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
5,082 questions , 2,232 unanswered
5,352 answers , 22,786 comments
1,470 users with positive rep
820 active unimported users
More ...

  Is my Summary of a Spinor Bundle Associated with a String Worldsheet Correct?

+ 1 like - 0 dislike
800 views

I've been having difficulty finding a source that lists all the properties of the spinor bundle of a string worldsheet explicitly, so I've had a go at creating my own description. I'd really appreciate it if someone could tell me if the following is true:


Take the worldsheet to be some 2d pseudo-Riemannian orientable manifold $M$.

One can associate with each point $x \in M$ a 2d tangent space $TM_{x}$. The disjoint union of $TM_{x}$ at all $x$ defines the total space $TM$ of a tangent bundle ($TM$, $\pi_{TM}$, $M$) whos projection is given by:

\begin{equation} \pi_{TM}: TM \rightarrow M \end{equation}

The worldsheet $M$ is the base space of the tangent bundle and each $TM_{x}$ is a fibre.

Since the tangent space is 2d, the bases that exist in each $TM_{x}$ are 2d also. Since the base space is pseudo-Riemannian, so is the tangent space and the ordered bases (frames) that exist on each $TM_{x}$ are 'pseudo-orthonormal'. This would mean that the bases transform under an $O(1,1)$ group. However, Since the base space $M$ is orientable, so is each $TM_{x}$ and that means that the frames are oriented pseudo-orthonormal and transform under $SO(1,1)$ instead.

This allows the oriented orthonormal frame bundle (a specific sub-class of principal bundle) to be written as $(F_{SO(1,1)}(M), \pi_{F}, M, SO(1,1))$, where the projection acts as:

\begin{equation} \pi_{F}: F_{SO(1,1)}(M) \rightarrow M \end{equation}

The fibre $F_{x}$ of this frame bundle at a point $x$ on $M$ is the set of all frames of $TM_{x}$ at the same point $x$. $F_{x}$ is homeomorphic to the gauge group $SO(1,1)$ and is said to be an $SO(1,1)$-torsor.

However, now one can define a lift of the group $SO(1,1)$ to $Spin(1,1)$. The corresponding frame bundle is now $(P, \pi_{P}, M, SO(1,1))$ with projection:

\begin{equation} \pi_{P}: P \rightarrow M \end{equation}

The fibre $P_{x}$ of this frame bundle at a point $x$ on $M$ is the set of all frames of $TM_{x}$ at the same point $x$. $P_{x}$ is homeomorphic to the gauge group $Spin(1,1)$ and is said to be an $Spin(1,1)$-torsor.

How can the set of all frames in $TM_{x}$ be homeomorphic to both $SO(1,1)$ and $Spin(1,1)$?

The spinor bundle can then be defined to be given by $(S, \pi_{S}, M, \Delta_{(1,1)} Spin(1,1))$, with projection that acts as:

\begin{equation} \kappa: S \rightarrow M \end{equation}

Here $S$ is given by:

\begin{equation} S = P \times_{\kappa} \Delta_{(1,1)} = (P \times \Delta_{(1,1)})/Spin(1,1) \end{equation}

The fibre is given by $\Delta_{(1,1)}$ which is the Hilbert space of all spinor states. Each section of this bundle then corresponds to a particular Majorana-Weyl spinor field configuration on the worldsheet.

This post imported from StackExchange Physics at 2015-03-04 12:46 (UTC), posted by SE-user Siraj R Khan
asked Mar 3, 2015 in Theoretical Physics by Siraj R Khan (105 points) [ no revision ]

1 Answer

+ 1 like - 0 dislike

It is nearly correct. First, $F_x$ is the space of oriented and orthonormal frames of $T_x M$. Furthermore, you gave the correct definition of the $Spin(1,1)$ bundle but confused a little bit its meaning. The $Spin(1,1,)$ bundle is defined as the lift of the $SO(1,1)$ bundle to $Spin(1,1)$. This means we have a principal $Spin(1,1)$-bundle $P$ and a principal bundle map $\phi: P \to F$ which is at the same time a covering (this literally lifts the double covering of $SO(1,1)$ by $Spin(1,1)$ to the bundle picture). For more details see http://en.wikipedia.org/wiki/Spin_structure#Spin_structures_on_Riemannian_manifolds. Now $P$ is not a frame bundle. More precisely, for every oriented, orthonormal frame of $T_xM$ there exists two points $p$ and $p'$ in the fiber $P_x$ which yield this frame by $\phi(p) = \phi(p')$. Thus $P_x$ is not homeomorphic to the frames on $T_xM$.

This post imported from StackExchange Physics at 2015-03-04 12:46 (UTC), posted by SE-user Tobias Diez
answered Mar 4, 2015 by Tobias Diez (90 points) [ no revision ]
Thanks for the reply Tobias. Wouldn't $F_{x}$ be the space of oriented and pseudo-orthonormal frames of $T_{x}M$, just to be a little more precise? (Since $T_{x}M$ has Lorentzian metric signature). And I understand the $Spin(1,1)$-bundle now, thank you.

This post imported from StackExchange Physics at 2015-03-04 12:46 (UTC), posted by SE-user Siraj R Khan

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$\hbar$ysicsOverf$\varnothing$ow
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
...