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%%from  Rudro Rana Biswas
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bookmarks={false}, pdfauthor={John McGreevy}, pdftitle={Yay, physics!}]{hyperref}
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\begin{document}
\begin{center}

University of California at San Diego -- 
Department of Physics --
Prof.~John McGreevy

{\Large\bf  Physics 215B QFT Winter 2025}\\
{\Large\bf Assignment 9 \answer{--~~~ Solutions} }
\end{center}


\noindent
%Posted September 17, 2015 
\hfill {\bf Due  11:59pm Tuesday, March 11, 2025}




\bigskip
\hrule
\begin{enumerate}


\item {\bf Gauge theory brain-warmers.} 

Please do 4 of the following 6 problems.  The rest are bonus material.

\begin{enumerate}
\item Show that the {\it adjoint} representation matrices
$$ \(T^{A}\)_{{B}C} \equiv - \ii f_{ABC} $$
furnish a $\dim \gG$-dimensional representation of the Lie algebra 
$$ 
[ T^A, T^B] = \ii f_{ABC} T^C ~~.
$$
Hint: commutators satisfy the Jacobi identity
$$ [A, [B,C]] + [B, [C,A]] + [C, [A,B]] = 0 . $$

\item{}  Show that if $(T_A)_{ij}$ are generators of a Lie algebra in some unitary representation $R$, then
so are $ - (T_A)^\star_{ij}$.  Convince yourselves that these are the generators of the complex conjugate representation ${\bar R}$.

\item{}  Show that in a basis of Lie algebra generators where $ \tr T^A T^B  =\lambda \delta^{AB}$, 
the structure constants $f_{ABC}$ are completely antisymmetric.




\item From the transformation law for $A$, show that the non-abelian field strength
transforms in the adjoint representation of the gauge group.



\item Show that 
$$ \tr F \wedge F =d \tr \( A \wedge dA + {2\over 3 } A \wedge A \wedge A \) .$$
Write out all the indices I've suppressed.



\item{} [Bonus] If you are feeling under-employed, find $\omega_{2n-1} $  such that 
$ \tr F^n = d \omega_{2n-1}$.


\end{enumerate}


\item {\bf The field of a magnetic monopole.}

We saw that $F = dA $ implies (when $A$ is a smooth, globally well-defined differential form) that $dF=0$, which means no magnetic charge.  
If $A$ is singular, $dF$ can be nonzero.  Moreover, by a gauge transformation we can move the singularity around and hide it, so that the field is everywhere non-singular.  

A magnetic monopole of magnetic charge $g$ is defined by the condition that $ \int_{S^2} F = g $,
where $S^2$ is any sphere surrounding the monopole.
If the system is spherically symmetric, 
we can write 
$$ F = {g \over 4 \pi} d \cos\theta d \varphi .$$
(In this problem, we'll work on a sphere at fixed distance from the monopole.)
\begin{enumerate}
\item Show that the vector potential 
$$A_N = {g \over 4 \pi} \(\cos \theta-1\) d\varphi $$
gives the correct $F= dA$.
Show that it is a well-defined one-form on the sphere except at the south pole $ \theta = \pi$.


\item Show that the one-form
$$A_S = {g \over 4 \pi} \(\cos \theta+1\) d\varphi $$
also gives the correct $F= dA$.
Show that it is well-defined except at the north pole $ \theta = 0$.

\item Near the equator both $A_{N,S}$ are well-defined.
Show that 
{\it as long as $ e g \in 2 \pi \IZ$}, 
these two one-forms differ by a gauge transformation
$$ A_S - A_N = {1\over \ii e} g^{-1}(\theta,\varphi) d g(\theta,\varphi) $$
for $g(\theta,\varphi)$ a $\gU(1)$-valued function on the sphere, well-defined away from the poles.


\end{enumerate}

\item {\bf Wilson loops in abelian gauge theory at weak and strong coupling.}

\begin{enumerate}

\item At weak coupling, the Wilson loop expectation value
is a gaussian integral.  
In $D=4$, study the continuum limit of a rectangular loop 
with time extent $T \gg R$, the spatial extent.
Show that this reproduces the Coulomb force.





\item Consider the weak coupling calculation again for a Wilson loop coupled to a massive
vector field.  Show that this reproduces an exponentially-decaying force between external static charges.

%\answer{The photon propagator is 

\item{} {[bonus problem]} Compute the combinatorial factors
in the first few terms of the strong-coupling expansion 
of the Wilson loop in $\gU(1)$ lattice gauge theory.



\item{} {[bonus problem]} Consider the case of lattice gauge theory in two spacetime dimensions.
In this case, show that the plaquette variables 
$W(\partial \Box) = \prod_{\ell \in \partial \Box} U_\ell$ 
are actually 
independent variables.
%(since the relations between them arise from the boundaries of 3-volumes).


%\item Using the strong coupling expansion, 
%estimate the mass of glueball 
%states 


\end{enumerate}



\end{enumerate}
\end{document}