TITLE: Harmonic function as real part of any analytic function
QUESTION [6 upvotes]: I denote with $\phi(x,y)$ a hamornic function. I would like to show that $\phi$ is the real part of any analytic function $f(z)$ of the form $$f(z)=2\phi\left(\frac{z+1}{2},\frac{z-1}{2i}\right)-\phi(1,0)+ic$$ where c is a real constant and provided that the RHS exists.
My idea was to consider $$g(z,\bar{z})=\phi\left(\frac{1}{2}(z+\bar{z}),\frac{1}{2i}(z-\bar{z})\right)+i\psi\left(\frac{1}{2}(z+\bar{z}),\frac{1}{2i}(z-\bar{z})\right)$$
and set this equal to $f(z)=g(z,\bar{z})$ with $\bar{z}=1$, but is this helpful?
Remark: In the end I would like to find an analytic function whose real part is $\tan^{-1}y/x$, my lecture notes states that the obvious function is $-i\log z$ but I do not see why this is so obvious.

REPLY [0 votes]: Hello I am very late but I 'd like to say something :)
Let $D\subset$ be a domain in the complex plan and $h$ be harmonic in $D$ then there is $f\in O(D)$ ( holomorphic functions) so that $h=\log|f|.$
\begin{proof}:
For $m>0$ let $(D_m)_m$ be a sequence of open disc so that $D_m\cap D_{m+1}\not=\varnothing$, $\bar D_m\subset D$ and $\cup_m D_m=D.$\
For each $m>0$ we construct (by induction) a holomorphic function $f_m$ defined on $ D_1\cup\cdots\cup D_m$ so that $\Re(f_m)=h.$\
For $m=1$, as $h$ is harmonic, there is $f_1\in O(D_1)$ so that $h=\Re(f_1)$ in  $D_1.$
For $m=2$ there is $g\in O(D_2)$ so that $h=\Re(g)$ in $D_2.$ Since $f_1$ and $g$ are holomorphic and $\Re(g)=\Re(f)$ on $D_1\cap D_2$ we infer that there is $c_1\in R$ so that $f_1=g+ic_1$ on $D_1\cap D_2.$ Set $f_2=f_1$ in $ D_1$ and $f_2=g+ic_1$ in $ D_2$ it follows that $f_2\in O( D_1\cup D_2).$ By the same way we construct $f_m,$ $m=3,4,\cdots$\
For $m>0$ set $F_m=e^{f_m}$. The squence $(F_m)_m$ has a subsequence that converges locally uniformly in $D.$ Indeed let $K\subset D$ be a compact ball. Observe that $|F_m|=e^h$ in $D_m$ for all $m>0$ we infer that $F_m$ is uniformly bounded on $K.$ By Montel's theorem $(F_m)_m$ has a subsequence $(F_{m_k})_k$ that converges on $K$. In fact $(F_{m_k})_k$ converges locally uniformly in $D.$ \ \
To prove that $(F_{m_k})_k$ is convergent, it is enough to show that :\
1) it does not contain divergent subsequences\
2) all its convergent subsequences converge to the same limit.  \ \
First assume that there are a compact set $K'$ and a subsequence $(F_{m_{k'}})$ of $(F_{m_{k}})$ which does not converge on $K'$. Let $U$ be a relatively compact domain  in $D$ containing $K'$ and $K$. By Montel theorem's theorem $(F_{m_{k}})$  has a subsequence which converges uniformly in $U$  to a holomorphic function $g$.  So there is $\epsilon>0$ so that :
(1) $\sup_{K'}|F_{m_{k'}}-g|>\epsilon$  for all $k'$ big enough.\
This is impossible since the sequence $(F_{m_{k'}})$ is uniformly bounded on $U$ and by Montel's theorem its has a subsequence which converges uniformly in $U$ to a function $g'$. As $g'=g$ on $K$ then they coincide in $U.$ Equality (1) can not hold in other words such a subsequence $(F_{m_{k'}})$ does not exist.
Second let $(F_{m_{k'}})$ be a subsequence of $(F_{m_{k}})$ which converges locally uniformly in $D$ to $g'$ and $(F_{m_{k''}})$ be a subsequence of $(F_{m_{k}})$ which converges locally uniformly in $D$ to $g''.$ As $g'$ and $g''$ are holomorphic in $D$ and coincide in $K$ then $g'\equiv g''$ in $D.$
For $z\in D$ set $f(z)=\lim_{k\rightarrow\infty} F_{m_k}(z)$ as the convergence is locally uniform we infer that $f$ is holomorphe and $|F_{m_k}|$ converges to $|f|$ as $k$ tends to infinite. Since $|F_{m_k}(z)|=e^{h(z)}$ for all $k$ we get $|f(z)|=e^{h(z)}$.
\end{proof}


