Mini Review Volume 2 Issue 4
The High Energy Accelerator Organization (KEK), Japan
Correspondence: Yoshimasa Kurihara, The High Energy Accelerator Organization (KEK), Oho 1?1, Tsukuba, Ibaraki 305?0801, Japan, Tel 8129 8796 088
Received: July 30, 2018 | Published: August 16, 2018
Citation: Kurihara Y. Gravitational theories with topological invariant. Phys Astron Int J. 2018;2(4):361-363 DOI: 10.15406/paij.2018.02.00110
This paper provides a brief review of the gravitational theories which have topological classes, such as the Chern–Simmons, BF Theory and Chern–Weil theory. For quantization of the Yang–Mills field theory, gauge symmetries and corresponding conserved charges play an essential role. These properties, symmetries and conserved charges, correspond to a structural group and topological classes of a principal bundle in differential geometry. Gravitational theories which have topological classes with respect to the structural group of the principal bundle are possible starting point of the still–unknown quantum gravitational theory.
Keywords: Chern–Weil field theory, gauge symmetries, conserved charges, topological classes, structural group, gravitational theories
A four–dimensional pseudo–Riemannian manifold(M,g)(M,g)withGL(n)GL(n)symmetry is introduced as a model of the universe. General relativity is constructed inMMas follows: The line element is defined using the metric tensor asds2:=gμ1μ2(x)dxμ1⊗dxμ2ds2:=gμ1μ2(x)dxμ1⊗dxμ2. (The Einstein convention for repeated indices is used throughout this report.) It is always possible to find a frame of reference whose affine connection asΓρμν(p)=0Γρμν(p)0=at any pointp∈Mp∈M. A local manifold with a vanishing affine connection displays Poincaré symmetryISO(n)=SO(n)⋉T(n), whereT(n)is a group ofn–dimensional translations. We introduce a n–dimensional Riemannian manifold(Mp,η)with vanishing affine connection atp∈Mand a bundleM:=∪pMp. The diffeomorphism mapE:M→M:g↦ηis represented using standard coordinates on any chart of M as
ηab=Eaμ1(xμ)Ebμ2(xμ)gμ1μ2(xμ).
The functionEaμ(xμ)=[E(xμ)]aμis referred to as a vierbein. The spin connectionw∈so(1,3)is introduced as a Lie–algebra–valued one–form, and is referred to as the spin form. It can be represented in standard coordinates as wab=ωaμcηcbdxμ, and is antisymmetric, i.e.,wab=−wba.
The Einstein–Hilbert gravitational action can be expressed as
I=12∫Σn(εa1⋯anℜa1a2∧ea3∧⋯∧ean),
whereℜab:=dwab+wac∧wcbis a curvature two–form. Theories which have topological (characteristic) classes are referred to as the topological theory in this article. We note that the topological theory does not means the so–called “topological field theory”, which has the metric independent correlation functions, and thus, it does not exhibit any dynamics. Three examples of the topological theories are reviewed.
On a n–dimensional Riemannian manifold(M,g), a principal bundleP(M,G)is introduced, whereGis the structural Lie–group. The connection one–formAand the corresponding curvature two–formF=dA+A∧Aare equipped on the base manifoldM. Ais a Lie–algebra valued one–form A∈g, wheregis a Lie–algebra of the structural group G.
Chern–Simons topological theory
In 1988, Witten shows that general relativity in the(1+2)–dimensional space can be considered as a Chern–Simons topological theory.1 In this theory, we employ a principal bundle consists of the base manifoldMwith(1+2)space–time dimension and the structural groupISO(1,2). The connection one–form and curvature two–form can be respectively introduced as
ACS:={Jab,Pc}×{wab,ec}=Jabwab+Pcec,
FCS:=dACS+ACS∧ACS=Jabℜab+PcTc,
whereJabandPaare respectively the Lie–algebra of a rotation and a translation, andTa:=dea+wab∧ebis a torsion two–form. The Chern–Simons action is thus obtained as
ICS:=14∫Σ4FCS∧FCS,
=12∫Σ4Tr[ACS∧dACS+23ACS∧ACS∧ACS],
=12∫Σ4d(εabcea∧ℜbc),
=12∫∂Σ4=Σ3(εabcea∧ℜbc),
whereΣ4is an appropriate simply connected and orientable four–dimensional manifold in which a three–dimensional manifoldΣ3⊂Mis immersed. This is nothing other than the Einstein–Hilbert action without the cosmological term, and thus, it is shown that general relativity can be constructed as the Chern–Simons topological theory in three–dimension. This coincidence between the Chern–Simons and Einstein–Hilbert actions is rather accidental only in the three dimensional space.2,3
Quantization of the theory can be performed using the canonical commutation–relations one. Whereas the Chern–Simins quantum gravity in three–dimensional space does not have any dynamical degree, this is not owing to a topological aspect of the theory. From a simple counting of the degree of freedom (DOF), one can understand that the dynamical DOF in three–dimensional general relativity is zero at the classical level. Detailed summary of Chern–Simons (super) gravity can be found in a textbook.4
BF topological theory
In 1977, the BF topological theory is introduced at first by Plebański,5 while the term “BF theory” did not exist yet at that time. In 1989, Horowitz first treated general relativity of the BF theory as the topological theory in general n–dimensional space–time.6 Review articles of the BF topological theory can be found.7,8
In the BF theory, the base manifoldMis chosen as a four–dimensional Riemannian manifold and the structural group is taken asSO(1,3)orSO(4). The connection one–form and curvature two–form can be respectively taken as the spin formwaband curvature formℜab. In addition, new Lie–algebra valued two–formBabis introduced, and an action in the BF theory is defined as
˜IBF:=12∫Σ4εabcdBab∧ℜcd,
whereΣ4is an appropriate simply connected and orientable four–dimensional manifold. This action cannot be simply recognized as the topological action of the structural groupSO(1,3), because the formBabdoes not belong to the principal bundle in general. Instead, the formBabmust be understood as a connection form on a 2–bundle, and it forms principal 2–bundle in the higher–gauge theory.9 Whereas the action˜IBFcan be topological by means of the 2–gauge theory, it is not coincide with the Einstein–Hilbert action. To convert the BF topological theory to a gravitational theory, additional constraints5,10 must be implement as, e.g., a Lagrange multiplier term such as
IBF:=12∫Σ4(εabcdBab∧ℜcd−12ϕabcdBab∧Bcd),
whereϕabcdis a Lagrange multiplier of a scalar symmetric traceless matrix. The equation of motion with respect toϕappears as the additional constraint, which is referred to as the simplicity condition. The simplicity condition is discussed in detail by Gielen & Oritti11 (linear constraints) and Celada et al.,12 (constraints onℂformalism).
