\input amstex
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\define\GL{\operatorname{GL}}
\define\ORTH{\operatorname{O}}
\define\UN{\operatorname{U}}
\define\SO{\operatorname{SO}}
\define\Hom{\operatorname{Hom}}
\define\End{\operatorname{End}}
\define\Aut{\operatorname{Aut}}
\define\Mat{\operatorname{Mat}}
\define\Ker{\operatorname{Ker}}
\define\Coker{\operatorname{Coker}}
\redefine\Im{\operatorname{Im}}
\define\Rank{\operatorname{Rank}}
\define\spin{\operatorname{Spin}}
\define\R{\Bbb R}
\define\C{\Bbb C}
\define\rp{\Bbb{RP}}
\define\cp{\Bbb{CP}}

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\title       The Geometry of Vector Bundles and an Introduction to Gauge
Theory \\
Lecture 39
\endtitle
\author      Professor Steven Bradlow \\ Class Notes From Math 433
\endauthor
\affil       University of Illinois at Urbana-Champaign
\endaffil
\address     273 Altgeld Hall, 1409 W. Green Street, Urbana, IL  61801
\endaddress
\email       bradlow\@math.uiuc.edu
\endemail
\date        May 4, 1998
\enddate

\endtopmatter

\document
\leftline{{\bf Spin Structures}}
\vskip.25in
We had a double cover, $p:\spin(n) \to \SO(n)$.  If $$P_{\SO(n)}=\coprod U_\alpha \times \SO(n)/g_{\alpha \beta}$$ is a principal $\SO(n)$-bundle, 
then a spin structure for the bundle is $$P_{\spin(n)}=\coprod U_\alpha \times \spin(n)/\tilde g_{\alpha \beta}$$ where $p \tilde g_{\alpha \beta}=g_{\alpha \beta}$.  This yields a bundle map, $\rho:P_{\spin(n)} \to P_{\SO(n)}$.  

If 
$P_{\SO(n)}$ is the bundle of oriented frames for $TM$, then $P_{\spin(n)}$ 
defines a spin structure on $M$.  $TM=P_{\SO(n)} \times_\sigma \R^n$, where 
$\sigma:\SO(n) \to \Aut(\R^n)=\GL(n,\R)$.  Sections of $TM$ are vector fields 
on $M$.  Given $P_{\spin(n)}$, denote the induced representation be 
$\tilde \sigma:\spin(n) \to \Aut(\R^n)$ and get $\Bbb V=P_{\spin(n)} \times_{\tilde \sigma} \R^n$.

What is the relation between $\SO(n)$-representations and $\spin(n)$-representations?  Since by definition $\tilde \sigma=\sigma \circ p:\spin(n) \to \SO(n) \to \Aut(\R^n)$, we have that every representation of 
$\SO(n)$ gives a representation of $\spin(n)$.

Observe that $\tilde \sigma(-I)=\tilde \sigma(I)=I$.  So, if a representation, $\tilde \sigma$ 
of $\spin(n)$ is such that $\tilde \sigma(-I)=\tilde \sigma(I)$, then we can define a representation of $\SO(n)$ in the natural way.  However, there are 
representations of $\spin(n)$ which do not have this property.  Infact, we can 
identify special irreducible representations.  This leads to $\spin$ bundles 
on $M$.  Sections of these bundles are called {\it spinors}.
\vskip.5in

\leftline{{\bf $\spin_\C$ Structures}}
\vskip.25in

$\spin(n)$ was defined by the exact sequence, 
$$1 \to \Bbb Z_2 \to \spin(n) \to \SO(n) \to 1$$
We now look to double cover $\SO(n) \times S^1=\SO(n) \times \UN(1)$.  $S^1$ 
is naturally a double cover of $S^1$ under the map $Z \mapsto z^2$.  We 
could thus look at the map, $\spin(n) \times S^1 \to \SO(n) \times S^1$ given 
by the product of the two covering maps. However, this is a four-fold, rather 
than two-fold, cover of $\SO(n) \times S^1$.  Observe that $\Bbb Z_2$ acts 
on both $\spin(n)$ and $S^1$ (antipodal action).  So we can examine the 
orbits of the action on $\spin(n) \times S^1$, 
$$\spin(n) \times_{\Bbb Z_2} S^1=\{[\tilde g,\lambda]:[\tilde g,\lambda]=[-\tilde g,-\lambda]\}$$
We thus make the following defintion,
\vskip.25in

\definition{Definition 1}  $\spin_\C(n)=\spin(n) \times_{\Bbb Z_2} S^1$.
\enddefinition
\vskip.25in

If we define $\spin_\C(n) \to \SO(n) \times S^1$ by $[\tilde g,\lambda] \mapsto (g,\lambda)$, then $\spin_\C$ is a double cover of $\SO(n) \times S^1$.  We 
now want an analogue of the bundle, $P_{\spin(n)} \to P_{\SO(n)}$. We have that 
$S^1=\UN(1)$ is the structure group of $P_{\UN(1)}$ - a principal $\UN(1)-$bundle (equivalently a complex line bundle).  Starting with 
$$P_{\SO(n)} \times L = \coprod U_\alpha \times (\SO(n) \times \UN(1))/\{g_{\alpha \beta},l_{\alpha \beta}\}$$ when can we lift to 
a $\spin_\C$ bundle?  Locally there is no problem to doing this, 
$$\sarrowlength=.42\harrowlength
\commdiag{&\spin_\C(n)\cr &\arrow(1,1)\lft{\tilde g_{\alpha \beta} \times l_{\alpha \beta}^{1/2}} \quad \arrow(1,-1)\cr
U_\alpha \cap U_\beta&\mapright_{(g_{\alpha \beta},l_{\alpha \beta})}&\SO(n) \times \UN(1)\cr}$$
just from basic covering space theory.  However, can we pick the lifts 
$\tilde g_{\alpha beta}$ and $l_{\alpha \beta}^{1/2}$ in a way so that 
the cocyle condition is satisfied?  That is, can we pick the lifts so that 
for all $\alpha,\beta,\gamma$ with $U_\alpha \cap U_\beta \cap U_gamma \neq \emptyset$, 
$$[\tilde g_{\alpha \beta} \tilde g_{\beta \gamma} \tilde g_{\gamma \alpha},l_{\alpha \beta}^{1/2} l_{\beta \gamma}^{1/2} l_{\gamma \alpha}^{1/2}]=1 \in \spin_\C(n)$$
i.e, we must get $[1,1]$ or $[-1,-1]$.
\enddocument
\end
