# Non-Riemannian gravity actions from double field theory

A.D. Gallegos, U. Gürsoy, S. Verma and N. Zinnato  Institute for Theoretical
Physics, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands

###### Abstract

Non-Riemannian gravitational theories suggest alternative avenues to
understand properties of quantum gravity and provide a concrete setting to
study condensed matter systems with non-relativistic symmetry. Derivation of
an action principle for these theories generally proved challenging for
various reasons. In this technical note, we employ the formulation of double
field theory to construct actions for a variety of such theories. This
formulation helps removing ambiguities in the corresponding equations of
motion. In particular, we embed Torsional Newton-Cartan gravity, Carrollian
gravity and String Newton-Cartan gravity in double field theory, derive their
actions and compare with the previously obtained results in literature.

###### Contents

  1. I Introduction and Summary
  2. II Doubled Gravity
    1. II.1 Generalized dilaton and Metric
    2. II.2 Connection and Curvature
  3. III Type I TNC geometry
    1. III.1 Basics
    2. III.2 Embedding in DFT
    3. III.3 Action and equations of motion
  4. IV Carrollian geometry
    1. IV.1 Basics
    2. IV.2 Embedding in DFT
    3. IV.3 Action and equations of motion
  5. V String Newton-Cartan geometry
    1. V.1 Basics
    2. V.2 Embedding in DFT
    3. V.3 Action and equations of motion
  6. VI Comparison with known results
    1. VI.1 TNC beta functions
    2. VI.2 SNC beta functions
    3. VI.3 Comparison of TNC and SNC equations of motion
  7. VII Discussion and Outlook
  8. A TNC identities
    1. A.1 Identities
    2. A.2 Variational calculus
  9. B Carroll identities
    1. B.1 Identities
    2. B.2 Variational calculus
  10. C SNC identities
    1. C.1 Identities
    2. C.2 Variational calculus
  11. D Actions in Einstein frame
    1. D.1 TNC
    2. D.2 Carroll
    3. D.3 SNC

##  I Introduction and Summary

Non-Riemannian gravitational theories have attracted renewed interest Kiritsis
; Hansen:2020pqs ; deBoer ; deBoer2  mainly due to the role they might play as
an alternative route to understanding properties of quantum gravity
Harmark2014 ; Hansen:2019vqf ; Niels2017 . However, despite various attempts,
providing actions for non-relativistic gravity based on local Galilean and
Carrollian symmetries proved difficult. It was not until recently that
successful attempts have been made Bergshoeff:2018vfn ; ActionPrinciple ;
Hansen:2020pqs  for certain classes and limits of non-relativistic geometries.
Importance of understanding dynamics of non-Riemannian gravity is underpinned
by the far-reaching possibilities this entails: theories based on Galilean
symmetry play a role on truncations of string theory Gomis:2000bd ;
Bergshoeff:2019pij ; Niels2018 , post-Newtonian physics Dautcourt_1997 ;
Dieter ; Dieter2 ; Tichy:2011te , and provide a natural setting to study
response in condensed matter systems with Galilean symmetry Son ; Hartong2014
; MTaylor ; Jay . Carrollian symmetry, on the other hand, is relevant to
description of excitations in the near horizon geometry of black holes
Duval:2014uoa ; Damour ; Donnay:2019jiz , and is instrumental in flat space
holography flatHol1 ; flatHol2 .

In the particular cases of the Bargmann and Carroll algebras it has been known
that the corresponding algebra of generators of the non-Riemannian space-time
symmetries can be constructed via null reduction of a parent relativistic
theory with a null isometry Null1 ; Null2 ; Duval:1990hj ; Duval:1984cj . In
the context of string theory they can be further related by means of T-duality
transformations Bergshoeff:2018yvt ; LEE2014134 ; Morand:2017fnv ;
Kluson:2018egd . When T-duality is applied in the null direction of the parent
relativistic theory a mapping between non-Riemannian geometries, in particular
Torsional Newton-Cartan (TNC) or Carroll type, can be established Gomis:2000bd
; Bergshoeff:2018yvt ; Morand:2017fnv . We note that this can be done for
algebras obtained via group contractions of the Poincaré group, while algebras
obtained from large speed of light expansions, i.e. expansions in $1/c$, will
fall out of the scope of this construction Hansen:2019vqf ; Hansen:2020pqs .

This seems to indicate that we should be able to describe both type of
geometries, Riemannian and non-Riemannian, in a T-invariant formulation of
gravity. Indeed, such a formulation exists. It is based on doubling the
degrees of freedom by treating D space-time coordinates and the corresponding
D space-time momenta of the compact directions on equal footing Hull_2009 ;
Hohm:2010jy ; Hohm:2010pp ; Siegel:1993xq ; Siegel:1993th . This results in a
local $O(D,D)$ invariant theory and the aforementioned T-duality
transformation becomes an $O(D,D)$ rotation. This double field theory (DFT) is
based on a generalized metric $\mathcal{H}$ and a generalized dilaton $d$
encompassing the degrees of freedom in the NS sector of string theory, namely
the matter content of the theory consists of metric, the Kalb-Ramond field and
dilaton. This generalized metric $\mathcal{H}$ is required to be an $O(D,D)$
tensor. Parametrization of the generalized metric in terms of the original
relativistic content $\\{g,B,\phi\\}$ is known since the postulation of double
field theory as a T-duality invariant generalization of string gravity
Hohm:2010pp . Importantly however, the generalized metric is not restricted to
this form and in particular admits non-Riemannian parametrizations, where the
TNC and Carroll limits appear as particular cases Morand:2017fnv . This means
that the gravitional dynamics of such non-Riemannian geometries can be
obtained by simply considering the double field equations of motion111These
equations can also be unified into a single master equation, as shown in
Park:2015bza ; Angus:2018mep . We would like to thank Jeong-Hyuck Park for
pointing this out.

$\displaystyle\mathcal{R}_{MN}$ $\displaystyle=0,$ $\displaystyle\mathcal{R}$
$\displaystyle=0$ (1)

with $\mathcal{R}\neq\eta^{MN}\mathcal{R}_{MN}$ being the generalization of
the Ricci tensor and Ricci scalar to a local $O(D,D)$ geometry. The tensor and
curvature scalar appearing in (1) can be written in terms of the generalized
metric and dilaton allowing us to fix a non-Riemannian parametrization of
$\mathcal{H}$ and obtaining the corresponding gravitational equations of
motion.

The goal of this work is to derive the actions and corresponding equations of
motion of certain type of non-Riemannian geometries by means of their
embedding in double field theory. The relation between DFT and non-Riemannian
geometries has already been explored in some detail in the literature, see
e.g. Ko:2015rha ; Berman:2019izh . We will employ the general parametrization
for $\mathcal{H}$ given by Park and Morand Morand:2017fnv  and derive the
corresponding general equations of motion. We will then specify to particular
cases of interest: Torsional Newton-Cartan (TNC) theory, Carrollian theory and
the string Newton-Cartan (SNC) theory. Some or part of these equations of
motion have been obtained from world-sheet beta-functions of string theory
Gomis:2019zyu ; Gallegos:2019icg ; YanYu  or from reductions of ordinary
Einstein’s equations. However, in all of these approaches some extra geometric
constraints arise. DFT formulation is free of these constraints which will
allow us to generalize and complete the existing studies.

The plan of the paper is as follows. In section II we give a brief
introduction to double field theory, presenting the basic tensors which will
be used to construct the non-relativistic actions. In sections III, IV and V
we determine, respectively, embeddings of TNC, Carroll and SNC theories in
double field theory. Using these embeddings we write down the respective
actions and compute the equations of motion. In section VI we compare and
discuss the equations of motion found in the current work with the ones
already found in the literature. Our work should be viewed as a compendium of
useful results on non-Riemannian gravity theories rather than an extensive
discussion of their physical properties.

### Notation and conventions

Throughout the paper greek indices $\left\\{\mu,\nu,\rho,\dots\right\\}$
denote curved spacetime directions of TNC, Carroll and SNC. The first letters
of the latin alphabet (both lowercase and uppercase) refer to flat directions,
e.g. the TNC inverse transverse metric is given by
$h^{\mu\nu}=e^{\mu}_{a}e^{\nu}_{b}\delta^{ab}$ and the SNC timelike vielbein
is $\tau_{\mu}^{A}$. Capital latin letters from the middle of the alphabet
$\left\\{M,N,\dots\right\\}$ are reserved for the DFT directions, e.g. for TNC
we will have $M=0,1,\dots,2d+1,$ where $d=D-1$ is the dimension of the TNC
spacetime. Their lowercase versions refer to half of the DFT directions, i.e.
$m=0,1,\dots,d$ for the TNC case.

The Riemann tensor is always defined via the action of the commutator of
covariant derivatives on an arbitrary vector:

$\left[D_{\mu},D_{\nu}\right]A_{\rho}=-R^{\lambda}_{\
\rho\mu\nu}A_{\lambda}-2\Gamma^{\lambda}_{[\mu\nu]}D_{\lambda}A_{\rho}$ (2)

where $\Gamma^{\rho}_{\mu\nu}$ is the connection associated to $D_{\mu}$.

##  II Doubled Gravity

In this section we review the necessary ingredients of double field gravity.
We will not be thorough in our discussion and will only discuss the key
ingredients. For a broader discussion see for example Hull_2009 ; Hohm:2010jy
; Hohm:2010pp ; Morand:2017fnv ; Hohm:2011si ; ParkRiemann . A double field
theory of gravity should be invariant under $O(D,D)$ rotations and under
double diffeomorphisms. As mentioned in the introduction the basic ingredients
are the generalized metric $\mathcal{H}$, generalized dilaton $d$, and the
$O(D,D)$ invariant metric $\eta$, given below. Coordinates on double geometry
are denoted by $X^{M}$ and can be decomposed as
$X^{M}=(X^{\mu},\bar{X}_{\nu})$ with the invariant metric

$\displaystyle\eta^{MN}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}\,.$ (3)

Double diffeomorphisms are generated via the generalized Lie derivative
$\mathcal{\hat{L}}_{\xi}$ acting on an arbitrary tensor density with weight
$\omega$ as Morand:2017fnv

$\displaystyle\mathcal{L}_{\xi}T_{M_{1}..M_{n}}$
$\displaystyle=\xi^{N}\partial_{N}T_{M_{1}...M_{n}}+\omega\partial_{N}\xi^{N}T_{M_{1}...M_{n}}+\sum_{i=1}^{n}\left(\partial_{M_{i}}\xi_{N}-\partial_{N}\xi_{M_{i}}\right)T_{M_{1}...M_{i-1}\hphantom{N}M_{i+1}...M_{n}}^{\hphantom{M_{1}...M_{i-1}}N}$
(4)

with $\xi^{M}=(\xi^{\mu},\tilde{\xi}_{\mu})$ a generalized vector. The form
(4) was originally devised such that it reduces to the one form symmetry for
the $B$-field in addition to the standard Lie derivative $\mathcal{L}_{\xi}$
on the $X^{\mu}$ coordinates after the so called “section” or strong
constraint $\bar{\partial}^{\mu}=0$ has been imposed DFTReview . This
condition arises from the covariant constraint
$\eta_{MN}\partial^{M}\partial^{N}=0$ introduced to reduce the degrees of
freedom to their original value and in fact follows from the requirement that
the Lie derivatives form a closed symmetry algebra222We thank Chris Blair for
pointing this out to us.. Under (4) it can be shown that after exponentiation
the generalized dilaton, $e^{-2d}$, will act as a scalar density of unit
weight and consequently as the integral measure, while, by construction,
$\mathcal{H}$ will be an $O(D,D)$ symmetric tensor satisfying

$\displaystyle\mathcal{H}_{AC}\eta^{CD}\mathcal{H}_{BD}$
$\displaystyle=\eta_{AB}.$ (5)

###  II.1 Generalized dilaton and Metric

When considering a Riemannian manifold the generalized dilaton has a simple
expression in terms of the usual dilaton $\phi$ and ordinary “undoubled”
metric $g_{\mu\nu}$ which will be useful for example when considering TNC and
Carrollian geometries:

$e^{-2d}=e^{-2\phi}\sqrt{-\det g_{\mu\nu}}\equiv e.$ (6)

The most general form of $\mathcal{H}$ that is symmetric and compatible with
(5) is classified by two non-negative integers, $(n,\bar{n})$ with
$n+\bar{n}<D$, and is of the form Morand:2017fnv

$\displaystyle\mathcal{H}_{MN}$
$\displaystyle=\begin{pmatrix}K_{\mu\nu}-\mathcal{B}_{\mu\rho}H^{\rho\sigma}\mathcal{B}_{\sigma\nu}+2x^{i}_{(\mu}\mathcal{B}_{\nu)\rho}y^{\rho}_{i}-2\bar{x}^{\bar{\imath}}_{(\mu}\mathcal{B}_{\nu)\rho}\bar{y}^{\rho}_{\bar{\imath}}&&-H^{\nu\rho}\mathcal{B}_{\rho\mu}+y^{\nu}_{i}x^{i}_{\mu}-\bar{y}^{\nu}_{\bar{\imath}}\bar{x}^{\bar{\imath}}_{\mu}\\\
-H^{\mu\rho}\mathcal{B}_{\rho\nu}+y^{\nu}_{i}x^{i}_{\mu}-\bar{y}^{\nu}_{\bar{\imath}}\bar{x}^{\bar{\imath}}_{\mu}&&H^{\mu\nu}\end{pmatrix}$
(7)

with $1\leq i\leq n$ and $1\leq\bar{\imath}\leq\bar{n}$. Here, $\mathcal{B}$
is a skew-symmetric matrix that is identified with the Kalb-Ramond field of
string theory, while $K$ and $H$ are symmetric matrices333These matrices are
identified with the target space metric and inverse metric respectively, when
the target space is Riemannian. They have $n+\bar{n}$ null eigenvalues each,

$\displaystyle
H^{\mu\nu}x^{i}_{\nu}=H^{\mu\nu}\bar{x}^{\bar{\imath}}_{\nu}=0,\qquad
K_{\mu\nu}y^{\nu}_{j}=K_{\mu\nu}\bar{y}^{\nu}_{\bar{\jmath}}=0\,,$ (8)

The corresponding null eigenvectors of $H$ and $K$ are denoted as $x$,
$\bar{x}$ and $y$, $\bar{y}$ respectively. They are subject to the following
completeness relation

$\displaystyle
H^{\mu\rho}K_{\rho\nu}+y^{\mu}_{i}x^{i}_{\nu}+\bar{y}^{\mu}_{\bar{\imath}}\bar{x}^{\bar{\imath}}_{\nu}=\delta^{\mu}_{\nu}$
(9)

from which the following identities can be inferred

$\displaystyle y^{\mu}_{i}x^{j}_{\mu}$
$\displaystyle=\delta^{j}_{i},\quad\bar{y}^{\mu}_{\bar{\imath}}\bar{x}^{\bar{\jmath}}_{\mu}=\delta^{\bar{\jmath}}_{\bar{\imath}},\quad
y^{\mu}_{i}\bar{x}^{\bar{\jmath}}_{\mu}=\bar{y}^{\mu}_{\bar{\imath}}x^{j}_{\mu}=0,\quad
H^{\rho\mu}K_{\mu\nu}H^{\nu\sigma}=H^{\rho\sigma},\quad
K_{\rho\mu}H^{\mu\nu}K_{\nu\sigma}=K_{\rho\sigma}$ (10)

Once the “section” condition is imposed, the generalized Lie derivative (4)
reduces to (up to $GL(n)\times GL(\bar{n})$ transformations and Milne shifts)

$\displaystyle\delta x^{i}_{\mu}$
$\displaystyle=\mathcal{L}_{\xi}x^{i}_{\mu}\,,\quad$
$\displaystyle\delta\bar{x}^{\bar{\imath}}_{\mu}$
$\displaystyle=\mathcal{L}_{\xi}\bar{x}^{\bar{\imath}}_{\mu}\,,\quad$
$\displaystyle\delta y^{\nu}_{j}$
$\displaystyle=\mathcal{L}_{\xi}y^{\nu}_{j}\,,\quad$
$\displaystyle\delta\bar{y}^{\nu}_{\bar{\jmath}}$
$\displaystyle=\mathcal{L}_{\xi}\bar{y}^{\nu}_{\bar{\jmath}}\,,$ (11)
$\displaystyle\delta H^{\mu\nu}$
$\displaystyle=\mathcal{L}_{\xi}H^{\mu\nu}\,,\quad$ $\displaystyle\delta
K_{\mu\nu}$ $\displaystyle=\mathcal{L}_{\xi}K_{\mu\nu}\,,$
$\displaystyle\delta\mathcal{B}_{\mu\nu}$
$\displaystyle=\mathcal{L}_{\xi}\mathcal{B}_{\mu\nu}+\partial_{\mu}\tilde{\xi}_{\nu}-\partial_{\nu}\tilde{\xi}_{\mu}$

Note that the trace $\mathcal{H}^{M}_{M}=2(n-\bar{n})$444In Park:2020ixf  it
was shown that, upon BRST quantization, a critical bosonic string theory can
only be anomaly-free if the trace of the generalized metric is zero, i.e. we
have to impose $n=\bar{n}$ at the quantum level for a critical theory to be
consistent.  is an $O(D,D)$ invariant scalar and also that the $B$-field acts
as an $O(D,D)$ transformation, i.e. its contribution to the generalized metric
can be factorized as follows:

$\displaystyle\mathcal{H}_{AB}=\begin{pmatrix}1&\mathcal{B}\\\
0&1\end{pmatrix}\begin{pmatrix}K&Z\\\
Z^{T}&H\end{pmatrix}\begin{pmatrix}1&0\\\ -\mathcal{B}&1\end{pmatrix}$ (12)

where we defined

$Z^{\mu}_{\ \nu}\equiv
y^{\mu}_{i}x_{\nu}^{i}-\bar{y}^{\mu}_{\bar{\imath}}\bar{x}_{\nu}^{\bar{\imath}}.$
(13)

The generalized metric (7) as well as the relations (8)-(10) are invariant
under the $GL(n)\times GL(\bar{n})$ rotations

$\displaystyle\left(x^{i}_{\mu},y^{\mu}_{i},\bar{x}^{\bar{\imath}}_{\mu},\bar{y}^{\nu}_{\bar{\imath}}\right)\rightarrow\left(x^{j}_{\mu}R^{i}_{j},\left(R^{-1}\right)_{i}^{\hphantom{i}j}y^{\nu}_{j},\bar{x}^{\bar{\jmath}}_{\mu}\bar{R}_{\bar{\jmath}}^{\hphantom{\bar{\jmath}}\bar{\imath}},\left(\bar{R}^{-1}\right)_{\bar{\imath}}^{\hphantom{\bar{\imath}}\bar{\jmath}}\bar{y}^{\nu}_{j}\right)\,,$
(14)

and under the generalized shift symmetry

$\displaystyle\begin{split}\left(y^{\mu}_{i}\right)^{\prime}&=y_{i}^{\mu}+V^{\mu}_{i}\\\
\left(\bar{y}^{\mu}_{\bar{\imath}}\right)^{\prime}&=\bar{y}_{\bar{\imath}}^{\mu}+\bar{V}^{\mu}_{\bar{\imath}}\\\
\left(K_{\mu\nu}\right)^{\prime}&=K_{\mu\nu}-2x^{i}_{(\mu}K_{\nu)\rho}V^{\rho}_{i}-2\bar{x}^{\bar{\imath}}_{(\mu}K_{\nu)\rho}\bar{V}^{\rho}_{\bar{\imath}}+\left(x_{\mu}^{i}V_{\rho
i}+\bar{x}^{\bar{\imath}}_{\mu}\bar{V}_{\rho\bar{\imath}}\right)\left(x^{i}_{\nu}V^{\rho}_{i}+\bar{x}^{\bar{\imath}}_{\nu}\bar{V}^{\rho}_{\bar{\imath}}\right)\\\
\left(\mathcal{B}_{\mu\nu}\right)^{\prime}&=\mathcal{B}_{\mu\nu}-2x^{i}_{[\mu}V_{\nu]i}+2\bar{x}^{\bar{\imath}}_{[\mu}\bar{V}_{\nu]\bar{\imath}}+2x^{i}_{[\mu}\bar{x}^{\bar{\jmath}}_{\nu]}\left(y_{i}^{\rho}\bar{V}_{\rho\bar{\jmath}}+\bar{y}_{\bar{\jmath}}^{\rho}V_{\rho
i}+V_{\rho i}\bar{V}^{\rho}_{\bar{\jmath}}\right)\end{split}$ (15)

with $V_{\mu i}$ and $\bar{V}_{\mu\bar{\imath}}$ being the transformation
parameters and we defined $V^{\mu}_{i}\equiv H^{\mu\rho}V_{\rho
i},\bar{V}^{\mu}_{\bar{\imath}}\equiv H^{\mu\rho}\bar{V}_{\rho\bar{\imath}}$.

Finally we note that in the present work we do not include a cosmological
constant term in the DFT action. However this generalization is
straightforward as one only needs to add a term proportional to
$e^{-2d}\Lambda_{DFT}$ to the action Park:2019hbc . This term could
potentially be important when considering non-relativistic holography.

###  II.2 Connection and Curvature

In analogy with general relativity we will introduce a connection
$\Gamma_{CAB}$ that will allow us to covariantize derivative interactions. The
following unique Christoffel connection is found ParkRiemann

$\displaystyle\begin{split}\Gamma_{CAB}&=2\left(P\partial_{C}P\bar{P}\right)_{[AB]}+2\left(\bar{P}_{[A}^{\hphantom{[A}D}\bar{P}_{B]}^{\hphantom{B]}E}-P_{[A}^{\hphantom{[A}D}P_{B]}^{\hphantom{B]}E}\right)\partial_{D}P_{EC}\\\
&\hphantom{=}-4\left(\frac{1}{P_{M}^{\hphantom{M}M}-1}P_{C[A}P_{B]}^{\hphantom{B]}D}+\frac{1}{\bar{P}_{M}^{\hphantom{M}M}-1}\bar{P}_{C[A]}\bar{P}_{B]}^{\hphantom{B]}D}\right)\left(\partial_{D}d+\left(P\partial^{E}P\bar{P}\right)_{[ED]}\right)\end{split}$
(16)

with $P_{MN}$ and $\bar{P}_{MN}$ the projector operators defined as

$\displaystyle P_{MN}$
$\displaystyle=\frac{1}{2}\left(\eta_{MN}+\mathcal{H}_{MN}\right)\,,$ (17)
$\displaystyle\bar{P}_{MN}$
$\displaystyle=\frac{1}{2}\left(\eta_{MN}-\mathcal{H}_{MN}\right)\,,$ (18)

and satisfying the standard properties
$P^{2}=P,\bar{P}^{2}=\bar{P},P\bar{P}=0,P+\bar{P}=1,$ and with the
corresponding traces

$\displaystyle
P_{M}^{\hphantom{M}M}=D+n-\bar{n}\,,\qquad\bar{P}_{M}^{\hphantom{M}M}=D-n+\bar{n}$
(19)

The connection (16) is determined uniquely after imposing compatibility with
$\mathcal{H}$, $d$, and the Lie derivative (4) as well as the additional set
of projection constraints555If these projections are not enforced the
connection can not be fully determined DFTReview . However the relevant
covariant curvatures constructed from it will be unique.

