DLCQ, Non-Lorentzian Supergravity, and T-Duality

We comment on the T-duality relation between non-Lorentzian string theory and the DLCQ of relativistic string theory. Particular focus will be put on the structure of the background geometries. We show how target space supersymmetry constrains the form of the respective supergravity multiplets. In the conclusions, we propose several natural extensions with the aim of eventually establishing a non-Lorentzian web of dualities

The Supersymmetric Neveu-Schwarz Branes of Non-Relativistic String Theory

We construct the basic Neveu-Schwarz (NS) brane solutions of non-relativistic string theory using longitudinal T-duality as a solution generating technique. Extending the NS background fields to a supergravity multiplet, we verify that all solutions we find are half-supersymmetric. The two perturbative solutions we find both have an interpretation as the background geometry outside a string-like object. Correspondingly, we refer to these non-Lorentzian backgrounds as winding string and unwound string solution. Whereas the winding string is part of the on-shell spectrum of non-relativistic string theory, the unwound string only makes sense off-shell where it mediates the instantaneous gravitational force. Seen from the nine-dimensional point of view, we find that the winding string solution is sourced by a non-relativistic massive particle and that the unwound string solution is sourced by a massless Galilean particle of zero colour and spin. We explain how these two string solutions fit into a discrete lightcone quantization of string theory. We shortly discuss the basic NS five-brane and Kaluza-Klein monopole solutions and show that they are both half-supersymmetric.Comment: 33 page

A Non-Relativistic Limit of NS-NS Gravity

We discuss a particular non-relativistic limit of NS-NS gravity that can be taken at the level of the action and equations of motion, without imposing any geometric constraints by hand. This relies on the fact that terms that diverge in the limit and that come from the Vielbein in the Einstein-Hilbert term and from the kinetic term of the Kalb-Ramond two-form field cancel against each other. This cancelling of divergences is the target space analogue of a similar cancellation that takes place at the level of the string sigma model between the Vielbein in the kinetic term and the Kalb-Ramond field in the Wess-Zumino term. The limit of the equations of motion leads to one equation more than the limit of the action, due to the emergence of a local target space scale invariance in the limit. Some of the equations of motion can be solved by scale invariant geometric constraints. These constraints define a so-called Dilatation invariant String Newton-Cartan geometry.Comment: 26 page

Non-Lorentzian IIB Supergravity from a Polynomial Realization of SL(2,R)

We derive the action and symmetries of the bosonic sector of non-Lorentzian IIB supergravity by taking the non-relativistic string limit. We find that the bosonic field content is extended by a Lagrange multiplier that implements a restriction on the Ramond-Ramond fluxes. We show that the SL(2,R) transformation rules of non-Lorentzian IIB supergravity form a novel, nonlinear polynomial realization. Using classical invariant theory of polynomial equations and binary forms, we will develop a general formalism describing the polynomial realization of SL(2,R) and apply it to the special case of non-Lorentzian IIB supergravity. Using the same formalism, we classify all the relevant SL(2,R) invariants. Invoking other bosonic symmetries, such as the local boost and dilatation symmetry, we show how the bosonic part of the non-Lorentzian IIB supergravity action is formed uniquely from these SL(2,R) invariants. This work also points towards the concept of a non-Lorentzian bootstrap, where bosonic symmetries in non-Lorentzian supergravity are used to bootstrap the bosonic dynamics in Lorentzian supergravity, without considering the fermions.Comment: 43 page

Branched SL(2,ℤ) duality

We investigate how SL(2,ℤ) duality is realized in nonrelativistic type IIB superstring theory, which is a self-contained corner of relativistic string theory. Within this corner, we realize manifestly SL(2,ℤ)-invariant (p, q)-string actions. The construction of these actions imposes a branching between strings of opposite charges associated with the two-form fields. The branch point is determined by these charges and the axion background field. Both branches must be incorporated in order to realize the full SL(2,ℤ) group. Besides these string actions, we also construct D-instanton and D3-brane actions that manifestly realize the branched SL(2,ℤ) symmetry

Non-Relativistic Supersymmetry on Curved Three-Manifolds

We construct explicit examples of non-relativistic supersymmetric field theories on curved Newton-Cartan three-manifolds. These results are obtained by performing a null reduction of four-dimensional supersymmetric field theories on Lorentzian manifolds and the Killing spinor equations that their supersymmetry parameters obey. This gives rise to a set of algebraic and differential Killing spinor equations that are obeyed by the supersymmetry parameters of the resulting three-dimensional non-relativistic field theories. We derive necessary and sufficient conditions that determine whether a Newton-Cartan background admits non-trivial solutions of these Killing spinor equations. Two classes of examples of Newton-Cartan backgrounds that obey these conditions are discussed. The first class is characterised by an integrable foliation, corresponding to so-called twistless torsional geometries, and includes manifolds whose spatial slices are isomorphic to the Poincar\'e disc. The second class of examples has a non-integrable foliation structure and corresponds to contact manifolds

