Note on the gravitational anomaly∗
Farhang Loran†
Department of Physics, Isfahan University of Technology,
Isfahan, 84156-83111, Iran
Abstract
I briefly discuss the gravitational anomaly in two dimensions.
∗
†
Based on a talk given at http://www.math-phys.ir/.
e-mail: loran@cc.iut.ac.ir
1
Contents
1 Introduction
2
2 Flat space
2
3 Coupling the CFT to gravity
3
4 Weyl anomlay
5
5 Diffeomorphism Invariance
5
6 Observing the anomaly
6
1
Introduction
In the context of AdS3 /CFT2 , gravitational anomalies are encountered in Topologically
Massive Gravity [1]. This anomaly implies that in the boundary CFT cL 6= cR [2]. In [3],
it is shown that in such models, the boundary stress-energy tensor can be reproduced by
integrating the conformal and gravitational anomalies.
Historically, Alvarez-Gaumé and Witten have shown that one-loop graphs describing the
propagation of chiral fermions in an external gravitational field in two-dimensions contain
gravitational anomalies [4]. Coupling the two-dimensional conformal matter with cL 6= cR
to two-dimensional gravity is studied in [5].
In this note, following [4] and [6], we study the gravitational anomalies that appear when
two-dimensional conformal matter with cL 6= cR is coupled to two-dimensional gravity. The
interested reader may consult [2] and [3] for further study.
2
Flat space
A key ingredient of CFT is the energy momentum tensor, which, classically is defined by,1
2 δS
Tµν = − √
.
g δ g µν
For a CFT on a cylinder,
(2.1)
hT i = 0,
­ ®
T̄ = 0
(2.2)
T = Tzz ,
T̄ = Tz̄z̄ ,
(2.3)
in which,
1
Note that our convention differs from that of [8] by a factor of −2π.
2
and,
hT (z) T (w)i =
1
c/2
,
4π 2 (z − w)4
­
®
1
c̄/2
T̄ (z̄) T̄ (w̄) = 2
,
4π (z̄ − w̄)4
(2.4)
Tµµ
= 0.
4
(2.5)
where c = cL and c̄ = cR . Furthermore,
Tzz̄ = Tz̄z =
These results can be found in [7, 8].
3
Coupling the CFT to gravity
Assume that,
gµν = δµν + hµν ,
where,
and hµν
(3.1)
1
δzz̄ = δz̄z = , δzz = δz̄z̄ = 0,
2
is a small fluctuation. Using Eq.(2.1), one obtains,
Z
δ hTµν (z, z̄)i = δ
D[φ] Tµν e−S
Z
√
1
=
d2 w δ hTµν (z, z̄) Tαβ (w, w̄)i hαβ (w, w̄) + O(h2 ).
2
(3.2)
(3.3)
Here, following [8], d2 z = 2dσ 1 dσ 2 , where z = σ 1 + iσ 2 and z̄ = σ 1 − iσ 2 . Using Eqs.(2.2),
(2.4) and,
1
1
(3.4)
∂¯ = ∂ = 2πδ 2 (z),
z
z̄
it can be verified that,
1 π c 3 zz
∂¯ T = − 2
∂ h ,
4π 24
∂ T̄ = −
1 π c̄ ¯3 z̄z̄
∂ h .
4π 2 24
(3.5)
where, e.g. T = hTzz i on the curved background (3.1). Eq.(2.5) implies that Tzz̄ = Tz̄z = 0,
i.e. the non-zero components of the energy momentum tensor on the curved background
(3.1), are T and T̄ . Consequently, Eq.(3.5) means that,
∇i T ij 6= 0,
(3.6)
where, ∇i denotes the covariant derivative with respect to the Levi-Civita connection corresponding to the metric (3.1), and we have noted that to lowest order in the gravitational
field, the covariant derivative can be replaced with ordinary ones.
Eq.(3.6) implies that the energy-momentum tensor is not covariantly conserved. This
phenomena is known as gravitational anomaly. To understand why it is called an anomaly,
note that an anomaly can be understood as an ultraviolet effect, so it should be a local
3
function. On the other hand the anomaly can not be removed by adding a local term [6]. In
our case, Eq.(3.5) implies that
1 π c ∂ 3 zz
1 π c̄ ∂¯3 z̄z̄
T =− 2
h .
(3.7)
h
,
T̄
=
−
4π 24 ∂¯
4π 2 24 ∂
Thus, (1) ∇i T ij is a local term, and (2) it can not be eliminated by adding a local term to
Eq.(3.7).
Eq.(3.7) can be given in a more suggestive way. Recall that the scalar curvature of the
background (3.1) is given by,
and consequently,
R = ∂ 2 hzz + ∂¯2 hz̄z̄ − 2∂ ∂¯ hzz̄ + O(h2 ),
(3.8)
∂ 3 zz
∂
¯ z̄z̄ − 2∂ 2 hzz̄ + O(h2 ).