$\displaystyle\mathcal{P}_{ABC}^{\hphantom{ABC}DEF}\Gamma_{DEF}$
$\displaystyle=0,$
$\displaystyle\mathcal{\bar{P}}_{ABC}^{\hphantom{ABC}DEF}\Gamma_{DEF}$
$\displaystyle=0$ (20)

with

$\displaystyle\mathcal{P}_{ABC}^{\hphantom{ABC}DEF}\equiv
P_{A}^{\hphantom{A}D}P_{[B}^{\hphantom{B]}[E}P_{C]}^{\hphantom{C]}F]}+\frac{2}{P_{M}^{\hphantom{M}M}-1}P_{A[B}P_{C]}^{\hphantom{C]}[E}P^{F]D}\,,$
(21)
$\displaystyle\mathcal{\bar{P}}_{ABC}^{\hphantom{ABC}DEF}\equiv\bar{P}_{A}^{\hphantom{A}D}\bar{P}_{[B}^{\hphantom{B]}[E}\bar{P}_{C]}^{\hphantom{C]}F]}+\frac{2}{\bar{P}_{M}^{\hphantom{M}M}-1}\bar{P}_{A[B}\bar{P}_{C]}^{\hphantom{C]}[E}\bar{P}^{F]D}\,.$
(22)

A field strength $R_{ABCD}$ for the connection $\Gamma$ can be constructed as
usual

$\displaystyle R_{CDAB}$
$\displaystyle=\partial_{A}\Gamma_{BCD}-\partial_{B}\Gamma_{ACD}+\Gamma_{AC}^{\hphantom{AC}E}\Gamma_{BED}-\Gamma_{BC}^{\hphantom{BC}E}\Gamma_{AED}$
(23)

However (23) is not a covariant object, in fact no fully covariant four-index
Riemann curvature can be constructed in DFT DFTReview ; Hohm:2011si .
Nevertheless we can build the semi-covariant curvature Hohm:2011si ; DFTReview
; ParkRiemann  $\mathcal{R}_{ABCD}$

$\displaystyle\mathcal{R}_{ABCD}$
$\displaystyle=\frac{1}{2}\left(R_{ABCD}+R_{CDAB}-\Gamma^{E}_{\hphantom{E}AB}\Gamma_{ECD}\right)\,,$
(24)

satisfying symmetry properties

$\displaystyle\mathcal{R}_{ABCD}=\mathcal{R}_{CDAB}=\mathcal{R}_{[AB][CD]}\,,\qquad\mathcal{R}_{A[BCD]}=0$
(25)

and the semi-covariant transformation rule

$\displaystyle\delta_{\xi}\mathcal{R}_{ABCD}$ $\displaystyle=$
$\displaystyle\mathcal{\hat{L}}_{\xi}\mathcal{R}_{ABCD}+2\nabla_{[A}\left(\left(\mathcal{P}+\mathcal{\bar{P}}\right)_{B][CD]}^{\hphantom{B][CD]}EFG}\partial_{E}\partial_{F}\xi_{G}\right)$
(26)
$\displaystyle+2\nabla_{[C}\left(\left(\mathcal{P}+\mathcal{\bar{P}}\right)_{D][AB]}^{\hphantom{D][CD]}EFG}\partial_{E}\partial_{F}\xi_{G}\right)\,.$

Even though the curvature defined in (25) is not covariant we can build the
following covariant contraction

$\displaystyle\mathcal{R}$
$\displaystyle\equiv\left(P^{AC}P^{BD}-\bar{P}^{AC}\bar{P}^{BD}\right)\mathcal{R}_{ABCD}$
(27)

which we will call the doubled Ricci scalar. It can be shown that it can be
written in terms of the double fields $\mathcal{H}$ and $d$ as

$\displaystyle\mathcal{R}$
$\displaystyle=4\mathcal{H}^{MN}\partial_{M}\partial_{N}d-\partial_{M}\partial_{N}\mathcal{H}^{MN}-4\mathcal{H}^{MN}\partial_{M}d\partial_{N}d+4\partial_{M}\mathcal{H}^{MN}\partial_{N}d$
$\displaystyle\quad+\frac{1}{8}\mathcal{H}^{MN}\partial_{M}\mathcal{H}^{KL}\partial_{N}\mathcal{H}_{KL}-\frac{1}{2}\mathcal{H}^{MN}\partial_{M}\mathcal{H}^{KL}\partial_{K}\mathcal{H}_{NL}$
(28)

Equipped with (27), we can write down the DFT equivalent of the Einstein-
Hilbert action

$\displaystyle S_{DFT}$ $\displaystyle=\int
d^{D}xd^{D}\tilde{x}\,e^{-2d}\mathcal{R}.$ (29)

The equations of motion are then found by varying (29), and given by666It is
important to note that the variation cannot be performed using the
parametrization (7) for fixed $(n,\bar{n})$. This would miss $n\times\bar{n}$
equations Cho:2019ofr .

$\displaystyle\delta S$ $\displaystyle=\int
d^{D}xd^{D}\tilde{x}e^{-2d}\left[\delta\mathcal{H}^{MN}\mathcal{K}_{MN}-2\mathcal{R}\delta
d\right]$ (30)

with $\mathcal{K}_{MN}$ defined as

$\displaystyle\begin{split}\mathcal{K}_{MN}&=\frac{1}{8}\partial_{M}\mathcal{H}^{KL}\partial_{N}\mathcal{H}_{KL}-\frac{1}{4}\left(\partial_{L}-2\left(\partial_{L}d\right)\right)\left(\mathcal{H}^{LK}\partial_{K}\mathcal{H}_{MN}\right)+2\partial_{M}\partial_{N}d\\\
&\hphantom{=}-\frac{1}{2}\partial_{(M}\mathcal{H}^{KL}\partial_{L}\mathcal{H}_{N)K}+\frac{1}{2}\left(\partial_{L}-2\left(\partial_{L}d\right)\right)\left(\mathcal{H}^{KL}\partial_{(M}\mathcal{H}_{N)K}+\mathcal{H}^{K}_{(M}\partial_{K}\mathcal{H}^{L}_{N)}\right)\,.\end{split}$
(31)

We need to ensure $\delta H^{MN}$ satisfies (5), this can be achieved as long
as we assume the variation takes the form Hohm:2010pp

$\displaystyle\delta H^{MN}$
$\displaystyle=P^{M}_{L}\delta\mathcal{M}^{LK}\bar{P}^{N}_{K}+\bar{P}^{M}_{L}\delta\mathcal{M}^{LK}P^{N}_{K}\,.$
(32)

After assuming the form (32) we find that the equations of motion associated
to the action (29) are nothing but777The same result could also be obtained by
using the following property of the Riemann tensor Park:2012xn :
$\delta\mathcal{R}_{MNPQ}=\nabla_{[M}\delta\Gamma_{N]PQ}+\nabla_{[P}\delta\Gamma_{Q]MN}.$

$\displaystyle\begin{split}\mathcal{R}_{MN}&=0\,,\\\
\mathcal{R}&=0\end{split}$ (33)

where the double Ricci tensor is now expressed in terms of $\mathcal{K}$:

$\displaystyle\mathcal{R}_{MN}$
$\displaystyle=\bar{P}^{L}_{M}\mathcal{K}_{LK}P^{K}_{N}+P^{L}_{M}\mathcal{K}_{LK}\bar{P}^{K}_{N}.$
(34)

Calling $\mathcal{R}_{MN}$ as generalized Ricci tensor is appropriate as it
can be related to the semi-covariant curvature via
$\mathcal{R}_{MN}=P^{R}_{M}\mathcal{R}^{A}_{\hphantom{A}RAS}\bar{P}^{S}_{N}$.
Note however, $\mathcal{R}\neq\eta^{MN}\mathcal{R}_{MN}$=0.

In the following sections we will consider action (29) and equations of motion
(33) for some choices of (7), in particular TNC, Carroll and SNC
parametrizations. This will allow us to write non-Riemannian gravitational
equations of motion by means of the parametrization (7). Our usage of the
relation between DFT and the previously mentioned non-relativistic geometries
will not go further than (7). A detailed analysis pertaining the symmetries
and related aspects of non-Riemann geometry following from (7) has been done
by Blair et al. in newDFT .

##  III Type I TNC geometry

###  III.1 Basics

A covariant treatment of Galilean symmetry was first obtained by Cartan
Cartan1923 ; Cartan2 ; Friedrichs  which later lead to the Newton-Cartan (NC)
geometry as the underlying structure of classical Newtonian gravity.
Subsequent work Inonu1952 ; Bargmann:1954gh ; LevyLeblond:1967zz ;
LeblondBook1 ; Duval_1993  clarified the algebra of spacetime transformations
and its representation theory that underlies NC geometry. In particular it was
shown in Andringa:2010it  that the NC geometry follows from gauging the
Bargmann algebra, that is, the U(1) central extension of Galilean boost
algebra, together with translations and rotations. Generalization of the
Newton-Carton geometry to include torsion was obtained in Niels2014 ;
connection1 , and this extension was coined “torsional Newton-Cartan” (TNC)
geometry888In fact, it was argued that it is necessary to include torsion in
this theory, see connection1 .. A crucial element in this geometry is the U(1)
gauge symmetry corresponding to the aforementioned central charge and
physically related to mass conservation. Non-relativistic gravity has recently
been studied in the context of non-relativistic effective actions Son , non-
relativistic holography Hartong2014 , post-Newtonian expansions of general
relativity Dieter , and more recently in the context of string theory
Niels2018 ; Gallegos:2019icg ; Harmark:2019upf . We remark that in this paper
we will not consider Type II Newton-Cartan geometries, which arise from a
$1/c$ expansion of General Relativity Dieter ; Hansen:2019vqf ; Hansen:2020pqs
.

TNC geometries are characterized by local Galilean symmetry of its tangent
space. The non-relativistic spacetime is embedded using a set of spatial
frames $e^{a}_{\mu}$ used to define a spatial (or transverse) metric

$\displaystyle h_{\mu\nu}$
$\displaystyle=e^{a}_{\mu}e^{b}_{\nu}\delta_{ab}\,,$ (35)

and a temporal frame $\tau_{\mu}$ used as a universal one-form clock. The
local Galilean symmetry can be extended by including a local $U(1)$
transformation through the introduction of a mass one-form $m_{\mu}$. This
centrally extended local Galilean symmetry acts on the metric complex
$\\{e^{a}_{\mu},\tau_{\mu}\\}$ and the $U(1)$ connection $m_{\mu}$ as

$\displaystyle\begin{split}\delta
e^{a}_{\mu}&=\mathcal{L}_{\xi}e^{a}_{\mu}+\lambda^{a}\tau_{\mu}+\lambda^{a}_{\hphantom{b}b}e^{b}_{\mu}\,,\\\
\delta\tau_{\mu}&=\mathcal{L}_{\xi}\tau_{\mu}\,,\\\ \delta
m_{\mu}&=\mathcal{L}_{\xi}m_{\mu}+\lambda_{a}e^{a}_{\mu}+\partial_{\mu}\sigma\,,\end{split}$
(36)

with $\lambda^{a}$ a local Galilean boost parameter, $\lambda^{ab}$ a local
parameter of rotations, $\sigma$ a $U(1)$ gauge transformation parameter, and
$\xi^{\mu}$ parametrizing diffeomorphisms. The square matrix
$(\tau_{\mu},e^{a}_{\mu})$ has an inverse $(-\upsilon^{\mu},e^{\mu}_{a})$
where these inverse frames are orthogonal

$\upsilon^{\mu}\tau_{\mu}=-1\,,\qquad e^{\mu}_{a}\tau_{\mu}=0\,,\qquad
e^{a}_{\mu}\upsilon^{\mu}=0\,,$ (37)

and complete

$e^{a}_{\nu}e_{a}^{\mu}-\upsilon^{\mu}\tau_{\nu}=\delta^{\mu}_{\nu}\,.$ (38)

Making use of inverse spatial frames $e^{\mu}_{a}$, one can define an inverse
spatial metric as

$\displaystyle h^{\mu\nu}$
$\displaystyle=e^{\mu}_{a}e^{\nu}_{b}\delta^{ab}\,,$ (39)

such that the completeness relation can be rewritten as

$h^{\mu\rho}h_{\rho\nu}-\upsilon^{\mu}\tau_{\nu}=\delta^{\mu}_{\nu}\,.$ (40)

It is well known that a geometry described by these fields and symmetries can
be obtained from null reduction of a relativistic theory described by the line
element

$ds^{2}=2\tau_{\mu}dx^{\mu}(du-m_{\nu}dx^{\nu})+h_{\mu\nu}dx^{\mu}dx^{\nu},$
(41)

where $u$ is the direction associated to the null isometry, i.e.
$\partial_{u}$ is a Killing vector. Note that the null reduction of the
relativistic connection will not give exactly the connection we use in this
text, (45), however the difference between the two is simply given by the
torsion component.

Instead of the set of variables
$\\{\tau_{\mu},\upsilon^{\nu},h^{\mu\nu},h_{\mu\nu},m_{\mu}\\}$ it will be
more convenient for us to work with the boost-invariant set
$\\{\tau_{\mu},\hat{\upsilon}^{\mu},\bar{h}_{\mu\nu},h^{\mu\nu},\Phi\\}$ where
$\\{\hat{\upsilon}^{\mu},\bar{h}_{\mu\nu},\Phi\\}$ are defined as

$\displaystyle\begin{split}\bar{h}_{\mu\nu}&\equiv
h_{\mu\nu}-\tau_{\mu}m_{\nu}-\tau_{\nu}m_{\mu},\\\
\hat{\upsilon}^{\mu}&\equiv\upsilon^{\mu}-h^{\mu\nu}m_{\nu}\\\
\Phi&\equiv-\upsilon^{\rho}m_{\rho}+\frac{1}{2}h^{\rho\sigma}m_{\rho}m_{\sigma}\end{split}$
(42)

which are the boost-invariant combinations with the physical interpretation of
a spatial metric, an “inverse clock”, and Newton gravitational potential (i.e.
the scalar which will appear in Poisson’s equation). While these variables are
explicitly invariant under Galilean boosts, they still transform under the
$U(1)$ extension as

$\delta_{\sigma}\bar{h}_{\mu\nu}=-2\tau_{(\mu}\partial_{\nu)}\sigma\,,\qquad\delta_{\sigma}\hat{\upsilon}^{\mu}=-h^{\mu\nu}\partial_{\nu}\sigma\,,\qquad\delta_{\sigma}\Phi=-\hat{\upsilon}^{\rho}\partial_{\rho}\sigma\,,$
(43)

whereas $\\{\tau_{\mu},h^{\mu\nu}\\}$ remain invariant under all local gauge
transformations. The Galilean invariant fields are subject to the identities

$\bar{h}_{\mu\rho}h^{\rho\nu}-\hat{\upsilon}^{\nu}\tau_{\mu}=\delta^{\nu}_{\mu}\,,\qquad\hat{\upsilon}^{\mu}\tau_{\mu}=-1\,,\qquad\hat{\upsilon}^{\mu}\bar{h}_{\mu\nu}=2\Phi\tau_{\nu}\,.$
(44)

The connection we will use is999See also Bernal:2002ph ; Bekaert:2014bwa  for
a classification of TNC connections.

$\Gamma^{\rho}_{\mu\nu}=-\hat{\upsilon}^{\rho}\partial_{\mu}\tau_{\nu}+\frac{1}{2}h^{\rho\sigma}\left(\partial_{\mu}\bar{h}_{\nu\sigma}+\partial_{\nu}\bar{h}_{\mu\sigma}-\partial_{\sigma}\bar{h}_{\mu\nu}\right)$
(45)

which is manifestly boost invariant, but not $U(1)$ invariant. Note that this
connection is compatible with $h^{\mu\nu}$ and $\tau_{\mu}$. The antisymmetric
part of this connection is poportional to the torsion tensor of TNC:

$F_{\mu\nu}\equiv\partial_{[\mu}\tau_{\nu]},$ (46)

in terms of which we define the acceleration

$a_{\mu}\equiv\hat{\upsilon}^{\rho}F_{\rho\mu}.$ (47)

In the following section we will reformulate this theory by embedding it in
DFT.

###  III.2 Embedding in DFT

By examining (41) we can identify the following metric and inverse
relativistic metrics:

$\displaystyle g_{\mu\nu}$
$\displaystyle=\begin{pmatrix}\bar{h}_{\mu\nu}&\tau_{\mu}\\\
\tau_{\mu}&0\end{pmatrix}\,,\qquad$ $\displaystyle g^{\mu\nu}$
$\displaystyle=\begin{pmatrix}h^{\mu\nu}&-\hat{\upsilon}^{\mu}\\\
-\hat{\upsilon}^{\mu}&2\Phi\end{pmatrix},$ (48)

which can be embedded in DFT as

$\displaystyle\mathcal{H}_{MN}$
$\displaystyle=\begin{pmatrix}\bar{h}_{\mu\nu}&\tau_{\mu}&0&0\\\
\tau_{\mu}&0&0&0\\\ 0&0&h^{\mu\nu}&-\hat{\upsilon}^{\mu}\\\
0&0&-\hat{\upsilon}^{\mu}&2\Phi\end{pmatrix}\,.$ (49)

We now apply a T-duality transformation to swap the null direction $u$ with
the dual null direction $\bar{u}$,

$\displaystyle\mathcal{T}^{M}_{\hphantom{M}N}$
$\displaystyle=\begin{pmatrix}\delta^{\mu}_{\hphantom{\mu}\nu}&0&0&0\\\
0&0&0&1\\\ 0&0&\delta_{\mu}^{\hphantom{\mu}\nu}&0\\\ 0&1&0&0\end{pmatrix},$
(50)

which results in the following TNC generalized metric:

$\displaystyle\mathcal{H}_{MN}$
$\displaystyle=\begin{pmatrix}\bar{h}_{\mu\nu}&0&0&\tau_{\mu}\\\
0&2\Phi&-\hat{\upsilon}^{\nu}&0\\\ 0&-\hat{\upsilon}^{\mu}&h^{\mu\nu}&0\\\
\tau_{\nu}&0&0&0\end{pmatrix}.$ (51)

The generalized metrics (49) and (51) will produce the same actions and
equations of motion, however note that the lower right block in (51) is now
degenerate meaning that the latter has a clearer non-relativistic
interpretation. As shown in Blair:2019qwi , after adding matter to this
parametrization we find that the embedding is given by101010Note that now we
are briefly using $m,n$ as indices of the DFT tensors, as a way to avoid
introducing new symbols. We have $m=0,1,\dots,d+1$ and $\mu=0,1,\dots,d$ where
$d$ is the dimension of the TNC spacetime and $d+1=D$ is the dimension of its
uplift, which includes the null direction $u$.

$\displaystyle K_{mn}$ $\displaystyle=\begin{pmatrix}h_{\mu\nu}&&0\\\
0&&0\end{pmatrix},$ $\displaystyle H^{mn}$
$\displaystyle=\begin{pmatrix}h^{\mu\nu}&&h^{\mu\rho}\aleph_{\rho}\\\
h^{\nu\rho}\aleph_{\rho}&&h^{\rho\sigma}\aleph_{\rho}\aleph_{\sigma}\end{pmatrix},$
$\displaystyle\mathcal{B}_{mn}$
$\displaystyle=\begin{pmatrix}\bar{B}_{\mu\nu}&&-m_{\mu}\\\
m_{\nu}&&0\end{pmatrix}$ (52)

and

$\displaystyle x_{m}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}\tau_{\mu}-\aleph_{\mu}\\\
1\end{pmatrix},$ $\displaystyle\bar{x}_{m}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}\tau_{\mu}+\aleph_{\mu}\\\
-1\end{pmatrix},$ $\displaystyle y^{m}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}-\upsilon^{\mu}\\\
1-\upsilon^{\mu}\aleph_{\mu}\end{pmatrix},$ $\displaystyle\bar{y}^{m}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}-\upsilon^{\mu}\\\
-1-\upsilon^{\mu}\aleph_{\mu},\end{pmatrix}.$ (53)

where $\aleph_{\mu}\equiv-B_{\mu u}$ appears from the dimensional reduction of
the $B$-field. We also define

$b_{\mu\nu}\equiv\partial_{[\mu}\aleph_{\nu]},\qquad\qquad\mathfrak{e}_{\mu}\equiv\hat{\upsilon}^{\rho}b_{\rho\mu}.$
(54)

The tensors $K_{\mu\nu}$ and $H^{\mu\nu}$ both have $2=1+1$ null eigenvectors
and the trace of the generalized metric is $\mathcal{H}^{M}_{M}=0$, implying
that a TNC geometry corresponds to a (1,1) theory in the DFT framework.

###  III.3 Action and equations of motion

Having obtained the embedding of TNC geometry in double field theory, we can
now immediately write down its action using the DFT action in equation (29).
We first introduce the following notation. Given an arbitrary tensor
$A_{\mu\nu\dots}$ we will define for brevity

$A^{\mu\nu\dots}\equiv A_{\rho\sigma\dots}h^{\mu\rho}h^{\nu\sigma}\dots$ (55)

Note that the only TNC fields which naturally have upper indices are
$\hat{\upsilon}^{\mu}$ and $h^{\mu\nu}$, hence any other tensor with upper
indices is to be understood as defined via (55).

Now, the TNC action is given by111111It is worth noting that this same action
can also be obtained as the dimensional reduction of the standard NS-NS sector
of supergravity  $S_{NS-NS}=\int d^{D}x\,e^{-2\phi}\sqrt{-\det
g}\left(\mathcal{R}-\frac{1}{12}H^{2}+4\left(\partial\phi\right)^{2}\right)$
where $g$ is the Riemannian metric appearing in (48), $H_{\mu\nu\rho}$ is the
field-strength of the B-field and $D=d+1$ with $d$ the dimension of the TNC
spacetime. Note however that this null reduction is not fully consistent, as
one needs to impose Poisson’s equation on-shell, rather than deriving it from
the (null-reduced) action.

$\displaystyle\begin{split}S=&\int
d^{d}x\,e\,\left[\mathcal{R}+\frac{1}{2}a^{\mu}a_{\mu}+\frac{1}{2}\mathfrak{e}^{\mu}\mathfrak{e}_{\mu}-4a^{\mu}D_{\mu}\phi+4D^{\mu}\phi
D_{\mu}\phi-\frac{1}{12}H^{\mu\nu\rho}H_{\mu\nu\rho}\right.\\\
&\hphantom{(((\int
d^{d}xe\,}\left.-\frac{1}{2}\hat{\upsilon}^{\rho}H_{\rho\mu\nu}b^{\mu\nu}-\frac{1}{2}\left(F^{\mu\nu}F_{\mu\nu}+b^{\mu\nu}b_{\mu\nu}\right)\Phi\right]\end{split}$
(56)

with $H=dB$, $b=d\aleph$ and

$e\equiv\sqrt{\frac{\text{det}\bar{h}}{2\Phi}}e^{-2\phi}.$ (57)

We choose the independent fields of our theory to be
$\Phi,\hat{\upsilon}^{\mu},h^{\mu\nu},B_{\mu\nu}$ and $\aleph_{\mu}$, see
Appendix A for useful identites.

