We establish a duality between massive fermions coupled to topologically massive gravity (TMG) in d = 3 space–time dimensions and a purely gravity theory which also will turn out to be a TMG theory but with different parameters: the original graviton mass in the TMG theory coupled to fermions picks up a contribution from fermion bosonization. We obtain explicit bosonization rules for the fermionic currents and for the energy–momentum tensor showing that the identifications do not depend explicitly on the parameters of the theory. These results are the gravitational analog of the results for 2 + 1 Abelian and non-Abelian bosonization in flat space–time.Facultad de Ciencias Exacta

Fradkin, Eduardo

Moreno, Enrique Francisco

Schaposnik, Fidel Arturo

Physics Letters B

English

SEDICI - Repositorio de la UNLP

Physics Letters B 730 (2014) 284–287Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Bosonization of fermions coupled to topologically massive gravity
Eduardo Fradkin a, Enrique F. Moreno b,∗, Fidel A. Schaposnik c,1
a Department of Physics and Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana,
IL 61801-3080, USA
b Department of Physics, Northeastern University, Boston, MA 02115, USA
c Departamento de Física, Universidad Nacional de La Plata, Instituto de Física La Plata, C.C. 67, 1900 La Plata, Argentina
a r t i c l e i n f o a b s t r a c t
Article history:
Received 11 January 2014
Accepted 27 January 2014
Available online 31 January 2014
Editor: L. Alvarez-Gaumé
We establish a duality between massive fermions coupled to topologically massive gravity (TMG) in d = 3
space–time dimensions and a purely gravity theory which also will turn out to be a TMG theory but
with different parameters: the original graviton mass in the TMG theory coupled to fermions picks up a
contribution from fermion bosonization. We obtain explicit bosonization rules for the fermionic currents
and for the energy–momentum tensor showing that the identifications do not depend explicitly on the
parameters of the theory. These results are the gravitational analog of the results for 2 + 1 Abelian and
non-Abelian bosonization in flat space–time. 
© 2014 The Authors. Published by Elsevier B.V. Open access under CC BY license. Funded by SCOAP3.Bosonization of fermion models is strongly connected to the ex-
istence of quantum anomalies associated with symmetries that ex-
ist at the classical level. Indeed, the well-honored fermions–boson
duality in d = 2 space–time dimensions [1] can be reproduced
within the path-integral approach once the chiral anomaly affect-
ing the fermionic measure [2] is taken into account [3,4]. In the
case of d = 3 fermions coupled to Abelian and non-Abelian gauge
fields it is the parity anomaly that plays a central role unveiling
the occurrence of a Chern–Simons term [5,6] in the bosonic dual
[7–9].
Interestingly enough, when fermions are coupled to a gravi-
tational field in odd-dimensional spaces the parity anomaly also
induces a Chern–Simons term which can be written in terms of the
spin connection or in terms of the Christoffel connection [10–12].
Recently this fact has been exploited to simulate the effects of
crystal defects on the electronic degrees of freedom of topologi-
cal insulators in condensed matter physics. Those systems exhibit
physical phenomena related to parity breaking that take place in
terms of Dirac fermion models in 2 + 1 space–time dimensions in
the presence of gravitational sources [13,14].
In view of the discussion above, it is natural to consider the
possibility of finding a bosonization recipe connecting fermions
with gravitons analogous to those arising for fermionic models
coupled to gauge fields in flat space. However, contrary to the
standard Fermi–Bose duality, in this case the Dirac fermions will
be coupled not to background gravitational sources but to a dy-
* Corresponding author.
1 Also at CICBA.http://dx.doi.org/10.1016/j.physletb.2014.01.057 
