We study classical closed bosonic strings probing the near-horizon region of a non-extremal black hole and show that this corresponds to understanding string theory in the Carroll regime. This is done by first performing a Carroll expansion and then a near-horizon expansion of a closed relativistic string, subsequently showing that they agree. Concretely, we expand the phase space action in powers of c 2, where c is the speed of light, assuming that the target space admits a string Carroll expansion (where two directions are singled out) and show that there exist two different Carroll strings: a magnetic and an electric string. The magnetic string has a Lorentzian worldsheet, whereas the worldsheet of the electric string is Carrollian. The geometry near the horizon of a four-dimensional (4D) Schwarzschild black hole takes the form of a string Carroll expansion (a 2D Rindler space fibred over a 2-sphere). We show that the solution space of relativistic strings near the horizon bifurcates and the two sectors precisely match with the magnetic/electric Carroll strings with an appropriate target space. Magnetic Carroll strings near a black hole shrink to a point on the two-sphere and either follow null geodesics or turn into folded strings on the 2D Rindler spacetime. Electric Carroll strings wrap the two-sphere and follow a massive geodesic in the Rindler space. Finally, we show that 4D non-extremal Kerr and Reissner-Nordström black holes also admit string Carroll expansions near their outer horizons, indicating that our formulation extends to generic non-extremal black holes.

India

Arjun Bagchi

South Korea

Aritra Banerjee

United Kingdom

Jelle Hartong

Denmark

Emil Have

China

Kedar Kolekar

Strings near black holes are Carrollian. Part II

The 1/c-expansion of general relativity appropriately sourced by matter can be used to derive an action principle for Newtonian gravity. The gravitational part of this action is known as non-relativistic gravity (NRG). It is possible to source NRG differently and in such a way that one can construct solutions that are not described by Newtonian gravity (as they do not admit a notion of absolute time). It is possible to include a negative cosmological constant such that NRG admits a non-relativistic AdS solution. This non-relativistic AdS vacuum has Killing vectors that form the Galilean conformal algebra and a boundary that admits a conformal class of Newton-Cartan geometries. This begs the question of whether there exists an analogue of the fluid/gravity correspondence for NRG. In this paper we derive a non-relativistic AdS brane solution of NRG and confirm that it corresponds to the 1/c 2-expansion of the AdS black brane geometry. We perform a Galilean boost of the non-relativistic AdS brane and derive the associated boundary energy-momentum tensor. We then show that this is the energy-momentum tensor of a massless Galilean fluid and explain how this is linked to the conformal isometries of the boundary. Along the way, we also present several new results for the theory of non-relativistic gravity itself. In particular we present a rewriting that greatly shortens and simplifies the equations of motion of the NRG action.

Aditya Mehra

Jørgen Musaeus

Galilean fluids from non-relativistic gravity

We demonstrate that strings near the horizon of a Schwarzschild black hole, when viewed by a stationary observer at infinity, probe a string Carroll geometry, where the effective light speed is given by the distance from the horizon. We expand the Polyakov action in powers of this light speed to find a theory of Carrollian strings. We show that the string shrinks to a point to leading order near the horizon, which follows a null geodesic in a two-dimensional Rindler space. At the next-to-leading order the string oscillates in the embedding fields associated with the near-horizon two-sphere.

Kedar S. Kolekar

Mangesh Mandlik

Strings near black holes are Carrollian

We expand the relativistic open bosonic string in powers of 1/c 2 where c is the speed of light. We perform this expansion to next-to-leading order in 1/c 2 and relate our results to known descriptions of non-relativistic open strings obtained by taking limits. Just as for closed strings the non-relativistic expansion is well-defined if the open string winds a circle in the target space. This direction must satisfy Dirichlet boundary conditions. It is shown that the endpoints of the open string behave as Bargmann particles in the non-relativistic regime. These open strings end on nrDp-branes with p ≤ 24. When these nrDp-branes do not fluctuate they correspond to (p + 1)-dimensional Newton-Cartan submanifolds of the target space. When we include fluctuations and worldvolume gauge fields their dynamics is described by a non-relativistic version of the DBI action whose form we derive from symmetry considerations. The worldvolume gauge field and scalar field of a nrD24-brane make up the field content of Galilean electrodynamics (GED), and the effective theory on the nrD24-brane is precisely a non-linear version of GED. We generalise these results to actions for any nrDp-brane by demanding that they have the same target space gauge symmetries that the non-relativistic open and closed string actions have. Finally, we show that the nrDp-brane action is transverse T-duality covariant. Our results agree with the findings of Gomis, Yan and Yu in [1].