The variation of the action (56) with respect to $\Phi$ imposes a generalized
twistlessness contraint on torsion:

$F_{\mu\nu}F^{\mu\nu}=b_{\mu\nu}b^{\mu\nu}.$ (58)

The equations for the matter fields are

$\displaystyle D_{\mu}D^{\mu}\phi+a^{\mu}D_{\mu}\phi-2D_{\mu}\phi D^{\mu}\phi$
$\displaystyle=\frac{1}{2}\mathfrak{e}^{2}-\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}-\frac{1}{2}\hat{\upsilon}^{\lambda}H_{\lambda\mu\nu}b^{\mu\nu}-\frac{1}{2}F_{\mu\nu}F^{\mu\nu}\Phi$
(59) $\displaystyle D^{\rho}b_{\rho\mu}-2D^{\rho}\phi\,b_{\rho\mu}$
$\displaystyle=\frac{1}{2}F^{\rho\sigma}H_{\rho\sigma\mu}$ (60) $\displaystyle
D^{\rho}H_{\rho\mu\nu}+a^{\rho}H_{\rho\mu\nu}-2D^{\rho}\phi\,H_{\rho\mu\nu}$
$\displaystyle=2D_{[\mu}\mathfrak{e}_{\nu]}-2\hat{\upsilon}^{\lambda}D_{\lambda}b_{\mu\nu}+2a_{[\mu}\mathfrak{e}_{\nu]}+4b^{\rho}_{\hphantom{\rho}[\mu}F_{\nu]\rho}\Phi$
$\displaystyle\quad+\left(2\hat{\upsilon}^{\rho}D_{\rho}\phi-
D_{\rho}\hat{\upsilon}^{\rho}\right)b_{\mu\nu},$ (61)

while the equation for $\tau_{\mu}$ is

$\displaystyle D^{\rho}F_{\rho\mu}+a^{\rho}F_{\rho\mu}-2D^{\rho}\phi
F_{\rho\mu}$
$\displaystyle=\frac{1}{2}b^{\rho\sigma}H_{\rho\sigma\mu}+\mathfrak{e}^{\rho}b_{\rho\mu}.$
(62)

The equation obtained from the variation with respect to $h^{\mu\nu}$ requires
more care. First of all, notice that we must have $\tau_{\mu}\tau_{\nu}\delta
h^{\mu\nu}=0$. This means that the most general variation is given by

$\delta
h^{\mu\nu}=\left(\Delta_{S}\right)_{\rho}^{\mu}\delta\mathcal{M}^{\rho\nu}+\delta\mathcal{M}^{\mu\rho}\left(\Delta_{S}\right)_{\rho}^{\nu}$
(63)

where $\delta\mathcal{M}^{\mu\nu}$ is an arbitrary symmetric tensor and
$\left(\Delta_{S}\right)_{\nu}^{\mu}=h^{\mu\rho}\bar{h}_{\rho\nu}$. It also
follows that the time-time projection of the equation obtained from this
variation will be trivially zero, i.e. we will not be able to obtain Newton’s
law from this variation. This will also be true for variation of the
Carrollian and SNC actions and is, in fact, a general property inherited from
double field theory Cho:2019ofr . By imposing an ansatz on the generalized
metric and then computing the variation of the resulting action we will end up
with $n\times\bar{n}$ equations less than we would initially expect. However,
this problem can be easily avoided by taking the variation of the DFT action
first and imposing the TNC ansatz on the resulting equations of motion (1).
With this in mind, we find that the variation of (56) with respect to
$h^{\mu\nu}$ produces the equation

$\displaystyle\begin{split}\mathcal{R}_{(\mu\nu)}+2D_{(\mu}D_{\nu)}\phi-\frac{1}{4}h^{\rho\sigma}h^{\lambda\kappa}H_{\mu\rho\lambda}H_{\nu\sigma\kappa}&=\frac{a_{\mu}a_{\nu}-\mathfrak{e}_{\mu}\mathfrak{e}_{\nu}}{2}-\hat{\upsilon}^{\rho}D_{(\mu}F_{\nu)\rho}+\hat{\upsilon}^{\lambda}h^{\rho\sigma}b_{\rho(\mu}H_{\nu)\lambda\sigma}\\\
&\quad-\left(F_{\mu\rho}F_{\nu\sigma}h^{\rho\sigma}-b_{\mu\rho}b_{\nu\sigma}h^{\rho\sigma}\right)\Phi\\\
&\quad+\mathfrak{R}_{\rho\sigma}\left(\Delta_{T}\right)^{\rho}_{\mu}\left(\Delta_{T}\right)^{\sigma}_{\nu}\end{split}$
(64)

where we defined

$\mathfrak{R}_{\mu\nu}\equiv\mathcal{R}_{\mu\nu}+2D_{\mu}D_{\nu}\phi-\frac{1}{4}h^{\rho\sigma}h^{\lambda\kappa}H_{\mu\rho\lambda}H_{\nu\sigma\kappa}+a^{2}\tau_{\mu}\tau_{\nu}\Phi-\mathfrak{e}^{2}\tau_{\mu}\tau_{\nu}\Phi.$
(65)

Note that because of the presence of $\mathfrak{R}_{\mu\nu}$ in (64) the time-
time projection of Einstein’s equations is identically zero. As we already
explained, Newton’s law can be found by imposing the TNC ansatz on the DFT
equations (1). The resulting equation is

$D^{\mu}D_{\mu}\Phi+3a^{\mu}D_{\mu}\Phi+m^{2}_{\Phi}\Phi-2F_{\mu\nu}F^{\mu\nu}\Phi^{2}=\rho_{\mathcal{K}}+\rho_{m}$
(66)

with

$\displaystyle m^{2}_{\Phi}$
$\displaystyle=a^{2}+\mathfrak{e}^{2}+4a^{\mu}D_{\mu}\phi-\hat{\upsilon}^{\rho}H_{\rho\mu\nu}b^{\mu\nu},$
(67) $\displaystyle\rho_{\mathcal{K}}$
$\displaystyle=\hat{\upsilon}^{\mu}D_{\mu}D_{\nu}\hat{\upsilon}^{\nu}+D_{\mu}\hat{\upsilon}^{\nu}D_{\nu}\hat{\upsilon}^{\mu}=-\hat{\upsilon}^{\mu}D_{\mu}\mathcal{K}^{\nu}_{\nu}+\mathcal{K}_{\mu\nu}\mathcal{K}^{\mu\nu},$
(68) $\displaystyle\rho_{m}$
$\displaystyle=\frac{1}{4}\hat{\upsilon}^{\mu}\hat{\upsilon}^{\nu}H_{\mu\rho\sigma}H_{\nu}^{\hphantom{\nu}\rho\sigma}-2\hat{\upsilon}^{\mu}\hat{\upsilon}^{\nu}D_{\mu}D_{\nu}\phi.$
(69)

Notice that by using the identities

$\displaystyle\hat{\upsilon}^{\mu}\hat{\upsilon}^{\nu}\mathcal{R}_{\mu\nu}$
$\displaystyle=-\hat{\upsilon}^{\mu}D_{\mu}D_{\nu}\hat{\upsilon}^{\nu}+\hat{\upsilon}^{\mu}D_{\nu}D_{\mu}\hat{\upsilon}^{\nu}+a^{\mu}D_{\mu}\Phi+2a^{2}\Phi,$
(70) $\displaystyle
D_{\mu}\hat{\upsilon}^{\rho}D_{\rho}\hat{\upsilon}^{\nu}+\hat{\upsilon}^{\rho}D_{\mu}D_{\rho}\hat{\upsilon}^{\nu}$
$\displaystyle=h^{\nu\sigma}\left(D_{\mu}D_{\sigma}\Phi+2D_{\mu}a_{\sigma}\,\Phi+2a_{\sigma}D_{\mu}\Phi\right),$
(71)

we can rewrite (66) as

$\hat{\upsilon}^{\mu}\hat{\upsilon}^{\nu}\mathcal{R}_{\mu\nu}=\frac{1}{4}\hat{\upsilon}^{\mu}\hat{\upsilon}^{\nu}H_{\mu\rho\sigma}H_{\nu}^{\hphantom{\nu}\rho\sigma}-2\hat{\upsilon}^{\mu}\hat{\upsilon}^{\nu}D_{\mu}D_{\nu}\phi-a^{2}\Phi+\mathfrak{e}^{2}\Phi$
(72)

so that the full Einstein’s equations can compactly be written as

$\displaystyle\begin{split}\mathcal{R}_{(\mu\nu)}+2D_{(\mu}D_{\nu)}\phi-\frac{1}{4}h^{\rho\sigma}h^{\lambda\kappa}H_{\mu\rho\lambda}H_{\nu\sigma\kappa}&=\frac{a_{\mu}a_{\nu}-\mathfrak{e}_{\mu}\mathfrak{e}_{\nu}}{2}-\hat{\upsilon}^{\rho}D_{(\mu}F_{\nu)\rho}+\hat{\upsilon}^{\lambda}h^{\rho\sigma}b_{\rho(\mu}H_{\nu)\lambda\sigma}\\\
&\quad-\left(F_{\mu\rho}F_{\nu\sigma}h^{\rho\sigma}-b_{\mu\rho}b_{\nu\sigma}h^{\rho\sigma}\right)\Phi.\end{split}$
(73)

In summary, the TNC equations of motion are given by

$\displaystyle F_{\mu\nu}F^{\mu\nu}$ $\displaystyle=b_{\mu\nu}b^{\mu\nu},$
(74) $\displaystyle D^{\rho}F_{\rho\mu}+a^{\rho}F_{\rho\mu}-2D^{\rho}\phi
F_{\rho\mu}$
$\displaystyle=\frac{1}{2}b^{\rho\sigma}H_{\rho\sigma\mu}+\mathfrak{e}^{\rho}b_{\rho\mu}$
(75) $\displaystyle D^{\rho}b_{\rho\mu}-2D^{\rho}\phi\,b_{\rho\mu}$
$\displaystyle=\frac{1}{2}F^{\rho\sigma}H_{\rho\sigma\mu},$ (76)
$\displaystyle D_{\mu}D^{\mu}\phi+a^{\mu}D_{\mu}\phi-2D_{\mu}\phi D^{\mu}\phi$
$\displaystyle=\frac{1}{2}\mathfrak{e}^{2}-\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}-\frac{1}{2}\hat{\upsilon}^{\lambda}H_{\lambda\mu\nu}b^{\mu\nu}-\frac{1}{2}F_{\mu\nu}F^{\mu\nu}\Phi,$
(77) $\displaystyle
D^{\rho}H_{\rho\mu\nu}+a^{\rho}H_{\rho\mu\nu}-2D^{\rho}\phi\,H_{\rho\mu\nu}$
$\displaystyle=2D_{[\mu}\mathfrak{e}_{\nu]}-2\hat{\upsilon}^{\lambda}D_{\lambda}b_{\mu\nu}+2a_{[\mu}\mathfrak{e}_{\nu]}+4b^{\rho}_{\hphantom{\rho}[\mu}F_{\nu]\rho}\Phi$
$\displaystyle\quad+\left(2\hat{\upsilon}^{\rho}D_{\rho}\phi-
D_{\rho}\hat{\upsilon}^{\rho}\right)b_{\mu\nu},$ (78)
$\displaystyle\mathcal{R}_{(\mu\nu)}+2D_{(\mu}D_{\nu)}\phi-\frac{1}{4}h^{\rho\sigma}h^{\lambda\kappa}H_{\mu\rho\lambda}H_{\nu\sigma\kappa}$
$\displaystyle=\frac{a_{\mu}a_{\nu}-\mathfrak{e}_{\mu}\mathfrak{e}_{\nu}}{2}-\hat{\upsilon}^{\rho}D_{(\mu}F_{\nu)\rho}+\hat{\upsilon}^{\lambda}h^{\rho\sigma}b_{\rho(\mu}H_{\nu)\lambda\sigma}$
$\displaystyle\quad-\left(F_{\mu\rho}F_{\nu\sigma}h^{\rho\sigma}-b_{\mu\rho}b_{\nu\sigma}h^{\rho\sigma}\right)\Phi,$
(79)

where we remind the reader of the short-hand notation (55), as well as the
following definitions

$\displaystyle\begin{split}a_{\mu}&=\hat{\upsilon}^{\rho}F_{\rho\mu}=2\hat{\upsilon}^{\rho}\partial_{[\rho}\tau_{\mu]},\\\
\mathfrak{e}_{\mu}&=\hat{\upsilon}^{\rho}b_{\rho\mu}=2\hat{\upsilon}^{\rho}\partial_{[\rho}\aleph_{\mu]}\end{split}$
(80)

and $\Phi$ is Newton’s potential, not to be confused with the dilaton $\phi$.

Note that these equations are manifestly invariant under almost all
transformations described by the Bargmann algebra. The only nontrivial
transformation corresponds to the $U(1)$ generator $m_{\mu}$. However a
straightforward (but tedious) computation shows that these equations are
indeed invariant under mass $U(1)$, although not manifestly so. We will
discuss some properties of these equations and their relation to known results
in section VI.

##  IV Carrollian geometry

###  IV.1 Basics

The Carroll algebra can be obtained by considering a particular contraction
($c\rightarrow 0$) of the Poincaré algebra Levy1965 ; Hartong:2015xda ;
Bergshoeff:2017btm . In Duval:2014uva  it was suggested that this algebra
could play an important role in flat space holography, hence it would be
interesting to study how field theories couple to Carrollian spacetime, see
e.g. Hofman:2014loa ; Bagchi:2019clu ; Bagchi:2019xfx . To this end, one needs
a gravitational action coupled to matter, and this is precisely what we will
compute below.

We will describe Carrollian geometry using the same set of symbols we already
introduced for the TNC case above. However, these fields will now transform
under the Carrollian boosts rather than the Galilean boosts:

$\delta_{\mathcal{C}}\tau_{\mu}=\lambda_{\mu},\qquad\qquad\delta_{\mathcal{C}}h^{\mu\nu}=2h^{\rho(\mu}\upsilon^{\nu)}\lambda_{\rho}.$
(81)

Moreover we replace the gauge field $m_{\mu}$ by a contravariant vector
$M^{\mu}$, which transforms as

$\delta
M^{\mu}=\mathcal{L}_{\xi}M^{\mu}+e^{\mu}_{a}\lambda^{a}=\mathcal{L}_{\xi}M^{\mu}+h^{\mu\nu}\lambda_{\nu}.$
(82)

This allows us to build the following manifestly boost-invariant tensors

$\displaystyle\hat{\tau}_{\mu}$ $\displaystyle=\tau_{\mu}-h_{\mu\nu}M^{\nu},$
$\displaystyle\hat{h}^{\mu\nu}$
$\displaystyle=h^{\mu\nu}-2M^{(\mu}\upsilon^{\nu)}+2\Phi\upsilon^{\mu}\upsilon^{\nu}\equiv\bar{h}^{\mu\nu}+2\Phi\upsilon^{\mu}\upsilon^{\nu},$
(83) $\displaystyle\Phi$
$\displaystyle=-M^{\mu}\tau_{\mu}+\frac{1}{2}h_{\mu\nu}M^{\mu}M^{\nu}=-M^{\mu}\hat{\tau}_{\mu}-\frac{1}{2}h_{\mu\nu}M^{\mu}M^{\nu}\,,$

which have the interpretation as the boost-invariant clock one-form, inverse
spatial metric and the Newton potential respectively. They satisfy the
following orthogonality and completeness relations

$\hat{h}^{\mu\rho}h_{\rho\nu}-\upsilon^{\mu}\hat{\tau}_{\nu}=\delta^{\mu}_{\nu},\qquad\qquad\hat{\tau}_{\mu}\hat{h}^{\mu\rho}=0=\upsilon^{\mu}h_{\mu\rho}.$
(84)

It is possible to embed a Carrollian geometry in a Lorentzian one just as we
did for TNC in (41):

$ds^{2}=du(2\Phi\,du-2\hat{\tau}_{\mu}dx^{\mu})+h_{\mu\nu}dx^{\mu}dx^{\nu}$
(85)

A connection compatible with $\hat{\tau}_{\mu},\upsilon^{\mu},h_{\mu\nu}$ and
$\hat{h}^{\mu\nu}$ can be constructed Hartong:2015xda ; Bekaert_2018 :

$\widetilde{\Gamma}^{\rho}_{\mu\nu}=-\upsilon^{\rho}\partial_{\mu}\hat{\tau}_{\nu}+\frac{1}{2}\hat{h}^{\mu\lambda}\left(\partial_{\mu}h_{\nu\lambda}+\partial_{\nu}h_{\mu\lambda}-\partial_{\lambda}h_{\mu\nu}\right)-\hat{h}^{\mu\lambda}\mathcal{K}_{\lambda\mu}\hat{\tau}_{\nu},$
(86)

where we introduced the extrinsic curvature

$\mathcal{K}_{\mu\nu}=-\frac{1}{2}\mathcal{L}_{\upsilon}h_{\mu\nu}=-\frac{1}{2}\left(\upsilon^{\rho}\partial_{\rho}h_{\mu\nu}+\left(\partial_{\mu}\upsilon^{\rho}\right)h_{\rho\nu}+\left(\partial_{\nu}\upsilon^{\rho}\right)h_{\mu\rho}\right).$
(87)

However, we find it more convenient to use a slightly different connection,

$\Gamma^{\rho}_{\mu\nu}\equiv\widetilde{\Gamma}^{\rho}_{\mu\nu}+\hat{h}^{\mu\lambda}\mathcal{K}_{\lambda\mu}\hat{\tau}_{\nu}.$
(88)

Using $\hat{\tau}_{\mu}\hat{h}^{\mu\nu}=0$, it is easy to show that this
connection is still compatible with $\hat{\tau}_{\mu}$ and $\hat{h}^{\mu\nu}$,
but now we have

$D_{\mu}\upsilon^{\nu}=-\hat{h}^{\nu\lambda}\mathcal{K}_{\lambda\mu},\qquad\qquad
D_{\rho}h_{\mu\nu}=-2\mathcal{K}_{\rho(\mu}\hat{\tau}_{\nu)}$ (89)

where we used

$\upsilon^{\mu}\mathcal{K}_{\mu\nu}=\upsilon^{\rho}D_{\rho}\upsilon^{\mu}=0.$
(90)

We also define the following tensors in analogy with TNC:

$F_{\mu\nu}\equiv\partial_{[\mu}\hat{\tau}_{\nu]},\qquad\qquad
a_{\mu}\equiv\upsilon^{\rho}F_{\rho\mu}.$ (91)

###  IV.2 Embedding in DFT

Given the relativistic geometry (85) we can construct the generalized metric

$\displaystyle\mathcal{H}_{MN}$
$\displaystyle=\begin{pmatrix}h_{\mu\nu}&-\hat{\tau}_{\mu}&0&0\\\
-\hat{\tau}_{\mu}&2\Phi&0&0\\\ 0&0&\bar{h}^{\mu\nu}&\upsilon^{\mu}\\\
0&0&\upsilon^{\mu}&0\end{pmatrix}.$ (92)

There are clearly some similarities between this metric and the TNC one (49).
In fact when the scalars $\Phi^{(\mathcal{C})}$ and $\Phi^{(TNC)}$ are both
zero (which implies $M^{\mu}=m_{\mu}=0$) it is easy to see that the two
generalized metrics become identical up to some obvious identifications. When
the two scalars are not zero we still have a relation between the two
geometries, but it is a bit more involved. To see this we can start from (92)
and then apply a T-duality transformation that swaps the Carrollian directions
$\mu$ with their dual ones $\bar{\mu}$ and arrive at

$\displaystyle\mathcal{H}_{MN}$
$\displaystyle\rightarrow\begin{pmatrix}\bar{h}^{\mu\nu}&0&0&\upsilon^{\mu}\\\
0&2\Phi&-\hat{\tau}_{\nu}&0\\\ 0&-\hat{\tau}_{\mu}&h_{\mu\nu}&0\\\
\upsilon^{\nu}&0&0&0\end{pmatrix}.$ (93)

This is equivalent to (51) if we make the identifications

$\displaystyle\bar{h}^{\mu\nu}_{(\mathcal{C})}$
$\displaystyle\leftrightarrow\bar{h}_{\mu\nu}^{(TNC)},$
$\displaystyle\hat{\tau}_{\mu}^{(\mathcal{C})}$
$\displaystyle\leftrightarrow\hat{\upsilon}^{\mu}_{(TNC)},$
$\displaystyle\Phi^{(\mathcal{C})}$
$\displaystyle\leftrightarrow\Phi^{(TNC)},$ (94) $\displaystyle
h_{\mu\nu}^{(\mathcal{C})}$ $\displaystyle\leftrightarrow h^{\mu\nu}_{(TNC)},$
$\displaystyle\upsilon^{\mu}_{(\mathcal{C})}$
$\displaystyle\leftrightarrow\tau_{\mu}^{(TNC)},$ $\displaystyle
M^{\mu}_{(\mathcal{C})}$ $\displaystyle\leftrightarrow m_{\mu}^{(TNC)}.$ (95)

This is the same duality that was proposed and discussed in Hartong:2015xda ;
Duval_2014 ; Bekaert_2018 , however when working in the DFT framework it is
clear what the interpretation of this duality is, i.e. it is simply a
T-duality following from the fact that this theory is embedded in string
theory! Note that this duality is mapping two theories that are in principle
really different, since one is a non-relativistic theory while the other is an
ultra-relativistic theory. In some sense it seems like this T-duality is
acting on these spacetimes as $c\leftrightarrow 1/c$ with $c$ being the speed
of light. It is also interesting to note that when acting with this
transformation on the Carrollian side we are effectively generating massive
particles, which will correspond to the eigenstates of the $U(1)$ generator on
the TNC side.

Moreover we remark that this duality only maps the TNC generalized metric to
the Carrollian one (and vice versa). To map the full actions to one another we
would need to transform the partial derivatives as well, i.e. we would need a
transformation of the form

$(\partial_{\mu})^{(Car)}\leftrightarrow(\partial^{\mu})^{(TNC)},$ (96)

but it is not clear what a partial derivative with upper index means in a non-
relativistic (or ultra-relativistic) theory. It will be interesting to explore
this duality in more detail in future works, to understand if the two actions
can indeed be mapped to each other.

###  IV.3 Action and equations of motion

Given an arbitrary tensor $A_{\mu\nu\dots}$ we define

$A^{\mu\nu\dots}\equiv
A_{\rho\sigma\dots}\hat{h}^{\mu\rho}\hat{h}^{\nu\sigma}\dots$ (97)

The only Carroll fields which naturally have upper indices are
$\upsilon^{\mu},\bar{h}^{\mu\nu},\hat{h}^{\mu\nu}$ and $M^{\mu}$, hence any
other tensor with upper indices is to be understood as defined via (97).

