This implies that the structure constants are neither functions of fieldA nor of positionx. If Ga(x) are the gauge generators in a YM theory, they obey an algebra of the form.
The Vielbein
More importantly, the collection of tangent spaces at each point of the manifold makes it possible to define the action of the symmetry group of the tangent space (the Lorentz group) at each point of M, thus providing it with a fiber bundle structure, the tangent bundle. This relation can be read as meaning that vielbein is, in this sense, the square root of the metric.
The Lorentz Connection
Given eaµ(x) one can find the metric, and therefore all the metric properties of spacetime are contained in vielbein. The connectionωabµ(x) defines the parallel transport of Lorentz tensors in the tangent space TxtoTx+dx, where xandx+dx are neighboring points.
Differential forms
In the following sections, we discuss the construction of the possible actions for gravity using these ingredients. For D= 2 this action reduces to a linear combination of the 2-dimensional Euler character, χ2, and the space-time volume (area).
Dynamical Content
The generalization to arbitrary D of the form (44) was obtained more than 30 years ago in [29] and is known as the Lovelock lagrangian, . The Einstein-Hilbert action, as well as its Lovelock generalization, both give second-order field equations and for the same reason: the second derivative of the metric enters through a total derivative in the Lagrangian and so the field equations remain second-order.
Adding Torsion
Dimensional parameters in action are potentially dangerous because they can cause uncontrolled quantum corrections. As shown below, in odd dimensions there is a unique combination of terms at work that can give the theory increased scale symmetry.
Extending the Lorentz Group
The resulting action can be seen as dependent on a unique multiplication coefficient (κ), analogous to Newton's constant. This means that ordinary gravity in three dimensions can be considered a measure theory of the Poincare group. The signs (±) in the transformation above can be traced back to the sign of the cosmological constant.
More Dimensions
Since the (A)dS group has an invariant tensor²ABCD, one can construct the 4-form invariant. 86) It is invariant under the (A)dS group and is readily recognized as the Euler density for a four-dimensional manifold12 whose tangent space is not Minkowski, but has the metric ΠAB =diag (ηab,∓1). The parameterl is a length scale – the Planck length – and cannot be determined by theoretical considerations. In fact, the lagrangian (90) can also be written in terms of WAB and its outer derivative, as
Chern-Simons
Torsional Chern-Simons
Characteristic Classes and Even D
Then the action for the geometry M can be expressed as the integral of the Euler density E2n over Ω multiplied by κ. The quantity in parentheses—with the right normalization—is the Euler number of the manifold obtained by gluing Ω and Ω0 along M in the right way to produce an orientable manifold, χ[Ω∪Ω0˙]. The Lagrangian BI is obtained by a certain choice of αp in the Lovelock series, so that the Lagrangian takes the form.
Finite Action and the Beauty of Gauge Invariance
With this definition it is clear that the Lagrangian (98) contains only one free parameter, l, which, as we have explained, can always be absorbed in a Vielbein redefinition. The choice of BI is in this aspect the best behavior since the degeneracies are limited to only one value of the radius of curvature (Rab±l12eaeb = 0). The resulting action reaches an extrema for the boundary conditions governing the outer curvature of the boundary.
Further Extensions
PART TWO
Supersymmetry
This is one of the most remarkable features of SUSY: local (measure) SUSY is not only compatible with gravity. The most important lesson from supersymmetry is not the unification of bosons and fermions, but the extension of the bosonic symmetry. SUSY is non-trivial because the algebra is not a direct sum of the spacetime and internal symmetries.
Superalgebras
Supergravity
Again, one can see that the standard form of SUGRA is not a gauge theory for a group or a supergroup, and that the local (super-)symmetry algebra naturally closes only on shell. The algebra can be made to close shell by force, at the cost of introducing auxiliary fields – which are not guaranteed to exist for alldandN [58] – and still the theory would not have a fiber bundle structure since the base manifold is identified with part of the fiber. However, it is certainly true that if GR could be formulated as a measure theory, the chances for its renormalizability would clearly increase.
From Rigid Supersymmetry to Supergravity
Whether the lack of fiber bundle structure is the ultimate cause of gravity's non-renormalizability remains to be proven. In any case, most high-energy physicists now regard supergravity as an effective theory obtained from string theory in a certain limit. In string theory, eleven-dimensional supergravity is seen as an effective theory obtained from ten-dimensional string theory by strong coupling [59].
Standard Supergravity
To construct a supergravity theory containing gravity with a cosmological constant, a mathematically oriented physicist would search for the smallest superalgebra containing the generators of the AdS algebra. In what follows, we present an explicit construction of the superalgebras containing AdS algebraso(D−1,2) along the lines of [55] where we extended the method to apply it to the cases D = 5, 7 and 9 if yes [61]. The crucial observation is that the Dirac matrices provide a natural representation of the AdS algebra in any dimension.
