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\begin{document}

\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, Wednesday May 5, 2010
\\[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]
 **********************************
\end{center}

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%$\hbar = 1$, $c=1$.}





% 1 *******************************************************************
\newpage

\nextq

\begin{enumerate}


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

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

\item Using `stereographic projection' explicitly construct a 2 chart atlas for the real manifold $S^n$, for general $n$, and confirm that it satisfies the definition of part i).

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

\item Give an explicit construction of the complex manifold $\mathbb{CP}^m$, for general $m$, using an $(m+1)$ chart atlas. Again confirm that it satisfies the definition given in part i).

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

\item Using your answers above, show that the real manifold $S^2$ can be thought of as the complex manifold $\mathbb{CP}^1$.

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



\vspace{1cm}


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

\end{enumerate}



% 1 *******************************************************************
\newpage

\nextq

\begin{enumerate}

\item When is a chain a \emph{cycle} and when is it a \emph{boundary}? How do cycles and boundaries give rise to \emph{homology}?

When is a form \emph{closed} and when is it \emph{exact}? How do closed and exact forms give rise to \emph{cohomology}?

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


\item Give a simplicial complex that triangulates the topological space $S^1$.

{\hspace{14cm} [2 marks]}

\item Construct the boundary operator $\partial_1$ as a matrix which acts on your triangulation of $S^1$. 
Confirm this is a nilpotent operator. 

Give the kernel of this matrix, and use this to compute the Betti numbers $b_0, b_1$ of $S^1$.

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


\item
Show by \emph{explicit} calculation with 0 and 1-forms that for $S^1$ the cohomology vector spaces $H^0$ and $H^1$ are both 1-dimensional. Hence give representatives for their non-trivial classes.

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


\item
Comment on the relation between the dimensions of $H^{0}$ and $H^{1}$ and on the Betti numbers $b_{0}$ and $b_1$ for $S^1$.

{\hspace{14cm} [1 marks]}


\vspace{1cm}


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



\end{enumerate}








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

\nextq


\begin{enumerate}
\item Give the definition of a \emph{Lie group}. Define its \emph{left invariant vector fields}.

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

\item Show that the Lie bracket closes on the left invariant vector fields. Give the definition of the \emph{structure constants}?

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


\item We may parameterize the matrix group $SU(2)$ as,
\begin{eqnarray}
M = \left( \begin{array}{cc} a & b \\ -\bar{b} & \bar{a} \end{array} \right) \nonumber
\end{eqnarray}
where $a = t + i x$, $b = y + i z$ with $t,x,y,z\in \mathbb{R}$, and $t^2 + x^2 + y^2 + z^2 = 1$.

Take coordinates $\{x,y,z\}$ on $SU(2)$, and elements $h,g,g' \in SU(2)$. If these are related by the left action, $g' = L_g h$, compute the coordinates of $g'$ in terms of the coordinates of $h$ and $g$.

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

\item Using your answer above, compute a basis of left invariant vector fields on $SU(2)$.

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



\item 
Choose 2 of your basis left invariant vector fields. Explicitly compute their Lie bracket. Confirm the result is again left invariant.

{\hspace{14cm} [2 marks]}



\vspace{1cm}

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



\end{enumerate}






% 4 *******************************************************************
\newpage

\nextq


\begin{enumerate}



\item Write down the equations of electromagnetism in terms of the field strength 2-form $F$ and the current 1-form $j$ in the language of differential forms. What condition must $j$ satisfy?

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

\item State Poincare's Lemma. Using this show that $F$ may be written in terms of a 1-form $A$ as $F = d A$ for electromagnetism on the manifold $\mathbb{R}^4$. 

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

\item 
Take the Minkowski metric on $\mathbb{R}^4$ in spherical polar coordinates, so,
\begin{equation}
g = - dt \otimes dt + dr \otimes dr + r^2 d\theta \otimes d\theta + r^2 \sin^2{\theta} d\phi \otimes d\phi . \nonumber
\end{equation}
Keeping the metric unchanged, let us now remove the line $r = 0$ from $\mathbb{R}^4$ to form a new manifold which we denote ${\mathcal{M}}$.

Consider the field strength $F = q \sin{\theta} \,d \theta \wedge d \phi$ for some constant $q$ defined on $\mathcal{M}$. Confirm that $F$ satifies the vacuum (ie. $j = 0$) electromagnetic equations on ${\mathcal{M}}$. 

\emph{(This is the solution for a static magnetic monopole with world-line $r=0$.)}

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

\item 
Compute the integral of $F$ over the 2-sphere $t = 0$, $r = r_0$ for some constant $r_0$.  

The answer should be non-zero. Why does this confirm that $F$ is \emph{not} exact?

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

\item 
Consider a field strength $F'$ solving the vacuum electromagnetic equations now on $\mathbb{R}^4$ rather than $\mathcal{M}$.

Using your answers to part ii) and iv) show that one cannot find a smooth 2-form field strength tensor field $F'$ on $\mathbb{R}^4$ such that $F' = q \sin{\theta} \,d \theta \wedge d \phi$ for $r > 1$. 

{\hspace{14cm} [2 marks]}



\vspace{1cm}

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


\end{enumerate}






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