Non-relativistic expansion of open strings and D-branes

It is well known that one can take an infinite speed of light limit that gives rise to non-relativistic strings with a relativistic worldsheet sigma model but with a non-relativistic target space geometry. In this work we systematically explore two further limits in which the worldsheet becomes non-Lorentzian. The first gives rise to a Galilean string with a Galilean structure on the worldsheet, extending previous work on Spin Matrix-related string theory limits. The second is a completely novel limit leading to a worldsheet theory with a Carrollian structure. We find the Nambu-Goto and Polyakov formulations of both limits and explore gauge fixing choices. Furthermore, we study in detail the case of the Galilean string for a class of target space geometries that are related to Spin Matrix target space geometries, for which the Nambu-Goto action (in static gauge) is quadratic in the fields.

Leo Bidussi

Troels Harmark

Sweden

Niels Obers

Gerben Oling

Longitudinal Galilean and Carrollian limits of non-relativistic strings

We study various aspects of the Carroll limit in which the speed of light is sent to zero. A large part of this paper is devoted to the quantization of Carroll field theories. We show that these exhibit infinite degeneracies in the spectrum and may suffer from non-normalizable ground states. As a consequence, partition functions of Carroll systems are ill-defined and do not lead to sensible thermodynamics. These seemingly pathological properties might actually be a virtue in the context of flat space holography.

Netherlands

Jan Boer

Iceland

Watse Sybesma

Stefan Vandoren

Carroll stories

We further develop the string 1/c 2 expansion of closed bosonic string theory, where c is the speed of light. The expansion will be performed up to and including the next-to-next-to-leading order (NNLO). We show that the next-to-leading order (NLO) theory is equal to the Gomis-Ooguri string, generalised to a curved target space, provided the target space geometry admits a certain class of co-dimension-2 foliations. We compute the energy of the string up to NNLO for a flat target space with a circle that must be wound by the string, and we show that it agrees with the 1/c 2 expansion of the relativistic energy. We also compute the algebra of Noether charges for a flat target space and show that this matches order-by-order with an appropriate expansion of the Poincaré algebra, which at NLO gives the string Bargmann algebra. Finally, we expand the phase space action, which allows us to perform the Dirac procedure and pass to the quantum theory. It turns out that the Poisson brackets change at each order, and we show that the normal ordering constant of the relativistic theory, which does not depend on c, can be reproduced by the NLO and NNLO theories.

Nonrelativistic approximations of closed bosonic string theory

We revisit the formulation of non-relativistic (NR) string theory and its target space geometry. We obtain a new formulation in which the geometry contains a two-form field that couples to the tension current and that transforms under string Galilei boosts. This parallels the Newton-Cartan one-form that couples to the mass current of a non-relativistic point particle. We show how this formulation of the NR string arises both from an infinite speed of light limit and a null reduction of the relativistic closed bosonic string. In both cases, the two-form originates from a combination of metric quantities and the Kalb-Ramond field. The target space geometry of the NR string is seen to arise from the gauging of a new algebra that is obtained by an İnönü-Wigner contraction of the Poincaré algebra extended by the symmetries of the Kalb-Ramond field. In this new formulation, there are no superfluous target space fields that can be removed by fixing a Stückelberg symmetry. Classically, there are no foliation/torsion constraints imposed on the target space geometry.

Torsional string Newton-Cartan geometry for non-relativistic strings

We develop a novel approach to nonrelativistic closed bosonic string theory that is based on a string <math><mn>1</mn><mo>/</mo><msup><mi>c</mi><mn>2</mn></msup></math> expansion of the relativistic string, where <math><mi>c</mi></math> is the speed of light. This approach has the benefit that one does not need to take a limit of a string in a near-critical Kalb-Ramond background. The <math><mn>1</mn><mo>/</mo><msup><mi>c</mi><mn>2</mn></msup></math>-expanded Polyakov action at next-to-leading order reproduces the known action of nonrelativistic string theory provided that the target space obeys an appropriate foliation constraint. We compute the spectrum in a flat target space, with one circle direction that is wound by the string, up to next-to-leading order and show that it reproduces the spectrum of the Gomis-Ooguri string.

Nonrelativistic Expansion of Closed Bosonic Strings

We reformulate the Palatini action for general relativity in terms of moving frames that exhibit local Galilean covariance in a large speed of light expansion. For this, we express the action in terms of variables that are adapted to a Galilean subgroup of the <math><mrow><mrow><mi>GL</mi></mrow><mo>(</mo><mrow><mi>n</mi><mo>,</mo><mi>R</mi></mrow><mo>)</mo></mrow></math> structure group of a general frame bundle. This leads to a novel Palatini-type formulation of general relativity that provides a natural starting point for a first-order nonrelativistic expansion. In doing so, we show how a comparison of Lorentzian and Newton-Cartan metric compatibility explains the appearance of torsion in the nonrelativistic expansion.