Chern-Simons formulation of three-dimensional gravity with torsion and nonmetricity

0.26 MB·English

by  Sergio L. Cacciatori

IFUM-839-FT
PreprinttypesetinJHEPstyle-HYPERVERSION
hep-th/0507200
Chern-Simons formulation of
three-dimensional gravity with torsion and
nonmetricity
7
0
0
2
Sergio L. Cacciatori

n
a Dipartimento di Matematica dell’Universita` di Milano
J
Via Saldini 50, I-20133 Milano and
9
2 INFN, Sezione di Milano, Via Celoria 16, I-20133 Milano.
E-mail: [email protected]
2
v
Marco M. Caldarelli, Alex Giacomini, Dietmar Klemm and Diego S. Mansi
0
0 Dipartimento di Fisica dell’Universita` di Milano
2
7 Via Celoria 16, I-20133 Milano and
0 INFN, Sezione di Milano, Via Celoria 16, I-20133 Milano.
5
0 E-mail: [email protected], [email protected],
/ [email protected], [email protected]
h
t
-
p
Abstract: We consider various models of three-dimensional gravity with torsion or
e
h
nonmetricity (metric affine gravity), and show that they can be written as Chern-
:
v Simons theories with suitable gauge groups. Using the groups ISO(2,1), SL(2,C) or
i
X SL(2,R) SL(2,R), and the fact that they admit two independent coupling constants,
r ×
a we obtain the Mielke-Baekler model for zero, positive or negative effective cosmological
constant respectively. Choosing SO(3,2) as gauge group, one gets a generalization of
conformal gravity that has zero torsion and only the trace part of the nonmetricity.
This characterizes a Weyl structure. Finally, we present a new topological model of
metric affine gravity in three dimensions arising froman SL(4,R) Chern-Simons theory.
Keywords: Chern-Simons Theories, Models of Quantum Gravity, Differential
Geometry.
Dedicated to the memory of Antonio Meucci
Contents
1. Introduction 1
2. Metric affine gravity 4
3. The Mielke-Baekler model as a Chern-Simons theory 6
4. Weyl structures from Chern-Simons theory 10
5. Metric affine gravity from SL(4,R) Chern-Simons theory 15
5.1 Simple solutions 18
5.2 Partial gauge fixing 20
6. Conclusions 21
A. Decomposition of curvature in metric affine gravity 23
1. Introduction
General relativity in four spacetime dimensions is a notoriously difficult theory, already
at the classical and in particular at the quantum level. This is one of the main reasons
why people are interested in simpler models that nevertheless retain almost all of the
essential features of four-dimensional general relativity. One such model is pure gravity
in2+1dimensions, withorwithoutcosmologicalconstant. Thistheoryhasbeenstudied
extensively in the past, in particular by Deser, Jackiw and ‘t Hooft [1,2]. The most
famous example where we learned something on general relativity by considering a
simpler toy model is perhaps the BTZ black hole [3], whose study revealed a lot on the
quantum structure and the statistical mechanics of black holes (for a review cf. [4]).
Major progress in 2+1 dimensional gravity came when Achu´carro and Townsend
[5] and Witten [6] showed that these systems can be written as Chern-Simons (CS)
theories, with gauge group ISO(2,1), SL(2,C) or SL(2,R) SL(2,R) for zero, positive
×
or negative cosmological constant respectively. In trying to write down a CS action for
– 1 –
the Poincar´e group, one encounters the problem that ISO(2,1) is not semisimple, and
therefore the Killing form is degenerate. As noted by Witten [6], the Poincar´e algebra
admits nevertheless a nondegenerate, Ad-invariant bilinear form given by
J ,P = η , J ,J = λη , P ,P = 0, (1.1)
a b ab a b ab a b
h i h i h i
where J and P denote the Lorentz and translation generators, and λ is an arbi-
a a
trary real constant. Mathematically, the existence of an Ad-invariant, nondegenerate
quadratic form on the Poincar´e algebra follows from the fact that iso(2,1) is the double
extension of a reductive Lie algebra (in this case the trivial algebra): Let be a reduc-
A
tive Lie algebra, i. e. , a direct sum of a semisimple and an abelian algebra. admits
A
an invariant nondegenerate bilinear form Ω , whose restriction to the semisimple part
ij
is simply given by the Killing form, and the restriction to the abelian subalgebra is
proportional to the identity. The generators τ of satisfy
i
A
[τ ,τ ] = f kτ .