The action for a Carrollian gravitational theory is given by121212As for TNC,
this action can be obtained as the dimensional reduction of the standard NS-NS
sector of the supergravity actions, with (relativistic) metric given by
$g_{\mu\nu}$ appearing in (85). Furthermore note that when
$\Phi^{(Carroll)}=0$ this action correctly reduces to (56) with
$\Phi^{(TNC)}=0$.

$\displaystyle\begin{split}S=&\int
d^{d}x\,e\,\left[\mathcal{R}+\frac{1}{2}a^{\mu}a_{\mu}+\frac{1}{2}\mathfrak{e}^{\mu}\mathfrak{e}_{\mu}-4a^{\mu}D_{\mu}\phi+4D^{\mu}\phi
D_{\mu}\phi+2\mathcal{K}\upsilon^{\mu}D_{\mu}\Phi+4\upsilon^{\mu}D_{\mu}\phi\,\upsilon^{\nu}D_{\nu}\Phi\right.\\\
&\hphantom{(((\int
d^{d}xe}\left.+2\Phi\left(\mathcal{K}^{\mu\nu}\mathcal{K}_{\mu\nu}-\mathcal{K}^{2}-4\upsilon^{\mu}D_{\mu}\phi\,\upsilon^{\nu}D_{\nu}\phi-4\mathcal{K}\upsilon^{\mu}D_{\mu}\phi+\frac{1}{4}\upsilon^{\rho}\upsilon^{\sigma}H^{\mu\nu}_{\hphantom{\mu\nu}\rho}H_{\mu\nu\sigma}\right)\right.\\\
&\hphantom{(((\int
d^{d}xe}\left.-\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}+\frac{1}{2}b^{\mu\nu}H_{\mu\nu\rho}\upsilon^{\rho}\right]\end{split}$
(98)

with
$\mathcal{K}\equiv\hat{h}^{\mu\nu}\mathcal{K}_{\mu\nu}=-D_{\mu}\upsilon^{\mu}$,
$\mathcal{R}\equiv\hat{h}^{\mu\nu}\mathcal{R}_{\mu\nu}$ and the measure is
defined as

$e=e^{-2\phi}\sqrt{\frac{2\Phi}{\det\bar{h}^{\mu\nu}}}=e^{-2\phi}\sqrt{2\Phi\det\bar{h}_{\mu\nu}},$
(99)

where
$\bar{h}_{\mu\nu}=h_{\mu\nu}-\frac{\hat{\tau}_{\mu}\hat{\tau}_{\nu}}{2\Phi}$.
We choose the independent fields of our theory to be
$\Phi,\upsilon^{\mu},\hat{h}^{\mu\nu},B_{\mu\nu}$ and $\aleph_{\mu}$, see
Appendix B for useful identites.

The variation of the action with respect to $\Phi$ gives

$\mathcal{R}_{\mu\nu}\upsilon^{\mu}\upsilon^{\nu}=\frac{1}{4}\upsilon^{\rho}\upsilon^{\sigma}H^{\mu\nu}_{\hphantom{\mu\nu}\rho}H_{\mu\nu\sigma}-2\upsilon^{\mu}\upsilon^{\nu}D_{\mu}D_{\nu}\phi.$
(100)

Note that contrary to what happened for TNC, this equation does not impose any
constraint on the antisymmetric part of the connection, however it does impose
a constraint on intrinsic torsion, which in the Carrollian case is given by
the extrinsic curvature Figueroa-OFarrill:2020gpr .

The equations for the matter fields are

$\displaystyle D_{\mu}D^{\mu}\phi+a^{\mu}D_{\mu}\phi-2D_{\mu}\phi D^{\mu}\phi$
$\displaystyle=2\upsilon^{\mu}\upsilon^{\nu}\left(\Phi
D_{\mu}D_{\nu}\phi-2\Phi D_{\mu}\phi D_{\nu}\phi+D_{\mu}\phi
D_{\nu}\Phi\right)-2\mathcal{K}\Phi\upsilon^{\mu}D_{\mu}\phi$
$\displaystyle\quad-\frac{1}{12}H^{\mu\nu\rho}H_{\mu\nu\rho}+\frac{1}{2}\upsilon^{\rho}\upsilon^{\sigma}\Phi
H_{\rho}^{\
\mu\nu}H_{\sigma\mu\nu}+\frac{\mathfrak{e}^{\mu}\mathfrak{e}_{\mu}}{2}+\frac{1}{2}\upsilon^{\rho}b^{\mu\nu}H_{\rho\mu\nu}$
(101) $\displaystyle D^{\rho}b_{\rho\mu}-2D^{\rho}\phi\,b_{\rho\mu}$
$\displaystyle=-2\Phi\upsilon^{\rho}D_{\rho}\mathfrak{e}_{\mu}-2\Phi\mathfrak{e}^{\rho}\mathcal{K}_{\rho\mu}+2\Phi\mathfrak{e}_{\mu}\mathcal{K}+4\Phi\upsilon^{\rho}D_{\rho}\Phi\,\mathfrak{e}_{\mu}$
$\displaystyle\quad+\frac{1}{2}F^{\rho\sigma}H_{\rho\sigma\mu}+2D^{\rho}\Phi\,\upsilon^{\sigma}H_{\rho\sigma\mu}-2\Phi\upsilon^{\rho}a^{\sigma}H_{\rho\sigma\mu}$
(102) $\displaystyle
D^{\rho}H_{\rho\mu\nu}+a^{\rho}H_{\rho\mu\nu}-2D^{\rho}\phi\,H_{\rho\mu\nu}$
$\displaystyle=\upsilon^{\rho}D_{\rho}b_{\mu\nu}-2\upsilon^{\rho}D_{\rho}\phi\,b_{\mu\nu}-2\mathcal{K}_{[\mu}^{\hphantom{\mu}\rho}b_{\nu]\rho}-\mathcal{K}b_{\mu\nu}$
$\displaystyle\quad+2\Phi\upsilon^{\rho}\upsilon^{\sigma}\left(D_{\rho}H_{\sigma\mu\nu}-2D_{\rho}\phi\,H_{\sigma\mu\nu}\right)+2\upsilon^{\rho}\upsilon^{\sigma}D_{\rho}\Phi\,H_{\sigma\mu\nu}$
$\displaystyle\quad+4\Phi\upsilon^{\sigma}\mathcal{K}_{[\mu}^{\hphantom{\mu}\rho}H_{\nu]\rho\sigma}-2\mathcal{K}\Phi\upsilon^{\rho}H_{\rho\mu\nu}.$
(103)

The equation for $\upsilon^{\mu}$ is

$\displaystyle\begin{split}D^{\rho}F_{\rho\mu}+a^{\rho}F_{\rho\mu}-2D^{\rho}\phi\,F_{\rho\mu}&=-4\Phi\upsilon^{\rho}D_{(\rho}a_{\mu)}+2\Phi
a_{\mu}\mathcal{K}+4\Phi\upsilon^{\rho}D_{\rho}\phi\,a_{\mu}-2\upsilon^{\rho}D_{\rho}\Phi\,a_{\mu}\\\
&\quad+2\upsilon^{\rho}D_{\rho}D_{\mu}\Phi+2\mathcal{K}_{\rho\mu}D^{\rho}\Phi-2\mathcal{K}D_{\mu}\Phi-4\upsilon^{\rho}D_{\rho}\phi\,D_{\mu}\Phi\\\
&\quad+\frac{1}{2}b^{\rho\sigma}H_{\rho\sigma\mu}-\mathfrak{e}^{\rho}b_{\rho\mu}-2\Phi\upsilon^{\rho}\mathfrak{e}^{\sigma}H_{\rho\sigma\mu}.\end{split}$
(104)

Einstein’s equations are given by

$\displaystyle\begin{split}\mathcal{R}_{(\mu\nu)}+2D_{(\mu}D_{\nu)}\phi-\frac{1}{4}H_{\mu}^{\hphantom{\mu}\rho\sigma}H_{\nu\rho\sigma}&=\frac{a_{\mu}a_{\nu}-\mathfrak{e}_{\mu}\mathfrak{e}_{\nu}}{2}+D_{(\mu}a_{\nu)}-F_{(\mu}^{\hphantom{\mu}\rho}\mathcal{K}_{\nu)\rho}+\upsilon^{\rho}b_{(\mu}^{\hphantom{\mu}\sigma}H_{\nu)\rho\sigma}\\\
&\quad+2\upsilon^{\rho}\mathcal{K}_{\mu\nu}D_{\rho}\Phi-2\Phi\mathcal{K}\,\mathcal{K}_{\mu\nu}+2\Phi\upsilon^{\rho}D_{\rho}\mathcal{K}_{\mu\nu}\\\
&\quad-4\Phi\upsilon^{\rho}\mathcal{K}_{\mu\nu}D_{\rho}\phi-\Phi\upsilon^{\lambda}\upsilon^{\kappa}H_{\mu\lambda}^{\hphantom{\mu\lambda}\rho}H_{\nu\kappa\rho}.\end{split}$
(105)

Once again one equation is missing, but it can be found directly from DFT. It
is found to be

$\displaystyle\begin{split}D^{\mu}D_{\mu}\Phi-a^{\mu}D_{\mu}\Phi-2D^{\mu}\phi
D_{\mu}\Phi+(a^{2}-\mathfrak{e}^{2})\Phi&=\frac{1}{4}\left(F^{\mu\nu}F_{\mu\nu}-b^{\mu\nu}b_{\mu\nu}\right)-2\mathcal{K}\Phi\upsilon^{\rho}D_{\rho}\Phi\\\
&\quad+2\Phi\upsilon^{\rho}\upsilon^{\sigma}\left(D_{\rho}D_{\sigma}\Phi-2D_{\rho}\phi
D_{\sigma}\Phi\right).\end{split}$ (106)

It is also possible to rewrite these equations in terms of
$\bar{h}^{\mu\nu}=\hat{h}^{\mu\nu}+2\Phi\upsilon^{\mu}\upsilon^{\nu}$ rather
than $\hat{h}^{\mu\nu}$. The new connection $\bar{\Gamma}^{\rho}_{\mu\nu}$ is
defined as

$\Gamma^{\rho}_{\mu\nu}=\bar{\Gamma}^{\rho}_{\mu\nu}+2\Phi\upsilon^{\rho}\mathcal{K}_{\mu\nu}$
(107)

where quantities with a bar on them are understood to be defined with
$\bar{h}^{\mu\nu}$ instead of $\hat{h}^{\mu\nu}$, i.e.

$\bar{\Gamma}^{\rho}_{\mu\nu}=-\upsilon^{\rho}\partial_{\mu}\hat{\tau}_{\nu}+\frac{1}{2}\bar{h}^{\mu\lambda}\left(\partial_{\mu}h_{\nu\lambda}+\partial_{\nu}h_{\mu\lambda}-\partial_{\lambda}h_{\mu\nu}\right).$
(108)

The new Ricci tensor is related to the old one via

$\mathcal{R}_{\mu\nu}={\mathcal{\bar{R}}}_{\mu\nu}-2\upsilon^{\rho}\bar{D}_{\rho}\Phi\,\mathcal{K}_{\mu\nu}+2\Phi\mathcal{K}\,\mathcal{K}_{\mu\nu}-2\Phi\upsilon^{\rho}\bar{D}_{\rho}\mathcal{K}_{\mu\nu}.$
(109)

The action and equations will look nicer when using this connection, however
$\bar{h}^{\mu\nu},\hat{h}^{\mu\nu}$ and $\hat{\tau}_{\mu}$ are not compatible
now:

$\displaystyle\begin{split}\bar{D}_{\mu}\bar{h}^{\rho\sigma}&=2\upsilon^{\rho}\upsilon^{\sigma}\bar{D}_{\mu}\Phi-8\mathcal{K}_{\mu\lambda}\bar{h}^{\lambda(\rho}\upsilon^{\sigma)}\Phi,\\\
\bar{D}_{\mu}\hat{h}^{\rho\sigma}&=-4\mathcal{K}_{\mu\lambda}\bar{h}^{\lambda(\rho}\upsilon^{\sigma)}\Phi,\\\
\bar{D}_{\mu}\hat{\tau}_{\nu}&=-2\mathcal{K}_{\mu\nu}\Phi.\end{split}$ (110)

The action can then be rewritten as

$\displaystyle\begin{split}S=&\int
d^{d}x\,e\,\left[\mathcal{\bar{R}}+\frac{1}{2}a^{\mu}a_{\mu}+\frac{1}{2}\mathfrak{e}^{\mu}\mathfrak{e}_{\mu}-4a^{\mu}\bar{D}_{\mu}\phi+4\bar{D}^{\mu}\phi\bar{D}_{\mu}\phi+4\upsilon^{\mu}\upsilon^{\nu}\bar{D}_{\mu}\phi\,\bar{D}_{\nu}\Phi\right.\\\
&\hphantom{(((\int
d^{d}xe}\left.-8\Phi\mathcal{K}\upsilon^{\rho}\bar{D}_{\rho}\phi-\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}+\frac{1}{2}b^{\mu\nu}\upsilon^{\rho}H_{\mu\nu\rho}\right],\end{split}$
(111)

where now the tensors with upper indices are defined using $\bar{h}^{\mu\nu}$
and we will always write any expression such that no derivatives act on
$\bar{h}^{\mu\nu}$, e.g.
$\bar{D}^{\mu}\phi=\bar{h}^{\mu\nu}\bar{D}_{\nu}\phi\neq\bar{D}_{\nu}\left(\bar{h}^{\mu\nu}\phi\right)$.
The resulting equations of motion for the geometric fields are

$\displaystyle\mathcal{\bar{R}}_{\mu\nu}\upsilon^{\mu}\upsilon^{\nu}$
$\displaystyle=\frac{1}{4}\upsilon^{\rho}\upsilon^{\sigma}H^{\mu\nu}_{\hphantom{\mu\nu}\rho}H_{\mu\nu\sigma}-2\upsilon^{\mu}\upsilon^{\nu}\bar{D}_{\mu}\bar{D}_{\nu}\phi$
(112)
$\displaystyle\bar{D}^{\rho}F_{\rho\mu}+a^{\rho}F_{\rho\mu}-2\bar{D}^{\rho}\phi\,F_{\rho\mu}$
$\displaystyle=\frac{1}{2}b^{\rho\sigma}H_{\rho\sigma\mu}+\mathfrak{e}^{\rho}b_{\mu\rho}+2\bar{D}^{\rho}\mathcal{K}_{\rho\mu}-2\mathcal{K}\bar{D}_{\mu}\Phi$
$\displaystyle\quad+2\upsilon^{\rho}\bar{D}_{\mu}\bar{D}_{\rho}\Phi-4\upsilon^{\rho}\bar{D}_{\rho}\phi\,D_{\mu}\Phi$
(113)
$\displaystyle\mathcal{\bar{R}}_{(\mu\nu)}+2\bar{D}_{(\mu}\bar{D}_{\nu)}\phi-\frac{1}{4}H_{\mu}^{\hphantom{\mu}\rho\sigma}H_{\nu\rho\sigma}$
$\displaystyle=\frac{a_{\mu}a_{\nu}-\mathfrak{e}_{\mu}\mathfrak{e}_{\nu}}{2}+D_{(\mu}a_{\nu)}-F_{(\mu}^{\hphantom{\mu}\rho}\mathcal{K}_{\nu)\rho}+\upsilon^{\rho}b_{(\mu}^{\hphantom{\mu}\sigma}H_{\nu)\rho\sigma}$
$\displaystyle\quad+4\Phi\upsilon^{\rho}\left(\bar{D}_{\rho}\mathcal{K}_{\mu\nu}-\mathcal{K}\,\mathcal{K}_{\mu\nu}+\frac{\bar{D}_{\rho}\Phi}{\Phi}\mathcal{K}_{\mu\nu}\right)$
(114)

The equations for the matter fields are

$\displaystyle\bar{D}_{\mu}\bar{D}^{\mu}\phi+a^{\mu}\bar{D}_{\mu}\phi-2\bar{D}_{\mu}\phi\bar{D}^{\mu}\phi$
$\displaystyle=2\upsilon^{\mu}\upsilon^{\nu}D_{\mu}\Phi
D_{\nu}\phi-\frac{1}{2}H^{\rho\mu\nu}H_{\rho\mu\nu}+\frac{1}{2}\mathfrak{e}^{\mu}\mathfrak{e}_{\mu}+\frac{1}{2}\upsilon^{\rho}b^{\mu\nu}H_{\rho\mu\nu}$
(115)
$\displaystyle\bar{D}^{\rho}b_{\rho\mu}-2\bar{D}^{\rho}\phi\,b_{\rho\mu}$
$\displaystyle=\frac{1}{2}F^{\rho\sigma}H_{\rho\sigma\mu}+2\upsilon^{\rho}\bar{D}^{\sigma}\Phi\,H_{\rho\sigma\mu}$
(116)
$\displaystyle\bar{D}^{\rho}H_{\rho\mu\nu}+a^{\rho}H_{\rho\mu\nu}-2\bar{D}^{\rho}\phi\,H_{\rho\mu\nu}$
$\displaystyle=2b_{[\mu}^{\hphantom{[\mu}\rho}\mathcal{K}_{\nu]\rho}-\mathcal{K}b_{\mu\nu}+\upsilon^{\rho}\left(\bar{D}_{\rho}b_{\mu\nu}-2\bar{D}_{\rho}\phi\,b_{\mu\nu}+2\upsilon^{\sigma}\bar{D}_{\sigma}\Phi\,H_{\rho\mu\nu}\right).$
(117)

The ”missing” equation is

$\displaystyle\bar{D}^{\mu}\bar{D}_{\nu}\Phi-a^{\mu}\bar{D}_{\mu}\Phi-2\bar{D}^{\mu}\phi\,\bar{D}_{\mu}\Phi$
$\displaystyle=\frac{1}{4}\left(F^{\mu\nu}F_{\mu\nu}-b^{\mu\nu}b_{\mu\nu}\right).$
(118)

As in the case of TNC, we note that the equations (112)-(117) are manifestly
invariant under transformations corresponding to the Carroll algebra. We also
recall the following definitions:

$\displaystyle\begin{split}a_{\mu}&=\upsilon^{\rho}F_{\rho\mu}=2\upsilon^{\rho}\partial_{[\rho}\hat{\tau}_{\mu]},\\\
\mathfrak{e}_{\mu}&=\upsilon^{\rho}b_{\rho\mu}=2\upsilon^{\rho}\partial_{[\rho}\aleph_{\mu]},\\\
\mathcal{K}_{\mu\nu}&=-\frac{1}{2}\mathcal{L}_{\upsilon}h_{\mu\nu}=-\frac{1}{2}\left(\upsilon^{\rho}\partial_{\rho}h_{\mu\nu}+\left(\partial_{\mu}\upsilon^{\rho}\right)h_{\rho\nu}+\left(\partial_{\nu}\upsilon^{\rho}\right)h_{\mu\rho}\right)\,,\end{split}$
(119)

which have the physical interpretation as the acceleration field, the electric
and the extrinsic curvature of the geometry.

##  V String Newton-Cartan geometry

###  V.1 Basics

In Seiberg:2000ms ; Gopakumar:2000na ; Klebanov:2000pp ; Gomis:2000bd ;
Danielsson:2000gi  a non-relativistic string theory was formulated, which was
then found to correspond to a target space geometry called String Newton-
Cartan (SNC) Gomis:2005pg ; Brugues:2004an ; Andringa:2012uz ; Brugues:2006yd
; Bergshoeff:2019pij ; Bergshoeff:2018yvt . A $D$-dimensional SNC spacetime
naturally splits into two longitudinal directions and $D-2$ transverse
directions, that are mapped to each other by means of string Galilean boosts.
Recent works have delved deeper into the quantum aspects of such non-
relativistic string theory, in particular studying the Weyl symmetry and
computing the beta functions, that are required to vanish for the theory to be
anomaly-free YanYu ; Gomis:2019zyu .

The basic geometric fields are the longitudinal and transverse vielbeins,
$\tau_{\mu}^{A},\upsilon^{\mu}_{A},E_{\mu}^{A^{\prime}},E^{\mu}_{A^{\prime}}$
where $A$ runs over the two longitudinal directions and $A^{\prime}$ runs over
the remaining transverse directions. Using the transverse vielbeins we can
build, as usual, two tensors $h^{\perp}_{\mu\nu}$ and $h^{\mu\nu}$. These
fields satisfy the following completeness relations

$h^{\mu\rho}h^{\perp}_{\rho\nu}-\upsilon^{\mu}_{A}\tau_{\nu}^{A}=\delta^{\mu}_{\nu},\qquad\qquad\upsilon^{\mu}_{A}\tau_{\mu}^{B}=-\delta_{A}^{B},\qquad\qquad\tau_{\mu}^{A}h^{\mu\rho}=\upsilon^{\mu}_{A}h^{\perp}_{\mu\rho}=0$
(120)

and transform under string boosts with parameter $\Sigma_{A}^{A^{\prime}}$ as

$\delta_{\Sigma}\upsilon^{\mu}_{A}=-E^{\mu}_{A^{\prime}}\Sigma_{A}^{A^{\prime}},\qquad\qquad\delta_{\Sigma}E_{\mu}^{A^{\prime}}=-\tau_{\mu}^{A}\Sigma_{A}^{A^{\prime}},\qquad\qquad\delta
E^{\mu}_{A^{\prime}}=\delta_{\Sigma}\tau_{\mu}^{A}=0.$ (121)

The metric of the longitudinal space is
$\eta_{AB}=\text{diag}\,\left(-1,1\right)$ and our convention for the
longitudinal Levi-Civita symbol is $\epsilon_{01}=+1$.

Furthermore we can introduce a $Z_{A}$ symmetry in the theory via the gauge
field $m_{\mu}^{A}$, which transforms under boosts as

$\delta_{\Sigma}m_{\mu}^{A}=E^{A^{\prime}}_{\mu}\Sigma_{A^{\prime}}^{A}.$
(122)

This allows us to build the following manifestly boost invariant quantities:

$\displaystyle\begin{split}\bar{h}_{\mu\nu}&=h^{\perp}_{\mu\nu}+2\eta_{AB}m_{(\mu}^{A}\tau_{\nu)}^{B},\\\
u^{\mu}_{A}&=\upsilon^{\mu}_{A}+h^{\mu\rho}m_{\rho A},\\\
\Phi^{AB}&=2u^{\rho(A}m_{\rho}^{\ B)}-h^{\mu\nu}m_{\mu}^{A}m_{\nu}^{B}=-u^{\mu
A}u^{\nu B}\bar{h}_{\mu\nu},\end{split}$ (123)

which satisfy

$h^{\mu\rho}\bar{h}_{\rho\nu}-u^{\mu}_{A}\tau_{\nu}^{A}=\delta^{\mu}_{\nu},\qquad\qquad
u^{\mu}_{A}\tau_{\mu}^{B}=-\delta_{A}^{B},\qquad
u^{\mu}_{A}\bar{h}_{\mu\rho}=\Phi_{AB}\tau_{\rho}^{B}$ (124)

The $Z_{A}$ transformations of the fields are

$\displaystyle\begin{split}\delta_{Z}m_{\mu}^{A}&=D_{\mu}\sigma^{A},\\\
\delta_{Z}\bar{h}_{\mu\nu}&=2\eta_{AB}\tau_{(\mu}^{AB}D_{\nu)}\sigma^{B}\\\
\delta_{Z}u^{\mu}_{A}&=h^{\mu\rho}D_{\rho}\sigma^{A}\\\
\delta_{Z}\Phi^{AB}&=2u^{\rho(A}D_{\rho}\sigma^{B)}.\end{split}$ (125)

Note that this transformation is a symmetry of SNC only if the foliation
constraint is imposed. However, one can avoid imposing this constraint by
requiring the $B$-field to transform as well Harmark:2019upf , the full
$Z_{A}$ transformations are then given by the upcoming (158).