TITLE: How to define quantum chaos?
QUESTION [3 upvotes]: I was told that quantum chaos is just a system whose Hamiltonian's classical version shows chaotic behavior. However, I just wondering

what happens when one eigenstate of this Hamiltonian evolves?
what the chaotic (nonlinear, of course) Hamiltonian looks like in quantum version (for Hamiltonian should be a linear one)?

I know that mean-field theory could reduce the linear Hamiltonian into a nonlinear form, but it is not an exact form.
Maybe a good start point is the BEC equation and GP equation. I hope you can help me here.

For the second question, after a whole year, I think the many-body interaction will directly introduce the nonlinearity. However I still have no idea of how the eigenstate will evolve under "quantum chaotic" Hamiltonian.

REPLY [2 votes]: Just apply the normal rules. If the Hamiltonian is time-independent, the eigenstate gains an overall phase related to its energy. If the Hamiltonian is time-dependent, then the time evolution operator can be obtained by exponentiation of the Hamiltonian, and that determines the evolution of the eigenstate.
It depends on the Hamiltonian. The coordinates in a classical Hamiltonian have corresponding operators. If all of the operators involved commute with each other, then you can just directly replace the classical coordinates with operators. If there are some non-commuting operators, you must re-derive the Hamiltonian, paying attention this time to the non-commutativity of the operators (whereas in the classical version, all coordinates commute).


TITLE: $N$ doors Monty hall type problem
QUESTION [0 upvotes]: In Peter office there are $N$ doors. Behind each door except
one hides a doll. Lucy know what is behind each door while Peter does not. Peter has to select a door and wins if there is no doll behind it. Peter selects a door, instead of opening that door, Lucy opens $k$ other doors behind each of which is a doll , with $1\le k\le N-2$.
Now Peter is given a choice to switch the door.

What is the probability for Peter to win if Peter does not switch
his initial choice?
probability if he switch his initial choice
prove that whatever are values of $N$ and $k$ , Peter should always
switch doors.

REPLY [0 votes]: Initial selection win probability is $\frac{1}{N}$.  What happens after that doesn't effect result of initial selection.  Win probability if he switches is then $\frac{N-1}{N}\times\frac{1}{(N-1-k)}$.


text:
\section{Applications}
\label{sec:examples}

In this section we show how to model different problems as {\rLCproblems{1}} with polynomial partial neighborhood systems. As a result, we obtain polynomial-time algorithms to solve these problems for bounded treewidth graphs. For double Roman domination, minimum chromatic violation and Grundy domination (and their variants), no such algorithms were previously known (until the date and to the best of our knowledge).
As regards the problems that were already known to be polynomial-time solvable for the before-mentioned classes, it is worth to mention how to restate these problems as {\rLCproblems{1}}, even when the time complexity of the proposed solution is worse than the best one known, because problems modeled this way can be easily modified or combined, adding global properties or more restrictions, or even dealing with some distance versions, and they can inspire the statement of other problems as {\rLCproblems{1}}.

Throughout this section, we will assume that, otherwise stated, the definitions
of $\NeighborhoodSet{v}{i}$ are for all $v \in V(G)$ and $i \in L_v$,
of $n \neighborhoodSum{v}{i} n'$ for all $v \in V(G)$, $i \in L_v$ and $n, n' \in \NeighborhoodSet{v}{i}$,
of $\toNeighborhood{v}{i}(u, j)$ for all $v \in V(G)$, $i \in L_v$, $u \in N_G(v)$ and $j \in L_u$,
and of $check_{v,i}(n)$ for all $v \in V(G)$, $i \in L_v$ and $n \in \NeighborhoodSet{v}{i}$.