The Fermionic Generators
A spinor satisfying this condition is called a Majorana, and C= (Cαβ) is called the charge conjugation matrix. Using the form (110) for the supersymmetry generator, we find that its Majoran conjugate is Q¯. The matrix C can be viewed as performing a change of basis ψ→ψT=Cψ, which corresponds to the change Γ→ΓT.
Closing the Algebra
So far, we have given only some of the constraints necessary to close the algebra so that AdS generators appear in the anticommutator of two supercharges. AS we have observed, generally other than Ja and Jab, other matrices will appear in the r.h.s. In the previous sections, we have seen how to construct CS actions for an AdS connection for any D = 2n+ 1.
Examples of AdS-CS SUGRAs
The cosmological constant is −l−2, and the U(2,2|N) covariant derivative∇ acting on ψr. 135) where D is the covariant derivative in the Lorentz connection. 18The first term in LG is the dimensional continuation of the Euler (or Gauss-Bonnet) density from two and four dimensions, exactly as the three-dimensional Einstein-Hilbert lagrangian is the continuation of the two-dimensional Euler density. It is the leading term in the limit of vanishing cosmological constant (l→ ∞), whose local supersymmetric expansion yields a non-trivial expansion of the Poincar'e group [60].
Wigner-In¨ on¨ uu Contractions
Minimal Super-Poincar´ e Theory
In the eleven-dimensional case, one can imagine this algebra as the result of a WI contraction of theosp(32|1) superalgebra with compound (136). However, it is not easy how to implement the limit in the lagrangian and some redefinitions of the field are needed to make contact between the two theories.
M-Algebra Extension of the Poincar´ e Group
The algebra (152) is known as M-algebra because it is the expected gauge invariance of M-theory [77]. The M-algebra has more generators than the minimal super Poincar'e algebra of the previous section because the new Zab has no match there. The action is invariant under local supersymmetry transformations obtained from a gauge transformation of the M connection (157) with parameter λ=²αQα,. 156) The field content is given by the components of the M-algebra compound, .
Field Equations
As already mentioned, these theories cannot be related by a WI contraction, because contractions cannot increase the number of generators in the algebra. In fact, the M-algebra can be obtained from the AdS-algebra by a more general singular transformation called a deformation[78]. These deformations are analytic mappings in the algebra with the only restriction that they respect the Maurer-Cartan structural equations.
Overview
For example, in the pure gravity (matter-free) sector there exist spherically symmetric, asymptotic AdS standard [42], as well as topological black holes [84]. In the extreme case, these black holes can be shown to be BPS states [85]. In this background, the sympectic form has maximum rank and the gauge superalgebra is realized in the Dirac brackets.
Hamiltonian Analysis
This reflects the usual characteristic that some coordinates are non-propagating gauge degrees of freedom and the corresponding constraints are the generators of gauge transformations. This means that the separation between coordinates and momentums cannot be made uniform throughout phase space, and that there are regions where the degrees of freedom of the theory change abruptly: while some of the φs are second class and some are first class, depending on the rank of the symplectic form Ω.
Degeneracy
It can be shown in the context of some simplified mechanical models that the degeneracy of a system generally occurs in the lower dimensional submanifolds of the phase space. As shown in this reference, if the system evolves along an orbit that reaches a degeneracy surface, Σ, it is trapped by the surface and loses degrees of freedom corresponding to displacements away from Σ. This is an irreversible process which has been observed in the dynamics of vortices described by the Burgers equation.
Counting of Degrees of Freedom
The additional degrees of freedom correspond to the inωµab propagation modes, which in CS theory are independent of the metric ones, containing ineaµ. It is a nice surprise in the CS supergravity cases discussed above that for some unique choice of N, the algebras develop an abelian subalgebra and make it possible to separate the first and second class constraints (e.g., N= 4 for D = 5, and N = 32 for d= 11). In some cases the algebra is not a direct sum, but an algebra with a central abelian extension (D = 5).
Irregularity
In other cases, the algebra is a direct sum, but the abelian subgroup is not entered manually, but is a subset of generators that separate from the rest of the algebra (D= 11). Feynman, Quantum Theory of Gravitation, Lecture at the Conference on Relativistic Gravitational Theories, Jablonna, Warsaw, July 1962. 76] M.Hassa¨ıne, R.Troncoso and J.Zanelli, Poincar´e- Invariant Gravity with Local Supersymmetry as a Gauge Theory for M-algebra, Phys.