Using these fields it is also possible to construct the following boost-
invariant connection:

$\displaystyle\begin{split}\Gamma^{\rho}_{\mu\nu}&=-u^{\rho}_{A}\left(\partial_{\mu}\tau_{\nu}^{A}+\omega_{\mu}\epsilon^{A}_{\
B}\tau_{\nu}^{B}\right)+\frac{1}{2}h^{\rho\sigma}\left(\partial_{\mu}\bar{h}_{\nu\sigma}+\partial_{\nu}\bar{h}_{\mu\sigma}-\partial_{\sigma}\bar{h}_{\mu\nu}\right)\\\
&=-u^{\rho}_{A}\nabla_{\mu}\tau_{\nu}^{A}+\frac{1}{2}h^{\rho\sigma}\left(\partial_{\mu}\bar{h}_{\nu\sigma}+\partial_{\nu}\bar{h}_{\mu\sigma}-\partial_{\sigma}\bar{h}_{\mu\nu}\right)\end{split}$
(126)

where we introduced the spin connection $\omega_{\mu}^{\
AB}\equiv\omega_{\mu}\epsilon^{AB}$, associated with the longitudinal
covariant derivative $\nabla_{\mu}$. The connection (126) is compatible with
$h^{\mu\nu}$ and $\tau_{\mu}^{A}$ and has an antisymmetric component

$2\Gamma^{\rho}_{[\mu\nu]}\equiv-u^{\rho}_{A}F_{\mu\nu}^{A},$ (127)

where we defined the “torsion” tensors as131313Note that $F_{\mu\nu}^{A}=0$ is
the so called foliation constraint.

$F_{\mu\nu}^{A}\equiv 2\partial_{[\mu}\tau_{\nu]}^{\ A}+2\epsilon^{A}_{\
B}\omega_{[\mu}\tau_{\nu]}^{\ B}=2\nabla_{[\mu}\tau_{\nu]}^{A}.$ (128)

We can also decompose these tensors as

$F_{\mu\nu}^{A}=f^{A}\epsilon_{BC}\tau^{B}_{\mu}\tau^{C}_{\nu}+2a_{[\mu}^{\ \
BA}\tau_{\nu]B}+\widetilde{F}_{\mu\nu}^{A}$ (129)

where we defined the acceleration $a_{\mu AB}$, the temporal part of torsion
$f^{A}$ and the transverse torsion tensor $\widetilde{F}_{\mu\nu}^{A}$ as

$\displaystyle u^{\rho}_{A}F_{\rho\mu B}\equiv a_{\mu AB}\,,\qquad u^{\mu
A}a_{\mu}^{\ BC}=\epsilon^{AB}f^{C}\,,\qquad u^{\mu A}\widetilde{F}_{\mu\nu
B}=0\,.$ (130)

Finally we define the extrinsic curvature in the usual way:

$\mathcal{K}_{\mu\nu}^{A}=-\frac{1}{2}\mathcal{L}_{u}\bar{h}_{\mu\nu}$ (131)

which satisfies in particular

$\displaystyle\begin{split}\mathcal{K}_{\mu\nu}^{A}h^{\mu\nu}&=-D_{\mu}u^{\mu
A},\\\
h^{\mu\nu}h^{\rho\sigma}\mathcal{K}_{\mu\rho}^{A}\mathcal{K}_{\nu\sigma}^{B}&=D_{\mu}u^{\nu
A}D_{\nu}u^{\mu B}+\frac{1}{4}h^{\mu\nu}h^{\rho\sigma}F_{\mu\nu
C}F_{\rho\sigma D}\Phi^{AC}\Phi^{BD}.\end{split}$ (132)

Below we discuss how this structure is embedded in the double field theory
framework.

###  V.2 Embedding in DFT

To find the embedding of SNC in DFT we will use the same approach used for TNC
Blair:2019qwi , namely we will compare the SNC worldsheet action Gomis:2019zyu
,

$\displaystyle\begin{split}S_{SNC}=&-\frac{1}{2}\int
d^{2}\sigma\left[e\,\partial X^{\mu}\partial
X^{\nu}\,\bar{h}_{\mu\nu}+\epsilon^{\alpha\beta}\left(\lambda
e_{\alpha}\tau_{\mu}+\bar{\lambda}\bar{e}_{\alpha}\bar{\tau}_{\mu}\right)\partial_{\beta}X^{\mu}\right]\\\
&-\frac{1}{2}\int
d^{2}\sigma\,\varepsilon^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}B_{\mu\nu},\end{split}$
(133)

with the worldsheet action written in terms of DFT fields141414We define
$e_{\alpha}\equiv e^{0}_{\alpha}+e^{1}_{\alpha}$, $\bar{e}_{\alpha}\equiv
e^{0}_{\alpha}-e^{1}_{\alpha}$, and similarly for $\tau$ and $\bar{\tau}$.
Notice that
$\gamma^{\alpha\beta}=\eta^{ab}e^{\alpha}_{a}e^{\beta}_{b}=-e^{\alpha}_{0}e^{\beta}_{0}+e^{\alpha}_{1}e^{\beta}_{1}=-4e^{(\alpha}\bar{e}^{\beta)}$
where $e^{\alpha}\equiv\frac{1}{2}(e^{\alpha}_{0}+e^{\alpha}_{1})$,
$\bar{e}^{\alpha}\equiv\frac{1}{2}(e^{\alpha}_{0}-e^{\alpha}_{1})$, such that
$e^{\alpha}e_{\alpha}=1=\bar{e}^{\alpha}\bar{e}_{\alpha}$ and
$e^{\alpha}\bar{e}_{\alpha}=0$. Moreover we have
$\varepsilon^{\alpha\beta}=e\epsilon^{ab}e^{\alpha}_{a}e^{\beta}_{b}$ with
$\varepsilon^{01}=-\varepsilon_{01}=+1$ and
$e^{\alpha}_{a}=e^{-1}\varepsilon^{\alpha\beta}e_{\beta}^{b}\varepsilon_{ba}$,
which implies
$e^{\alpha}=-\frac{1}{2e}\varepsilon^{\alpha\beta}\bar{e}_{\beta}$ and
$\bar{e}^{\alpha}=\frac{1}{2e}\varepsilon^{\alpha\beta}e_{\beta}$. Note that
our convention for the worldsheet Levi-Civita symbol
$\varepsilon^{\alpha\beta}$ is the opposite of the one for the longitudinal
SNC Levi-Civita symbol $\epsilon^{AB}$. ,

$\displaystyle\begin{split}S_{DFT}=&-\frac{1}{2}\int
d^{2}\sigma\left[e\,\partial X^{\mu}\partial
X^{\nu}\,K_{\mu\nu}+2\epsilon^{\alpha\beta}\left(\beta_{\alpha
a}x^{a}_{\mu}+\bar{\beta}_{\alpha\bar{a}}\bar{x}^{\bar{a}}_{\mu}\right)\partial_{\beta}X^{\mu}\right]\\\
&-\frac{1}{2}\int
d^{2}\sigma\left[\varepsilon^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\mathcal{B}_{\mu\nu}-2\gamma^{\alpha\beta}\left(\beta_{\alpha
a}x^{a}_{\mu}-\bar{\beta}_{\alpha\bar{a}}\bar{x}^{\bar{a}}_{\mu}\right)\partial_{\beta}X^{\mu}\right].\end{split}$
(134)

To make the identifications needed to define the SNC embedding we need to
rewrite (133) in a more suitable way. In particular there are two issues we
need to address:

  1. 1.

We need to have a term proportional to
$\gamma^{\alpha\beta}\lambda\partial_{\beta}X^{\mu}$ and
$\gamma^{\alpha\beta}\bar{\lambda}\partial_{\beta}X^{\mu}$ to be able to
properly identity the null eigenvectors of DFT;

  2. 2.

We need to make sure that $K_{\mu\nu}$ is a degenerate matrix, which is not
the case if we naively identify $K_{\mu\nu}=\bar{h}_{\mu\nu}$.

The first issue is simply solved by the following field redefinition:

$A_{\alpha}=\frac{1}{2}(\lambda-\bar{\lambda})e^{0}_{\alpha}+\frac{1}{2}(\lambda+\bar{\lambda})e^{1}_{\alpha}=\frac{1}{2}\lambda
e_{\alpha}-\frac{1}{2}\bar{\lambda}\bar{e}_{\alpha}.$ (135)

In terms of these new Lagrange multipliers the SNC action becomes

$\displaystyle\begin{split}S_{SNC}=&-\frac{1}{2}\int
d^{2}\sigma\left[e\,\partial X^{\mu}\partial
X^{\nu}\,\bar{h}_{\mu\nu}+2\epsilon^{\alpha\beta}\tau_{\mu}^{1}A_{\alpha}\partial_{\beta}X^{\mu}\right]\\\
&-\frac{1}{2}\int
d^{2}\sigma\left[\epsilon^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}B_{\mu\nu}+2e\gamma^{\alpha\beta}\tau_{\mu}^{0}A_{\alpha}\partial_{\beta}X^{\mu}\right],\end{split}$
(136)

This form of the action makes it easy for us to identify the null eigenvectors
of DFT, however we still have to solve problem (2), i.e. the fact that
$K_{\mu\nu}=\bar{h}_{\mu\nu}$ is not a degenerate matrix. To solve this issue
we can make use of the Stueckelberg symmetry of the SNC action. The action
(133) is invariant under the following transformations with parameters $C^{A}$
Harmark:2019upf ; Bergshoeff:2019pij :

$\displaystyle\delta\bar{h}_{\mu\nu}$
$\displaystyle=2C_{(\mu}^{A}\tau_{\nu)}^{B}\eta_{AB},$ $\displaystyle\delta
B_{\mu\nu}$ $\displaystyle=-2C_{[\mu}^{A}\tau_{\nu]}^{B}\epsilon_{AB}$ (137)
$\displaystyle\lambda^{\prime}$
$\displaystyle=\lambda+e^{-1}\epsilon^{\alpha\beta}\bar{e}_{\alpha}\bar{C}_{\beta},$
$\displaystyle\bar{\lambda}^{\prime}$
$\displaystyle=\bar{\lambda}+e^{-1}\epsilon^{\alpha\beta}e_{\alpha}C_{\beta},$
(138)

where $C_{\mu}=C^{0}_{\mu}+C^{1}_{\mu}$ and
$\bar{C}_{\mu}=C_{\mu}^{0}-C_{\mu}^{1}$. This means we can rewrite the action
in terms of $h^{\perp}_{\mu\nu}$ and $\bar{B}_{\mu\nu}$ by choosing
$C_{\mu}^{A}=-m_{\mu}^{A}$. After redefining the Lagrange multipliers one more
time we arrive at the action

$\displaystyle\begin{split}S_{SNC}=&-\frac{1}{2}\int
d^{2}\sigma\left[e\,\partial X^{\mu}\partial
X^{\nu}\,h^{\perp}_{\mu\nu}+2\epsilon^{\alpha\beta}\tau_{\mu}^{1}A_{\alpha}\partial_{\beta}X^{\mu}\right]\\\
&-\frac{1}{2}\int
d^{2}\sigma\left[\epsilon^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\bar{B}_{\mu\nu}+2e\gamma^{\alpha\beta}\tau_{\mu}^{0}A_{\alpha}\partial_{\beta}X^{\mu}\right],\end{split}$
(139)

where now all the tensors appearing in (139) are manifestly $Z_{A}$ invariant.
This allows us to make the following identifications:

$K_{\mu\nu}=h^{\perp}_{\mu\nu},\qquad\qquad\mathcal{B}_{\mu\nu}=\bar{B}_{\mu\nu}$
(140)

and

$\displaystyle\begin{split}\left(\beta_{\alpha}x_{\mu}+\bar{\beta}_{\alpha}\bar{x}_{\mu}\right)&=\left(\tau_{\mu}+\bar{\tau}_{\mu}\right)A_{\alpha}=2\tau_{\mu}^{0}A_{\alpha}\\\
\left(\beta_{\alpha}x_{\mu}-\bar{\beta}_{\alpha}\bar{x}_{\mu}\right)&=-\left(\tau_{\mu}-\bar{\tau}_{\mu}\right)A_{\alpha}=-2\tau_{\mu}^{1}A_{\alpha},\end{split}$
(141)

This is solved by (note the mismatch of bars between DFT and SNC fields)

$\displaystyle\begin{split}x_{\mu}&=\frac{1}{\sqrt{2}}\bar{\tau}_{\mu},\qquad\qquad\beta_{\alpha}=\frac{1}{\sqrt{2}}A_{\alpha}\\\
\bar{x}_{\mu}&=\frac{1}{\sqrt{2}}\tau_{\mu},\qquad\qquad\bar{\beta}_{\alpha}=\frac{1}{\sqrt{2}}A_{\alpha}.\end{split}$
(142)

By requiring $x_{\mu}y^{\mu}=1=\bar{x}_{\mu}\bar{y}^{\mu}$ we find

$y^{\mu}_{a}=-\sqrt{2}\,\bar{\upsilon}^{\mu},\qquad\qquad\bar{y}^{\mu}_{\bar{a}}=-\sqrt{2}\,\upsilon^{\mu}.$
(143)

Moreover we have

$H^{\mu\nu}=h^{\mu\nu}$ (144)

and

$Z_{\mu}^{\nu}=\tau_{\mu}\upsilon^{\nu}-\bar{\tau}_{\mu}\bar{\upsilon}^{\nu}.$
(145)

In summary, string Newton-Cartan can be embedded in DFT as

$\displaystyle K_{\mu\nu}$ $\displaystyle=h^{\perp}_{\mu\nu},$ $\displaystyle
H^{\mu\nu}$ $\displaystyle=h^{\mu\nu}$ (146)
$\displaystyle\mathcal{B}_{\mu\nu}$ $\displaystyle=B_{\mu\nu},$ $\displaystyle
Z_{\mu}^{\nu}$
$\displaystyle=-\tau_{\mu}\upsilon^{\nu}+\bar{\tau}_{\mu}\bar{\upsilon}^{\nu}$
(147)

and the eigenvectors are

$\displaystyle\begin{split}x_{\mu}&=\frac{1}{\sqrt{2}}\bar{\tau}_{\mu},\qquad\qquad
y^{\mu}=-\sqrt{2}\,\bar{\upsilon}^{\mu}\\\
\bar{x}_{\mu}&=\frac{1}{\sqrt{2}}\tau_{\mu},\qquad\qquad\bar{y}^{\mu}=-\sqrt{2}\,\upsilon^{\mu}.\end{split}$
(148)

An issue with this parametrization is that it does not include the gauge field
$m_{\mu}^{A}$. To reinstate it we can once again make a Stueckelberg
transformation (or a shift transformation (15) from the point of view of DFT)

$\displaystyle\begin{split}\left(y^{\mu}\right)^{\prime}&=y^{\mu}+H^{\mu\rho}V_{\rho}\\\
\left(\bar{y}^{\mu}\right)^{\prime}&=\bar{y}^{\mu}+H^{\mu\rho}\bar{V}_{\rho}\\\
\left(K_{\mu\nu}\right)^{\prime}&=K_{\mu\nu}-2x_{(\mu}K_{\nu)\rho}V^{\rho}-2\bar{x}_{(\mu}K_{\nu)\rho}\bar{V}^{\rho}+\left(x_{\mu}V_{\rho}+\bar{x}_{\mu}\bar{V}_{\rho}\right)\left(x_{\nu}V^{\rho}+\bar{x}_{\nu}\bar{V}^{\rho}\right)\\\
\left(\mathcal{B}_{\mu\nu}\right)^{\prime}&=\mathcal{B}_{\mu\nu}-2x_{[\mu}V_{\nu]}+2\bar{x}_{[\mu}\bar{V}_{\nu]}+2x_{[\mu}\bar{x}_{\nu]}\left(y^{\rho}\bar{V}_{\rho}+\bar{y}^{\rho}V_{\rho}+V_{\rho}\bar{V}^{\rho}\right)\end{split}$
(149)

where $V_{\mu},\bar{V}_{\mu}$ are two arbitrary local parameters and we set
$V^{\rho}\equiv H^{\rho\mu}V_{\mu}$ for brevity. The choice
$V_{\mu}=\frac{1}{2}m_{\mu},\bar{V}_{\mu}=\frac{1}{2}\bar{m}_{\mu}$ gives

$\displaystyle\begin{split}\left(y^{\mu}\right)^{\prime}&=-\bar{\upsilon}^{\mu}+\frac{1}{2}m^{\mu}\\\
\left(\bar{y}^{\mu}\right)^{\prime}&=-\upsilon^{\mu}+\frac{1}{2}\bar{m}^{\mu}\\\
\left(K_{\mu\nu}\right)^{\prime}&=h_{\mu\nu}-\frac{1}{2}\bar{\varphi}\tau_{\mu}\tau_{\nu}-\frac{1}{2}\varphi\bar{\tau}_{\mu}\bar{\tau}_{\nu}-\mathcal{T}\tau_{(\mu}\bar{\tau}_{\nu)}\\\
\left(\mathcal{B}_{\mu\nu}\right)^{\prime}&=B_{\mu\nu}+\mathcal{T}\tau_{[\mu}\bar{\tau}_{\nu]}\end{split}$
(150)

with

$\displaystyle\begin{split}\varphi&\equiv
2\bar{u}^{\mu}m_{\mu}+\frac{1}{2}m^{2}\\\ \bar{\varphi}&\equiv
2u^{\mu}\bar{m}_{\mu}+\frac{1}{2}\bar{m}^{2}\\\ \mathcal{T}&\equiv
u^{\mu}m_{\mu}+\bar{u}^{\mu}\bar{m}_{\mu}+\frac{1}{2}m\cdot\bar{m}\end{split}$
(151)

and

$\displaystyle u^{\mu}$
$\displaystyle\equiv\upsilon^{\mu}-\frac{1}{2}h^{\mu\rho}\bar{m}_{\rho},$
$\displaystyle\bar{u}^{\mu}$
$\displaystyle\equiv\bar{\upsilon}^{\mu}-\frac{1}{2}h^{\mu\rho}m_{\rho}$ (152)
$\displaystyle\bar{h}_{\mu\nu}$
$\displaystyle=h^{\perp}_{\mu\nu}-\tau_{(\mu}\bar{m}_{\nu)}-\bar{\tau}_{(\mu}m_{\nu)},$
$\displaystyle B_{\mu\nu}$
$\displaystyle=\bar{B}_{\mu\nu}+\tau_{[\mu}\bar{m}_{\nu]}-\bar{\tau}_{[\mu}m_{\nu]}.$
(153)

Changing back from lightcone coordinates we finally find

$\displaystyle\begin{split}K_{\mu\nu}&=\bar{h}_{\mu\nu}+\Phi_{AB}\tau_{\mu}^{A}\tau_{\nu}^{B},\\\
H^{\mu\nu}&=h^{\mu\nu},\\\ Z^{\mu}_{\nu}&=-\epsilon_{AB}u^{\mu
A}\tau_{\nu}^{B},\\\
\mathcal{B}_{\mu\nu}&=B_{\mu\nu}-\frac{1}{2}\Phi\epsilon_{AB}\tau_{\mu}^{A}\tau_{\nu}^{B},\end{split}$
(154)

where we defined $\Phi\equiv\Phi^{A}_{A}$.

###  V.3 Action and equations of motion

Given an arbitrary tensor $A_{\mu\nu\dots}$ we will define

$A^{\mu\nu\dots}\equiv A_{\rho\sigma\dots}h^{\mu\rho}h^{\nu\sigma}\dots$ (155)

The only SNC fields which naturally have upper curved indices are
$u^{\mu}_{A}$ and $h^{\mu\nu}$ hence any other tensor with upper indices is to
be understood as defined via (155).

Using the parametrization (154) we find that the SNC action is given
by151515Close to the completion of this work we became aware of the work of
Bergshoeff et al. Bergshoeff:2020  where they obtained a similar action by
taking a particular non-relativistic limit of NS-NS gravity.

$\displaystyle\begin{split}S=&\int d^{D}x\,e\,\left[\mathcal{R}-a^{\mu
AB}(a_{\mu(AB)}-\frac{1}{2}\eta_{AB}a_{\mu})+\left(a^{\mu}-2D^{\mu}\phi\right)\left(a_{\mu}-2D_{\mu}\phi\right)\right.\\\
&\hphantom{(((\int d^{d}xe\,}\left.-\frac{1}{2}F^{\mu\nu
A}F_{\mu\nu}^{B}(\Phi_{AB}-\frac{1}{2}\eta_{AB}\Phi)+\frac{1}{2}\epsilon_{AB}u^{\rho
A}F^{\mu\nu
B}H_{\rho\mu\nu}-\frac{1}{12}H^{\rho\mu\nu}H_{\rho\mu\nu}\right]\\\
\end{split}$ (156)

where we defined $a_{\mu}\equiv a_{\mu A}^{\ \ A}$, $\Phi\equiv\Phi^{A}_{A}$
and the invariant measure is given by

$e\equiv e^{-2\phi}\sqrt{\frac{\det\bar{h}}{\det\Phi}}.$ (157)

This action is invariant under the $Z_{A}$ transformations

$\displaystyle\begin{split}\delta\bar{h}_{\mu\nu}=2\tau_{(\mu}^{\
A}D_{\nu)}\sigma_{A},\qquad\qquad\delta
B_{\mu\nu}=2\epsilon_{AB}\tau_{[\mu}^{A}D_{\nu]}\sigma^{B},\\\ \delta u^{\mu
A}=h^{\mu\nu}D_{\nu}\sigma^{A},\qquad\qquad\delta\Phi^{AB}=2u^{\mu(A}D_{\mu}\sigma^{B)}.\end{split}$
(158)

The equations of motion can be found by varying the action with respect to the
independent fields $h^{\mu\nu},u^{\mu}_{A}$ and $\Phi_{AB}$161616Note that,
despite appearances, the action does not depend on the spin connection, i.e.
$\delta S/\delta\omega_{\mu}=0$., also see Appendix C for useful identities.
Sometimes it will be useful to further decompose the acceleration in its
antisymmetric and traceless symmetric components:

$a_{\mu}^{\
AB}=\mathcal{S}_{\mu}^{AB}+\frac{1}{2}\eta^{AB}a_{\mu}+\epsilon^{AB}\mathcal{A}_{\mu}$
(159)

with $\mathcal{S}_{\mu}^{\ AB}\eta_{AB}=0$ and we recall $a_{\mu}=a_{\mu}^{\
AB}\eta_{AB}$.