i j ij k
∗
The double extension of is obtained by adding the new generators H and H such
A a ¯b
that
[τ ,τ ] = f kτ +h a¯H∗,
i j ij k ij a¯
[H ,τ ] = h jτ ,
a i ai j
[H ,H ] = g cH , (1.2)
a b ab c
[H ,H∗] = g c¯H∗,
a ¯b a¯b c¯
∗ ∗ ∗
[τ ,H ] = [H ,H ] = 0,
i a¯ a¯ ¯b
where h jΩ = h a¯δ . If furthermore g c = g c¯, there exists an Ad-invariant,
ai jk ik a¯a ab a¯b
nondegenerate quadratic form on the double extension of , given by [7]
A
Ω 0 0
ij
ΩIJ =  0 λab δa¯b , (1.3)
0 δ 0
 a¯b 
where I = i,a,a¯, andλ denotes anyinvariant quadraticformonthealgebragenerated
ab
by the H . If the algebra is trivial (no generators τ ), (1.2) has exactly the structure
a i
A
of the Poincar´e algebra
[J ,J ] = ǫ cJ , [J ,P ] = ǫ cP , [P ,P ] = 0, (1.4)
a b ab c a b ab c a b
∗
if we identify the generators H with J and H with P . The invariant quadratic form
a a ¯b b
(1.3) reduces then to (1.1). Retaining a nonvanishing λ in a CS formulation of three-
dimensional gravity leads to the inclusion of a gravitational Chern-Simons action (i. e. ,
– 2 –
a CS term for the spin connection) [6]. This does not change the classical equations of
motion, but leads to modifications at the quantum level [6].
One can now try to depart from pure gravity, rendering thus the model less trivial,
while maintaining at the same time its integrability. A possible way to introduce ad-
ditional structure is to permit nonvanishing torsion and/or nonmetricity (metric affine
gravity) [8]. We would like to do this in such a way that the resulting model can still be
written as a CS theory for some gauge group. There are several reasons that motivate
theintroductionoftorsionornonmetricity. Letusmentionhereonlyafewofthem. For
a more detailed account we refer to [8]. First of all, nonmetricity is a measure for the
violation of local Lorentz invariance [8], which has become fashionable during the last
years. Second, the geometrical concepts of nonmetricity and torsion have applications
in the theory of defects in crystals, where they are interpreted as densities of point de-
fects and line defects (dislocations) respectively, cf. [9] and references therein. Finally,
nonmetric connections or connections with torsion are interesting from a mathematical
point of view. For example, a torsionless connection that has only the trace part of
the nonmetricity characterizes a so-called Weyl structure. If, moreover, the symmet-
ric part of the Ricci tensor is proportional to the metric, one has an Einstein-Weyl
structure (cf. e. g. [10]). Einstein-Weyl manifolds represent the analogue of Einstein
spaces in Weyl geometry, and are less trivial than the latter, which have necessarily
constant curvature in three dimensions. Einstein-Weyl structures are interesting also
due to their relationship to certain integrable systems, like the SU( ) Toda [11] or the
∞
dispersionless Kadomtsev-Petviashvili equation [10].
In this paper, we consider various models of three-dimensional metric-affine gravity and
show that they can be written as CS theories. This is accomplished either by using
gauge groups larger than ISO(2,1), SL(2,C) or SL(2,R) SL(2,R), or by using the
×
fact that these groups admit two independent coupling constants, as was explained
above for the case of the Poincar´e group.
The remainder of our paper is organized as follows: In the next section, we briefly
summarize the basic notions of metric affine gravity. In section 3, we show that the
Mielke-Baekler model, which is characterized by nonvanishing torsion and zero non-
metricity, can be written as a CS theory for arbitrary values of the effective cosmolog-
ical constant. In section 4, a CS action for the conformal group SO(3,2) is considered,
and it is shown that this leads to a generalization of conformal gravity with a Weyl
connection. Finally, insection 5, weproposeatopologicalmodelofmetricaffinegravity
based on an SL(4,R) CS theory and discuss some of its solutions. In the last section
we summarize the results and draw some conclusions.
– 3 –
2. Metric affine gravity
In order to render this paper self-contained, we summarize briefly the basic notions of
metric affine gravity. For a detailed review see [8].
The standard geometric setup of Einstein’s general relativity is a differential ma-
nifold , of dimension D, endowed with a metric g and a Levi-Civita connection ˜,
M ∇
which is uniquely determined by the requirements of metricity (˜g = 0) and vanishing
∇
torsion. This structure is known as a semi-Riemannian space ( ,g).
