is a multiplet of m differential one-forms on the base manifold M . Secondly, inspecting (5.2.9), we observe that the same metric tensor can be constructed from a multitude of equivalent vielbein one-forms. Indeed, consider arbitrary smooth mappings from open neighborhoods of the (pseudo-)Riemannian manifold into the pseudo-orthogonal group

produce the same metric tensor. How do we geometrically interpret this fact? Over the base manifold M , let us construct a principle bundle P (M , SO(p, m − p)) (or,

π

in equivalent notation Pso(p,m−p) → M ), with structural group SO(p, m − p) and let us consider the associated vector bundle

provided by the m-dimensional, defining representation of SO(p, m − p). Mathematically the vielbein Ea is a one-form with values in sections of the bundle Vso(p,m−p) or, if you prefer, is a section of the product bundle T M Vso(p,m−p), namely:

This observation suggests an obvious idea: why do we not introduce a principle

π

connection on the bundle Pso(p,m−p) → M ? Let us do it.

5.2.2 The Spin-Connection

The Lie algebra so(p, m − p) is made by matrices ωab satisfying the defining condition:

If we raise the second index by means of the flat metric ηab , the rank two tensor ωab ≡ ωacηcb obtained in this way is antisymmetric by construction ωab = −ωba .

π

Hence a principle connection on the bundle Pso(p,m−p) → M is locally described, in each coordinate patch Uα M , by a one-form with a pair of antisymmetric indices:

ωab = ωμab(x) dxμ; ωab = −ωba (5.2.16)

which, under SO(p, m − p) transition functions from one local trivialization to another one, transforms in the canonical way established by (3.4.14):

This transformation of ω, named the spin-connection is paired with the already introduced transformation of the vielbein (5.2.12) which was our starting point.

Once we have a principle connection, according to the general principles extensively discussed in Chap. 3, we can introduce the covariant exterior differential of any section of any associated bundle. In particular we can write the covariant differential of the vielbein one-form, to which we assign the name of torsion two-form:

At the same time, specializing to the case under consideration the general formula (3.7.6) for the curvature two-form of a principle connection we can write that of the spin-connection as follows:

The attentive reader will recognize that (5.2.18) and (5.2.19) are the same as (3.2.25) and (3.2.27) anticipated in Sect. 3.2.5. The same attentive reader will also recognize that considered together, the torsion two-form and the curvature two-form can be given a further inspiring interpretation spelled out in the next subsection.

5.2.3 The Poincaré Bundle

Given any flat metric ηab in m-dimensions with signature (p, m − p), the corresponding Poincaré group is the semidirect product of the m-dimensional translation group with the pseudo-orthogonal group SO(p, m − p). It is denoted ISO(p, m − p) and, by definition we have:

Naming Pa the generators of the translations and Jab , the generators of the subgroup SO(p, m − p), which is the Lorentz group in the case p = 1, the standard commutation relation of the Lie algebra iso(p, m − p) are the following ones:

which are just the generalization to higher dimensions and with an arbitrary flat η- metric of (1.6.6)–(1.6.7) introduced in Chap. 1. From (5.2.23) we easily read off the structure constants: