$Spin_{\mathbb{C}}$-Connection for Physicists

Question

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. 

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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?

Answer 1

I do not know in what context is the $spin_c$ connection discussed in that paper but generically you will need such a structure when you try to put a theory that contains spinors in a generic curved background. Generically, such a background will admit spinors if its frame bundle can be lifted to a Spin bundle. If not, and the obstruction is measured by $w_2 \in H_2(X,\mathbb{Z}_2)$ then you are in trouble but as long as you couple the spinor to a gauge field then it is ok. Mathematically the spinor bundle $V$ wont exist for a generic background but $V \otimes L$ for a line bundle $L$ will exist. To define a $spin_c$ structure we need a line bundle as explained, not a rank $k$ bundle with $k>1$.

Comments

Thank you conformal_gk. What is the bundle for spinor coupled with non-abelian gauge field $k>1$ over a generic manifold?

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My understanding is that you need a line bundle in order to define the $spin_c$ structure. Thinking of Vafa-Witten theory when one tries to mass deform it in an arbitrary background the only solution I know of is to tensor with the U(1) baryon symmetry despite the fact that the R-symmetry is so "rich". I think you always need a line bundle. You can see this from the definition:

$$Spin_c(n) \to Spin(n) \times_{\mathbb{Z}_2} U(1)$$

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Thank you conformal_gk. Is that related with the "splitting principle"?

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In pseudo-classical limit, the spinors are Grassmann number valued objects. Is there a generalization of $spin_{\mathbb{C}}$ structure for supermanifold? 

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Spin c structure tells you if you can write spinor bundle which is a specific vector bundle over your manifold. Supermanifold is a usual manifold with a specific sheaf of algebras on it. I guess you allow that sections of your spinor bundle to take values in that sheaf.