Non-Lorentzian Geometry and its Applications

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Background

This Research Topic inhabits a relatively unexplored landscape for physical theories lying beyond Lorentzian geometry. Lorentzian geometry is the natural arena for Quantum Field Theory and General Relativity: two theories that are highly successful in explaining a vast range of phenomena while simultaneously ignoring each other. Theoretical physics predicts that at some scales quantum and gravitational effects will be commensurate and this would seem to require a quantum theory of gravity. Present attempts to quantize gravity (e.g., String Theory) are also based on Lorentzian geometry, but they have met with limited success. A possible approach out of this impasse is to take limits of Lorentzian theories (e.g., GR) which might prove easier to quantize. A common feature of many of these limits is that the Lorentzian metric degenerates and hence one is compelled to consider geometrical structures that go beyond Lorentzian manifolds: e.g., Newton-Cartan geometry. Such geometrical structures are also finding applications in other areas of theoretical physics (e.g., hydrodynamics, condensed matter physics) or even mathematical biology.

A full classification of physically relevant non-Lorentzian space-time (super)symmetry algebras and (super)geometries remains an open problem and the issue will address new results. The issue will also focus on diverse, novel applications of non-Lorentzian geometry. A non-exhaustive list includes

1) The application of novel types of Newton-Cartan geometry to organize the post-Newtonian expansion of General Relativity in the strong gravity regime. The issue will focus on obtaining a better understanding of the structure and symmetry principles of this framework, as well as the exploration of the physical phenomena it captures.

2) Applications of Newton-Cartan geometry with torsion or with a codimension-two foliation to non-relativistic string theory and novel limits of the AdS/CFT correspondence. Open problems here include e.g. finding new examples and applications of non-Lorentzian string backgrounds, as well as constructing (super)gravity theories that they are solutions of.

3) Applications of non-Lorentzian geometry to effective field theory and holographic approaches to e.g. strongly coupled condensed matter systems and hydrodynamics of Galilean systems, with recent applications to the description of fluid membranes.

4) The use of non-Lorentzian geometry as a tool to study exact, non-perturbative results in non-Lorentzian supersymmetric QFT and solvable irrelevant deformations of two-dimensional non-Lorentzian QFT.

The first aim of this Research Topic will be to put into place a much-needed set of reviews of the new developments of the last decade, extending the much earlier foundational work. At the same time, we want to include a selection of Original Research at the cutting edge of present knowledge. The aim is to create a document that will serve as an entry point into this fascinating subject, providing the necessary background, as well as exhibiting a snapshot of the exciting new developments in this active research area.

• Foundational material:
 o Non-Lorentzian (super)symmetries
 o Non-Lorentzian (super)geometries
• Applications
 o Newton-Cartan gravity and cosmology
 o Condensed matter theory
 o Hydrodynamics and effective field theories
 o String theory and doubled field theory
 o Flat space holography
 o Non-lorentzian supersymmetric theories and supergravities
 o Newton-Cartan geometry in mathematical biology.

Keywords: Newton-Cartan geometry, flat space holography, non-relativistic string theory, post-Newtonian gravity, non-relativistic quantum field theories

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