Properties of non-relativistic string theory

We show how Newton-Cartan geometry can be generalized to String Newton-Cartan geometry which is the geometry underlying non-relativistic string theory. Several salient properties of non-relativistic string theory in this geometric background are presented and a discussion of possible research for the future is outlined. Copyright E. A. Bergshoeff et al. This work is licensed under the Creative Commons Attribution 4.0 International License. Published by the SciPost Foundation. Received 28-10-2020 Accepted 22-12-2020 Published 13-08-2021 Check for updates doi:10.21468/SciPostPhysProc.4.001


Introduction
Newtonian gravity. Combining two of these extensions leads to general relativity, quantum field theory or NR quantum gravity. Ultimately, combining all three extensions leads to relativistic quantum gravity.
small distances certain physical quantities get quantized in units of the reduced Planck's con- 14 stant h corresponding to quantum mechanics and (3) a gravitational force can be introduced 15 via Newton's constant G leading to Newtonian gravity. There are two well-known ways to 16 combine two of these extensions: (1) extending classical mechanics with high velocities and 17 gravity leads to general relativity and (2) extending classical mechanics to high velocities and 18 small distances leads to quantum field theory. Logically speaking, however, there is a third 19 way, namely extending classical mechanics to small distances and gravity. This would lead 20 to a theory of non-relativistic (NR) quantum gravity. Finally, the maximal extension to high 21 velocities, small distances and gravity leads to the long sought for theory of quantum gravity.
This situation can nicely be summarized via the the so-called Bronstein cube [1] in Figure 1. 23 Usually, the issue of finding a consistent theory of quantum gravity is approached either by 24 adding gravity to quantum field theory or by quantizing general relativity. The Bronstein cube 25 suggests a third way to approach this issue: can quantum gravity be viewed as the relativistic 26 extension of a self-consistent NR theory of quantum gravity? This leads to the related question 27 of how essential relativity is in constructing a theory of quantum gravity or, put differently, 28 whether one can take in a consistent way the NR limit of quantum gravity. Motivated by this 29 we wish to address the following intriguing question: 30 can one define a consistent NR theory of quantum gravity?
This question can be asked for each approach to define a consistent theory of quantum 31 gravity: is relativity essential for the construction, yes or no? String theory is one approach to 32 define a theory of quantum gravity. In this talk we wish to discuss the definition of a NR string 33 theory including its underlying geometry and some of its basic properties. In particular, we will 34 show how the geometry corresponding to NR string theory can be viewed as a generalization 35 of the well-known Newton-Cartan (NC) geometry that underlies NC gravity. The independent fields of D-dimensional NC geometry are given by (a = 1, · · · , D − 1) Here, τ µ is the time-like Vierbein acting as the clock function and E µ a is the spatial Vierbein (2) The spin-connection fields ω µ a b corresponding to spatial rotations and ω µ a corresponding to 44 Galilean boosts are functions of τ µ , E µ a and M µ .

45
In NC gravity one cannot define a single non-degenerate metric for the full spacetime like 46 the Riemannian metric in general relativity. Instead, one defines two degenerate metrics is not invariant under Galilean boosts and, for this reason, cannot be used as a metric. In order 51 to make a boost-invariant combination one often considers the combination However, this combination is not invariant under central charge transformations. Neverthe-53 less, it is used in the construction of a NR particle action coupled to NC gravity in such a way 54 that the central charge gauge field M µ couples to the particle via a Wess-Zumino (WZ) term It is instructive to give some details here. To define the NR limit we first express the Rie-69 mannian metric of general relativity and the gauge fieldM µ in terms of the NC fields (1) and 70 a contraction parameter ω. Next, after substituting these expressions into the action of the 71 relativistic particle coupled to general relativity, we take the limit ω → ∞. This leads to a 72 divergence linear in ω coming form the kinetic term that is cancelled by a similar divergent 73 term coming from the WZ term by expressingM µ in terms of the NC fields as follows: Given the fact that a vector field only couples via a WZ term to a particle, it is clear that 75 one cannot apply the same procedure to define the NR limit of a string. In this case, it is the 76 Kalb-Ramond 2-form gauge fieldB µν that couples to the relativistic string via a WZ term of where ∂ α (α = 0, 1) is the derivative with respect to the world-sheet coordinates σ α and 79 x µ (σ α ) are the string embedding coordinates. It turns out that taking the NR limit of a string where B µν is the NR Kalb-Ramond field. This leads to a new so-called String Newton-Cartan 86 (SNC) geometry that is characterized by two special directions instead of the single Newto-87 nian time direction in NC gravity. The difference between particles and strings is that a particle 88 sweeps out a one-dimensional time direction whereas a sting sweeps out two directions lon-89 gitudinal to the string: one time direction and one spatial direction, see Figure 2.
Ignoring matter fields, like the Kalb-Ramond 2-form field, the independent string NC fields are For the construction of a NR string action we need both a longitudinal metric τ µν and a trans-

