# How large is the subset of flat spin connections?

In Poincare Gauge Theories spin connections take the role of the Levi-Civitta connection of GR for defining covariant derivatives. The GR can be formulated as a Poincare Gauge Theory which is called the Teleparallel equivalent of GR. In this theory the spin connection is decomposed into a flat spin connection $A_{ab\mu}=-A_{ba\mu}$ and a contortion $K_{ab\mu}=1/2(T_{ba\mu}+T_{\mu ab}-T_{ab\mu})$.

The Weitzenböck connection $\Gamma^{\beta}_{\phantom{\beta}\mu\nu} = h^{\beta}_i\delta_{\nu}h^i_{\mu}$ is normally used for the flat spin connection. However, a general flat spin connection is only defined by the condition that its curvature is zero $$R^a_{\phantom{a}b\mu\nu} = \partial_{\mu}A^a_{\phantom{a}b\nu} - \partial_{\nu}A^a_{\phantom{a}b\mu} + A^a_{\phantom{a}c\mu}A^c_{\phantom{c}b\nu} - A^a_{\phantom{a}c\nu}A^c_{\phantom{c}b\mu} = 0.$$ These equations do not define a single connection. Instead, a set of spin connections comply with this condition.

To understand how large this subset is, the number of independent curvature components have to be counted. The curvature tensor has at most 36 independent components. However, the Bianchi identities show that not all are independent. The first Bianchi identities $$D_{\rho}T^a_{\phantom{a}\mu\nu} + D_{\mu}T^a_{\phantom{a}\nu\rho} + D_{\nu}T^a_{\phantom{a}\rho\mu} = R^a_{\phantom{a}\rho\mu\nu} + R^a_{\phantom{a}\mu\nu\rho} + R^a_{\phantom{a}\nu\rho\mu}$$ sets constraints for the Torsion components, but the second Bianchi identities $$D_{\rho}R^a_{\phantom{a}b\mu\nu} + D_{\mu}R^a_{\phantom{a}b\nu\rho} + D_{\nu}R^a_{\phantom{a}b\rho\mu} = 0$$ with $D_{\mu}$ a Weitzenböck covariant derivative, do set constraints in the components of the curvature.

Which is the final number of independent curvature components? How large is the resulting subset of flat spin connections?

• What do you mean "How large is the set"? Obviously it's infinite, so is there a more precise notion of "size" you're interested in...? – Alex Nelson Jun 16 '17 at 18:45
• If the spin connection in 4 dimensions has 24 components, a generic flat spin connection will have fewer independent components because of the constraints. I am trying to understand how much is reduced. – asierzm Jun 16 '17 at 23:26
• If it's flat, it's equivalent to having zero components...because you can always change coordinates (via diffeomorphism invariance) to the coordinates where the connection is all zeroes... – Alex Nelson Jun 17 '17 at 1:01
• Flat spin connections, unlike the Levi-Civitta connection of GR, are not trivial. Applying local Lorentz transformations or diffeomorphisms neither exhausts the possible flat spin connections, but how many extra degrees of freedom there are? – asierzm Jun 17 '17 at 5:48

Well, any (linear/principal) connection transforms as $\omega'=A^{-1}\omega A+A^{-1}dA$, which means that is $\omega$ is a flat $G$-connection, then there must exist such $A:\mathcal{U}\rightarrow G$ functions (local sections of the principal fiber bundle, $\mathcal{U}\subset M$ open), such that $\omega=A^{-1}dA$. Now let $g\in G$, and we assume that $G$ is a matrix Lie group (which we implicitly assumed so far as well), then $w=g^{-1}dg$ is the Maurer-Cartan form of $G$, which is unique. We see then, that $\omega=A^*w$.
Which means that locally there are as many flat connections as there are local sections of the principal $G$-bundle the connection belongs to, as every local section will produce a flat connection via pulling back the Maurer-Cartan form, and every flat connection is of this form.
• Thank you for your answer. I understand that the problem can be reformulated as the study of the local sections of the principal $G$-bundle of the connection. For a 4D Riemann spacetime as a base space and a Minkowski frame bundle, how can the local sections studied? Sorry for the wording but I am not sure that I can write this in a precise way. Can the "embeddings" of the local sections be related using the Lorentz group? In the sense, all local sections can be obtained using Lorentz "rotations". – asierzm Jun 19 '17 at 10:34
• I am trying to reformulate the question in a geometrically more appropriate manner: Can the principal $G$-bundle of the flat connections be an orbit of the Lorentz Group? Knowing this is my main motivation to start this discussion. – asierzm Jun 19 '17 at 14:19