I have been studying the paper "A Duality Web in 2+1 Dimensions and Condensed Matter Physics".

https://arxiv.org/abs/1606.01989

On page 10, it shows something about $spin_{\mathbb{C}}$-connection. I checked some maths books talking about spin structures, but they are quite difficult to understand for physics students like me.

Could anybody give me some physicists friendly references introducing:

1. spin-connection

2. $spin_{\mathbb{C}}$-connection

3. spin Chern-Simons theory

Could anybody please explain the equation 1.10 for me?

Thank you in advance.

Now I understand that for $spin_{\mathbb{C}}$) connection, the first Chern class is related with the second Stieffel Whitney class. This is explained in the lecture notes of Professor, Steven Bradlow

http://cwillett.imathas.com/bundles/

I also found a good introduction of spin and $spin_{\mathcal{C}}$ structure from the book "Structural Aspects of Quantum Field Theory" Vol.2, section 46.7, page 1214.

https://www.amazon.com/Structural-Aspects-Quantum-Noncommutative-Geometry/dp/9814472697

It shows that the $U(1)$ part in the $spin_{\mathbb{C}}$ connection in physics represents a spinor field coupling to electromagnetic field.

My new question is that do we need a generalization of $spin_{\mathbb{C}}$ structure to the case when a spinor field is coupled to non-Abelian gauge fields?