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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, 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]
 **********************************
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% 1 *******************************************************************
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\nextq

\begin{enumerate}


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

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


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


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

\item 
Consider a compact manifold $\mathcal{M}$ without boundary. Write down the 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 used in the definition.


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


\item Poincar\'e proved that this inner product is non-degenerate. Explicitly check that this inner product is not degenerate for the example of $S^1$. Using your basis for the homology vector spaces from part i), give the dual basis for the cohomology with respect to this inner product in terms of your basis from part ii).



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



\vspace{1cm}


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

\end{enumerate}



% 2 *******************************************************************
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\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} [6 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} [5 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 *******************************************************************
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\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, define 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

Using your previous answer to part iv), show that $T^2$ can be made into a complex manifold by explicitly giving an atlas that satisfies the definition 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 the Hodge star, $\star$, and the inner product $( \cdot , \cdot )$ between two $n$-forms on $\mathcal{M}$.


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

\item

Ignoring any overall $\pm$ signs, give an expression for the adjoint exterior derivative, $d^\dagger$, and 
explicitly confirm that $( d \beta, \alpha) = (  \beta, d^\dagger \alpha)$ for some $\alpha \in \Omega^n$ and $\beta \in \Omega^{n-1}$ on $\mathcal{M}$.


{\hspace{14cm} [4 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 $\delta_{\mu\nu}$. 

\vspace{0.2cm}
\emph{You may assume that given a real 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 the coordinates from part iii) 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 from parts ii) and iv) 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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