Nonrelativistic supergravity

Man kennt und versteht Supergravitation seit ungefähr 40 Jahren, Newtonsche Gravitation sogar seit über 300 Jahren. Trotzdem ist es bis heute niemandem gelungen, eine supersymmetrische Erweiterung der Newtonschen Gravitationstheorie zu konstruieren. Das ist, gelinde gesagt, beachtenswert. In dieser Arbeit legen wir einige Forschungsergebnisse über nichtrelativistische Geometrie -- sogenannte Newton-Cartan Geometrie -- dar, die den Weg zu Newtonscher Supergravitation in vier Dimensionen bereiten. Newton-Cartan Geometrie ist charakterisiert durch eine Blätterung der Raumzeit, der eine absolute Zeitrichtung entspricht. Insbesondere betrachten und verallgemeinern wir Resultate der letzten Jahre, in denen die Theorie durch ein sogenanntes Eichprinzip aus einer nichtrelativistischen Raumzeitsymmetrie Superalgebra abgeleitet wurde. Wir haben eine alternative Methode angewandt, die auf einer speziellen Art der Dimensionsreduktion -- sogenannter Null Reduktion -- beruht. Dadurch war es möglich, die Theorie besser zu verstehen, sodass der Übergang zu allgemeinen Dimensionen in Reichweite scheint.Supergravity has been known for about 40 years, Newtonian gravity for over 300 years. Nevertheless, to this day nobody has constructed a supersymmetric extension of Newtonian gravity. This is remarkable, to say the least. In this work we present some developments in nonrelativistic geometry -- so-called Newton-Cartan geometry -- leading the way to Newtonian supergravity in four dimensions. Newton-Cartan geometry is characterized by a foliation of spacetime, corresponding to an absolute time direction. In particular, we review and generalize recent results, where the theory has been obtained via a gauging of a nonrelativistic spacetime superalgebra. We applied an alternative technique building on a special type of dimensional reduction called Null Reduction. In the process we gained a better understanding of the theory making the transition to general dimensions seem within range

Non-lorentzian supergravity and dualities

In this thesis, I investigate aspects of non-Lorentzian string theory. This theory is a self-consistent model realizing a UV completion of Newtonian gravity. That is, it describes the instantaneous gravitational interaction of winding states in the theory. More specifically, I develop an effective supergravity description of the targetspace dynamics of this theory. This is in agreement with results coming from Weyl anomaly calculations. The construction of an effective supergravity description naturally leads to several extensions and applications. I investigate two of these in this thesis. Firstly, I construct background solutions and prove that they preserve half supersymmetry. Secondly, I study dualities relating different theories and speculate on the question of a non-Lorentzian web of dualities and an eleven-dimensional completion

Non-relativistic Limits of General Relativity

We discuss non-relativistic limits of general relativity. In particular, we define a special fine-tuned non-relativistic limit, inspired by string theory, where the Einstein-Hilbert action has been supplemented by the kinetic term of a one-form gauge field. Taking the limit, a crucial cancellation takes place, in an expansion of the action in terms of powers of the velocity of light, between a leading divergence coming from the spin-connection squared term and another infinity that originates from the kinetic term of the one-form gauge field such that the finite invariant non-relativistic gravity action is given by the next subleading term. This non-relativistic action allows an underlying torsional Newton-Cartan geometry as opposed to the zero torsion Newton-Cartan geometry that follows from a more standard limit of General Relativity but it lacks the Poisson equation for the Newton potential. We will mention extensions of the model to include this Poisson equation.</p

Non-Lorentzian IIB supergravity from a polynomial realization of SL(2, ℝ)

Abstract We derive the action and symmetries of the bosonic sector of non-Lorentzian IIB supergravity by taking the non-relativistic string limit. We find that the bosonic field content is extended by a Lagrange multiplier that implements a restriction on the Ramond-Ramond fluxes. We show that the SL(2, ℝ) transformation rules of non-Lorentzian IIB supergravity form a novel, nonlinear polynomial realization. Using classical invariant theory of polynomial equations and binary forms, we will develop a general formalism describing the polynomial realization of SL(2, ℝ) and apply it to the special case of non-Lorentzian IIB supergravity. Using the same formalism, we classify all the relevant SL(2, ℝ) invariants. Invoking other bosonic symmetries, such as the local boost and dilatation symmetry, we show how the bosonic part of the non-Lorentzian IIB supergravity action is formed uniquely from these SL(2, ℝ) invariants. This work also points towards the concept of a non-Lorentzian bootstrap, where bosonic symmetries in non-Lorentzian supergravity are used to bootstrap the bosonic dynamics in Lorentzian supergravity, without considering the fermions