\input{8-DoubleRoman.tex}

\input{8-MinimumChromaticViolation.tex}

\input{8-Grundy.tex}

\input{8-AdditiveColoring.tex}

\input{8-DistanceDom.tex}

\input{8-ProblemsEdges.tex}

config: arxiv


TITLE: Evaluate the integral over all circles centered at the origin.
QUESTION [2 upvotes]: Find all possible values for
$$\int_\gamma \frac{e^z}{z^3+1} dz,$$
where $\gamma$ varies over all circles centered at the origin not passing through a zero of the denominator traversed once in the counterclockwise direction.
Solution attempt: I rewrote the original integral as
$$\int_\gamma \frac{e^z}{(z+1)(z^2-z+1)} dz,$$
then applied the Cauchy integral formula with $f(z) = \frac{e^z}{z^2-z+1}$, and $z = -1$
Is that correct?

REPLY [4 votes]: Your approach is incorrect. Detailed description is in Omnomnomnom's post. Instead, you can apply residue theorem.
Let $R$ be the radius of $\gamma$. If $R<1$, then $\dfrac{e^z}{z^3+1}$ is analytic inside $\gamma$. Thus we get
$$
\int_{\gamma} \dfrac{e^z}{z^3+1} dz = 0.
$$
If $R>1$, then $\dfrac{e^z}{z^3+1}$ is analytic inside $\gamma$ except $z=e^{\frac{i\pi}{3}}$, $z=e^{-\pi i}$, $z=e^{-\frac{i\pi}{3}}$. Using residue theorem, we get
$$
\int_{\gamma} \dfrac{e^z}{z^3+1} dz=2\pi i (\operatorname{Res}(f;e^{\frac{i\pi}{3}})+\operatorname{Res}(f;e^{-\pi i})+\operatorname{Res}(f;e^{-\frac{i\pi}{3}})).
$$
If you can compute residue of $f$, then you will solve the problem.


TITLE: Compute $\lim_{u\to 0}\frac{(u+1)^\tau-1}{u}$ without l'Hopital ($\tau>0$).
QUESTION [0 upvotes]: How can I compute $$\lim_{u\to 0}\frac{(u+1)^\tau-1}{u}$$ without l'Hopital ($\tau>0$) ?
I wrote it as
$$\lim_{u\to 0}\frac{(u+1)^\tau-1}{u}=\lim_{u\to 0}\frac{e^{\tau\frac{\ln(u+1)}{u}u}-1}{u},$$
and tried to us that $$\lim_{u\to 0}\frac{\ln(1+u)}{u}=1,$$
but I can't conclude.

REPLY [1 votes]: Note that $(u+1)^\tau=\exp(\tau\log(u+1))$; if you set $\tau\log(u+1)=v$, then
$$
(u+1)^\tau=e^v
$$
and
$$
u=e^{v/\tau}-1
$$
so your limit becomes
$$
\lim_{v\to0}\frac{e^v-1}{e^{v/\tau}-1}=
\tau\lim_{v\to0}\frac{e^v-1}{v}\frac{v/\tau}{e^{v/\tau}-1}
$$
If you consider $\lim_{v\to0}\frac{e^v-1}{v}=1$ as a known limit, you have what you're looking for.

The derivative of $f(x)=x^\tau$ can be more easily computed with the chain rule:
$$
f(x)=e^{\tau\log x}
$$
so
$$
f'(x)=e^{\tau\log x}\cdot\frac{\tau}{x}=\tau x^\tau x^{-1}=\tau x^{\tau-1}
$$


TITLE: Four-dimensional generalization of the equation $\vec{D} = \epsilon \vec{E}$
QUESTION [0 upvotes]: In "The electrodynamics of moving dielectrics" (p. 262) Landau and Lifshitz have defined four-tensors and four-vector
$F^{\mu \nu} =
\begin{pmatrix}
0&-E_x&-E_y&-E_z\\
E_x&0&-B_z&B_y\\
E_y&B_z&0&-B_x\\
E_z&-B_y&B_x&0\\
\end{pmatrix}$
$H^{\mu \nu} =
\begin{pmatrix}
0&-D_x&-D_y&-D_z\\
D_x&0&-H_z&H_y\\
D_y&H_z&0&-H_x\\
D_z&-H_y&H_x&0\\
\end{pmatrix}$
$u^{\mu} =
\begin{pmatrix}
\frac{1}{\sqrt{1-v^2/c^2}},&\frac{\vec{v}}{c\sqrt{1-v^2/c^2}}
\end{pmatrix}$
Then they write

