Classical integrability in the presence of a cosmological constant: analytic and machine learning results

TL;DR

This work analyzes classical integrability of 2D theories arising from dimensional reduction of 4D gravity with Maxwell fields and neutral scalars in the presence of a potential $V(\phi)$. It establishes that while the BM linear system underpins integrability when $V=0$, a suitable solution subspace admits a modified linear system whose compatibility reproduces the reduced equations even for nonzero $V$, and it connects this to a complementary 1D Liouville integrability framework. The authors develop both analytic Lax-pair descriptions and ML-driven searches for numerical Lax pairs and for conserved currents, with symbolic regression used to extract interpretable currents. They demonstrate explicit AdS$_4$-based examples, illustrate the 1D reduction to a setting with Hamiltonian constraints, and show that certain subcases admit a full set of independent Poisson-commuting integrals, signaling Liouville integrability. The combination of analytic, 1D Liouville, and ML-based techniques provides a multifaceted toolkit to probe integrability in gravitational reductions and highlights both the potential and the limitations of ML methods in identifying integrable structures.

Abstract

We study the integrability of two-dimensional theories that are obtained by a dimensional reduction of certain four-dimensional gravitational theories describing the coupling of Maxwell fields and neutral scalar fields to gravity in the presence of a potential for the neutral scalar fields. For a certain solution subspace, we demonstrate partial integrability by showing that a subset of the equations of motion in two dimensions are the compatibility conditions for a linear system. Subsequently, we study the integrability of these two-dimensional models from a complementary one-dimensional point of view, framed in terms of Liouville integrability. In this endeavour, we employ various machine learning techniques to systematise our search for numerical Lax pair matrices for these models, as well as conserved currents expressed as functions of phase space variables.

Figures (9)

• Figure 1: Flow chart of the Lax pair training process. The neural network takes points in phase space as its inputs, and in training, minimises the loss function \ref{['eq:Loss']}, using the EOMs, to output a numerical approximation for a Lax pair $L$, $M$ at each point.
• Figure 2: Training process for three different potentials. Left: Liouville potential $V(q)=e^{-q}$. Center: de Alfaro-Fubini-Furlan (DFF) potential $V(q)=\frac{1}{2} q^2 +1/q$. Right: A linear potential $V(q)=q$.
• Figure 3: Numerical Lax pair training in the no scalar field model described by \ref{['eq:rescaled']}. Left: Case (a) $h_0=0$, $Q_0=1$. Center: Case (b) $h_0=1$, $Q_0=0$. Right: Case (c) $h_0=Q_0=1$.
• Figure 4: Numerical Lax pair training in the one scalar field model (see \ref{['actuppsi']}). Left: Case $h_1=0$, $h_0=Q_0=1$. Right: $h_0=h_1=Q_0=1$.
• Figure 5: ML training of diagonal matrix $L$ in the no scalar field model described by \ref{['eq:rescaled']}. Left: Case (a) $h_0=0$, $Q_0=1$. Center: Case (b) $h_0=1$, $Q_0=0$. Right: Case (c) $h_0=Q_0=1$.