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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, Tuesday May 3, 2011
\\[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 What is \emph{homology}? What is \emph{cohomology}? 

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


\item Show in detail how to compute the homology vector spaces for $S^1$ and give representatives for a basis using an explicit triangulation. 

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


\item Show in detail how to compute the cohomology vector spaces for $S^1$ using forms and give explicit representatives for a basis.


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

\item 
Consider a compact manifold $\mathcal{M}$ without boundary. Write down an inner product between the $r$-th homology and $r$-th cohomology vector spaces on $\mathcal{M}$. Explicitly show that it is independent of the choice of representatives.


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


\item Poincar\'e proved that this inner product is non-degenerate. Explicitly check that $\Lambda$ is not degenerate for the example of $S^1$.



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



\vspace{1cm}


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

\end{enumerate}



% 2 *******************************************************************
\newpage

\nextq


\begin{enumerate}


\item Give the definition of a \emph{Lie group}, its \emph{left invariant vector fields}, and its \emph{structure constants}. 

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


\item Define the General Linear matrix group $GL(n,\mathbb{R})$. Using the elements of the $GL(n,\mathbb{R})$ matrix as coordinates, show how to write a vector field in the coordinate basis. Using this coordinate basis compute the components of the Lie bracket $[A, B]$  in terms of the components of the vector fields $A$ and $B$.


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

\item

Again take these coordinates and consider three elements $h,g,g' \in GL(n,\mathbb{R})$. 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} [2 marks]}

\item Using your answer above, compute a basis of left invariant vector fields on $GL(n,\mathbb{R})$.

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

\item Explicitly confirm the Lie bracket closes on the left invariant vector fields you computed in part iv).


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



\vspace{1cm}

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



\end{enumerate}








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

\nextq


\begin{enumerate}

\item


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

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

\item 

Give an altas on the real manifold $S^1$. Explicitly show your atlas indeed satisfies the conditions needed for a real manifold.


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

\item 

Given two manifolds $\mathcal{M}$ and $\mathcal{N}$ and atlases on these, describe the product manifold $\mathcal{M} \times \mathcal{N}$ and give an atlas for it.


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


\item

By describing the real $2$-torus, $T^2$,  as a product manifold $S^1 \times S^1$, explicitly give an altas for it in terms of your previous $S^1$ altas, checking it satisfies the conditions needed for a real manifold.

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


\item

Show that $T^2$ can be made into a complex manifold by explicitly giving an atlas that satisfies the requirements for a complex manifold that you gave in part i).

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


\vspace{1cm}

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



\end{enumerate}






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

\nextq


\begin{enumerate}

\item

Consider an $m$-dimensional  Riemannian manifold $\mathcal{M}$ (ie. it has a metric with Euclidean signature) which is orientable, compact and without boundary.

Define i) the Hodge star $\star$, ii) the inner product $( \cdot , \cdot )$ between two $n$-forms, and iii) the adjoint exterior derivative $d^\dagger$ on $\mathcal{M}$.

\vspace{0.2cm}
\emph{In these definitions you may ignore any overall $\pm$ signs.}


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

\item

Ignoring any overall $\pm$ signs, explicitly show that $( d \beta, \alpha) = (  \beta, d^\dagger \alpha)$ for some $\alpha \in \Omega^n$ and $\beta \in \Omega^{n-1}$.


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

\item 

Consider the metric tensor $g$ and a chart so that we may write $g$ in a coordinate basis, $g = g_{\mu\nu}(x) dx^\mu \otimes dx^\nu$. Show how the components change if we describe the tensor in the coordinates basis of a second overlapping chart.
Show that at a single point $p$ we may choose coordinates so that the metric components at $p$ are equal to the Euclidean metric $\delta_{\mu\nu}$. These are called `Riemann normal' coordinates.

\vspace{0.2cm}
\emph{You may assume that given a symmetric matrix $M$ with positive eigenvalues there always exists a matrix $b$ such that $b^T \cdot M \cdot b = \mathbf{1}$.}


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

\item 
Use `Riemann normal' coordinates to argue that for a form $\omega \in \Omega^n$  then $(\omega, \omega) \ge 0$ with equality if and only if $\omega$ vanishes. 

\vspace{0.2cm}
\emph{You may find the following identities useful;}
\begin{eqnarray} 
\epsilon^{\mu_1\ldots \mu_m} & = & (\det g)^{-1} \; \epsilon_{\mu_1 \ldots \mu_m} \nonumber \\
\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)!  (\det g_{\mu\nu})^{-1} \; \delta^{[ \alpha_{r+1}}_{~\beta_{1+r}}\delta^{ \alpha_{r+2}}_{~\beta_{r+2}} \ldots \delta^{\alpha_m]}_{~\beta_m} \nonumber
\end{eqnarray}


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


\item 

Define the Laplacian operator $\triangle$ in terms of $d$ and $d^\dagger$. Consider the Laplace equation $\triangle \omega = 0$ for an $n$-form $\omega \in \Omega^n$ on $\mathcal{M}$. Using the previous results show that solutions to the Laplace equation obey $d^\dagger \omega = d \omega = 0$.


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

\vspace{1cm}

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


\end{enumerate}






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