From this four-vector and the four-tensors $F^{\mu \nu}$ and $H^{\mu \nu}$ we form combinations which become $\vec{E}$ and $\vec{D}$ in a medium at rest. These combinations are the four-vectors $F^{\lambda \mu}u_{\mu}$ and $H^{\lambda \mu}u_{\mu}$; for $\vec{v} = 0$ their time components are zero and their space components are $\vec{E}$ and $\vec{D}$ respectively. The four-dimensional generalization of the equation $\vec{D} = \epsilon \vec{E}$ is therefore evidently
$$ H^{\lambda \mu} u_{\mu} = \epsilon F^{\lambda \mu}u_{\mu} $$

How is that evident?
I understand that with the assumption $\vec{v} = 0$ that generalized equation gives us $\vec{D} = \epsilon \vec{E}$. But couldn't that just be a lucky coincidence?
Or is the reasoning something along these lines: "We found this equation, it gives good results, all the experiments work with that too, so it's probably right." ?

REPLY [0 votes]: We know that $\vec{D}=\epsilon\vec{E}$ in a medium at rest, assuming Einstein's principle of relativity to hold not only for Newtonian mechanics but for Maxwell's electrodynamics as well. Landau and Lifshitz manage to express this in a manifestly covariant way (as one says) as $H^{\lambda\mu}u_\mu=\epsilon F^{\lambda\mu}u_\mu$. Equations of the same form will then automatically be valid in all other inertial frames, since both sides of the equation transform as contravariant four-vectors under the Lorentz group.


TITLE: Counting trees with different limitations
QUESTION [0 upvotes]: I'm struggling with a homework assignment, and would like for a review of how I'm doing so far. I'm trying to answer this question:

A. Count the number of trees for vertices $1 \dots n$, where $1$ is a
leaf
B. Count the number of trees for vertices $1 \dots n$, where $1$
is a leaf and $(1,2)$ is an edge.

Here's what I have so far:
A. Say I cut the leaf ($1$) and I'm left with vertices $2 \dots n$ and their degree sequence is $d_2,\dots,d_j-1,\dots\,d_n$, where $j$ is the vertex that was connected to $1$ before I cut it. By Cayley's theorem $T_{n-1}(d_2,...,d_j-1,...,d_n) = (n-1)^{n-3}$. $J$ could be anything from $2$ to $n$, but whatever it is $T_{n-1}$ stays the same, so to get $T_n$ I just multiply $T_{n-1}$ by the $(n-1)$ possible values of $j$ and I get $T_n = (n-1)^{n-2}$.
Is that true? Somehow I feel that its not, but I can't think of anything better.
B. This time I know that the leaf I cut is connected to $2$, so there's no need to multiply by $(n-1)$ which means that $T_n = (n-1)^{n-3}$.
If I was right about the first question I think I should be right here as well, but I'm not sure at all.
Any hints or ideas would be welcomed, thanks!

REPLY [1 votes]: If you're not sure about this sort of thing, it's always a good idea to check your results with some examples for small numbers. The numbers for $n=1,2,3,4$ are readily counted to be $1,1,2,9$, so your formula seems to work out (except for $n=1$, which was to be expected since Cayley's formula doesn't hold for $n=0$).
I think your argument is correct, but I don't know why you mention the degree sequence and what you mean when you write it as an argument for $T_{n-1}$. The number $(n-1)^{n-3}$ is the number of all trees with $n-1$ vertices, not just the ones with a particular degree sequence.
Your result for B) is also correct; this is simply the number of trees with $n-1$ vertices, since there is exactly one way to attach a leaf $1$ to vertex $2$ in such a tree.


TITLE: Limits of joint distribution
QUESTION [0 upvotes]: Hopefully a simple question to clear up some understanding.

Goal: calculate $f_Y(y)$
My thoughts: to calculate $f_Y(y)$ we need to integrate with respect to $dx$ so in my mind from calculus this is like considering (infinitesimally) thin vertical slices of width $dx$ where each slice has area of $dA = (x+1) - x\: dx$. In other words constant height 1 and integrating over all such $dx$.
If we imagine starting at $x=0$ and sliding the thin vertical slice from left to right we can see the left and right bounds of $x$ now have two equations for them indicating we need to split $f_Y(y)$ up.
My question is whether this is a valid way to think of setting up the question by relating to calculus with areas and slicing or whether I should be setting things up differently. The reason I am asking is that when I first did this question I set it up with $x$ limits from 0 to 1, which is incorrect.
Solution here in case needed (Q5).

