双引力对奇异3D重力的统一扩展
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Unitary Extension of Exotic Massive 3D Gravity from Bigravity
Mehmet Ozkan,
1,*
Yi Pang,
2,†
and Utku Zorba
1,‡
1
Department of Physics, Istanbul Technical University, Maslak 34469, Istanbul, Turkey
2
Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, United Kingdom
(Received 8 May 2019; published 17 July 2019)
We obtain a new 3D gravity model from two copies of parity-odd Einstein-Cartan theories. Using
Hamiltonian analysis, we demonstrate that the only local degrees of freedom are two massive spin-2 modes.
Unitarity of the model in anti–de Sitter and Minkowski backg rounds can be satisfied for vast choices of the
parameters without fine-tuning. The recent “exotic massive 3D gravity” model arises as a limiting case of
the new model. We also show that there exist trajectories on the parameter space of the new model which
cross the boundary between unitary and nonunitary regions. At the crossing point, one massive graviton
decouples resulting in a unitary model with just one bulk degree of freedom but two positive central charges
at odds with the usual expectation that the critical model has at least one vanishing central charge. Given the
fact that a suitable nonrelativistic version of bigravity has been used as an effective theory for gapped spin-2
fractional quantum Hall states, our model may have interesting applications in condensed matter physics.
DOI: 10.1103/PhysRevLett.123.031303
Three-dimensional massive gravity possesses many sur-
prises that are not shared by its higher-dimensional cousins.
There exists a particular combination of linear and quad-
ratic curvature terms such that the theory is perturbatively
unitary and power-counting renormalizable about the
Minkowski background [1]. At the linearized level, this
theory reproduces the Fierz-Pauli theory propagating a pair
of massive gravitons. It is general coordinate invariant,
differing from the background dependent de Rham–
Gabadadze–Tolley massive gravity [2] and thus gaining
its name “new massive gravity” (NMG). The maximally
symmetric vacua of NMG also contains the anti–de Sitter
(AdS) space. Unitarity in the AdS background requires in
addition to the ghost-free dynamical degrees of freedom
(d.o.f.) the positivity of the central charges present in the
asymptotic symmetry algebra. This is a requirement from
the correspondence between quantum gravity in the AdS
3
background and two-dimensional conformal field theory
(CFT) in which the central charges in the gravity model are
identified with those in the dual CFT. NMG, however, is
not a satisfactory model to apply the AdS
3
=CFT
2
corre-
spondence, as the condition under which the massive
gravitons are unitary implies negative central charges.
Another surprise in 3D gravity is that many interesting
models can be reformulated using frame-field formalism as
a multiflavor Chern-Simons- (CS) like theory [3]. Different
from usual CS theories, CS-like theories can bear dynami-
cal d.o.f., as the structure constants disobey Jacobi identity.
The resolution of the bulk-boundary clash for NMG in
AdS
3
becomes straightforward, once adopting the frame-
field formalism and losing the requirement that the massive
gravitons interact only through a finite number of curvature
terms. As a consequence, NMG is extended to the “zwei-
dreibein gravity” (ZDG) [4] and further generalizations [5]
which are also parity even. The main building blocks of
ZDG are two copies of parity-even cosmological Einstein-
Cartan theories glued together by a cubic potential involv-
ing the two dreibeins. Thus ZDG can be viewed as a
variant of bigravity. By properly choosing parameters,
ZDG can satisfy both criteria for a unitary AdS
3
quantum
gravity, thus serving as an attractive toy model of lower-
dimensional quantum gravity defined via AdS
3
=CFT
2
correspondence.
Besides the more mathematically orientated goal of
constructing a well-defined lower-dimensional quantum
gravity model, 3D spatially covariant bigravity has recently
been utilized as a tool to build effective theories for the
spin-2 gapped collective excitations observed in certain
fractional quantum Hall states (FQHS) [6–8]. This latest
application of bigravity in condensed matter physics is of
particular interest, as it directly links two broad areas in
physics, gravity, and condensed matter physics.
NMG and its two-frame fields extension ZDG are
standard gravity models in the sense that the same parity
property is shared by the equations of motion and the
Lagrangian. Interestingly, there also exist “exotic" 3D
gravity models of which the equations of motion are parity
even, while the actions are parity odd. The recently
proposed “exotic massive gravity” (EMG) [9] is of just
Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article’s title, journal citation,
and DOI . Funded by SCOAP
3
.
PHYSICAL REVIEW LETTERS 123, 031303 (2019)
0031-9007=19=123(3)=031303(6) 031303-1 Published by the American Physical Society
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