R
=
h + ∂ ∂h
∂¯
∂¯
(3.9)
Thus,
1 πc
4π 2 24
1 π c̄
T̄ = − 2
4π 24
T = −
∂
R + ∂X ,
∂¯
∂¯
¯ .
R + ∂X
∂
(3.10)
(3.11)
in which,
¢
1 π c ¡ ¯ z̄z̄
z z̄
∂h
−
2∂h
,
(3.12)
4π 2 24
¢
1 π c̄ ¡ zz
¯ zz̄ ,
X̄ =
∂h
−
2
∂h
(3.13)
4π 2 24
are local terms. These terms can be canceled by means of local counter-terms. To see this,
note that by an infinitesimal diffeomorphism,
X =
δhµν = ∇µ ξν + ∇ν ξµ ,
(3.14)
R → R and,
c 2
c̄ ¯2
∂ ξz̄ , X̄ → X̄ 0 = X̄ −
∂ ξz .
12π
12π
One can solve X 0 = X̄ 0 = 0 for ξz and ξz̄ . For example,
Z
6
z−w
X(w, w̄)d2 w.
ξz̄ =
c
z̄ − w̄
X → X0 = X −
Such diffeomorphism renders,
R = 4∂¯2 h0zz = ∂ 2 h0z̄z̄ .
(3.15)
(3.16)
(3.17)
Consequently,
¯ = − 1 π c ∂R,
∂T
4π 2 24
Tzz̄ = 0,
∂ T̄ = −
1 π c̄ ¯
∂R,
4π 2 24
Tz̄z = 0.
Exercise: Verify that the corresponding counter-terms are given by δξ S.
Hint: Use Eqs.(3.3) and (3.14).
4
(3.18)
(3.19)
4
Weyl anomlay
Consider adding the following non-covariant counter-term [9],
Z
√
√
ct
S =β
gR ln g,
(4.1)
which gives,
ct
Tµν
= β R gµν .
(4.2)
Thus, for the background (3.1),
ct
= Tz̄ctz̄ = 0,
Tzz
By choosing β =
1 π
(c
4π 2 24
ct
=
Tzctz̄ = Tz̄z
β
R.
2
(4.3)
+ c̄) one obtains the following standard result [3],
T̃µµ
1 π
= 2 (c + c̄) R,
4
4π 48
1 π
= − 2 (c − c̄) ²µν ∂µ R.
4π 24
T̃zz̄ = T̃z̄z =
(4.4)
∂µ T̃ µν
(4.5)
ct
where, T̃µν = Tµν + Tµν
. Eq.(4.4) gives the Weyl anomaly. This result is important: if c = c̄,
then the gravitational anomaly (3.7) can be eliminated at the cost of Weyl invariance [6].
5
Diffeomorphism Invariance
Assume that, after adding a suitable local counter-term to the action one obtains Eq.(3.18)
and,
∂i T ij = 0.
(5.1)
This implies that,
∂¯ T + ∂ Tz̄ z = 0,
which, gives,
Tz̄ z =
1 πc
R,
4π 2 24
∂ T̄ + ∂¯ Tzz̄ = 0,
(5.2)
1 π c̄
R.
4π 2 24
(5.3)
Tzz̄ =
Consequently,
1 π
(c − c̄) R.
(5.4)
4π 2 24
Thus, if c 6= c̄, diffeomorphism invariance results in anomalous Lorentz symmetry. The
exchange of gravitational anomaly for local Lorentz anomaly is discussed in [2].
Tz̄ z − Tzz̄ =
5
6
Observing the anomaly
One can show that the gravitational anomaly can be determined in term of the Hawking
effect of a 2d Schwarzschild black hole,
Ttt =
π
(c + c̄) TH2 ,
12
Txt =
π
(c − c̄) TH2 .
12
(6.1)
See [3] for details.
References
[1] S. Deser, R. Jackiw and S. Templeton, Annals Phys. 140, 372 (1982) [Erratum-ibid.
185, 406 (1988)] [Annals Phys. 185, 406 (1988)] [Annals Phys. 281, 409 (2000)].
[2] P. Kraus and F. Larsen, JHEP 0601, 022 (2006) [hep-th/0508218].
[3] S. N. Solodukhin, Phys. Rev. D 74, 024015 (2006) [hep-th/0509148].
[4] L. Alvarez-Gaume and E. Witten, Nucl. Phys. B 234, 269 (1984).
[5] R. C. Myers and V. Periwal, Nucl. Phys. B 397, 239 (1993) [hep-th/9207117].
[6] M. B. Green, J. H. Schwarz and E. Witten, Cambridge, Uk: Univ. Pr. (1987) 469 P.
(Cambridge Monographs On Mathematical Physics)
[7] P. H. Ginsparg, hep-th/9108028.
[8] J. Polchinski, Cambridge, UK: Univ. Pr. (1998) 402 p.
[9] D. R. Karakhanian, R. P. Manvelyan and R. L. Mkrtchian, Phys. Lett. B 329, 185
(1994) [hep-th/9401031].
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