0370-2693 © 2014 The Authors. Published by Elsevier B.V. Open access under CC BY license.namical gravity theory, topologically massive gravity (TMG) theory.
Dualities involving gravity theories and fermion models have re-
cently been discussed within the gauge/gravity correspondence but
in that case the gravity theory (a Vasiliev high-spin theory [15])
is defined in an AdS4 bulk while the fermion dual — consisting
of k fermions coupled to a U (k)N Chern–Simons term — lives in
the three-dimensional boundary of AdS4 [16]. In fact, since also
a bosonic theory coupled to a U (N)k Chern–Simons gauge theory
defined on the same boundary can be seen to be dual to Vasiliev
theory in the AdS4 bulk, it has been conjectured that these bosonic
and fermionic theories are dual to each other in the large N limit.
The aim of this work is to establish a duality between fermions
coupled to topologically massive gravity (TMG) in d = 3 space–
time dimensions and a purely gravity theory, which also will turn
out to be a TMG theory but with different parameters.
1. The action
Our model will be defined on a three-dimensional space–time
manifold M3 with local coordinates xμ (μ = 0,1,2) and a metric
of Minkowski signature. Vierbeins (or, rather, dreibeins in 2 + 1
dimensions) are denoted as eaμ where “a” is the frame index (a =
0,1,2). The spin connection ωμ ,
ωμ = −1 
4 
ωabμγ
aγ b = − i
4 
ωabμ²
abcγ c (1)
is an SO(2,1)-valued 1-form on the manifold M3 satisfying the
standard relation with the dreibeins
ωabμ = eaνeνb;μ (2) Funded by SCOAP3.
E. Fradkin et al. / Physics Letters B 730 (2014) 284–287 285where the semicolon refers to covariant differentiation using the
Christoffel symbol.
We shall consider massive fermions coupled to topologically
massive gravity (TMG) [5] with dynamics governed by the action S
S[ψ¯,ψ, e; s]
=
Z
d3x (det e)ψ¯
¡
iγ μ(∂μ + ωμ + sμ) +m
¢
ψ + STMG (3)
where the topological massive gravity action STMG reads
STMG = 1
64πκ2m
SCS
£
ω[e]¤ + 1
κ2
SEH
£
e,ω[e]¤ (4)
Here det e is the determinant of the dreibein fields, and sμ is an
external source
sμ = −1
4
sabμγ
aγ b = − i
4
sabμ²
abcγ c (5)
which transforms covariantly under local Lorentz transformations
so that ω + s transforms as a connection.
The Chern–Simons term SCS[ω] reads
SCS[ω] =
Z
d3x²μνα
µ
ωμab∂νωαba + 23ωμabωνbcωαca
¶
(6)
and the Einstein–Hilbert action SEH[e,ω[e]] is written as
SEH
£
e,ω[e]¤ =
Z
d3x²μναeaμ
¡
∂νω
a
α − ∂αωaν + ²abcωbνωcα
¢
(7)
where
ωaα =
1
2
²abcωαbc (8)
Note that in d = 3 space–time dimensions the Newton constant
κ−2 has dimensions of mass, [κ2] =m−1, in fundamental units in
which the spin connection has dimension one, [ωμ] = 1. The mass
parameter m in SCS is required by dimensional consistency and can
be replaced by a dimensionless effective coupling constant κ2m
which, alternatively, can be regarded as the inverse of the level k
of the Chern–Simons theory, κ2m = 1/k. In topologically massive
gravity there is a single propagating mode with mass m and spin 2
[5].
The generating functional Z [s] of the source fields sμ is defined
by
Z [s] =
Z
DμψDμG exp
¡
i S[ψ¯,ψ, e; s]¢ (9)
where Dμψ is the appropriate fermionic path-integral measure
defined in terms of Fujikawa variables [2]:
Dμψ =D
¡
(det e)1/2ψ¯
¢D¡(det e)1/2ψ¢ (10)
Concerning the measure of the gravitational sector DμG , one
could also follow Fujikawa prescription introducing Faddeev–Popov
ghosts fields and auxiliary fields arising in the anti-ghost superfield
introduced in the BRST supersymmetry approach associated with
general coordinate transformations [17]. In this respect one should
note that in Ref. [18] it has been shown that TMG has no uni-
tarity and ghost problems and is power-counting renormalizable.
Moreover, arguments in [19] based on the existence of a functional
integration measure make appear TMG to be finite as a quantum
theory.2. Bosonization
As it is well-known the d = 3 Dirac operator determinant aris-
ing from the fermionic path-integral in Eq. (9) can be written as
