Understanding the Geometry Spawned from Quotient Spaces $GL^+(4,R)/SO(3,1)$, $GL^+(4,R)/Spin(3,1)$, and $GL^+(4,R)/Spin^c(3,1)$

Question

I'm working on a theoretical framework where I explore different quotient spaces formed with GL$^+$(4,R) and various groups. Specifically, I'm interested in the types of geometry that arise from the following quotient spaces:

1. GL$^+$(4,R)/SO(3,1)
2. GL$^+$(4,R)/Spin(3,1)
3. GL$^+$(4,R)/Spin$^c$(3,1)

For GL$^+$(4,R)/SO(3,1), the situation is relatively clear as it leads to the usual symmetric, non-degenerate metric tensors commonly used in theories of gravity. However, I encounter difficulties when trying trying to understand the GL$^+$(4,R)/Spin(3,1) and GL$^+$(4,R)/Spin$^c$(3,1) cases. In these scenarios, what geometry is described by these?

To provide some context, I am using the Majorana representation of the Dirac matrices to construct a representation of 4x4 matrices. Furthermore in this representation Spin$^c$(3,1) can be represented by exp (f+b), where f is a bivector and b a pseudo-scalar which is in GL$+$(4,R). My question is:

For instances GL+(4,R)/SO(3,1)xR spawns Weyl conformal geometry. But what do GL$^+$(4,R)/Spin(3,1) and GL$^+$(4,R)/Spin$^c$(3,1) spawn?

Any insights, references, or suggestions on how to approach these constructions would be greatly appreciated.

Answer 1

The group $Spin(3,1)$ is nothing but $SL(2, {\mathbb C})$. The latter has a natural embedding in $GL(4, {\mathbb R})$. The subgroup $SL(2, {\mathbb C})< GL(4, {\mathbb R})$ preserves two things:

1. The standard complex structure on ${\mathbb R}^4\cong {\mathbb C}^2$.

2. The complex volume form on ${\mathbb C}^2$, i.e. the alternating complex 2-form $dz_1 \wedge dz_2$.

Accordingly, suppose that $M$ is a smooth 4-dimensional manifold whose frame bundle admits a reduction to a vector bundle with the structure group $Spin(3,1)$. Then this reduction yields an almost complex structure $J$ on $M$, together with a complex volume form $\omega$ (where "complex" is understood with respect to the above almost complex structure $J$), i.e. $\omega$ is a nowhere vanishing complex 2-form. I can spell out the technical meaning of the above terms if you are interested. Things are a bit more intuitive if the almost complex structure $J$ is integrable, i.e. corresponds to a complex structure on the manifold $M$. Then, in local holomorphic coordinates, a form $\omega$ as above can be written as $f(z)dz_1 \wedge dz_2$, where $f(z)=f(z_1,z_2)$ is a smooth complex-valued function (which need not be holomorphic).

There is no "(semi)metric" of any kind on $M$ corresponding to such a reduction, since $Spin(3,1)< GL(4, {\mathbb R})$ does not preserve any nonzero real symmetric bilinear form on ${\mathbb R}^4$ (nor a complex symmetric bilinear form on on ${\mathbb C}^2$).

How useful it is, I am not sure.

I do not have a good geometric interpretation of a $Spin^c(3,1)$-structure. Algebraically speaking, $Spin^c(3,1)= S^1\cdot SL(2, {\mathbb C})$ (it is not a direct product since $S^1$, identified with the group of unitary scalar matrices of the form $Diag(e^{i\theta}, e^{i\theta})$, has nontrivial intersection with $SL(2, {\mathbb C})$). Accordingly, there exists a $Spin^c(3,1)$-invariant density, $|dz_1\wedge dz_2|$.

A reduction of the frame bundle to the structure group $Spin^c(3,1)$ does yield an almost complex structure on the 4-manifold $M$ and a real-valued density.