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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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 \put(1,1){\framebox(148,8){\bf TP.2
--- Differential Geometry}}
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\\[3cm]
 2:00 -- 5:00, Wednesday May 2, 2007
\\[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}




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

\nextq

This question concerns manifolds and fiber bundles.

\begin{enumerate}

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

{\hspace{15cm} [6 marks]}

\item Show that $S^2$ is a manifold by using the `stereographic projection' to explicitly construct a 2 chart  atlas, and show the transition functions satisfy any requirements stated in part i) above.

{\hspace{15cm} [5 marks]}


\item Recall that the complex manifold $\mathbb{CP}^1$ can be thought of as the set of undirected complex lines in $\mathbb{C}^2$ that pass through the origin.  Give a 2 chart atlas on $\mathbb{CP}^1$ 
and show the transition functions satisfy any requirements stated in part i) above.

{\hspace{15cm} [5 marks]}


\item Using your answers above and combining real coordinates into a complex coordinate appropriately, confirm that $S^2$ can be thought of as the complex manifold $\mathbb{CP}^1$.

{\hspace{15cm} [4 marks]}



\item Explain what the tangent bundle is for a general real manifold. In particular you should explain the projection map,  the local trivializations and the transition functions. 

{\hspace{15cm} [5 marks]}





{\hspace{14cm}[Total 25 marks]}

\end{enumerate}


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

\nextq

This question discusses Lie groups and Riemannian geometry.

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

{\hspace{15cm} [4 marks]}

\item We may parameterize the matrix group $SU(2)$ as,
\begin{equation}
M = \left( \begin{array}{cc} a & b \\ -\bar{b} & \bar{a} \end{array} \right) 
\end{equation}
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{15cm} [3 marks]}

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

{\hspace{15cm} [7 marks]}

\item For a general Lie group $G$, we denote the basis of left invariant vector fields as $\{ L_a \}$. 
A set of one-form fields, $\{ \theta^a \}$, can be constructed such that $< \theta^a , L_b > |_g = \delta^a_{~b}$ for all $g \in G$. Show that these one-forms satisfy,
\begin{equation}
L_g^* ( \theta^a |_g ) = \theta^a |_e .
\end{equation}

{\hspace{15cm} [4 marks]}

\item 
Using the expression $d\omega(X,Y) = X[\omega(Y)] - Y[\omega(X)] - \omega([X,Y])$ for vector fields $X,Y$ and one-form field $\omega$, derive the Cartan-Maurer equation,
\begin{equation}
d \theta^a + K_{ab}^{~~c} \theta^a \wedge \theta^b = 0
\end{equation}
where you must find the constants $K_{ab}^{~~c}$. 

{\hspace{15cm} [4 marks]}

\item 
Take the Riemannian metric, $g = \delta_{ab} \theta^a \otimes \theta^b$ on G, so $\{ \theta^a \}$ form a non-coordinate basis. Give the Cartan structure equation relating $\theta^a$, the torsion 2-form, $T^a$, and the connection 1-form, $\omega^a_{~b}$. Find the connection 1-form such that the torsion vanishes.

{\hspace{15cm} [3 marks]}


{\hspace{14cm} [Total 25 marks]}



\end{enumerate}



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

\nextq

This question concerns the formulation of Electromagnetism using differential forms.

\begin{enumerate}

\item Take $\mathcal{M}$ to be an $m$-dimensional pseudo-Riemannian manifold with signature (1,m-1). For  $\omega \in \Omega^r(\mathcal{M})$, define $\star \omega$ by giving its components in a coordinate basis. Show that $\star \star \omega = \pm \omega$ where you should determine the sign $\pm$ in terms of $m$, $r$.

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} $

{\hspace{15cm} [7 marks]}

\item We define $d^\dagger \omega \equiv (-1)^{m r + m} \star d \star \omega$ for $\omega \in \Omega^r(\mathcal{M})$. For $F \in \Omega^2(\mathcal{M})$ and $j \in \Omega^1(\mathcal{M})$, the equations,
\begin{equation}
d F = 0 , \quad d^\dagger F = j
\end{equation}
define electromagnetism on $\mathcal{M}$. What are the physical interpretation of $F$ and $j$?

Show that $d^\dagger j = 0$ - what does this represent physically? Show that for an $(m-1)$-chain $V$, with boundary $S = \partial V$, $Q = \int_S \star F$ can be written in terms of an integral involving $j$ rather than $F$.