Taking the variation of (156) with respect to $\Phi_{AB}$ we find the
equation171717The action actually only depends on the trace of the tensor
$\Phi_{AB}$, as can be seen by expanding the Ricci scalar and noticing that
the $F^{\mu\nu A}F_{\mu\nu}^{B}\Phi_{AB}$ terms cancel out. This is why the
corresponding equation is a singlet and not a tensor.

$F^{\mu\nu A}F_{\mu\nu A}=0,$ (160)

which we will use when writing the remaining equations. The space projection
of the equation of motion for $u^{\mu}_{A}$ is

$\displaystyle D^{\rho}F^{\mu}_{\ \rho A}+a_{\rho A}^{\ \ B}F^{\mu\rho}_{\ \
B}-2F^{\mu}_{\ \rho A}D^{\rho}\phi$
$\displaystyle=\frac{1}{2}\epsilon_{AB}F^{\rho\sigma B}H^{\mu}_{\
\rho\sigma},$ (161)

while the time projection is

$\displaystyle D^{\rho}\mathcal{S}_{\rho}^{\ AB}+a^{\rho}\mathcal{S}_{\rho}^{\
AB}-2\mathcal{S}^{\rho AB}D_{\rho}\phi$ $\displaystyle=-\frac{1}{4}F^{\mu\nu
A}F_{\mu\nu}^{B}\Phi+2\epsilon^{(A}_{\ \ C}\mathcal{F}^{B)C},$ (162)

where we defined

$\displaystyle\mathcal{F}^{BC}$
$\displaystyle\equiv-\mathcal{A}^{\rho}\mathcal{S}_{\rho}^{\
BC}+\frac{1}{4}u^{\rho(B}F_{\mu\nu}^{C)}H_{\rho}^{\ \mu\nu}.$ (163)

The equation for the B-field is

$\displaystyle D^{\rho}H_{\rho}^{\ \mu\nu}+a^{\rho}H_{\rho}^{\
\mu\nu}-2D^{\rho}\phi\,H_{\rho}^{\ \mu\nu}$
$\displaystyle=2\Omega^{\mu\nu}+\epsilon_{AB}\mathcal{H}^{\mu\nu AB},$ (164)

where we defined

$\displaystyle\begin{split}\mathcal{H}^{\mu\nu AB}&\equiv 2u^{\rho
A}D_{\rho}F^{\mu\nu B}-2u^{\rho A}F^{\mu\nu
B}D_{\rho}\phi-2D^{[\mu}a^{\nu]AB}-2\mathcal{S}^{\mu A}_{\ \
C}\mathcal{S}^{\nu BC}\\\ &\quad-\mathcal{K}^{A}F^{\mu\nu B}+2F^{[\mu}_{\ \
\rho C}F^{\nu]\rho A}\Phi^{BC}\end{split}$ (165)

as well as $\Omega_{\mu\nu}\equiv 2\partial_{[\mu}\omega_{\nu]}$ and
$\mathcal{K}^{A}\equiv h^{\mu\nu}\mathcal{K}_{\mu\nu}^{A}$.

The dilaton equation is

$\displaystyle
D^{\mu}D_{\mu}\phi-\frac{1}{2}D^{\mu}a_{\mu}+2a_{\mu}D^{\mu}\phi-2D^{\mu}\phi\,D_{\mu}\phi-\frac{1}{2}a_{\mu}a^{\mu}$
$\displaystyle=\frac{1}{4}\epsilon_{AB}u^{\rho A}F^{\mu\nu
B}H_{\rho\mu\nu}-\frac{1}{12}H^{\rho\mu\nu}H_{\rho\mu\nu}.$ (166)

The space-space projection of Einstein’s equations is

$\displaystyle\begin{split}\mathcal{R}^{(\mu\nu)}+2D^{(\mu}D^{\nu)}\phi-\frac{1}{4}H^{\mu}_{\
\rho\sigma}H^{\nu\rho\sigma}&=u^{\rho}_{A}D^{(\mu}F_{\rho}^{\
\nu)A}+\mathcal{S}^{\mu AB}\mathcal{S}^{\nu}_{\ AB}\\\ &\ \ \
-\epsilon^{AB}u^{\sigma}_{\ A}F^{\rho(\mu}_{\ \ \ B}H^{\nu)}_{\ \
\rho\sigma}-\frac{1}{2}F^{\mu\rho A}F^{\nu}_{\ \rho A}\Phi.\end{split}$ (167)

The time-space projection is

$\displaystyle\begin{split}h^{\mu\rho}u^{\sigma
A}\mathcal{R}_{(\rho\sigma)}+&2u^{\rho A}D^{\mu}D_{\rho}\phi+a^{\mu}u^{\rho
A}D_{\rho}\phi-\frac{1}{4}H^{\mu}_{\ \rho\sigma}H_{\lambda}^{\
\rho\sigma}u^{\lambda A}=\frac{1}{2}u^{\rho A}u^{\sigma}_{\
B}D_{\rho}F_{\sigma}^{\ \mu B}-u^{\rho}_{\ B}D^{\mu}a_{\rho}^{\ (AB)}+u^{\rho
A}D^{\mu}a_{\rho}\\\ &-\frac{1}{2}u^{\rho
B}D_{\rho}a^{\mu}_{BA}+\frac{1}{2}u^{\rho
A}D_{\rho}a^{\mu}-2a^{\mu[AB]}u^{\rho}_{\ B}D_{\rho}\phi-a^{\mu}u^{\rho
A}D_{\rho}\phi-\frac{1}{2}a^{\mu}\mathcal{K}^{A}+\frac{1}{2}a^{\mu
BA}\mathcal{K}_{B}\\\ &-F^{\mu}_{\ \rho C}a^{\rho B(A}\Phi_{B}^{\
C)}+a^{\rho}F^{\mu}_{\ \rho C}\Phi^{AC}+\frac{1}{2}a^{\rho}F_{\rho}^{\ \mu
A}\Phi+\frac{1}{2}F^{\mu\rho B}D_{\rho}\Phi^{A}_{\ B}-\frac{1}{2}F^{\mu\rho
A}D_{\rho}\Phi\\\ &+\epsilon^{AB}\mathcal{P}^{\mu}_{\ B},\end{split}$ (168)

where

$\displaystyle\begin{split}\mathcal{P}^{\mu B}&=-\frac{1}{2}u^{\sigma
B}D^{\rho}H^{\mu}_{\ \rho\sigma}+u^{\rho B}H^{\mu}_{\
\rho\sigma}D^{\sigma}\phi+a^{\rho}u^{\lambda B}H^{\mu}_{\
\rho\lambda}+\frac{1}{2}F^{\mu\rho C}u^{\sigma
B}u^{\lambda}_{C}H_{\rho\sigma\lambda}\\\ &\quad-\frac{1}{2}a^{\rho
CB}u^{\sigma}_{\ C}H^{\mu}_{\
\rho\sigma}-D^{\mu}f^{B}-\frac{1}{2}a^{\mu}f^{B}+a^{\mu
BC}f_{C}+\frac{1}{2}a^{\mu CB}f_{C}+a^{\mu BC}f_{C}\end{split}$ (169)

Finally the time-time component (found directly from DFT) is given by

$\displaystyle\begin{split}u^{\mu A}u^{\nu}_{\ A}\mathcal{R}_{\mu\nu}+&2u^{\mu
A}u^{\nu}_{\ A}D_{\mu}D_{\nu}\phi-\frac{1}{4}u^{\mu A}u^{\nu}_{\
A}H_{\mu\rho\sigma}H_{\nu}^{\ \rho\sigma}=\frac{3}{2}\mathcal{S}^{\rho
AB}D_{\rho}\Phi_{AB}-\frac{3}{4}a^{\mu}D_{\mu}\Phi\\\
&-\frac{3}{4}a^{\mu}a_{\mu}\Phi+\mathcal{S}^{\mu AB}\mathcal{S}_{\mu
AC}\Phi_{B}^{\ C}-2f^{A}f_{A}+\epsilon^{AB}\mathcal{Q}_{AB}\end{split}$ (170)

where

$\displaystyle\begin{split}\mathcal{Q}_{AB}&=-\frac{1}{2}u^{\rho}_{\
A}u^{\sigma}_{\
B}D^{\lambda}H_{\rho\sigma\lambda}+u^{\mu}_{A}u^{\nu}_{B}H_{\mu\nu\rho}D^{\rho}\phi+u^{\mu}_{A}u^{\nu}_{B}D_{\mu}\mathcal{A}_{\nu}+2f_{A}u^{\mu}_{B}D_{\mu}\phi+\frac{3}{2}u^{\mu}_{A}D_{\mu}f_{B}\\\
&\quad-u^{\mu}_{A}u^{\nu C}\mathcal{S}^{\rho}_{\
CB}H_{\mu\nu\rho}-u^{\mu}_{A}u^{\nu}_{B}a^{\rho}H_{\mu\nu\rho}+f_{A}\mathcal{K}_{B}+3\mathcal{A}^{\mu}\mathcal{S}_{\mu
AC}\Phi_{B}^{\ C}\\\ &\quad-\frac{1}{4}F^{\mu\nu}_{\ \ A}u^{\rho
C}H_{\rho\mu\nu}\Phi_{BC}-\frac{1}{4}F^{\mu\nu}_{\ \
B}u^{\rho}_{A}H_{\rho\mu\nu}\Phi+\frac{1}{4}F^{\mu\nu
C}u^{\rho}_{A}H_{\rho\mu\nu}\Phi_{BC}\end{split}$ (171)

We remind the reader of the definitions of some of the fields appearing in
these equations:

$\displaystyle\begin{split}F_{\mu\nu}^{A}&=2\nabla_{[\mu}\tau_{\nu]}^{A}\equiv
f^{A}\epsilon_{BC}\tau^{B}_{\mu}\tau^{C}_{\nu}+2a_{[\mu}^{BA}\tau_{\nu]B}+\widetilde{F}_{\mu\nu}^{A}\\\
a_{\mu}^{AB}&=u^{\rho
A}F_{\rho\mu}^{B}\equiv\mathcal{S}_{\mu}^{AB}+\frac{1}{2}\eta^{AB}a_{\mu}+\epsilon^{AB}\mathcal{A}_{\mu}.\end{split}$
(172)

These equations are clearly more complicated than the ones found for TNC or
Carroll, as there is no obvious way to combine the time and space projections
of an equation into a single simpler equation. Despite this technical
difficulty, it is possible to show that these equations are invariant under
the SNC algebra as expected. In the next section we will discuss these
equations and their relations to previous results .

##  VI Comparison with known results

###  VI.1 TNC beta functions

In Gallegos:2019icg , three of us computed the beta functions for string
theory describing a Type I TNC target space. The equations obtained by setting
those beta functions to zero were the two twistlessness contraints

$\displaystyle F_{\mu\nu}$
$\displaystyle=a_{\mu}\tau_{\nu}-\tau_{\mu}a_{\nu}\,,$ (173) $\displaystyle
b_{\mu\nu}$
$\displaystyle=\mathfrak{e}_{\mu}\tau_{\nu}-\tau_{\mu}\mathfrak{e}_{\nu}\,,$
(174)

two scalar equations

$\displaystyle D\cdot a+a^{2}$ $\displaystyle=2\mathfrak{e}^{2}+2\left(a\cdot
D\phi\right)\,,$ (175) $\displaystyle D\cdot\mathfrak{e}$
$\displaystyle=2\left(\mathfrak{e}\cdot D\phi\right)\,,$ (176)

and two tensor equations

$\displaystyle R_{(\mu\nu)}-\frac{1}{4}H^{\rho\sigma}_{\ \
(\mu}H_{\nu)\rho\sigma}+2D_{(\mu}D_{\nu)}\phi$
$\displaystyle=\frac{\mathfrak{e}^{2}\left(2\Phi\tau_{\mu}\tau_{\nu}-\bar{h}_{\mu\nu}\right)-\mathfrak{e}_{\mu}\mathfrak{e}_{\nu}}{2}-\mathfrak{e}^{\sigma}\left(\Delta_{T}\right)^{\rho}_{(\mu}H_{\nu)\rho\sigma}$
$\displaystyle\hphantom{=}+\left(\Delta_{S}\right)^{\rho}_{(\mu}D_{\nu)}a_{\rho}+\frac{a_{\mu}a_{\nu}-2a^{2}\,\Phi\tau_{\mu}\tau_{\nu}}{2}\,,$
(177) $\displaystyle
D^{\rho}H_{\rho\mu\nu}+a^{\rho}H_{\rho\mu\nu}-2H^{\rho}_{\
\mu\nu}D_{\rho}\phi$
$\displaystyle=2\left(\Delta_{S}\right)^{\rho}_{[\mu}D_{\nu]}\mathfrak{e}_{\rho}-2\left(\Delta_{T}\right)^{\rho}_{[\mu}D_{\nu]}\mathfrak{e}_{\rho}-2a_{[\mu}\mathfrak{e}_{\nu]}$
$\displaystyle\hphantom{=}+\left(2\hat{\upsilon}^{\rho}D_{\rho}\phi-
D_{\rho}\upsilon^{\rho}\right)b_{\mu\nu}\,$ (178)

where we remind the reader of the definitions of the projectors
$\left(\Delta_{T}\right)^{\mu}_{\nu}\equiv-\hat{\upsilon}^{\mu}\tau_{\nu}$ and
$\left(\Delta_{S}\right)^{\mu}_{\nu}\equiv h^{\mu\rho}\bar{h}_{\rho\nu}$ which
satisfy

$\left(\Delta_{S}\right)^{\mu}_{\nu}+\left(\Delta_{T}\right)^{\mu}_{\nu}=\delta^{\mu}_{\nu}\,.$
(179)

We now want to compare these equations with the ones obtained from DFT. First
of all, notice that in Gallegos:2019icg  twistless torsion was assumed from
the start which explains the difference between the equations (173)-(174) and
(58). To compare the rest of the equations, we impose the twistlessness
constraints in equations (59)-(62) and (73). We then find two scalar equations

$\displaystyle D\cdot a+a^{2}$ $\displaystyle=\mathfrak{e}^{2}+2\left(a\cdot
D\phi\right)\,,$ (180) $\displaystyle D\cdot\mathfrak{e}$
$\displaystyle=2\left(\mathfrak{e}\cdot D\phi\right)\,,$ (181)

and two tensor equations

$\displaystyle R_{(\mu\nu)}-\frac{1}{4}H^{\rho\sigma}_{\ \
(\mu}H_{\nu)\rho\sigma}+2D_{(\mu}D_{\nu)}\phi$
$\displaystyle=\frac{2\mathfrak{e}^{2}\Phi\tau_{\mu}\tau_{\nu}-\mathfrak{e}_{\mu}\mathfrak{e}_{\nu}}{2}-\frac{2a^{2}\,\Phi\tau_{\mu}\tau_{\nu}-a_{\mu}a_{\nu}}{2}$
$\displaystyle\hphantom{=}+\left(\Delta_{S}\right)^{\rho}_{(\mu}D_{\nu)}a_{\rho}-\mathfrak{e}^{\sigma}\left(\Delta_{T}\right)^{\rho}_{(\mu}H_{\nu)\rho\sigma}\,,$
(182) $\displaystyle
D^{\rho}H_{\rho\mu\nu}+a^{\rho}H_{\rho\mu\nu}-2H^{\rho}_{\
\mu\nu}D_{\rho}\phi$
$\displaystyle=2\left(\Delta_{S}\right)^{\rho}_{[\mu}D_{\nu]}\mathfrak{e}_{\rho}-2\left(\Delta_{T}\right)^{\rho}_{[\mu}D_{\nu]}\mathfrak{e}_{\rho}-2a_{[\mu}\mathfrak{e}_{\nu]}$
$\displaystyle\hphantom{=}+\left(2\hat{\upsilon}^{\rho}D_{\rho}\phi-
D_{\rho}\upsilon^{\rho}\right)b_{\mu\nu}\,$ (183)

where in the last equation we also used the Bianchi identity for $b_{\mu\nu}$.
The main difference between these two sets of equations is the factor in front
of $\mathfrak{e}^{2}$ in (175) and (180). The further difference in equations
(VI.1) and (VI.1) arise precisely because of this factor of 2, since one needs
to use the scalar equations to prove that the remaining equations are $U(1)$
invariant. In other words, the form of Einstein’s equations depends on the
factor in front of $\mathfrak{e}^{2}$ in the scalar equation, because of the
requirement of $U(1)$-mass invariance, also see upcoming equations (184) and
(185). Hence it looks like the DFT and beta functions computations produce
almost the same result with the difference being a factor of two in (175) and
(180). It is worth noting that this difference goes away when we consider
torsionless geometries, since in that case we have $\mathfrak{e}_{\mu}=0$ and
world-sheet beta functions and double field theory give rise to the same set
of equations in the target space.

Despite this difference, both sets of equations describe spacetimes with the
same symmetries. In fact, we could generalize them further by modifying
equations (180) and (VI.1) by introducing an arbitrary constant $\lambda$:

$\displaystyle D\cdot a+a^{2}$
$\displaystyle=(1+\lambda)\mathfrak{e}^{2}+2\left(a\cdot D\phi\right)\,,$
(184) $\displaystyle R_{(\mu\nu)}-\frac{1}{4}H^{\rho\sigma}_{\ \
(\mu}H_{\nu)\rho\sigma}+2D_{(\mu}D_{\nu)}\phi$
$\displaystyle=\frac{\mathfrak{e}^{2}\left(2\Phi\tau_{\mu}\tau_{\nu}-\lambda\,\bar{h}_{\mu\nu}\right)-\mathfrak{e}_{\mu}\mathfrak{e}_{\nu}}{2}-\frac{2a^{2}\,\Phi\tau_{\mu}\tau_{\nu}-a_{\mu}a_{\nu}}{2}$
$\displaystyle\hphantom{=}+\left(\Delta_{S}\right)^{\rho}_{(\mu}D_{\nu)}a_{\rho}-\mathfrak{e}^{\sigma}\left(\Delta_{T}\right)^{\rho}_{(\mu}H_{\nu)\rho\sigma}$
(185)

while keeping the other equations unchanged. These equations are $U(1)$
invariant for any choice of the constant $\lambda$ and they are written in
terms of manifestly boost-invariant quantities. Therefore, they describe
theories with extended Galilean symmetry. The only difference between the
theories with $\lambda=0$ and $\lambda=1$ is that the former can be found by a
variational principle starting from the action $\eqref{TNCaction}$, while the
latter can be found by requiring vanishing of Weyl anomaly in the string
embedding.

Finally, note that to arrive at the parametrization (52)-(53) one needs to
perform field redefinitions involving both the $B$-field and $\aleph$. While
such field redefinitions are allowed at the classical level, it is not clear
whether this will produce the same actions at the quantum level as the path
integral measure may, in principle, transform as well. This question is beyond
the scope of this work and should be addressed separately in future.

###  VI.2 SNC beta functions

In Gomis:2019zyu ; YanYu  the beta functions for a worldsheet with SNC target
space have been computed. They imposed the so called foliation constraint
through their computation:

$\nabla_{[\mu}\tau_{\nu]}^{A}=0=F_{\mu\nu}^{A}.$ (186)

If we impose this geometrical constraint, the equations (161)-(171) become

$\displaystyle D^{\rho}H_{\rho}^{\ \mu\nu}-2D^{\rho}\phi\,H_{\rho}^{\ \mu\nu}$
$\displaystyle=0,$ $\displaystyle
D^{\mu}D_{\mu}\phi-2D^{\mu}\phi\,D_{\mu}\phi$
$\displaystyle=-\frac{1}{12}H^{\rho\mu\nu}H_{\rho\mu\nu},$
$\displaystyle\mathcal{R}^{(\mu\nu)}+2D^{(\mu}D^{\nu)}\phi-\frac{1}{4}H^{\mu}_{\
\rho\sigma}H^{\nu\rho\sigma}$ $\displaystyle=0,$ (187) $\displaystyle
h^{\mu\rho}u^{\sigma A}\mathcal{R}_{(\rho\sigma)}+2u^{\rho
A}D^{\mu}D_{\rho}\phi-\frac{1}{4}H^{\mu}_{\ \rho\sigma}H_{\lambda}^{\
\rho\sigma}u^{\lambda A}$ $\displaystyle=\frac{1}{2}\epsilon^{A}_{\
B}u^{\sigma B}\left(D^{\rho}H^{\mu}_{\ \rho\sigma}-2u^{\rho B}H^{\mu}_{\
\rho\sigma}D^{\rho}\phi\right),$ $\displaystyle u^{\mu A}u^{\nu}_{\
A}\mathcal{R}_{\mu\nu}+2u^{\mu A}u^{\nu}_{\
A}D_{\mu}D_{\nu}\phi-\frac{1}{4}u^{\mu A}u^{\nu}_{\
A}H_{\mu\rho\sigma}H_{\nu}^{\ \rho\sigma}$
$\displaystyle=-\frac{1}{2}\epsilon^{AB}u^{\rho}_{\ A}u^{\sigma}_{\
B}\left(D^{\lambda}H_{\rho\sigma\lambda}-2H_{\rho\sigma\lambda}D^{\lambda}\phi\right)$

where in the first equation we used the fact that the foliation constraint
implies181818Note that this is a sort of ”twistlelss condition” on the field
strength of $\omega_{\mu}$.

$\Omega_{\mu\nu}=u^{\rho}_{A}\tau_{[\mu}^{A}\Omega_{\nu]\rho}\qquad\Longrightarrow\qquad\Omega^{\mu\nu}=0.$
(188)

Note that the connection used in the present work is in general different from
the one used in the literature, however they are actually equal when the
foliation constraint is imposed. The connection used in Gomis:2019zyu ; YanYu
is

$\bar{\Gamma}^{\rho}_{\mu\nu}=\frac{1}{2}h^{\rho\sigma}\left(\partial_{\mu}\bar{h}_{\nu\sigma}+\partial_{\nu}\bar{h}_{\mu\sigma}-\partial_{\sigma}\bar{h}_{\mu\nu}\right)+\frac{1}{2}u^{\rho
A}u^{\sigma}_{A}\left(\partial_{\mu}\tau_{\nu\sigma}+\partial_{\nu}\tau_{\mu\sigma}-\partial_{\sigma}\tau_{\mu\nu}\right)$
(189)

where $\tau_{\mu\nu}\equiv\tau_{\mu}^{A}\tau_{\nu A}$. One can check that when
$F_{\mu\nu}^{A}=0$ we have

$\Gamma^{\rho}_{\mu\nu}=\bar{\Gamma}^{\rho}_{\mu\nu}.$ (190)

This means that the equations (VI.2) are exactly the same as the ones that are
obtained by requiring the cancellation of the Weyl anomaly in the worldsheet
theory.

