An action principle for the Einstein–Weyl equations
Introduction
A remarkable generalization of Riemannian geometry was first proposed in 1918 by Weyl (see e.g. [1], [5], [14], [30]), who introduced an additional symmetry in an attempt of geometrically unifying electromagnetism with gravity [36], [37]. In this theory, both the direction and the length of vectors are allowed to vary under parallel transport.1 The trace part of the connection introduced by Weyl is known as the Weyl vector. When it is exact, it can be gauged away by a local scale transformation. In this case, Weyl geometry is said to be integrable (parallel transported vectors along closed paths return with unaltered lengths), and there exists a subclass of global gauges in which the geometry is Riemannian.
Mathematically, a Weyl structure on an -dimensional manifold consists of a conformal structure , together with a torsion-free connection which is compatible with in the sense that for some one-form on . This compatibility condition is invariant under the transformation where is a function on .
A Weyl structure is said to be Einstein–Weyl (EW) [20] if the symmetrized Ricci tensor of is proportional to some metric . This conformally invariant condition is equivalent to with the Ricci scalar of .
A longstanding and still unresolved open problem has been that of finding an action principle for (3). Here we shall present for the first time an action principle for the EW equations in dimensions2 in which the Weyl vector is not exact. Notice that three-dimensional EW geometry [29] is particularly interesting, since it is related to dispersionless integrable systems [9], [13], [35]. Moreover, Jones and Tod [25] showed that selfdual conformal four-manifolds with a conformal vector field are in correspondence with abelian monopoles on Einstein–Weyl three-manifolds. In Ref. [15], Gauduchon and Tod studied the structure of four-dimensional hyper-Hermitian Riemannian spaces admitting a tri-holomorphic Killing vector, i.e., a Killing vector that is compatible with the three complex structures on the hyper-Hermitian space. It turned out that the latter is fibred over a specific type of three-dimensional EW spaces, called hyper-CR or Gauduchon–Tod.
Note also that 3d EW manifolds and their generalizations play an important role also in high energy physics, for instance in the context of supersymmetric solutions to fake supergravity [16], [17], [28], in the classification of four-dimensional Euclidean gravitational instantons [11], [12], or supersymmetric near-horizon geometries [10], [26]. Moreover, Weyl connections were considered recently in holography [8]. For a general review of Einstein manifolds with nonmetric and torsionful connections cf. [27].
Our construction of an action that leads to (3) involves a metric affine gravity action [19], [21], [23], [31], [32], [34] plus additional terms containing Lagrange multipliers and gravitational Chern–Simons contributions. We work in a first order formalism, where the metric and the connection are treated as independent variables, and make no a priori assumptions on the metricity and the torsion of the connection.
The remainder of this paper is organized as follows: In Section 2 we briefly review metric affine gravity in dimensions, following [21], [23]. Subsequently, in Section 3 we construct an action that yields the EW equations with nonexact Weyl vector in three dimensions. We conclude our work with some final remarks.
Section snippets
Metric affine gravity in dimensions
Consider the gravitational action where denotes the gravitational coupling constant and is an arbitrary function of the scalar curvature , with a general affine connection.3 We work in a first order (Palatini) formalism, where the metric and the connection are treated as independent variables. Variation of (4) w.r.t. gives
An action principle for the Einstein-Weyl equations in three dimensions
In this section, we present an action principle for the Einstein–Weyl equations in dimensions. To this end, we consider the contribution (9) plus additional terms involving Lagrange multipliers and gravitational Chern–Simons contributions.
Let us first recall the decomposition of the nonmetricity and torsion in a trace and traceless part. In three dimensions, one has [21] where the traces (the Weyl vector),
Final remarks
A longstanding open mathematical problem has been the construction of an action principle for the Einstein–Weyl equations. In this paper, we presented for the first time such an action in three dimensions, given by a metric affine gravity contribution plus additional pieces involving Lagrange multipliers and gravitational Chern–Simons terms. To be more precise, our model contains, in addition to the Weyl nonmetricity, also a traceless part. However, the latter can be consistently set to
Acknowledgements
This work was supported partly by INFN and by MIUR-PRIN contract 2017CC72MK003. The authors would like to thank L. Andrianopoli, R. D’Auria and M. Trigiante for inspiring discussions. L. R. would like to thank the Department of Applied Science and Technology of the Polytechnic University of Turin, and in particular F. Dolcini and A. Gamba, for financial support.
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