the product of two factors according to its behavior under parity
transformation [10,12]. The odd-parity contribution can be com-
puted exactly leading to a Chern–Simons effective action
det
¡
iγ μ(∂μ + ωμ + sμ) +m
¢
odd = exp
µ
± i
64π
SCS[ω + s]
¶
(11)
The parity-even contribution can be computed within a ∂/m ap-
proximation. Being the lowest parity-even term subleading to the
parity-odd term, it will be disregarded in what follows.
Using this result we can write Eq. (9) as a purely bosonic gen-
erating functional of the form
Z [s] =
Z
DμG exp
µ
± i
64π
SCS[ω + s]
¶
× exp
µ
i
64πκ2m
SCS[ω] + i
κ2
SEH[ω, e]
¶
(12)
One can calculate from the generating functional of Eq. (12) the
v.e.v. of the current Jμab
­
Jμab
® = − 1
Z
δZ [S]
δsμab
¯¯¯
¯
s=0
(13)
Jμ ab = 1
4
ψ¯γ μ
£
γ a, γ b
¤
ψ = 1
4
eμc ψ¯γ
c£γ a, γ b¤ψ (14)
Using Eq.(12) the fermionic current v.e.v. takes form
­
Jμab
®
= ∓1
64π
1
Z
Z
DμG exp
µ
i
64πκ2m
SCS[ω] + i
κ2
SEH
£
ω[e], e¤
¶
×
µ
δSCS[ω + s]
δsμab
¶¯¯¯
¯
s=0
= ∓ 1
256π
²μνβ
­¡
Rνβab[ω˜]
¢ba®
bos (15)
where
­
G[ω]®Sbos ≡
Z
DμGG[ω]exp
¡
i Sbos[ω]
¢
(16)
Here the curvature tensor Rμνab with two frame and two coordi-
nate indices is given by
Rμνab[ω] = ∂μωνab − ∂νωμab + ωμacωνcb − ωνacωνcb (17)
and the bosonized action Sbos reads
Sbos = 164πκ2m b SCS
£
ω[e]¤ + 1
κ2
SEH
£
e,ω[e]¤ (18)
with
m b = m1 ± κ2m (19)
This result implies that the Chern–Simons level of the bosonized
theory is shifted from k = 1/(κ2m) to kb = 1/(κ2m b) = k ± 1. No-
tice that if the level of the TMG is k = 1, it is possible then to have
kb = 0 (although formally the mass parameter m b diverges.) In this
case the parity anomaly cancels out and the effective action of the
bosonized theory reduces to a pure Einstein–Hilbert action in 2+1
dimensions.
286 E. Fradkin et al. / Physics Letters B 730 (2014) 284–287We can summarize these results establishing the following
bosonization recipes for action and current
S[ψ¯,ψ, e; s] 7→ Sbos
£
e,ω[e]¤ (20)
ψ¯γ μ
£
γ a, γ b
¤
ψ 7→ ∓ 1
256π
²μνβ Rνβab[ω] (21)
We conclude that the bosonized action Sbos corresponds to topo-
logically massive gravity action with a modified graviton mass
given by m b instead of the original m so that the original graviton
mass picked up a contribution from fermion bosonization increas-
ing or decreasing according to the choice of sign when regularizing
determinant (11).2
3. The energy–momentum “tensor”
One can complete the bosonization recipe (20)–(21) by relating
the energy–momentum tensor associated with the original matter-
TMG model to the dual action (18). To define the topologically
massive gravity energy–momentum “tensor”3 we shall follow [22,
5] and consider that the path integral measure DμG is restricted
to asymptotically flat space–times. We then decompose space–time
metric in the form
gμν = ημν + hμν (22)
where ημν is the flat-space–time metric with Euclidean signature
and hμν a deviation which is not necessarily small everywhere but
vanishes at infinity.
In the case of topologically massive gravity coupled to matter
Einstein equations can be written as
Gαβ + 1m Cαβ = −
κ2
2
Tαβ (23)
where Gαβ and Cαβ are the Einstein and Cotton–York tensors
respectively and Tαβ is the matter (i.e. the Dirac field) energy–
momentum tensor,
Gαβ = Rαβ − 1
2
gαβ Rγ γ
Cαβ = 1
2
√
g
¡
²αρσ Dρ R
β
σ + ²βρσ Dρ Rασ
¢
Tαβ = 1
2
(ψ¯γβ∇αψ − ∇αψγβψ) (24)
Here Rαβ is the Ricci curvature tensor (Rαβ = gγ δRαγ βδ), R the
scalar curvature (R = gαβ Rαβ ) and Tαβ the energy–momentum
tensor associated with matter. We can write the previous equa-
tion as
Gαβ + 1m Cαβ +
µ
G(1)αβ +
1
m
C (1)αβ
¶
−
µ
G(1)αβ +
1
m
C (1)αβ
¶
= −κ
2
2
Tαβ
where G(1) , C (1) are the part of Einstein and Cotton–York tensors
which are linear in hαβ . Passing the first and third terms in the
l.h.s. to the r.h.s. one can rewrite the previous equation in the form
2 The double sign ambiguity is inherent to any regularization in odd-dimensional