###  VI.3 Comparison of TNC and SNC equations of motion

It was found in Harmark:2019upf  that, under certain assumptions which we
review below, the SNC worldsheet action reduces to that of TNC. It is then
natural to ask whether the equations of motion of SNC also reduce to the ones
of TNC. The basic condition under which SNC reduces to TNC is the presence of
an isometry along a compact longitudinal direction. We can then split the SNC
spacetime directions as $m=\left(\mu,u\right)$, where $u$ is the compact
direction and $\mu$ will describe the TNC directions. Then we impose the
following gauge choice on the SNC fields:

$\displaystyle m_{M}^{A}$ $\displaystyle=0,$ $\displaystyle\tau_{\nu}^{0}$
$\displaystyle=0,$ $\displaystyle\tau^{1}_{u}$ $\displaystyle=1,$
$\displaystyle E_{u}^{A^{\prime}}=0.$ (191)

Substituting this ansatz in the equations of motion would be quite tedious.
Luckily, since the equations of motion are obtained from an action, which in
turn is obtained from the generalized metric of DFT, it will suffice to
compare the generalized metrics rather than the equations or the actions.
Comparing (52)-(53) with (154) we see that the two parametrizations are indeed
the same once we impose the following identifications:

$\displaystyle\upsilon^{\mu}_{1}$ $\displaystyle=0,$
$\displaystyle\upsilon_{1}^{u}$ $\displaystyle=1,$
$\displaystyle\upsilon_{0}^{\mu}$ $\displaystyle=\upsilon^{\mu},$
$\displaystyle\upsilon_{0}^{u}$ $\displaystyle=\upsilon^{\mu}\aleph_{\mu},$
$\displaystyle h^{\mu\nu}_{(SNC)}$ $\displaystyle=h^{\mu\nu}_{(TNC)},$
$\displaystyle h^{\mu u}_{(SNC)}$ $\displaystyle=h^{\mu\rho}\aleph_{\rho},$
(192) $\displaystyle h^{uu}_{(SNC)}$
$\displaystyle=h^{\rho\sigma}\aleph_{\rho}\aleph_{\sigma},$ $\displaystyle
B^{(SNC)}_{\mu\nu}$ $\displaystyle=B^{(TNC)}_{\mu\nu}$ $\displaystyle
B^{(SNC)}_{\mu u}$ $\displaystyle=-m_{\mu}.$

In addition, it is not hard to check that the invariant measure of SNC (157)
correctly reduces to the one of TNC (57).

This identification implies that the SNC equations of motion will match the
ones obtained for TNC, at least in the DFT formulation. Unfortunately, the SNC
beta functions have only been computed under the assumption that the foliation
constraint is satisfied191919The SNC beta functions without the requirement of
the foliation constraint were also computed, but only at the linearized level.
, and similarily the TNC beta functions have been computed assuming twistless
torsion, so we cannot compare the full SNC and TNC equations obtained from DFT
with the ones obtained by setting the beta functions to zero. However we
already showed that when $F_{\mu\nu}^{A}=0$ the SNC equations (VI.2) do indeed
match the ones computed in Gomis:2019zyu ; YanYu . Using the identification
between the SNC and TNC generalized metrics this also implies that the SNC
beta functions with $F_{\mu\nu}^{A}=0$ match the TNC beta functions with
$F_{\mu\nu}=0$.

Finally, we have not found any existing results in the literature on actions
of Carrollian geometry to compare with our result in section IV.

##  VII Discussion and Outlook

In this work we studied a family of non-relativistic gravity theories in the
framework of Double Field Theory. In particular we studied three spacetimes
which are related to each other: Type I Torsional Newton Cartan, String
Newton-Cartan and Carroll. The non-relativistic actions we found are
respectively given by equations (56), (156) and (98) . As far as we know,
these actions are explicitly presented for the first time in the literature.
They should correspond to consistent truncations of the related Supergravity
actions obtained from string theory and hence are fundamental ingredients in
the study of non-relativistic quantum gravity.

These three geometries correspond to a DFT embedding with $n=1=\bar{n}$, so it
is natural to ask how they are related to each other. Relation between
Carrollian and TNC geometries was already proposed in Hartong:2015xda ;
Duval:2014uoa . We found that this connection becomes more clear in the
context of DFT: to relate TNC to Carroll we simply have to perform a T-duality
on the generalized metric. This is quite a curious result by itself, in that,
it implies T-duality in this case transforms a non-relativistic geometry to
the null202020In some sense, ultra-relativistic, as it is obtained by the
limit $c\to 0$. Carrolian geometry. It would be interesting to figure out
physical implications of this. Moreover, we also explicitly showed how one can
recover the TNC generalized metric from the SNC one by using the prescriptions
proposed in Harmark:2019upf . This means that the three non-relativistic
geometries, Torsional Newton-Cartan, Carrollian, and String Newton-Cartan are
all related to each other by either a T-duality or a null reduction/uplift.

By varying the actions we also obtained a set of equations of motion for the
three aforementioned non-relativistic geometries. In particular we
consistently found that each set of equations misses a projected component. As
we discussed, this is a general property of DFT: $n\times\bar{n}$ number of
equations — which equals one equation in our case — would be missing in the
final set of equations. The full DFT framework Morand:2017fnv  resolves this
issue as (1) provides the complete set. This means that theories with
$n\times\bar{n}\neq 0$ are to be considered as particular sectors of Double
Field Theories rather than independent theories by themselves. When working at
the level of low energy effective actions one should then append the missing
constraints to the equations of motion derived from the actions (56), (98) and
(156). For example, to define a TNC theory we need to provide the action (56)
together with an extra equation, Newton’s law, which should be imposed on-
shell. This resembles the self-duality condition of the 5-form field in the
effective action for Type IIB String Theory, albeit the origin of the issue is
clearly different here.

Once equations of motion were found, we asked whether these equations match
the ones already known in the literature. For TNC we showed that the equations
in the present work match the ones obtained from world-sheet beta functions
Gallegos:2019icg  when $\mathfrak{e}\cdot d\aleph$, that is, contraction of
the electric field with the field strength of the Kalb Ramond vector, in (62)
vanishes. In its presence the difference between the two sets of equations
turns out to be merely a factor of $2$. The origin of this discrepancy is not
clear to us, see section VI.1 for a more detailed discussion. On the other
hand, we showed that the SNC equations of motion obtained from DFT match
precisely the ones obtained in Gomis:2019zyu ; YanYu  from the world-sheet
beta functions when the foliation constraint is imposed. As already mentioned,
the generalized SNC metric reduces to the TNC one in a particular limit, which
implies that the equations of motion will also reduce to the TNC ones.
Recalling that the foliation constraint reduces to the torsionless limit on
the TNC side we conclude that the SNC beta functions Gomis:2019zyu ; YanYu
reduce to the TNC ones Gallegos:2019icg , which is consistent with what we
found by embedding both theories in the DFT framework. It is worth stressing
that the discrepancy between the TNC equations and beta functions disappears
in the absence of $d\aleph$, which is also implied by the foliation constraint
on the SNC side. The full (non-linearized) SNC beta functions with
$\nabla_{[\mu}\tau_{\nu]}^{A}\neq 0$ are not known, therefore we are unable to
tell if the discrepancy we find in TNC theory will also appear for SNC.

The present work can be extended in a number of ways. The most obvious
direction is studying solutions to the non-relativistic actions (56), (98) and
(156). Of particular interest would be solutions that are equivalent to black
holes in general relativity. The definition of a black hole in a non-
Riemannian manifold is unclear Bergshoeff:2016soe ; Grumiller:2017sjh ;
Grumiller:2020elf ; Arvanitakis:2016zes , and we believe that the actions
presented in this work would help clarify their physical interpretation. In
particular, variations of on-shell actions (note that the DFT on-shell action
is trivially zero up to boundary terms) would allow for investigation of the
thermodynamical properties of such hypothetical black holes and possibly
relate them to holographically dual theories in non-relativistic quantum
plasmas. Another interesting direction would be to study the relation between
TNC and Carroll (94)-(95), i.e. understanding in detail the T-duality map
between a non-relativistic limit and an ultra-relativistic one.

###### Acknowledgements.

We wish to thank Eric Bergshoeff, Chris Blair, Troels Harmark, Jelle Hartong,
Johannes Lahnsteiner, Kevin Morand, Niels Obers, Gerben Oling, Jeong-Hyuck
Park, Luca Romano, Jan Rosseel, Ceyda Şimşek and Ziqi Yan for useful comments
and discussions. DG is supported in part by CONACyT through the program
Fomento, Desarrollo y Vinculacion de Recursos Humanos de Alto Nivel. This work
is partially supported by the Delta-Institute for Theoretical Physics (D-ITP)
funded by the Dutch Ministry of Education, Culture and Science (OCW).

##  Appendix A TNC identities

###  A.1 Identities

The connection we use is

$\Gamma^{\rho}_{\mu\nu}=-\hat{\upsilon}^{\rho}\partial_{\mu}\tau_{\nu}+\frac{1}{2}h^{\rho\sigma}\left(\partial_{\mu}\bar{h}_{\nu\sigma}+\partial_{\nu}\bar{h}_{\mu\sigma}-\partial_{\sigma}\bar{h}_{\mu\nu}\right).$
(193)

Integration by parts is not as straightforward as in usual general relativity.
Instead we have

$D_{\mu}A^{\mu}=e^{-1}\partial_{\mu}\left(e\,A^{\mu}\right)-a_{\mu}A^{\mu}+2A^{\mu}D_{\mu}\phi,$
(194)

where $A^{\mu}$ is an arbitrary vector.

Many geometric identities can be derived from the completeness relation

$-\hat{\upsilon}^{\mu}\tau_{\nu}+h^{\mu\rho}\bar{h}_{\rho\nu}=\delta^{\mu}_{\nu}.$
(195)

For example we can take the derivative of this relation and then multiply with
$\bar{h}$. This gives

$D_{\rho}\bar{h}_{\mu\nu}=2\tau_{(\mu}\bar{h}_{\nu)\lambda}D_{\rho}\hat{\upsilon}^{\lambda}-2\tau_{\mu}\tau_{\nu}D_{\rho}\Phi$
(196)

and

$D_{\sigma}D_{\rho}\bar{h}_{\mu\nu}=2\tau_{(\mu}\bar{h}_{\nu)\lambda}D_{\sigma}D_{\rho}\hat{\upsilon}^{\lambda}+2\tau_{(\mu}\bar{D}_{\sigma}h_{\nu)\lambda}D_{\rho}\hat{\upsilon}^{\lambda}-2\tau_{\mu}\tau_{\nu}D_{\sigma}D_{\rho}\Phi.$
(197)

More useful identities can be derived by the definition of the connection:

$\displaystyle\begin{split}h^{\rho\sigma}D_{\sigma}\bar{h}_{\mu\nu}&=4h^{\rho\sigma}F_{\sigma(\mu}\tau_{\nu)}\Phi+2\tau_{(\mu}D_{\nu)}\hat{\upsilon}^{\rho}\\\
\hat{\upsilon}^{\rho}D_{\rho}\upsilon^{\mu}&=h^{\rho\sigma}\left(D_{\sigma}\Phi+2a_{\sigma}\Phi\right)\\\
h^{\mu\rho}D_{\rho}\hat{\upsilon}^{\nu}&=h^{\nu\rho}D_{\rho}\hat{\upsilon}^{\mu}+2\Phi
F_{\rho\sigma}h^{\mu\rho}h^{\nu\sigma}.\end{split}$ (198)

###  A.2 Variational calculus

We choose the independent geometric fields to be
$h^{\mu\nu},\hat{\upsilon}^{\mu}$ and $\Phi$. The variations of the dependant
fields are given by

$\displaystyle\begin{split}\delta\bar{h}_{\mu\nu}&=-2\tau_{\mu}\tau_{\nu}\delta\Phi+2\tau_{(\mu}\bar{h}_{\nu)\rho}\delta\hat{\upsilon}^{\rho}-\bar{h}_{\mu\rho}\bar{h}_{\nu\sigma}\delta
h^{\rho\sigma}\\\
\delta\tau_{\mu}&=\tau_{\mu}\tau_{\rho}\delta\hat{\upsilon}^{\rho}-\bar{h}_{\mu\rho}\tau_{\sigma}\delta
h^{\rho\sigma}.\end{split}$ (199)

The variation of the measure is then

$\delta
e=\frac{e}{2}\left(\bar{h}^{\rho\sigma}\delta\bar{h}_{\rho\sigma}-\frac{\delta\Phi}{\Phi}-4\delta\phi\right)=\frac{e}{2}\left(-\bar{h}_{\rho\sigma}\delta
h^{\rho\sigma}+2\tau_{\rho}\delta\hat{\upsilon}^{\rho}-4\delta\phi\right)$
(200)

The variation of the acceleration is

$\delta
a_{\mu}=\delta\hat{\upsilon}^{\rho}F_{\rho\mu}+2\hat{\upsilon}^{\rho}D_{[\rho}\delta\tau_{\mu]}-a_{\mu}\hat{\upsilon}^{\rho}\delta\tau_{\rho}.$
(201)

The variation of the connection is

$\displaystyle\begin{split}\delta\Gamma^{\rho}_{\mu\nu}=&-\frac{1}{2}\delta\hat{\upsilon}^{\rho}F_{\mu\nu}-\hat{\upsilon}D_{\mu}\delta\tau_{\nu}-\frac{1}{2}\hat{\upsilon}F_{\mu\nu}\tau_{\sigma}\delta\hat{\upsilon}^{\sigma}+\frac{1}{2}h^{\rho\sigma}\left(D_{\mu}\delta\bar{h}_{\nu\sigma}+D_{\nu}\delta\bar{h}_{\mu\sigma}-D_{\sigma}\delta\bar{h}_{\mu\nu}\right)\\\
&+\frac{1}{2}\delta
h^{\rho\sigma}\left(D_{\mu}\bar{h}_{\nu\sigma}+D_{\nu}\bar{h}_{\mu\sigma}-D_{\sigma}\bar{h}_{\mu\nu}\right)-2\Phi\tau_{(\mu}F_{\nu)\sigma}\delta
h^{\rho\sigma}-2\tau_{(\mu}F_{\nu)\sigma}h^{\rho\sigma}\delta\Phi\\\
&-\delta\tau_{(\mu}F_{\nu)\sigma}h^{\rho\sigma}\Phi+h^{\rho\sigma}\bar{h}_{\lambda(\mu}F_{\nu)\sigma}\delta\hat{\upsilon}^{\lambda}.\end{split}$
(202)

The Palatini identity in the presence of torsion is

$\delta\mathcal{R}_{\mu\nu}=D_{\rho}\delta\Gamma^{\rho}_{\nu\mu}-D_{\nu}\delta\Gamma^{\rho}_{\rho\mu}-2\Gamma^{\sigma}_{[\nu\rho]}\delta\Gamma^{\rho}_{\sigma\mu}=D_{\rho}\delta\Gamma^{\rho}_{\nu\mu}-D_{\nu}\delta\Gamma^{\rho}_{\rho\mu}+\hat{\upsilon}^{\sigma}F_{\nu\rho}\delta\Gamma^{\rho}_{\sigma\mu}.$
(203)

The indipendent matter fields are $\aleph_{\mu},B_{\mu\nu}$ and $\phi$. The
variation of $b_{\mu\nu}$ is

$\delta
b_{\mu\nu}=2D_{[\mu}\delta\aleph_{\nu]}-2\Gamma^{\rho}_{[\mu\nu]}\delta\aleph_{\rho}$
(204)

so that the variation of the electric field is

$\displaystyle\delta\mathfrak{e}_{\mu}=\delta\hat{\upsilon}^{\rho}b_{\rho\mu}+2D_{[\mu}\delta\aleph_{\nu]}-2\hat{\upsilon}^{\nu}\Gamma^{\rho}_{[\mu\nu]}\delta\aleph_{\rho},$
(205)

while that of $H_{\mu\nu\rho}$ can be written as

$\delta H_{\rho\mu\nu}=3D_{[\rho}\delta
B_{\mu\nu]}-2\Gamma^{\sigma}_{[\rho\mu]}\delta
B_{\nu\sigma}-2\Gamma^{\sigma}_{[\nu\rho]}\delta
B_{\mu\sigma}-2\Gamma^{\sigma}_{[\mu\nu]}\delta B_{\rho\sigma}.$ (206)

The Kalb-Ramond matter satisfies the usual Bianchi identities

$dH=db=0.$ (207)

In the twistless case we have
$b_{\mu\nu}=\mathfrak{e}_{\mu}\tau_{\nu}-\mathfrak{e}_{\nu}\tau_{\mu}$, then
we can express the Bianchi identity in terms of $\mathfrak{e}_{\mu}$ rather
than $b_{\mu\nu}$. We find

$\partial_{[\rho}b_{\mu\nu]}=2D_{[\mu}\mathfrak{e}_{\nu}\tau_{\rho]}+2a_{[\mu}\mathfrak{e}_{\nu}\tau_{\rho]}=0.$
(208)

A similar identity can be derived for the acceleration, yielding

$\partial_{[\rho}F_{\mu\nu]}=2D_{[\mu}a_{\nu}\tau_{\rho]}=0.$ (209)

##  Appendix B Carroll identities

###  B.1 Identities

The connection we utilize when varying the action is

$\Gamma^{\rho}_{\mu\nu}=-\upsilon^{\rho}\partial_{\mu}\hat{\tau}_{\nu}+\frac{1}{2}\hat{h}^{\rho\sigma}\left(\partial_{\mu}h_{\nu\sigma}+\partial_{\nu}h_{\mu\sigma}-\partial_{\sigma}h_{\mu\nu}\right),$
(210)

from which we find the following identitiy

$D_{\mu}A^{\mu}=e^{-1}\partial_{\mu}\left(e\,A^{\mu}\right)-a_{\mu}A^{\mu}+2A^{\mu}D_{\mu}\phi,$
(211)

valid for any arbitrary vector $A^{\mu}$. From the completeness relation and
the definition of the connection it is possible to derive the following
identities:

$\displaystyle\begin{split}D_{\rho}h_{\mu\nu}&=-\mathcal{K}_{\rho\mu}\hat{\tau}_{\nu}-\mathcal{K}_{\rho\nu}\hat{\tau}_{\mu}\\\
D_{\mu}\upsilon^{\nu}&=-\hat{h}^{\nu\rho}\mathcal{K}_{\rho\mu},\\\
\upsilon^{\rho}D_{\rho}\upsilon^{\mu}&=0,\end{split}$ (212)

where the extrinsic curvature was defined in the main text as

$\mathcal{K}_{\mu\nu}=-\frac{1}{2}\mathcal{L}_{\upsilon}h_{\mu\nu}=-\frac{1}{2}\left(\upsilon^{\rho}\partial_{\rho}h_{\mu\nu}+\left(\partial_{\mu}\upsilon^{\rho}\right)h_{\rho\nu}+\left(\partial_{\nu}\upsilon^{\rho}\right)h_{\mu\rho}\right).$
(213)

###  B.2 Variational calculus

We will use $\hat{h}^{\mu\nu},\upsilon^{\mu},\Phi$ as indipendent geometric
fields. The other geometric fields are related to them via

$\displaystyle\begin{split}\delta
h_{\mu\nu}&=2\tau_{(\mu}h_{\nu)\rho}\delta\upsilon^{\rho}-h_{\mu\rho}h_{\nu\sigma}\delta\hat{h}^{\rho\sigma}\\\
\delta\hat{\tau}_{\mu}&=\tau_{\mu}\tau_{\rho}\delta\upsilon^{\rho}-h_{\mu\rho}\tau_{\sigma}\delta\hat{h}^{\rho\sigma}\\\
\delta\bar{h}^{\mu\nu}&=\delta\hat{h}^{\mu\nu}-4\Phi\upsilon^{(\mu}\delta\upsilon^{\nu)}-2\delta\Phi\,\upsilon^{\mu}\upsilon^{\nu}.\end{split}$
(214)

The variation of the measure is then

$\delta
e=\frac{e}{2}\left(\bar{h}^{\rho\sigma}\delta\bar{h}_{\rho\sigma}+\frac{\delta\Phi}{\Phi}-4\delta\phi\right)=\frac{e}{2}\left(-h_{\mu\nu}\delta\hat{h}^{\mu\nu}+2\hat{\tau}_{\mu}\delta\upsilon^{\mu}-4\delta\phi\right).$
(215)

The variation of the acceleration is

$\delta
a_{\mu}=\delta\upsilon^{\rho}F_{\rho\mu}+2\upsilon^{\rho}D_{[\rho}\delta\hat{\tau}_{\mu]}-a_{\mu}\upsilon^{\rho}\delta\hat{\tau}_{\rho}.$
(216)

The variation of the connection is

$\displaystyle\begin{split}\delta\Gamma^{\rho}_{\mu\nu}=&-\frac{1}{2}\delta\upsilon^{\rho}F_{\mu\nu}-\upsilon
D_{\mu}\delta\hat{\tau}_{\nu}-\frac{1}{2}\upsilon^{\rho}F_{\mu\nu}\hat{\tau}_{\sigma}\delta\upsilon^{\sigma}+\frac{1}{2}\hat{h}^{\rho\sigma}\left(D_{\mu}\delta
h_{\nu\sigma}+D_{\nu}\delta h_{\mu\sigma}-D_{\sigma}\delta
h_{\mu\nu}\right)\\\
&+\frac{1}{2}\delta\hat{h}^{\rho\sigma}\left(D_{\mu}h_{\nu\sigma}+D_{\nu}h_{\mu\sigma}-D_{\sigma}h_{\mu\nu}\right)+\hat{h}^{\rho\sigma}h_{\lambda(\mu}F_{\nu)\sigma}\delta\upsilon^{\lambda}.\end{split}$
(217)

The Palatini identity is

$\delta\mathcal{R}_{\mu\nu}=D_{\rho}\delta\Gamma^{\rho}_{\nu\mu}-D_{\nu}\delta\Gamma^{\rho}_{\rho\mu}-2\Gamma^{\sigma}_{[\nu\rho]}\delta\Gamma^{\rho}_{\sigma\mu}=D_{\rho}\delta\Gamma^{\rho}_{\nu\mu}-D_{\nu}\delta\Gamma^{\rho}_{\rho\mu}+\upsilon^{\sigma}F_{\nu\rho}\delta\Gamma^{\rho}_{\sigma\mu}.$
(218)

The variations of the Kalb-Ramond matter fields are given by (204)-(206).

