This question arises from a personal misunderstanding about a conversation with a friend of mine. He asked me a question about the "truly nature" of spinors, i.e., he asked a question to me about what is an spinor object. After a few lines of dialogue, he asked something quite alien to me:
"So, spinors are Levi-Civita connections?"
The relationship between a mathematical object which models physical entities in field theory (a Dirac spinor for example) and a purely mathematical entity like a Levi-Civita connection, still intrigues me.
Now, today I encountered this question here:
and in the second answer, the user made another relationship between field theory and connections:
"The "Christoffel symbols" are now just the components of a principal connection on that bundle, where a "connection form" is better known to physicists as a gauge field"
I'm asking this question because, from the point of view of elementary general relativity, we are taught that we need a pseudo-riemanninan manifold and a (Levi-Civita) connection to, roughly speaking, make a well-defined notion of derivative of tensor fields. From this point of view a connection is nothing more than a linear map.
So, what are Connections in physics, INDEED?