spaces as can be seen by using a ζ -function regularization where the sign depends
on the choice of upper or lower half plane to close the curve where one integrates
the Seeley coefficients [20]. The choice of a particular sign corresponds to a sign
of the parity anomaly. In Pauli–Villars regularization this sign is determined by the
sign of the mass of the Weyl fermion [6].
3 Quotes indicate that gauge systems with spin greater than one do not possess
gauge-invariant stress tensors, but only integrated Poincaré generators [21].G(1)αβ +
1
m
C (1)αβ
= −
µ
κ2
2
Tαβ +
µ
Gαβ + 1m Cαβ
¶
−
µ
G(1)αβ −
1
m
C (1)αβ
¶¶
(25)
Then the quantity
tαβ = 2
κ2
µ
Gαβ − G(1)αβ +
1
m
¡
Cαβ − C (1)αβ
¢¶
(26)
can be interpreted as the energy–momentum “tensor” of the grav-
itational field itself and then refer to
ταβ = Tαβ + tαβ (27)
as the total energy–momentum “tensor” of matter and gravitation.
In turn, the exact equations of motion for TMG coupled to grav-
ity can then be written in the form
G(1)αβ +
1
m
C (1)αβ = −
κ2
2
¡
Tαβ + tmαβ
¢
(28)
with tmαβ the energy–momentum “tensor” of the gravitational field
for TMG with graviton mass m . In the same vein, the equations of
motion of the bosonized action (18) can be written as
G(1)αβ +
1
mb
C (1)αβ = −
κ2
2
¡
tm
b
αβ
¢
(29)
Using the definition (26) and the fact that the two actions are
equivalent, their equations of motion can be related through the
bosonization recipe for the matter energy–momentum tensor
κ2
2
Tαβ →
µ
1
m b
− 1
m
¶
Cαβ = ± κ
2
64π
Cαβ (30)
which is thus mapped onto the Cotton–York tensor of the TMG.
In summary, the complete set of bosonization recipes reads
S
£
ψ¯,ψ, e,ω[e]¤ 7→ Sbos£e,ω[e]¤ (31)
ψ¯γ μ
£
γ a, γ b
¤
ψ 7→ ± 1
256π
²μνβ Rνβab[ω] (32)
Tαβ = 1
2
(ψ¯γβ∇αψ − ∇αψγβψ) 7→ ± 1
32π
Cαβ (33)
Notice the important fact that these identifications do not depend
explicitly on the parameters of the theory. These results are the
gravitational analog of the results of Refs. [8,9] in terms of gauge
fields. As usual they should be interpreted as identities inside ex-
pectation values.
Several interesting results can be drawn form the identifications
of Eqs. (18), (19), (31), (32) and (33). Let us consider the case
in which κ → 0. In this limit the TMG of Eq. (4) trivializes and
the gravitational degrees of freedom reduce to diffeomorphisms of
2 + 1-dimensional Minkowski space–time. In this limit the theory
reduces to a theory of free massive Dirac fermions (averaged over
trivial diffeomorphisms which amount to an average over bound-
ary conditions). From the results of Eq. (18) and Eq. (19), we see
that, in this limit, the bosonized theory is classical. Thus, the ex-
pectation values and correlators of the stress-tensor of the Dirac
field are trivial in this regime (as it is expected since if the Dirac
mass is arbitrarily large there is no energy or momentum). We
note that in this limit, in which gravity acts as a probe field, in or-
der to obtain non-trivial stress energy currents it is necessary to
consider a gravity theory with torsion [13,14]. On the other hand,
in the opposite limit κ → ∞, the TMG sector of the action drops
out and the original theory is a theory of massive Weyl fermions
with strongly fluctuating gravitational degrees of freedom. In this
limit we see that the bosonized TMG theory now is a Chern–
Simons theory for the spin connection with CS level kb = 1.
E. Fradkin et al. / Physics Letters B 730 (2014) 284–287 287Acknowledgements
We would like to thank G. Giribet for helpful comments. This
work is supported in part by the National Science Foundation
through the grant DMR-1064319 at the University of Illinois,
USA (E.F.) and by CONICET, ANPCYT, CIC, and UNLP, Argentina
(F.A.S.).
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Bosonization of fermions coupled to topologically massive gravity

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Bosonization of fermions coupled to topologically massive gravity

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