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\begin{center} {\Large
\underline{\bf Exam}}
\\[1.5cm]
{\large\bf M.Sc.\ in Quantum Fields and Fundamental Forces}
\\[2cm]
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\\[3cm]
 2:00 -- 5:00, Monday 30 April, 2012
\\[2.6cm]
 **********************************
\\[0.5cm]
 Answer {\bf THREE} out of the four questions.
%\\[1cm]
Use a separate booklet for each question. Make sure that each
booklet carries your name, the course title, and the number of the
question attempted.
\\[0.5cm]
 **********************************
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% 1 *******************************************************************
\newpage

\nextq



\begin{enumerate}


\item Consider the Lie group $GL(n,\mathbb{R})$ so that $g \in GL(n,\mathbb{R})$ is an $n \times n$ matrix (with non-zero determinant). We may take the matrix components $\{ x^{ij}(g) \}$ as coordinates. Using such coordinates show that the vector field
\begin{eqnarray*}
V =  v^{ij}  x^{ki}(g) \frac{\partial}{\partial x^{kj}(g) } 
\end{eqnarray*} 
is left invariant, where $v^{ij}$ are components of an $n \times n$ matrix. Use this to give a basis for the left invariant vector fields.


{\hspace{14cm} [5 marks]}




\item Use your answer to compute the Lie bracket of two left invariant vector fields, $V$ and $W$ in $GL(n,\mathbb{R})$, giving the components of $[ V, W ]$ in the above basis.  


{\hspace{14cm} [4 marks]}




\item Consider a matrix group $G$ embedded into $GL(n,\mathbb{R})$ by an embedding $f: G \to GL(n,\mathbb{R})$. The embedding is chosen so that the left action $L$ obeys $f \cdot L_g = L_{f \cdot g} \cdot f$ for $g \in G$ so that the group structure of $G$ is faithfully embedded into $GL$. Hence show that for $V \in T_eG$ then $f_\star (X_V) = X_{f_\star V}$. 


{\hspace{14cm} [4 marks]}



\item Show that under a map $f: \mathcal{M} \to \mathcal{N}$  the push-forward of the Lie bracket of two vector fields $A, B$ on $\mathcal{M}$ is given by the Lie bracket of their push-forwards $f_\star A, f_\star B$, ie. show $f_\star [ A, B ] = [ f_\star A, f_\star B ]$.

{\hspace{14cm} [4 marks]}



\item  Use your answers above to show how to compute the structure constants of a matrix group $G$ that may be embedded in $GL(n,\mathbb{R})$ using the Lie bracket of $GL(n,\mathbb{R})$.

{\hspace{14cm} [3 marks]}




\vspace{1cm}

{\hspace{13cm}[Total 20 marks]}

\end{enumerate}




% 2 *******************************************************************
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\nextq


\begin{enumerate}

\item Define \emph{cycle} and \emph{boundary} chains? What is \emph{homology}?


{\hspace{14cm} [5 marks]}


\item Give a simplicial complex that triangulates the 2-sphere.\\  ( \emph{Hint:} you may find a complex containing only \emph{four} 2-simplices. )

{\hspace{14cm} [3 marks]}

\item Construct the boundary operators $\partial_2$ and $\partial_1$ as \emph{matrices} for your triangulation of $S^2$. Use these to confirm that the boundary operator is nilpotent. 


{\hspace{14cm} [5 marks]}



\item Explicitly compute the Betti numbers $b_0, b_1$ and $b_2$ of $S^2$ using your triangulation and the matrices $\partial_2$ and $\partial_1$ you have constructed. Give a basis for the homology vector spaces. 

{\hspace{14cm} [7 marks]}


\vspace{1cm}


{\hspace{13cm} [Total 20 marks]}



\end{enumerate}










% 3 *******************************************************************
\newpage

\nextq


\begin{enumerate}


\item
Give the definition of a \emph{real} manifold.

{\hspace{14cm} [3 marks]}



\item Use Stereographic projection to give an Atlas on the $n$-sphere, $S^n$. Check this satisfies the definition of a manifold you gave above.