REPLY [1 votes]: So we can model $f(x,y)$ as
\begin{equation}
 f(x,y) = K  \qquad \text{if } (x,y) \in A
\end{equation}
and $0$ elsewhere. So, let's find $K$,
\begin{equation}
 \int_{\mathbb{R}^2} f(x,y) \ dxdy = \int_{A} f(x,y) \ dxdy = 1
\end{equation}
i.e.
\begin{equation}
K
 \int\limits_0^1
 \int\limits_{x}^{x+1}
   \ dydx = 1
\end{equation}
i.e.
\begin{equation}
 K = 1
\end{equation}
An alternative is that the area of the trapezoid is $A = \int_{A} \ dy dx = 1$. Now, let's find $f(y)$ by marginalizing out $x$,i.e.
\begin{equation}
 f(y) = \int f(x,y) \ dx
\end{equation}
Notice that $y$ is defined between $0$ and $2$ so we must cover all this range to define $y$. If $0 < y < 1$, $x$ will vary from  $x = 0$ till  the lower side of the trapezoid defined by $x = y$, so we should get
\begin{equation}
 f(y) = \int_0^y \ dx = y, \qquad \text{ if } 0 < y <1
\end{equation}
On the other hand, if $1 < y < 2$, then $x$ will vary from the upper side of the trapezoid defined by $y = x+ 1$ (or $x = y-1$) till $x = 1$, i.e.
\begin{equation}
 f(y) = \int_{y-1}^1 \ dx = 2-y, \qquad \text{ if } 1 < y <2
\end{equation}


TITLE: Distinguishing between two coins
QUESTION [14 upvotes]: It is well known that the complexity of distinguishing an $\epsilon$ biased coin from a fair one is $\theta(\epsilon^{-2})$. Are there results for distinguishing a $p$ coin from a $p+\epsilon$ coin? I can see that for the special case of $p=0$, the complexity will be $\epsilon^{-1}$. I have a hunch that the complexity will depend on whether $p$ is of the order of $\epsilon$, but can't prove so rigorously. Any hints/references?

REPLY [15 votes]: I suggest you use the framework found in the following paper:
How Far Can We Go Beyond Linear Cryptanalysis?, Thomas Baignères, Pascal Junod, Serge Vaudenay, ASIACRYPT 2004.
The crucial result says that you need $n \sim 1/D(D_0 \,||\, D_1)$, where $D(D_0 \,||\, D_1)$ is the Kullback-Leibler distance between the two distributions $D_0$ and $D_1$.  Expanding out the definition of the K-L distance, we see that in your case
$$D(D_0 \,||\, D_1) = p \log \frac{p}{p+\epsilon} + (1-p) \log \frac{1-p}{1-p-\epsilon},$$
with the convention that $0 \log \frac0p = 0$.
When $p \gg \epsilon$, we find $D(D_0 \,||\, D_1) \approx \epsilon^2/(p(1-p))$.  Thus, when $p \gg \epsilon$, we find that you need $n \sim p(1-p)/\epsilon^2$ coin flips.  When $p = 0$, we find $D(D_0 \,||\, D_1) = -\log(1-\epsilon) \approx \epsilon$, so you need $n \sim 1/\epsilon$ coin flips.  Thus, this formula is consistent with the special cases you already know about... but it generalizes to all $n,\epsilon$.
For justification, see the paper.

When $p \gg \epsilon$, the justification is easy to work through by hand.  With $n$ observations, the number of heads is either $\text{Binomial}(n,p)$ or $\text{Binomial}(n,p+\epsilon)$, so you want to find the smallest $n$ such that these two distributions can be distinguished.
You can approximate both of these by a Gaussian with the right mean and variance, and then use standard results on the difficulty of distinguishing two Gaussians, and the answer should fall out.  The approximation is fine if $p \ge 5/n$ or so.
In particular, this comes down to distinguishing $\mathcal{N}(\mu_0,\sigma_0^2)$ from $\mathcal{N}(\mu_1,\sigma_1^2)$ where $\mu_0 = pn$, $\mu_1 = p+\epsilon)n$, $\sigma_0^2 = p(1-p)n$, $\sigma_1^2 = (p+\epsilon)(1-p-\epsilon)n$.  You'll find that the probability of error in the optimal distinguisher is $\text{erfc}(z)$ where $z = (\mu_1-\mu_0)/(\sigma_0+\sigma_1) \approx \epsilon \sqrt{n / 2p(1-p)}$.  Thus, we need $z \sim 1$ to distinguish with constant success probability.  This amounts to the condition that $n \sim 2p(1-p)/\epsilon^2$ (up to a constant factor)... when $p \gg \epsilon$.
For the general case... see the paper.


