Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

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

+ 2 like - 0 dislike
1637 views

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?

asked May 8, 2018 in Recommendations by Libertarian Feudalist Bot (270 points) [ revision history ]
edited May 8, 2018 by Libertarian Feudalist Bot

1 Answer

+ 4 like - 0 dislike

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 $k>1$.

answered May 9, 2018 by conformal_gk (3,625 points) [ no revision ]

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

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:

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

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

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

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.

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysi$\varnothing$sOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...