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

Aug 5, 2026

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We study the integrability of two-dimensional theories obtained from dimensional reduction of four-dimensional gravitational systems where Maxwell fields and neutral scalar fields couple to gravity, together with a scalar potential. On a particular solution subspace, a subset of the two-dimensional equations of motion becomes the compatibility conditions for a linear system. Using a complementary one-dimensional framework, the work formulates Liouville integrability through Lax-pair data. Machine learning techniques are employed to systematise the search for numerical Lax-pair matrices and conserved currents depending on phase-space variables.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages38

Reading Time1 h 36 min

AccessFree to download

Introduction

Dimensional reduction and classical integrability

Effect of a cosmological constant or scalar potential

Linear-system and Lax-pair formulation

Liouville integrability in 1D

Machine learning strategy for Lax pairs and currents

arXiv :2404 . 18247v3 [hep-th] 21 Dec 2024 Classical integrability in the presence of a cosmological constant: analytic and machine learning results Gabriel Lopes Cardoso, Dami´an Mayorga Pe˜na, Suresh Nampuri Center for Mathematical Analysis, Geometry and Dynamical Systems, Department of Mathematics, Instituto Superior T´ecnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal [gabriel.lopes.cardoso@tecnico.ulisboa.pt](gabriel.lopes.cardoso@tecnico.ulisboa.pt), [damian.mayorga.pena@tecnico.ulisboa.pt](damian.mayorga.pena@tecnico.ulisboa.pt), [nampuri@gmail.com](nampuri@gmail.com) 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. Keywords: Integrability, Lax pair, linear system, neural network, machine learning. 1 Introduction It has long been established [1–4] that certain 4D gravitational theories at the two-derivative level, describing the coupling of Maxwell fields and neutral scalar fields to gravity in the absence of a cosmological constant or, more generally, a potential for the scalar fields, when dimensionally reduced to two dimensions, yield 2D non-linear sigma-models coupled to gravity, whose equations of motion are classically integrable, i.e. that can be viewed as compatibility equations of a linear system. This has recently been the subject of revived interest [5–13] . The dimensional reduction is performed by means of a two-step procedure. In the first step one reduces over an isometry direction down to three dimensions, and then Hodge dualises all Maxwell field strengths into neutral scalar fields. We will only consider the reduction of 4D gravitational theories satisfying the condition that the resulting target space for the enlarged set of neutral scalar fields is a symmetric space G/H. A further reduction along a second commuting isometry direction results in a 2D gravitational model whose field equations are the compatibility equations for a linear system. The latter, introduced by Belinski-Zakharov [1], has an equivalent description given by Breitenlohner-Maison [2,4], as shown in [14] . In this paper we will work with the Breitenlohner-Maison (BM) linear system. The addition of a cosmological constant or a potential for the neutral scalar fields to these four-dimensional gravitational theories will, in general, not preserve the integrable nature of the dimensionally reduced equations of motion. Aspects of this were discussed in [15,16] . In this paper we show that in the presence of a scalar potential, for a certain solution subspace defined by (14), a subset of the equations of motion in two dimensions can still be viewed as being the compatibility conditions of a linear system. We further show that this linear system admits a Lax pair description in terms of differential operators [17], and reduces to the BM system when the scalar potential vanishes. Hence, this linear system framework for studying integrability only applies to solutions that continue to exist in the absence of a scalar potential. Further, these solutions admit a complementary 1D description, thereby enabling us to independently study Liouville integrability in 1D expressed in terms of Lax pair

FAQs

What is the starting point for studying integrability in this work?

The analysis begins with certain four-dimensional gravitational theories coupled to Maxwell fields and neutral scalar fields, then performs a dimensional reduction to two dimensions while including a scalar potential.

How is partial integrability established for the two-dimensional models?

For a specified solution subspace, a subset of the two-dimensional equations of motion is shown to act as compatibility conditions for a linear system.

What role do machine learning methods play in the paper?

Machine learning is used to systematise the search for numerical Lax pair matrices and conserved currents expressed as functions of phase-space variables, framed in a way that supports symbolic regression.

Aug 5, 2026

Upload by Liam