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We are now in a position to construct the action of NR string theory in a general SNC gravity 103 background. For flat spacetime the action was already given a long time ago and reads [2, 3] with and similar for the Lagrange multipliers λ ,λ. A special feature of NR string theory is that the 106 (perturbative) spectrum only contains winding strings along the compact x 1 direction [2].

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The presence of the Lagrange multipliers can be understood as the result of taking the NR 108 limit of the relativistic string action in Polyakov form. 3 This is best understood by comparing relativity in Polyakov form: Here e is the worldline Einbein and M is a mass parameter. Expanding the general relativity 112 fields in terms of the Newton-Cartan background fields one encounters the following quadratic 113 divergence that is not cancelled by the vector field in the Wess-Zumino term: It should be noted that this is an artefact of the Polyakov formulation. In the Nambu-Goto 115 formulation there is no quadratic divergence left. The quadratic divergence given in (16) is 116 not fatal. The reason for this is that it is the square of something and therefore can be re-117 written, using a Lagrange multiplier λ as follows: Written in this form, the limit that ω → ∞ can be taken and one ends up with the following 119 NR Polyakov action: Integrating out the Lagrange multiplier λ one finds that Substituting this back into the Polyakov action (18) one obtains the following NR particle 122 action in Nambu-Goto form: One can now take a similar limit of the relativistic Polyakov string. We thus find the fol-124 lowing expression for a NR string in a (matter-coupled) SNC background [4, 5]: 4 where T is the string tension, σ α are the world-sheet coordinates, h αβ = e α a e β b η ab is the 126 worldsheet metric with Zweibeine e α a , R (2) is the Ricci scalar defined with respect to h αβ and 127 x µ (σ), µ = 0, 1, · · · , D−1 are the string embedding coordinates. The action (21) also describes 128 the coupling to the background Kalb-Ramond field B µν and the dilaton Φ. Furthermore, λ 129 and λ are two world-sheet Lagrange multiplier fields whose equations of motion allow us to longitudinal metric τ µν as follows: As mentioned in the previous section, the so-called transverse metric H µν is given in terms of 133 the SNC background fields by 5 The definition of G occurring in the string sigma model action (21) in terms of H µν and τ µ A 135 is given by Finally, the lightcone components τ µ , τ µ of τ µ A and e α ,ē α of e α a are defined in [4,5]. Here Part of this constraint contains the spin-connection field ω µ AB , enabling one to solve this con-146 nection field in terms of τ µ A and its derivative. The remaining part is a geometric constraint 147 given by the projection of (26) that does not contain the spin-connection: An important feature of the NR action (21), which is absent in the relativistic case, is that the NR Nambu-Goto particle coupled to a vector gauge field B µ : in which case the Stückelberg symmetries are given by where we have used flat indices and where we have defined Similarly, one finds that, after integrating out the Lagrange multipliers, the NR string action 156 (21) is invariant under the following (infinitesimal) Stueckelberg symmetries, with parameters 157 C µ A , given by This Stueckelberg symmetry is a reducible symmetry in the sense that the transformation rule 159 (32) of B µν is formally invariant under a gauge symmetry, with singlet parameter C, given by

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Once the action for the NR string in a curved background has been constructed several research This would open the way to start discussing NR D-branes and NR holography from the per-175 spective of a NR gravity theory in the bulk. We hope to come back to these interesting research 176 equations in the nearby future.