# The dimension of the subspace of flat spin connections

I am interested in the the flat spin connections in a Riemann spacetime of dimension 4. They appear in the context of the frame formalism of metric gravity theories. I believe that they form a subspace, homologous to the algebra of the Lorentz group, within the affine space of spin connections. The dimension of the subspace would be then 6. However, I cannot prove this idea myself. Does anybody can help (or correct) me with this? Sorry if the wording of the question is poor, but I am not a specialist in differential geometry.

• In what sense do you use the word "homologous" here? Also, I don't see why the space of flat spin connections should be finite dimensional. Given any such flat connection, a local frame rotation will give you another connection, yet still flat. This already gives you an infinite dimensional space. Or I just don't understand what you are asking. – Igor Khavkine Jul 10 '17 at 23:42
• I am here referring to spin connections of pseudo Riemann spacetimes. The affine space of spin connections has dimension 24 (for a Riemann spacetime of dimension 4). The flat spin connections form a subspace. You are right that a local frame rotation or boost (for the Lorentz group) gives another flat spin connection. But, for affine spaces of spin connections there exist also translations of the original flat spin connection. The translations are done by tensors generated by the algebra of the Lorentz group (dimension 6). The subspace of the affine translations is what I am interested about. – asierzm Jul 11 '17 at 0:00
• The space of connections is infinite dimensional. If you do not impose any further restrictions on topology of your manifold, the space of flat connections is also infinite dimensional, even modulo gauge group. – Misha Jul 12 '17 at 17:06
• The spin connections is defined from the affine connection in a spinor bundle on a pseudo-Riemann manifold (signature 2). In tensor notation, a general spin connection on a dimension 4 Riemann manifold (spacetime) locally has 24 independent components. The dimension I refer to is defined locally like the dimension of the affine space containing it. But how much of these components are necessary for a general flat spin connection? – asierzm Jul 12 '17 at 21:02