{\hspace{14cm} [5 marks]}



\item Let a Lie group $G$ have a transitive action on a compact manifold $\mathcal{M}$. Take a point $p_0 \in \mathcal{M}$, and its stabilizer subgroup $H_{p_0}$. Construct a smooth map from the coset manifold $G/H_{p_0}$ to  $\mathcal{M}$ which is \emph{invertible}. Be sure to explain \emph{why} it is invertible. What is the relationship between $\mathcal{M}$ and the coset $G/H_{p_0}$?


{\hspace{14cm} [6 marks]}


\item Show that the matrix group $U(n+1)$ has a \emph{transitive} group action on $S^{2n+1}$. \\
( \emph{Hint:} You may assume the fact that for vectors $\mathbf{u}, \mathbf{v}$ in $\mathbb{C}^{n+1}$ such that $\mathbf{v}^\dagger \cdot \mathbf{v} = \mathbf{u}^\dagger \cdot \mathbf{u}$ one may always find a matrix $\mathbf{M} \in U(n+1)$ so that $\mathbf{v} = \mathbf{M}\cdot \mathbf{u}$. )

{\hspace{14cm} [3 marks]}


\item Use this group action to write the manifold $S^{2n+1}$ as a coset $G/H$ where you should determine $G$ and $H$.

{\hspace{14cm} [3 marks]}


\vspace{1cm}


{\hspace{13cm} [Total 20 marks]}


\end{enumerate}






% 4 *******************************************************************
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\nextq



\begin{enumerate}


\item Using a coordinate basis define the exterior derivative $d$ of an $r$-form $\omega$. Show explicitly that $d$ is nilpotent.


{\hspace{14cm} [4 marks]}


\item Take $\mathcal{M}$ to be an $m$-dimensional Riemannian manifold. Using a coordinate basis define the Hodge star, $\star \omega$, of an $r$-form $\omega$. Show that $\star \star \omega = \pm \omega$ where you should determine the sign $\pm$ in terms of $m$ and $r$.


( \emph{Hint:} $(\det{g_{\mu\nu}}) \epsilon^{\alpha_1 \ldots \alpha_r \alpha_{r+1} \ldots \alpha_m} \epsilon_{\alpha_1 \ldots \alpha_r \beta_{r+1} \ldots \beta_m} = r! (m-r)! \delta^{[ \alpha_{r+1}}_{~\beta_{1+r}}\delta^{ \alpha_{r+2}}_{~\beta_{r+2}} \ldots \delta^{\alpha_m]}_{~\beta_m} \quad$ )



{\hspace{14cm} [5 marks]}


\item Use differential forms to write the equations of electromagnetism (EM) in terms of the field strength 2-form $F$ and current 1-form  $j$. Show these equations imply the current is co-closed - what does this imply physically?

{\hspace{14cm} [3 marks]}



\item Given an $(m-2)$-chain $b$ on the manifold $\mathcal{M}$ we may define a quantity $Q \equiv \int_b \star F$ in terms of the EM field strength $F$. Suppose $b$ is a \emph{boundary} chain and that $Q$ vanishes. Then what condition is placed on the current $j$?


{\hspace{14cm} [3 marks]}


\item Given a $2$-chain $c$ we may define a quantity $Q' \equiv  \int_c  F$  in terms of the EM field strength $F$. Suppose we take $\mathcal{M} = \mathbb{R}^m$, a general field strength $F$ obeying the EM equations, and $c$ to be a \emph{cycle} chain. Then show that $Q'$ always vanishes. 

\noindent
Suppose now we remove the origin point so that $\mathcal{M} = \mathbb{R}^m - \{ 0 \}$. Then for which dimensions $m$ does $Q'$ still vanish for a general $F$ solving the EM equations and for all choices of cycle $c$?

{\hspace{14cm} [5 marks]}



{\hspace{13cm} [Total 20 marks]}


\end{enumerate}







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