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
Q&A (4870)