M
One can now consider more complicated non-Riemannian geometries, where a new
generic connection is introduced on T which is, in general, independent of the
∇ M
metric. In this way one defines a new mathematical structure called a metric-affine
space ( ,g, ).
M ∇
One can measure the deviation from the standard geometric setup by computing
the difference ( ˜)v between the action of the two connections on a vector field v
∇−∇
defined on T . To be more specific one can choose a chart, so that the action of the
M
connection is described by its coefficients,1
vν = ∂ vν +Γν vλ, (2.1)
µ µ µλ
∇
and the deviation can be written as
( ˜ )vν = Nν vλ. (2.2)
µ µ µλ
∇ −∇
The tensor
Nλ = Γλ Γ˜λ (2.3)
µν µν µν
−
is called distortion and measures the deviation of from the Levi-Civita connection.
∇
This object can be decomposed in different parts, depending essentially on two quan-
tities: the torsion and the nonmetricity.
The torsion tensor Ta is defined by the first Cartan structure equation
Ta dea +ωa eb, (2.4)
b
≡ ∧
where ωa is the spin connection acting on (flat) tangent space indices a,b,..., and
b
ea denotes the vielbein satisfying ea eb gµν = ηab, with ηab the flat Minkowski metric.
µ ν
A priori, ωa is independent of the connection coefficients Γλ . Both objects become
b µν
dependent of each other by the tetrad postulate
ea = 0, (2.5)
µ ν
∇
1The connection coefficients for the Levi-Civita connection are called Christoffel symbols and are
denoted by Γ˜ν .
µλ
– 4 –
implying
ωa = ea Γλ e ρ e λ∂ ea , (2.6)
µb λ µρ b − b µ λ
so that the spin connection ωa is the gauge transform of Γλ with transformation
µb µρ
matrix ea .
λ
For nonvanishing torsion, the connection coefficients are no more symmetric in
their lower indices, as can be seen from
0 = 2 ea = ∂ ea ∂ ea +ωa eb ωa eb 2Γλ ea , (2.7)
∇[µ ν] µ ν − ν µ µb ν − νb µ − [µν] λ
(cid:0) (cid:1)
which yields
Tλ e λTa = 2Γλ , (2.8)
µν a µν [µν]
≡
or, equivalently,
Tλ = 2Nλ , (2.9)
µν [µν]
since the Levi-Civita connection has zero torsion, Γ˜λ = 0.
[µν]
The nonmetricity Q is a tensor which measures the failure of the metric to be
covariantly constant,
Q g . (2.10)
λµν µ νλ
≡ −∇
Using ˜g = 0 and the definition (2.3), one gets
∇
Q = N +N , (2.11)
λµν λµν νµλ
where N = g Nσ . For nonzero nonmetricity, the spin connection ωab is no more
λµν λσ µν
antisymmetric in a,b: By computing the covariant derivative ηab one obtains Qab =
∇µ µ
2ω(ab), with Qab eaρebλQ . This means that the spin connection takes values in
µ µ ≡ ρµλ
gl(D,R) instead of the Lorentz algebra so(D 1,1).
−
Notice that in presence of nonmetricity, the scalar product of two vectors u, v can
change when u, v are parallel transported along a curve. Let t be the tangent vector
of an infinitesimal curve c. The variation of the scalar product is then given by
δg(u,v) = (g uµvν) = Q tλuµvν. (2.12)
t µν µλν
∇ −
Physically, this states that if we enlarge the Lorentz group, the interval is not any
longer an invariant and in fact, for generic nonmetricity, the very concept of light cone
is lost.
The two tensors T and Q uniquely determine the distortion and, as a result, the
connection. This can also be seen by counting the degrees of freedom: the distortion
is a generic tensor with three indices, so it has D3 independent components. The
torsion and the non-metricity, due to their symmetry properties, have respectively
– 5 –
D2(D 1)/2 and D2(D+1)/2 independent components; their sum gives precisely the
−
expected number of degrees offreedom. To obtainthedistortioninterms of torsionand
nonmetricity one has to solve the equations (2.9) and (2.11). Considering all possible
permutations one obtains
1 1
N = (T T T )+ (Q +Q Q ) , (2.13)
λµν νλµ λνµ µνλ λµν λνµ µλν
2 − − 2 −
which is the expected decomposition of the distortion. The Levi-Civita connection is
obtained setting Ta = 0 and Q = 0. The combination
ab
1
K = (T T T ) , (2.14)
νλµ νλµ λνµ µνλ
2 − −
which is antisymmetric in the first two indices, is also called contorsion.
