I am exploring a geometric framework where the usual metric tensor role (as in $GL^+(4,\mathbb{R})/\text{SO}(3,1) $) is replaced by a structure defined by the quotient $ GL^+(4,\mathbb{R})/\text{Spin}^c(3,1) $. My objective is to understand how to construct a $ \text{Spin}^c(3,1) $ connection within this setup, derive the associated curvature tensor, and subsequently define the Ricci scalar, all in the absence of a traditional metric tensor.
Here are my key questions and considerations:
Spin^c(3,1) Connection in GL+(4,R)/Spin^c(3,1) Context:
- How can one construct a $ \text{Spin}^c(3,1) $ connection on a manifold where the geometric structure is given by $ GL^+(4,\mathbb{R})/\text{Spin}^c(3,1) $, rather than a standard metric? I am interested in the mathematical formalism that aptly describes this connection, considering the complexities introduced by the $\text{Spin}^c $ framework.
Curvature Tensor Without Conventional Metric:
- Given the $ \text{Spin}^c(3,1) $ connection in this context, what approach should be taken to derive a curvature tensor that encapsulates the geometry dictated by the $ GL^+(4,\mathbb{R})/\text{Spin}^c(3,1) $ structure? I seek to understand how the curvature tensor can reflect both the Lorentzian aspects and the additional $ U(1) $ gauge interactions inherent in $ \text{Spin}^c(3,1) $.
Ricci Scalar and Geometric Interpretation:
- Lastly, how can this curvature tensor be used to define a Ricci scalar in the absence of a conventional metric? What are the theoretical implications and challenges of interpreting curvature and gravitational effects in a manifold structured by $ GL^+(4,\mathbb{R})/\text{Spin}^c(3,1) $?
I am exploring these questions as part of a broader effort to integrate concepts from geometry and gauge theory, particularly in scenarios where traditional metric-based geometry is not directly applicable. Insights, references, or mathematical guidance on these topics would be greatly appreciated.