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%%from  Rudro Rana Biswas
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\begin{document}
\begin{center}

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

{\Large\bf  Physics 220 Symmetries Fall 2024}\\
{\Large\bf Assignment 7 \answer{--~~~ Solutions} }
\end{center}




\noindent
\hfill {\bf Due 11:59pm { Thursday, November 14, 2024}} 


\hrule


\def\jj{{\bf j}}
\def\kk{{\bf k}}

\def\gsu{{\mathsf{su}}}
\def\gSU{{\mathsf{SU}}}
\def\gso{{\mathsf{so}}}
\def\gSO{{\mathsf{SO}}}


\begin{enumerate}

\item {\bf Non-degeneracy of braiding in the 3d quantum double model.} 

The title of this problem is a description of the physics situation where I encountered the following nice group theory fact.  
You don't need to know what it means to do the problem!

\begin{enumerate}
\item Prove the following identity using character orthogonality: 
for any finite group $G$,
$$ \sum_\alpha |Z_\alpha|^{n-2} = \sum_{a_1 \cdots a_n} N_{a_1 \cdots a_n}^1 N_{\bar a_1 \cdots \bar a_n}^1 $$
where $Z_\alpha$ is the centralizer of an element of conjugacy class $\alpha$, 
$a_1 \cdots a_n$ are irreps of $G$, 
and $N_{a_1 \cdots a_n}^1$ is the number of times $R_{a_1} \otimes R_{a_2} \otimes \cdots R_{a_n}$ fuses to the identity\footnote{More precisely, 
$N_{a_1 \cdots a_n}^1$ is the dimension 
of the `fusion space' $V_{a_1 \cdots a_n}^1$ in 
$$R_{a_1} \otimes R_{a_2} \otimes \cdots R_{a_n} = {\bf 1} \otimes V_{a_1 \cdots a_n}^1 \oplus \cdots $$
where ${\bf 1}$ is the trivial rep of $G$.}.
The symbol $\sum_{a_1\cdots a_n}$ means $n$ independent sums over all the irreps of $G$.

[Cultural remark about the title of the problem, ignore if you want: 
The quantum double model is an exactly solvable lattice model with topological order.  
There is one for every finite group, and it can be put on any graph with the extra information of which faces (2-cells) are bounded by which edges of the graph.
If we put this model on a 3d lattice,
the LHS of this formula has an interpretation as the number of excitations shaped like an $n$-hole donut.
The RHS has an interpretation as the number of collections of particles that can be braided non-trivially around such an object.  
The idea of ``braiding non-degeneracy" in a system with topological order is that each topologically non-trivial excitation should 
be detectable from a distance by moving some {\it other} excitation around it.]


\item What is $N_{abc}^1$ for the group $S_3$?  
Find a group where $N_{abc}^1$ takes a value other than $0$ or $1$ for some choice of $abc$.



\end{enumerate}

\item {\bf Control-Z algebra.} [Bonus problem]


Recall that 
$$\CZ_{ij} \equiv e^{ \ii {\pi\over 4 }  (1 - Z_i) (1-Z_j) }  = \CZ_{ji}$$ 
is the control-$Z$ operation.
\begin{enumerate}
\item 
Check that in the $Z$ basis ($Z \ket{s} = s\ket{s}, s=0,1$) $\CZ$ acts as $Z$ on the second bit only if the first bit is 1:
$$ \CZ\ket{0s} = \ket{0s}, \CZ \ket{1s} = (-1)^s \ket{1s}, ~~\bar 0 \equiv 1, \bar 1 \equiv 0.$$


\item 
Check that 
\be\label{eq:CZ} X_i \CZ_{ij} X_i = \CZ_{ij} Z_j .\ee


\end{enumerate}

\item {\bf Practice at exponentiating matrices.}

\begin{enumerate}

\item Check that the matrices 
$ (J_i)^j{}_k = -\ii \eps_{ijk}$ satisfy the $\gsu(2)$ algebra.
These are the generators of the ${\bf 3}$-dimensional representation of $\gSU(2)$.



\item What are the eigenvalues of $\hat n \cdot \vec J$, where $\hat n$ is a unit vector?


\item 
Find an explicit expression for $ e^{ \ii \theta \hat n \cdot \vec J } $,
where $\vec J$ is the vector of matrices above.
Check that your answer is reasonable when $\hat n = \hat z$.

Hint: Show that $(\hat n \cdot \vec J)^3 = \hat n \cdot \vec J$ and use this to sum the exponential series.
One way to see this is to use the previous part to conclude that  $(J^3)^2$ is a projector.


\end{enumerate}
\item{\bf Conjugacy classes of rotations.}

Consider a rotation matrix $O \in \gSO(n)$.
For purposes of this problem, take as given the fact that conjugation $O \to V O V^{-1}$ changes the axis of rotation, but not the angle.

\begin{enumerate}
\item What are the eigenvalues of a rotation matrix?  


\item What are the eigenvectors of a rotation matrix?

\item Given a rotation matrix, how can you determine the angle of rotation?



\item Given a rotation matrix, how can you determine the plane of rotation?


\end{enumerate}

Hint: solve this problem in two dimensions first.  Then for general $n$ use conjugation 
to put the matrix in a canonical form.



\item {\bf Counting solutions to algebraic equations in a finite group.} [Bonus problem]

\begin{enumerate}
\item{} {\cor [This part you did already on the previous homework]} Starting from the definition of the Frobenius-Schur indicator for a representation $R_a$, 
find a formula for $\sigma(h)$, the number of solutions of the equation $g^2 = h$ in a finite group $G$.
Hint: multiply the definition by $\chi_a^\star(h)$, sum over $a$, and use character orthogonality.

Show that the number of square roots $\sigma(h)$ is a class function.

\item Redo the previous problem to count solutions of $g^3 = h$ in $G$. 
Write the answer in terms of the object 
$$\eta^{(3)}_a \equiv {1\over |G|} \sum_{g \in G} \chi_a(g^3) .$$
What's the number of cube roots of $(12)$ in $S_3$?  


\item{} [Super-bonus problem] Show that $\eta^{(3)}_a$ defined above is nonzero only when there is a cubic invariant of the representation $R_a$,
that is $R_a \otimes R_a \otimes R_a = {\bf 1} \oplus \cdots$.
Does it say something more specific (\eg~about how many cubic invariants there are)?  
Is $\eta^{(3)}_a$ always an integer?


\item{} Count homomorphisms of the trefoil knot group $\vev{g, h| h^2 = g^3}$ to $S_3$.
[Hint: $ {1\over |G|} \sum_g D^a(g^2) $ is an intertwiner.]


\end{enumerate}
\end{enumerate}

\end{document}