Note that in metric affine gravity, the local symmetry group is the affine group
A(D,R) = GL(D,R) ⋉RD insteadofthePoincar´e groupISO(D 1,1). Theassociated
∼
−
gauge fields are ωab and ea. In what follows, we shall specialize to the case D = 3.
3. The Mielke-Baekler model as a Chern-Simons theory
Let us first consider thecase ofRiemann-Cartan spacetimes, characterized by vanishing
nonmetricity, but nonzero torsion. A simple three-dimensional model that yields non-
vanishing torsion was proposed by Mielke and Baekler (MB) [12] and further analyzed
by Baekler, Mielke and Hehl [13]. The action reads [12]
I = aI +ΛI +α I +α I , (3.1)
1 2 3 3 4 4
where a, Λ, α and α are constants,
3 4
I = 2 e Ra,
1 a
Z ∧
1
I = ǫ ea eb ec,
2 abc
−3 Z ∧ ∧
1
I = ω dωa + ǫ ωa ωb ωc,
3 a abc
Z ∧ 3 ∧ ∧
I = e Ta,
4 a
Z ∧
and
1
Ra = dωa + ǫa ωb ωc,
bc
2 ∧
Ta = dea +ǫa ωb ec,
bc
∧
– 6 –
denote the curvature and torsion two-forms respectively. ωa is defined by ωa = 1ǫabcω
2 bc
with ǫ = 1. I yields the Einstein-Hilbert action, I a cosmological constant, I is a
012 1 2 3
Chern-Simons term for the connection, and I represents a translational Chern-Simons
4
term. Note that, in order to obtain the topologically massive gravity of Deser, Jackiw
and Templeton (DJT) [14] from (3.1), one has to add a Lagrange multiplier term that
ensures vanishing torsion. The field equations following from (3.1) take the form
2aRa Λǫa eb ec +2α Ta = 0,
bc 4
− ∧
2aTa +2α Ra +α ǫa eb ec = 0.
3 4 bc
∧
In what follows, we assume α α a2 = 02. Then the equations of motion can be
3 4
− 6
rewritten as
2Ta = Aǫa eb ec, 2Ra = Bǫa eb ec, (3.2)
bc bc
∧ ∧
where
α Λ+α a aΛ+α2
A = 3 4 , B = 4 .
α α a2 −α α a2
3 4 3 4
− −
Thus, the field configurations are characterized by constant curvature and constant
torsion. The curvature Ra of a Riemann-Cartan spacetime can be expressed in terms
of its Riemannian part R˜a and the contorsion one-form Ka by
1
Ra = R˜a dKa ǫa ωb Kc ǫa Kb Kc, (3.3)
bc bc
− − ∧ − 2 ∧
where Ka = 1ǫa ebβecγK , and K denotes the contorsion tensor given by (2.14).
µ 2 bc βγµ βγµ
Using the equations of motion (3.2) in (3.3), one gets for the Riemannian part
2R˜a = Λ ǫa eb ec, (3.4)
eff bc
∧
with the effective cosmological constant
A2
Λ = B .
eff
− 4
This means that the metric is given by the (anti-)de Sitter or Minkowski solution,
depending on whether Λ is negative, positive or zero. It is interesting to note that
eff
Λ can be nonvanishing even if the bare cosmological constant Λ is zero [13]. In this
eff
simple model, dark energy (i. e. , Λ ) would then be generated by the translational
eff
Chern-Simons term I .
4
In [15] it was shown that for Λ < 0, the Mielke-Baekler model (3.1) can be
eff
written as a sum of two SL(2,R) Chern-Simons theories. We will now shew that this
2For α3α4 a2 =0 the theory becomes singular [13].
−
– 7 –
can be generalized to the case of arbitrary effective cosmological constant. For positive
Λ , the action I becomes a sum of two SL(2,C) Chern-Simons theories with complex
eff
coupling constants, whereas for vanishing Λ , I can be written as CS theory for the
eff
Poincar´e group.
To start with, we briefly summarize the results of [15]. For Λ < 0 the geometry
eff
is locally AdS , which has the isometry group SO(2,2) = SL(2,R) SL(2,R), so if the
3 ∼
×
MB model is equivalent to a Chern-Simons theory, one expects a gauge group SO(2,2).