##  Appendix C SNC identities

###  C.1 Identities

The connection is

$\Gamma^{\rho}_{\mu\nu}=-u^{\rho}_{A}\nabla_{\mu}\tau_{\nu}^{A}+\frac{1}{2}h^{\rho\sigma}\left(D_{\mu}\bar{h}_{\nu\sigma}+D_{\nu}\bar{h}_{\mu\sigma}-D_{\sigma}\bar{h}_{\mu\nu}\right)$
(219)

where we introduced the spin connection via

$\nabla_{\mu}\tau_{\nu}^{A}=\partial_{\mu}\tau_{\nu}^{A}+\Omega_{\mu}^{\
AB}\tau_{\nu B}\equiv\partial_{\mu}\tau_{\nu}^{A}+\omega_{\mu}\epsilon^{A}_{\
B}\tau_{\nu}^{\ B}$ (220)

and our convention for the longitudinal epsilon symbol is $\epsilon_{01}=+1$.
This connection is boost invariant and compatible with $h^{\mu\nu}$ and
$\tau_{\mu}^{A}$,

$D_{\rho}h^{\mu\nu}=D_{\rho}\tau_{\mu}^{A}=0,$ (221)

where we are using the symbol $D$ (not to be confused with the dimensionality
of spacetime) to denote the full covariant derivative, i.e.

$D_{\mu}\tau_{\nu}^{A}=\partial_{\mu}\tau_{\nu}^{A}-\Gamma^{\rho}_{\mu\nu}\tau_{\rho}^{A}+\omega_{\mu}\epsilon^{A}_{\
B}\tau_{\nu}^{B}=0.$ (222)

Integration by parts is then performed with the use of the identity

$D_{\mu}A^{\mu}=e^{-1}\partial_{\mu}\left(e\,A^{\mu}\right)-a_{\mu}A^{\mu}+2A^{\mu}D_{\mu}\phi,$
(223)

where we recall $a_{\mu}=a_{\mu}^{AB}\eta_{AB}$.

From the completeness relation we find the following decomposition

$D_{\rho}\bar{h}_{\mu\nu}=2\tau_{(\mu}^{\ \
A}\bar{h}_{\nu)\sigma}D_{\rho}u^{\sigma}_{\
A}-\tau_{\mu}^{A}\tau_{\nu}^{B}D_{\rho}\Phi_{AB}.$ (224)

From the definition of the connection we find

$\displaystyle\begin{split}h^{\rho\sigma}\left(D_{\mu}\bar{h}_{\nu\sigma}+D_{\nu}\bar{h}_{\mu\sigma}-D_{\sigma}\bar{h}_{\mu\nu}\right)&=-2h^{\rho\sigma}F_{\sigma(\mu}^{\
\ \ A}\tau_{\nu)}^{\ B}\Phi_{AB}.\end{split}$ (225)

From the projections of this identity it follows that

$h^{\rho[\mu}D_{\rho}u^{\nu]}_{A}=\frac{1}{2}F^{\mu\nu B}\Phi_{BA}$ (226)

and

$u^{\nu}_{(A}D_{\nu}u^{\mu}_{B)}=h^{\mu\nu}\left(a_{\nu(A}^{\ \ \
\,C}\Phi_{B)C}+\frac{1}{2}D_{\nu}\Phi_{AB}\right).$ (227)

Since $\Phi_{AB}$ is a $2\times 2$ symmetric matrix, we can write its inverse
as

$(\Phi^{-1})_{AB}=\frac{\Phi_{AB}-\eta_{AB}\Phi}{\det\Phi}$ (228)

where $\det\Phi=\frac{1}{2}\epsilon^{AB}\epsilon^{CD}\Phi_{AC}\Phi_{BD}$ and
$\Phi=\Phi^{A}_{A}$. Moreover since the longitudinal indices can only take two
different values, we have that

$T^{[\alpha\beta\gamma\dots]}=0$ (229)

for any tensor $T$ with three or more longitudinal indices.

The field strength of $\tau_{\mu}^{A}$ can be decomposed as

$F_{\mu\nu}^{A}=f^{A}\tau_{\mu}^{B}\tau_{\nu}^{C}\epsilon_{BC}+2a_{[\mu}^{BA}\tau_{\nu]B}+\mathcal{F}_{\mu\nu}^{A}$
(230)

with

$u^{\mu}_{A}\mathcal{F}_{\mu\rho B}=0,\qquad\qquad u^{\mu A}a_{\mu}^{\
BC}=\epsilon^{AB}f^{C}.$ (231)

###  C.2 Variational calculus

The indipendent fields are $u^{\mu}_{A},h^{\mu\nu},\Phi^{AB}$ and
$B_{\mu\nu}$. The other SNC fields are related via

$\displaystyle\begin{split}\delta\tau_{\mu}^{A}&=\tau_{\mu}^{B}\tau_{\rho}^{A}\delta
u^{\rho}_{B}-\tau_{\rho}^{A}\bar{h}_{\mu\sigma}\delta h^{\rho\sigma}\\\
\delta\bar{h}_{\mu\nu}&=2\tau_{(\mu}^{A}\bar{h}_{\nu)\rho}u^{\rho}_{A}-\bar{h}_{\mu\rho}\bar{h}_{\nu\sigma}\delta
h^{\rho\sigma}-\tau_{\mu}^{A}\tau_{\nu}^{B}\delta\Phi_{AB}.\end{split}$ (232)

The variation of the connection is

$\displaystyle\begin{split}\delta\Gamma^{\rho}_{\mu\nu}&=-u^{\rho}_{A}\left(D_{\mu}\delta\tau_{\nu}^{A}+\Gamma^{\sigma}_{[\mu\nu]}\delta\tau_{\sigma}^{A}+\epsilon^{A}_{\
B}\tau_{\nu B}\delta\omega_{\mu}\right)-\delta
u^{\rho}_{A}\Gamma^{\sigma}_{[\mu\nu]}\tau_{\sigma}^{A}\\\
&\hphantom{=}+\frac{1}{2}h^{\rho\sigma}\left(D_{\mu}\delta\bar{h}_{\nu\sigma}+D_{\nu}\delta\bar{h}_{\mu\sigma}-D_{\sigma}\delta\bar{h}_{\mu\nu}\right)+\frac{1}{2}\delta
h^{\rho\sigma}\left(D_{\mu}\bar{h}_{\nu\sigma}+D_{\nu}\bar{h}_{\mu\sigma}-D_{\sigma}\bar{h}_{\mu\nu}\right)\\\
&\hphantom{=}+h^{\rho\sigma}\left(\Gamma^{\lambda}_{[\mu\sigma]}\delta\bar{h}_{\lambda\nu}+\Gamma^{\lambda}_{[\nu\sigma]}\delta\bar{h}_{\lambda\mu}\right)+\delta
h^{\rho\sigma}\left(\Gamma^{\lambda}_{[\mu\sigma]}\bar{h}_{\lambda\nu}+\Gamma^{\lambda}_{[\nu\sigma]}\bar{h}_{\lambda\mu}\right).\end{split}$
(233)

The variation of the Ricci scalar can be found as usual through the Palatini
identity

$\delta
R_{\mu\nu}=D_{\rho}\delta\Gamma^{\rho}_{\nu\mu}-D_{\nu}\delta\Gamma^{\rho}_{\rho\mu}-2\Gamma^{\rho}_{[\nu\sigma]}\delta\Gamma^{\sigma}_{\rho\mu}=D_{\rho}\delta\Gamma^{\rho}_{\nu\mu}-D_{\nu}\delta\Gamma^{\rho}_{\rho\mu}-2u^{\rho}_{A}\nabla_{[\nu}\tau_{\sigma]}^{A}\delta\Gamma^{\sigma}_{\rho\mu}.$
(234)

The variation of the field strength of the $B$-field can be read off from
(206).

##  Appendix D Actions in Einstein frame

In this section we will perform a conformal redefinition of the metric complex
to rewrite the non-relativistic string frame actions (56), (98), (111) and
(156) in Einstein frame.

###  D.1 TNC

The action and equations of motion can be written in terms of the basic fields
$h^{\mu\nu}$ , $\hat{v}^{\mu}$ and $\Phi$. The aforementioned fields transform
as

$\displaystyle\begin{split}h^{\mu\nu}&\rightarrow
e^{-\alpha\phi}h^{\mu\nu},\\\ \upsilon^{\mu}&\rightarrow
e^{-\alpha\phi}\upsilon^{\mu},\\\ \Phi&\rightarrow
e^{-\alpha\phi}\Phi,\end{split}$ (235)

where $e^{-\alpha\phi}$ is the conformal factor and $\alpha=\frac{4}{d-1}$.
Then, in Einstein frame, the TNC action (56) can be rewritten as

$\displaystyle\begin{split}S=&\int
d^{d}x\,\sqrt{\frac{\det\bar{h}_{\mu\nu}}{2\Phi}}\,\left[\mathcal{R}+\frac{1}{2}a^{\mu}a_{\mu}+\frac{1}{2}e^{-2\alpha\phi}\mathfrak{e}^{\mu}\mathfrak{e}_{\mu}-\alpha
D^{\mu}\phi
D_{\mu}\phi-\frac{1}{12}e^{-2\alpha\phi}H^{\mu\nu\rho}H_{\mu\nu\rho}\right.\\\
&\hphantom{(((\int
d^{d}x\,\sqrt{\frac{\det\bar{h}_{\mu\nu}}{2\Phi}}}\left.-\frac{1}{2}e^{-2\alpha\phi}\hat{\upsilon}^{\rho}H_{\rho\mu\nu}b^{\mu\nu}-\frac{1}{2}\left(F^{\mu\nu}F_{\mu\nu}+e^{-2\alpha\phi}b^{\mu\nu}b_{\mu\nu}\right)\Phi\right]\end{split}$
(236)

###  D.2 Carroll

The action and equations of motion can be written in terms of the basic fields
$\hat{h}^{\mu\nu}$ , ${v}^{\mu}$ and $\Phi$. The aforementioned fields
transform as

$\displaystyle\begin{split}\hat{h}^{\mu\nu}&\rightarrow
e^{-\alpha\phi}\hat{h}^{\mu\nu},\\\ \upsilon^{\mu}&\rightarrow
e^{-\alpha\phi}\upsilon^{\mu},\\\ \Phi&\rightarrow
e^{+\alpha\phi}\Phi,\end{split}$ (237)

where once again we have $\alpha=\frac{4}{d-1}$. In Einstein frame the action
(98) is

$\displaystyle\begin{split}S=&\int
d^{d}x\,\sqrt{2\Phi\det\bar{h}_{\mu\nu}}\,\left[\mathcal{R}+\frac{1}{2}a^{\mu}a_{\mu}+\frac{1}{2}e^{-2\alpha\phi}\mathfrak{e}^{\mu}\mathfrak{e}_{\mu}-\alpha
D^{\mu}\phi D_{\mu}\phi+2\mathcal{K}\upsilon^{\mu}D_{\mu}\Phi\right.\\\
&\hphantom{(((\int
d^{d}x\sqrt{2\Phi\det\bar{h}_{\mu\nu}}}\left.+2\Phi\left(\mathcal{K}^{\mu\nu}\mathcal{K}_{\mu\nu}-\mathcal{K}^{2}+\alpha\,\upsilon^{\mu}D_{\mu}\phi\,\upsilon^{\nu}D_{\nu}\phi+\frac{1}{4}e^{-2\alpha\phi}\upsilon^{\rho}\upsilon^{\sigma}H^{\mu\nu}_{\hphantom{\mu\nu}\rho}H_{\mu\nu\sigma}\right)\right.\\\
&\hphantom{(((\int
d^{d}x\sqrt{2\Phi\det\bar{h}_{\mu\nu}}}\left.-\frac{1}{12}e^{-2\alpha\phi}H_{\mu\nu\rho}H^{\mu\nu\rho}+\frac{1}{2}e^{-2\alpha\phi}b^{\mu\nu}H_{\mu\nu\rho}\upsilon^{\rho}\right]\end{split}$
(238)

Similarily we can transform the action (111):

$\displaystyle\begin{split}S=&\int
d^{d}x\,\sqrt{2\Phi\det\bar{h}_{\mu\nu}}\,\left[\mathcal{\bar{R}}+\frac{1}{2}a^{\mu}a_{\mu}+\frac{1}{2}e^{-2\alpha\phi}\mathfrak{e}^{\mu}\mathfrak{e}_{\mu}-\alpha\,\bar{D}^{\mu}\phi\bar{D}_{\mu}\phi\right.\\\
&\hphantom{(((\int
d^{d}x\sqrt{2\Phi\det\bar{h}_{\mu\nu}}}\left.-\frac{1}{12}e^{-2\alpha\phi}H_{\mu\nu\rho}H^{\mu\nu\rho}+\frac{1}{2}e^{-2\alpha\phi}b^{\mu\nu}\upsilon^{\rho}H_{\mu\nu\rho}\right],\end{split}$
(239)

###  D.3 SNC

The action and equations of motion can be written in terms of the basic fields
$h^{\mu\nu}$ , ${v}^{\mu}_{A}$ and $\Phi_{AB}$.The aforementioned fields
transform as

$\displaystyle\begin{split}h^{\mu\nu}&\rightarrow
e^{-\alpha\phi}h^{\mu\nu},\\\ u^{\mu}_{A}&\rightarrow
e^{-\alpha\phi}u^{\mu}_{A},\\\ \Phi_{AB}&\rightarrow
e^{-\alpha\phi}\Phi_{AB},\end{split}$ (240)

where $\alpha=\frac{4}{D}$ for SNC. The action (156) can be written in
Einstein frame as

$\displaystyle\begin{split}S=&\int
d^{D}x\,\sqrt{\frac{\det\bar{h}_{\mu\nu}}{\det\Phi_{AB}}}\,\left[\mathcal{R}-a^{\mu
AB}(a_{\mu(AB)}-\frac{1}{2}\eta_{AB}a_{\mu})+a^{\mu}a_{\mu}+\alpha\,a^{\mu}D_{\mu}\phi+\alpha(\alpha-1)D^{\mu}\phi\,D_{\mu}\phi\right.\\\
&\hphantom{(((\int d^{d}xe\,}\left.-\frac{1}{2}F^{\mu\nu
A}F_{\mu\nu}^{B}(\Phi_{AB}-\frac{1}{2}\eta_{AB}\Phi)+\frac{1}{2}e^{-\alpha\phi}\epsilon_{AB}u^{\rho
A}F^{\mu\nu
B}H_{\rho\mu\nu}-\frac{1}{12}e^{-2\alpha\phi}H^{\rho\mu\nu}H_{\rho\mu\nu}\right]\\\
\end{split}$ (241)

## References

  * (1) J. Hartong, E. Kiritsis, and N. Obers, “Field theory on newton-cartan backgrounds and symmetries of the lifshitz vacuum,” Journal of High Energy Physics, vol. 2015, 02 2015.
  * (2) D. Hansen, J. Hartong, and N. A. Obers, “Non-Relativistic Gravity and its Coupling to Matter,” 2020.
  * (3) J. de Boer, J. Hartong, E. Have, N. A. Obers, and W. Sybesma, “Non-Boost Invariant Fluid Dynamics,” SciPost Phys., vol. 9, p. 18, 2020.
  * (4) J. de Boer, J. Hartong, N. A. Obers, W. Sybesma, and S. Vandoren, “Perfect fluids,” SciPost Phys., vol. 5, p. 3, 2018.
  * (5) T. Harmark and M. Orselli, “Spin matrix theory: a quantum mechanical model of the ads/cft correspondence,” Journal of High Energy Physics, vol. 2014, p. 134, Nov 2014.
  * (6) D. Hansen, J. Hartong, and N. A. Obers, “Gravity between Newton and Einstein,” 2019.
  * (7) T. Harmark, J. Hartong, and N. A. Obers, “Nonrelativistic strings and limits of the ads/cft correspondence,” Phys. Rev. D, vol. 96, p. 086019, Oct 2017\.
  * (8) E. A. Bergshoeff, K. T. Grosvenor, C. Simsek, and Z. Yan, “An Action for Extended String Newton-Cartan Gravity,” JHEP, vol. 01, p. 178, 2019.
  * (9) D. Hansen, J. Hartong, and N. A. Obers, “Action principle for newtonian gravity,” Phys. Rev. Lett., vol. 122, p. 061106, Feb 2019.
  * (10) J. Gomis and H. Ooguri, “Nonrelativistic closed string theory,” J. Math. Phys., vol. 42, pp. 3127–3151, 2001.
  * (11) E. A. Bergshoeff, J. Gomis, J. Rosseel, C. Şimşek, and Z. Yan, “String Theory and String Newton-Cartan Geometry,” J. Phys. A, vol. 53, no. 1, p. 014001, 2020.
  * (12) T. Harmark, J. Hartong, L. Menculini, N. A. Obers, and Z. Yan, “Strings with non-relativistic conformal symmetry and limits of the ads/cft correspondence,” Journal of High Energy Physics, vol. 2018, p. 190, Nov 2018.
  * (13) G. Dautcourt, “Post-newtonian extension of the newton - cartan theory,” Classical and Quantum Gravity, vol. 14, pp. A109–A117, jan 1997.
  * (14) D. V. den Bleeken, “Torsional newton–cartan gravity from the large c expansion of general relativity,” Classical and Quantum Gravity, vol. 34, p. 185004, aug 2017.
  * (15) M. Ergen, E. Hamamc, and D. Bleeken, “Oddity in nonrelativistic, strong gravity,” The European Physical Journal C, vol. 80, 06 2020.
  * (16) W. Tichy and E. E. Flanagan, “Covariant formulation of the post-1-Newtonian approximation to General Relativity,” Phys. Rev. D, vol. 84, p. 044038, 2011.
  * (17) D. Son, “Newton-cartan geometry and the quantum hall effect,” arXiv:1306.0638 [cond-mat.mes-hall], 2013.
  * (18) M. H. Christensen, J. Hartong, N. A. Obers, and B. Rollier, “Torsional newton-cartan geometry and lifshitz holography,” Phys. Rev. D, vol. 89, p. 061901, Mar 2014.
  * (19) M. Taylor, “Non-relativistic holography,” 01 2009.
  * (20) J. Armas, J. Hartong, E. Have, B. F. Nielsen, and N. A. Obers, “Newton-cartan submanifolds and fluid membranes,” Phys. Rev. E, vol. 101, p. 062803, Jun 2020.
  * (21) C. Duval, G. Gibbons, P. Horvathy, and P. Zhang, “Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time,” Class. Quant. Grav., vol. 31, p. 085016, 2014.
  * (22) T. Damour, M. Henneaux, and H. Nicolai, “Cosmological billiards,” Classical and Quantum Gravity, vol. 20, p. R145, 04 2003.
  * (23) L. Donnay and C. Marteau, “Carrollian Physics at the Black Hole Horizon,” Class. Quant. Grav., vol. 36, no. 16, p. 165002, 2019.
  * (24) L. Ciambelli, C. Marteau, A. Petkou, P. Petropoulos, and K. Siampos, “Flat holography and carrollian fluids,” Journal of High Energy Physics, vol. 2018, 02 2018.
  * (25) J. Hartong, “Holographic reconstruction of 3d flat space-time,” Journal of High Energy Physics, vol. 2016, 11 2015.
  * (26) B. Julia and H. Nicolai, “Null-killing vector dimensional reduction and galilean geometrodynamics,” Nuclear Physics B, vol. 439, no. 1, pp. 291 – 323, 1995.
  * (27) M. Christensen, J. Hartong, N. Obers, and B. Rollier, “Boundary stress-energy tensor and newton-cartan geometry in lifshitz holography,” Journal of High Energy Physics, vol. 2014, 11 2013.
  * (28) C. Duval, G. W. Gibbons, and P. Horvathy, “Celestial mechanics, conformal structures and gravitational waves,” Phys. Rev. D, vol. 43, pp. 3907–3922, 1991.
  * (29) C. Duval, G. Burdet, H. Kunzle, and M. Perrin, “Bargmann Structures and Newton-cartan Theory,” Phys. Rev. D, vol. 31, pp. 1841–1853, 1985.
  * (30) E. Bergshoeff, J. Gomis, and Z. Yan, “Nonrelativistic String Theory and T-Duality,” JHEP, vol. 11, p. 133, 2018.
  * (31) K. Lee and J.-H. Park, “Covariant action for a string in doubled yet gauged spacetime,” Nuclear Physics B, vol. 880, pp. 134 – 154, 2014.
  * (32) K. Morand and J.-H. Park, “Classification of non-Riemannian doubled-yet-gauged spacetime,” Eur. Phys. J. C, vol. 77, no. 10, p. 685, 2017.  [Erratum: Eur.Phys.J.C 78, 901 (2018)].
  * (33) J. Klusoň, “Remark About Non-Relativistic String in Newton-Cartan Background and Null Reduction,” JHEP, vol. 05, p. 041, 2018.
  * (34) C. Hull and B. Zwiebach, “Double field theory,” Journal of High Energy Physics, vol. 2009, pp. 099–099, sep 2009.
  * (35) O. Hohm, C. Hull, and B. Zwiebach, “Background independent action for double field theory,” JHEP, vol. 07, p. 016, 2010.
  * (36) O. Hohm, C. Hull, and B. Zwiebach, “Generalized metric formulation of double field theory,” JHEP, vol. 08, p. 008, 2010.
  * (37) W. Siegel, “Two vierbein formalism for string inspired axionic gravity,” Phys. Rev. D, vol. 47, pp. 5453–5459, 1993.
  * (38) W. Siegel, “Superspace duality in low-energy superstrings,” Phys. Rev. D, vol. 48, pp. 2826–2837, 1993.
  * (39) J.-H. Park, S.-J. Rey, W. Rim, and Y. Sakatani, “O(D, D) covariant Noether currents and global charges in double field theory,” JHEP, vol. 11, p. 131, 2015.
  * (40) S. Angus, K. Cho, and J.-H. Park, “Einstein Double Field Equations,” Eur. Phys. J. C, vol. 78, no. 6, p. 500, 2018.
  * (41) S. M. Ko, C. Melby-Thompson, R. Meyer, and J.-H. Park, “Dynamics of Perturbations in Double Field Theory \& Non-Relativistic String Theory,” JHEP, vol. 12, p. 144, 2015.
  * (42) D. S. Berman, C. D. A. Blair, and R. Otsuki, “Non-Riemannian geometry of M-theory,” JHEP, vol. 07, p. 175, 2019.
  * (43) J. Gomis, J. Oh, and Z. Yan, “Nonrelativistic String Theory in Background Fields,” 2019.
  * (44) A. Gallegos, U. Gürsoy, and N. Zinnato, “Torsional Newton Cartan gravity from non-relativistic strings,” JHEP, vol. 09, p. 172, 2020.
  * (45) Z. Yan and M. Yu, “Background Field Method for Nonlinear Sigma Models in Nonrelativistic String Theory,” JHEP, vol. 03, p. 181, 2020.
  * (46) O. Hohm and B. Zwiebach, “On the Riemann Tensor in Double Field Theory,” JHEP, vol. 05, p. 126, 2012.
  * (47) I. Jeon, K. Lee, and J.-H. Park, “Stringy differential geometry, beyond riemann,” Phys. Rev. D, vol. 84, p. 044022, Aug 2011.
  * (48) G. Aldazabal, D. Marqués, and C. Núñez, “Double field theory: a pedagogical review,” Classical and Quantum Gravity, vol. 30, p. 163001, jul 2013.