Non-Riemannian geometry of M-theory

Published for SISSA by Springer Received: March 4, 2019 Revised: June 24, 2019 Accepted: July 16, 2019 Published: July 30, 2019 David S. Berman,a Chris D.A. Blairb and Ray Otsukia a Centre for Research in String Theory, School of Physics and Astronomy, Queen Mary University of London, 327 Mile End Road, London E1 4NS, U.K. b Theoretische Natuurkunde, Vrije Universiteit Brussel, and the International Solvay Institutes, Pleinlaan 2, B-1050 Brussels, Belgium E-mail: , , Abstract: We construct a background for M-theory that is moduli free. This background is then shown to be related to a topological phase of the E8(8) exceptional field theory (ExFT). The key ingredient in the construction is the embedding of non-Riemannian geometry in ExFT. This allows one to describe non-relativistic geometries, such as NewtonCartan or Gomis-Ooguri-type limits, using the ExFT framework originally developed to describe maximal supergravity. This generalises previous work by Morand and Park in the context of double field theory. Keywords: M-Theory, String Duality ArXiv ePrint: 1902.01867 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP07(2019)175 JHEP07(2019)175 Non-Riemannian geometry of M-theory Contents 2 2 Generalised metrics, projectors and the topological E8(8) vacuum 2.1 Generalised metrics and diffeomorphisms 2.2 The action and equations of motion 2.3 The E8(8) ExFT and its topological phase 6 6 9 10 3 Non-Riemannian backgrounds in O(D, D) DFT 3.1 Generalised metric and coset projectors 3.2 Review of Morand-Park classification 3.3 Examples: Gomis-Ooguri limit and timelike duality 3.4 Example: Newton-Cartan non-relativistic geometry from null duality 15 15 17 19 21 4 Riemannian backgrounds and exotic supergravities in SL(5) ExFT 4.1 The SL(5) ExFT 4.2 M-theory parametrisations 4.3 IIB parametrisations 23 23 25 26 5 Non-Riemannian backgrounds in SL(5) ExFT 5.1 Examples: Gomis-Ooguri and timelike U-duality 5.2 Non-Riemannian little metrics 5.3 Reduction to O(3,3) 5.4 Embedding Newton-Cartan 28 28 31 36 39 6 Discussion 42 A Reduction from SL(5) to O(3,3) A.1 M-theory/IIA parametrisation A.2 IIB parametrisation A.3 DFT non-Riemannian parametrisation from SL(5) non-Riemannian parametrisation 43 43 43 B U-duality of Newton-Cartan uplift 45 C The usual cosets 46 –1– 44 JHEP07(2019)175 1 Introduction 1 Introduction –2– JHEP07(2019)175 General relativity describes the geometry of gravity in terms of a dynamical (pseudo-)Riemannian metric. String theory and M-theory provide a route towards a quantum mechanical understanding of gravity. At low energies, the classical geometry of string theory/Mtheory is again described by a metric, whose dynamics is governed by a supergravity theory in which the metric is accompanied by a collection of scalars and p-form gauge fields, plus fermions. The presence of duality in these theories means that they exhibit a (hidden) symmetry which mixes metric and form field components. Inspired in large part by a desire to capture and explain this symmetry more fundamentally, and to find new notions of intrinsically “stringy” or “M-theoretic” geometry treating all the massless states of the theory on a more egalitarian footing, reformulations of the dynamics of supergravity have been found in which the geometry and the fields living in the geometry are united and covariance under the duality groups of string/M-theory is made manifest. These efforts have led to the modern development of double field theory (DFT) and exceptional field theory (ExFT) [1–9], building on pioneering earlier work such as [10–15] and on the introduction of generalised geometry [16, 17]. The starting point for these theories is to observe that the bosonic degrees of freedom of supergravity in a certain (n + d)-dimensional split can be recombined into multiplets of the groups O(d, d) (when n + d = 10 [18] with the original construction applicable to n = 0 [1, 19]) or Ed(d) (when n + d = 11, and so far allowing for d = 2, . . . , 9 [7–9, 20–24]). Then the full dynamics and local symmetries of 10- or 11-dimensional supergravity can be encoded in a formulation with a manifest covariance under O(d, d) or Ed(d) . The usual diffeomorphism symmetry, which is associated to the group GL(d), is extended to a notion of generalised diffeomorphisms involving local G (= O(d, d) or Ed(d) ) transformations and realised using an extended set of coordinates Y M transforming under a particular representation of G. The original theories are recovered by solving a constraint known as the “section condition” which restricts the dependence of all fields in the theory to a subset of the Y M . Different solutions of the section condition lead to different parametrisations of the fundamental DFT/ExFT variables in terms of standard supergravity fields, depending on different choices of the physical coordinates amongst the Y M . In this way, for instance, ExFT admits inequivalent solutions of the section condition giving either 11-dimensional supergravity or the 10-dimensional type IIB supergravity [7, 25]. One can think of the usual supergravity theories as following from the single unifying ExFT formulation on solving the consistency conditions of the latter. A more ambitious interpretation of the geometry of DFT/ExFT is to allow for solutions of the section condition, or parametrisations of the fields, which do not reduce to conventional supergravity. A number of avenues have been explored, often involving notions of “non-geometry” in one form or another (for lots on non-geometry, see the review [26]). This includes the possibility of relaxing the section condition in order to carry out Scherk-Schwarz type reductions where the twist matrices may depend on dual coordi- The examples of the previous paragraph are still based on the idea that there is some spacetime description involving (possibly only locally) a Riemannian metric and some set of forms or bivectors. The novelty in the exotic backgrounds arises from global data as we “glue” patches using Ed(d) transformations, rather than traditional diffeomorphisms, but locally there is a supergravity description of some sort though perhaps gauged or “generalised” due to a Scherk-Schwarz twist. These non-geometric aspects of DFT/ExFT are of course of crucial importance. There is, however, a further generalisation we can make that will be the subject of this paper. This is the, perhaps rather unexpected, observation that DFT/ExFT also accommodates descriptions of non-Riemannian geometry. These are backgrounds where there is not an invertible spacetime metric but instead a non-relativistic geometry, or even no intrinsic geometric structure at all. Examples of such geometries go back to Newton-Cartan geometry [41, 42], and include the non-relativistic limit of string theory studied by Gomis and Ooguri [43] and Danielsson, Guijosa and Kruczenski [44, 45]. The exploration of non-Riemannian string theory (...truncated)