Indeed, if one defines the SL(2,R) connections
Aa = ωa +qea, A˜a = ωa +q˜ea,
then the SL(2,R) SL(2,R) Chern-Simons action3
×
t 2 t˜ 2
I = A dA+ A A A + A˜ dA˜+ A˜ A˜ A˜ (3.5)
CS
8π Z h ∧ 3 ∧ ∧ i 8π Z h ∧ 3 ∧ ∧ i
coincides (uptoboundary terms) withI in(3.1), if theparameters q,q˜andthecoupling
constants t,t˜are given by
A A
q = + Λ , q˜= Λ (3.6)
eff eff
−2 − −2 − −
p p
and
t 2a+α A t˜ 2a+α A
3 3
= 2α + , = 2α . (3.7)
3 3
2π √ Λ 2π − √ Λ
eff eff
− −
Weseethatq,q˜,andthustheconnectionsAa,A˜a arerealfornegativeΛ . Thecoupling
eff
constants t,t˜are also real, but in general different from each other due to the presence
of I .
3
For Λ > 0, q and q˜become complex, with q˜= q¯and thus A˜a = A¯a. As the con-
eff
nections are no more real, we must consider the complexification SL(2,C) of SL(2,R).
Then (3.5) becomes a sum of two SL(2,C) Chern-Simons actions, with complex cou-
pling constants t,t˜, where t˜= t¯. Again, (3.5) is equal (modulo boundary terms) to the
Mielke-Baekler action (3.1). This makes of course sense, since the isometry group of
three-dimensional de Sitter space is SO(3,1) = SL(2,C). The usual CS formulation of
∼
dS gravity [6] is recovered for α = α = 0.
3 3 4
Therealpart oft, i.e. , uptoprefactors, α , is subject toa topologicalquantization
3
condition coming fromthe maximal compact subgroup SU(2) ofSL(2,C) [16]. As t˜= t¯,
the action (3.5) leads to a unitary quantum field theory [16].
3In (3.5), τ ,τ =2Tr(τ τ )=η , and the SL(2,R) generators τ satisfy [τ ,τ ]=ǫ cτ .
a b a b ab a a b ab c
h i
– 8 –
Finally, we come to the case of vanishing Λ . The condition B A2/4 = 0 implies
eff
−
that Λ can be expressed in terms of the other parameters according to
2a3 3aα3α4 2(a2 α3α4)32
Λ = − ± − . (3.8)
α2
3
As we want Λ to be real, we assume a2 α α > 0. Let us consider the CS action
3 4
−
k 2
I = A dA+ A A A , (3.9)
CS
4π Z h ∧ 3 ∧ ∧ i
where A denotes an iso(2,1) valued connection, and the quadratic formon the Poincar´e
algebra is given by (1.1). According to what was said in the introduction, this (non-
degenerate) bilinear form is Ad-invariant for any value of the parameter λ. If we
decompose the connection as
A = eaP +(ωa +γea)J , (3.10)
a a
then the CS action (3.9) coincides, up to boundary terms, with (3.1) (where now Λ is
not independent, but determined by (3.8)), if the constants k, λ and γ are chosen as
k α a √a2 α α
= a2 α α , λ = 3 , γ = ± − 3 4 . (3.11)
3 4
4π ∓ − ∓√a2 α α α
p 3 4 3
−
In conclusion, we have shown that the Mielke-Baekler model can bewritten as a Chern-
Simons theory for any value of the effective cosmological constant Λ , whose sign
eff
determines the gauge group. This was accomplished by a nonstandard decomposition
of the CS connection in terms of the dreibein and the spin connection, and by using
the fact that the considered gauge groups admit two independent coupling constants.
As the CS connection is flat, and thus entirely determined by holonomies, there are no
propagating local degrees of freedom; hence there cannot be any gravitons in the MB
model, contrary to the claim in [13].
It would be interesting to study the asymptotic dynamics of the Mielke-Baekler
model in the case Λ < 0, where the spacetime is locally AdS . According to the
eff 3
AdS/CFT correspondence [17], (3.1) should then be equivalent to a two-dimensional
conformal field theory on the boundary of AdS , where the bulk fields ea and ωa are
3
sources for the CFT energy-momentum current and spin current respectively. It was
claimed in [15] that in general the putative CFT has two different central charges.
(Unlike the case α = α = 0, a = 1/16πG, Λ = 1/l2, where c = c = 3l/2G [18]).
3 4 L R
−
It would be interesting to compute these central charges explicitely, and to see whether
the entropy of the Riemann-Cartan black hole [19] (which represents a generalization
– 9 –