牛顿-卡坦几何学中的变形变换
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Eur. Phys. J. C (2016) 76:436
DOI 10.1140/epjc/s10052-016-4282-7
Regular Article - Theoretical Physics
Disformal transformation in Newton–Cartan geometry
Peng Huang
1,2,a
, Fang-Fang Yuan
3,b
1
Present address: Department of Information, Zhejiang Chinese Medical University, Hangzhou 310053, China
2
School of Physics and Astronomy, Sun Yat-Sen University, Guangzhou 510275, China
3
School of Physics, Nankai University, Tianjin 300071, China
Received: 21 June 2016 / Accepted: 25 July 2016 / Published online: 4 August 2016
© The Author(s) 2016. This article is published with open access at Springerlink.com
Abstract Newton–Cartan geometry has played a central
role in recent discussions of the non-relativistic hologra-
phy and condensed matter systems. Although the conformal
transformation in non-relativistic holography can easily be
rephrased in terms of Newton–Cartangeometry, we showthat
it requires a nontrivial procedure to arrive at the consistent
form of anisotropic disformal transformation in this geom-
etry. Furthermore, as an application of the newly obtained
transformation, we use it to induce a geometric structure
which may be seen as a particular non-relativistic version
of the Weyl integrable geometry.
1 Introduction
Newton–Cartan geometry (NCG) was proposed by Élie Car-
tan as a geometrical description for Newtonian gravity in
the spirit of general relativity [1,2] (see also [3]). The recent
renewed interest in NCG is well motivated in view of its use-
fulness in the investigations of condensed matter systems and
non-relativistic holography. Concretely speaking, it has been
found that, together with the non-relativistic diffeomorphism
invariance, NCG provides a natural geometrical background
for the effective field theory description of fractional quan-
tum Hall effect [4].
Moreover, the progress in gauge/gravity duality attaches
increasing importance to non-Riemannian geometries while,
in particular, NCG acts an important role in various realiza-
tions of the non-AdS holography; see e.g. [5–9].
Similar to the extension of Riemannian geometry to
Weyl geometry, the conformal extension of NCG has also
been investigated from different perspectives. Following
the experience of “deriving” Einstein gravity (more prop-
erly, Einstein–Cartan theory) and Riemann–Cartan geome-
try through gauging the relativistic Poincaré algebra, NCG
a
e-mail: huangp46@mail.sysu.edu.cn
b
e-mail: ffyuan@nankai.edu.cn
is obtained by gauging the Bargmann algebra, which is the
centrally extended Galilean algebra [10]. Then the confor-
mal generalization of NCG is also got by performing a sim-
ilar gauging procedure to the Schrödinger algebra, the con-
formal version of Bargmann algebra [8,11]. On the other
hand, inspired by the celebrated Poincaré gauge theory [12–
14](see[15] for a detailed exposition) in which Riemann–
Cartan geometry emerges naturally by localizing the global
Poincaré symmetry of a field theory in Minkowski spacetime,
it has been shown that NCG can be obtained by localizing
the global Galilean symmetry of a field theory in 3d Galilean
space with universal time [16]. Such a method also provides
a systematic way for reformulating an originally Galilean-
invariant theory in Euclidean space with universal time into
a diffeomorphism-invariant theory in curved space [16–18].
Following this train of thought, the conformal extension of
NCG is also obtained by localizing the global Galilean and
scale symmetries [19].
The above two methods are not merely straightforward
repetitions of the corresponding process in the relativistic
case. There the space and time are on an equal footing
and thus can be treated uniformly which makes the whole
procedure relatively clear and concise. However, for non-
relativistic cases, the space is relative while the time is not
[20,21]. To preserve the absoluteness of universal time and
the relative character of space, special attention is needed
throughout the whole procedure; see [10,16–18] for details.
Such different concepts of space and time are also reflected in
the consistent form of the conformal transformation. In rel-
ativistic cases, conformal transformation must be isotropic
due to the equal footing of space and time. Nevertheless, in
non-relativistic case, the absolute nature of the universal time
makes the concept of anisotropic conformal transformation
emerge naturally; see [8,11,19] for further details.
On the other hand, by noting the existence of confor-
mally related Riemannian geometries in scalar-tensor theory,
Bekenstein considered the possibility of more general phys-
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