{\hspace{15cm} [4 marks]}

\item Consider the manifold $\mathcal{M} = \mathbb{R}^4$. Using Poincare's lemma explain why we may always write $F = dA$, where $A \in \Omega^1(\mathbb{R}^4)$. 


{\hspace{15cm} [3 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 $\tilde{\mathcal{M}}$.

Compute $F$ for $F = d A$ with $A = (q/r) dt$ and constant $q$. Confirm it satisfies the \emph{vacuum} electromagnetic equations, $dF = d^\dagger F = 0$ on $\tilde{\mathcal{M}}$. This is the solution for a static point electric charge whose world-line is $r=0$.

{\hspace{15cm} [5 marks]}

\item

Show that any $F_{vac}$ that solves the vacuum equations on $\tilde{\mathcal{M}}$ gives a new `dual' solution $F_{dual} = \star F_{vac}$. Write down the dual solution to the point electric charge solution above. This is the field of a magnetic monopole with world-line $r=0$. 


{\hspace{15cm} [3 marks]}

\item 
By considering the integral of F over an appropriate 2 sphere, show that $F$ is not an exact 2 form for the magnetic monopole solution.

{\hspace{15cm} [3 marks]}


{\hspace{14cm} [Total 25 marks]}


\end{enumerate}




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

\nextq

This question concerns homology and de Rham cohomology.

\begin{enumerate}

\item Given an $r$-chain $c\in C_r$ explain the conditions for $c$ to be a) an $r$-cycle, ie. $c \in Z_r$, and b) an $r$-boundary ie. $c \in B_r$. Show that an $r$-boundary is always an $r$-cycle.

Likewise for an $r$-form $\omega\in\Omega^r$ explain the conditions for it to be a) an $r$-cocycle ie. $\omega \in Z^r$ or b) an $r$-coboundary  ie. $\omega \in B^r$. Show that an $r$-coboundary is always an $r$-cocycle.


{\hspace{15cm} [4 marks]}


\item Explain what is meant by the equivalence class $[ \omega ]$ for $\omega \in Z^r$ and define the cohomology vector space $H^r$.


{\hspace{15cm} [4 marks]}


\item We define the inner product;
\begin{eqnarray}
\Lambda: H_r \times H^r & \rightarrow & \mathbb{R} \\ \nonumber
[c]\; , \;\;[\omega] & \rightarrow &  \Lambda([c],[\omega]) \equiv \int_c \omega 
\end{eqnarray}
Show this definition does not depend on the choice of representatives $c,\omega$, ie. $\Lambda([c],[\omega]) = \Lambda([c'],[\omega'])$ if $[c]=[c']$ and $[\omega]=[\omega']$.


{\hspace{15cm} [4 marks]}


\item Define a 3-torus as $\{ (e^{i \theta_1},e^{i \theta_2},e^{i \theta_3}) \in \mathbb{C}^3 \; \forall \; \theta_1,\theta_2,\theta_3 \in \mathbb{R} \}$. Explicitly give a basis for $H_1(T^3)$ and a \emph{dual} basis for $H^1(T^3)$ with respect to the inner product $\Lambda$.


{\hspace{15cm} [5 marks]}


\item Assume that $\mathcal{M}$ is compact and hence without boundary, so $\partial \mathcal{M} = 0$. We may define a new inner product;
\begin{eqnarray}
<\cdot,\cdot>: H^r \times H^{m-r} & \rightarrow & \mathbb{R} \\ \nonumber
[\omega]\; , \;\;[\eta] & \rightarrow &  <[\omega],[\eta]> \equiv \int_\mathcal{M} \omega \wedge \eta
\end{eqnarray}
Show this definition does not depend on the choice of representatives. (Poincar\'e has shown that on a compact manifold this inner product is non-degenerate, and hence the dimension of $H^r$ is equal to the dimension of $H^{m-r}$.)

{\hspace{15cm} [4 marks]}

\item You have specified a basis for $H^1(T^3)$ in part iv) above. Now compute a \emph{dual} basis for $H^2(T^3)$ with respect to the inner product $<\cdot,\cdot>$.

{\hspace{15cm} [4 marks]}

{\hspace{14cm} [Total 25 marks]}



\end{enumerate}


\newpage






\end{document}