We note that the actionIBFdoes not have any topological characteristic–classes any more after implementing the constraint term. One of possible choices of the simplicity condition is to identify the formea∧ebasBab, and thus, a shape of the BF action coincides with the Einstein–Hilbert action. This coincidence of the shape of the action is true only for solutions of the equation of motion (on–shell condition) in the classical level, and the on–shell condition cannot be simply fulfilled after quantization of the BF theory. The BF gravitational theory is complicated12 owing to this fact.
Chern–Weil topological theory
The four–dimensional Einstein–Hilbert gravitational action can be constructed using the Chern–Weil topological theory whose principal bundle consists of the co–Poincaré group3 as the structural group. The co–Poincaré principal bundle Θ=(wab,Scd)is introduced into four–dimensional space–time manifoldM, whereSab:=12εabcdec∧edis the surface form. Corresponding connection form ACW and curvature form FCW are respectively expresses as follows:
ACW:=(Jab,Pcd)×(wab,Scd)=Jabwab+PabSab,
FCW:=dACW+ACW∧ACW=Jabℜab+PvdwSab,
wheredwis the covariant derivative with respect toSO(1,3), andJabandPcdare respectively generators ofSO(1,3)and the co–translation, whose Lie algebra is expresses as
[Pab,Pcd]=0,
[Jab,Pcd]=−ηacPbd+ηbcPad,
[Jab,Jcd]=−ηacJbd+ηbcJad−ηbdJac+ηadJbc.
It is proven in Kurihara3 that the Einstein–Hilbert gravitational action without the cosmological constant can be written asL=Tr[F∧F], and thus, it has a topological invariant as Chern–classes due to the Chern–Weil theory. This result suggests that appropriate fundamental forms (phase space) of the symplectic geometry for general relativity can be identified as(w,S).
The base manifold(1+3)is taken as Riemannian manifold with(1+3)space–time dimension and the structural group is (1+3)–dimensional co–Poincaré group.3 The action integral of the Chern–Weil topological theory can be introduced as
ICW:=14∫Σ5Tr[F∧F]=12∫Σ4S⋅⋅∧ℜ⋅⋅.
In contrast with the BF gravitational theory, both formsSabandℜab re directly obtained from the principal bundle, and thus, the Einstein–Hilbert action itself preserves the characteristic class (second Chern–class), which is ensured by the Chern–Weil theory.3,13 Even though formsBabandSabare equivalent when the simplicity condition is required in the BF theory, the Chern–Weil gravitaional theory based on the different principal bundle with different structural group from those of the BH theory, and thus, they are completely different each other. The simplicity condition of the BF gravitational theory corresponds to the definition of the surface form in the Chern–Weil topological theory, and thus, it is exact after quantization. Quantization of the Chern–Weil topological theory can be performed using gauge fixing terms with respect to the globalGL(1,3)and localISO(1,3).14
In contrast with the gravitational theory in a three–dimensional space–time, the Chern–Weil theory in four–dimension has the dynamical DOF after quantization. Among 10 DOF on the symplectic fields of the vierbein and spin forms, 2 physical degrees are remaining as the dynamical DOF, corresponding to 2 spin–states of graviton. Quantization of constrained system can be performed using the Kugo–Ojima formalism.15 At first, the auxiliary, ghost and anti–ghost fields are introduced to fix the gauge and unphysical DOF in the system. Then the BRS transformations are required on all of physical and unphysical fields. At this stage, number of constraints due to the BRS transformations is the same as a total DOF in the system, and thus, there is no dynamical degree if all constraints are independent each other. In reality, as explained in Kurihara14 all constraints are not independent and there are two conserved BRS charges in the system. Therefore, the system still has two dynamical DOF after quantization. At the end, principal bundle and its structural group for three topological theories are summarized in Table 1.
Topological theory |
Principal bundle |
Structural group |
Chern–Simons |
(e,w) |
SO(1,2) |
BF |
(B,w) |
SO(1,3) |
Cherm–Weil |
(w,S) |
Co–Poincarè (1,3) |
Table 1 A summary of topological theories.
None.
The author declares no conflict of interest.
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