, while outlining uplift pathways to higher dimensions via exceptional field theory. The approach demonstrates robustness and scalability, suggesting a path toward systematic classification of conformal manifolds across the supergravity landscape and their CFT duals.

Abstract

We investigate the structure of conformal manifolds around AdS$_3 \times S^3$ which lift from continuous flat directions in the scalar potential of gauged supergravity resulting from six-dimensional $\mathcal{N}=(1,1)$ supergravity. Our approach combines numerical exploration and symbolic inference. For the latter, we develop a symbolic regression algorithm based on Annealed Sequential Monte Carlo samplers, a combination of Annealed Importance Sampling and Sequential Monte Carlo samplers, well-suited to uncovering polynomial constraints in high-dimensional parameter spaces. The algorithm reconstructs a set of polynomial relations that provides an explicit analytic parametrization of a new family of solutions.

Figures (9)

• Figure 1: Flow chart of the Annealed Sequential Monte Carlo sampler algorithm.
• Figure 2: Evolution of the loss function $\mathcal{L}$ in \ref{['eq:lossgraddes']} during gradient descent (right axis, in blue) and corresponding learning rate schedule $\alpha$ (left axis in orange), both plotted in logarithmic scale. Dashed lines indicate epochs at which the optimizer was reinitialized.
• Figure 3: (a) Histogram of the norm of the gradient for each point ($x$-axis is in $\log$ scale). (b) Histogram of the distance of the value of the potential to $V(0)$ for each point ($x$-axis is in $\log$ scale).
• Figure 4: Triangular plot showing all $2d$ projections and $1d$ histograms of the data after the gradient descent.
• Figure 5: Results of the local PCA analysis. The $x$-axis shows the dimension inferred by the algorithm, and the $y$-axis indicates the proportion of points for which that dimension was found. Each curve corresponds to a different value of $k$ in the $k$-nearest neighbours.