text:
\begin{document}

\title[On the kernel of the maximal flat Radon transform]
    {On the kernel of the maximal flat Radon transform on symmetric spaces
    of compact type}
\author{Eric L. Grinberg}
\author{Steven Glenn Jackson}
    \address{Department of Mathematics \\
             University of Massachusetts \\
             100 Morrissey Boulevard \\
             Boston, MA 02125 \\
             USA}
    \email{\href{mailto:jackson@math.umb.edu}{jackson@math.umb.edu}}
    \urladdr{\href{http://www.math.umb.edu/~jackson/}{http://www.math.umb.edu/~jackson/}}
\subjclass[2010]{Primary 44A12; Secondary 53C35, 53C65, 22E46}
\keywords{Integral geometry, Radon transform, symmetric space}

\begin{abstract}
    Let $M$ be a Riemannian globally symmetric space of compact type, $M'$ its
    set of maximal flat totally geodesic tori, and $\ad(M)$ its adjoint space.
    We show that the kernel of the maximal flat Radon transform $\tau:L^2(M)
    \rightarrow L^2(M')$ is precisely the orthogonal complement of the image
    of the pullback map $L^2(\ad(M))\rightarrow L^2(M)$.  In particular, we
    show that the maximal flat Radon transform is injective if and only if $M$
    coincides with its adjoint space.
\end{abstract}
\maketitle

\section{Introduction}

As part of his 1911 doctoral dissertation (published as \cite{funk13}), Paul
Funk considered the problem of recovering a function $f$ on the sphere $S^2$
from its integrals over great circles.  He showed that precisely the even part
$f^+(x)=\frac{1}{2} \left( f(x) + f(-x) \right)$ of $f$ can be recovered from
these integrals, while the odd part is annihilated by them.  Thus was born the
Funk transform---integration over closed geodesics in a Riemannian manifold.

Funk's original motivation was a problem in differential geometry suggested to
him by his advisor David Hilbert: the study of surfaces all of whose geodesics
are closed.  The sphere is the obvious example of such a surface, and Darboux
around 1884 had given a necessary condition for other surfaces of revolution
to possess this property.\footnote{It is difficult now to assign an exact date
to Darboux's condition.  In \cite{tan92}, Tannery writes:
\foreignlanguage{french}{\emph{``J'y ai \'et\'e conduit en \'etudiant la note XV
de la M\'ecanique de Despayrons, o\`u M.~Darboux a donn\'e une r\`egle
pour trouver les surfaces de r\'evolution admettant des lignes
g\'eod\'esiques ferm\'ees.''}}  However, the authors of the present paper
were unable to obtain this work; indeed, we could locate no reference to
it but this and an entry in the June 24, 1905 issue of the
\emph{Publisher's Circular} dating it to 1884--85.  The earliest
publication of Darboux's result still readily obtainable seems to be
\cite{darb94}.}
In 1892, Tannery had constructed an explicit, though singular, example
(\cite{tan92}), and in 1901 Otto Zoll, another of Hilbert's students, had
constructed smooth examples (published as \cite{zoll03}).  Funk's dissertation
sought to identify smooth deformations of the standard metric on $S^2$, all of
whose geodesics remain closed at each stage of the deformation.

Funk's result on integration over great circles implied that no such
deformation could exist if the deformed metrics were required to remain even
(nowadays we say that the standard metric on $\mathbb{RP}^2$ admits no such
deformations).  Conversely, given any odd function $h$, Funk tried to
construct by power series a conformal deformation of the standard metric with
initial derivative $h$, but was unable to prove convergence of his series.
Convergence of such deformations was finally proved by Guilleman in 1976
(\cite{guillemin76}).  Thus, for $S^2$ and $\mathbb{RP}^2$ at least, the
kernel of the Funk transform controls the ``Zoll rigidity'' of the
manifold.\footnote{To be precise, let us say that $g$ is a \emph{$P_l$-metric}
if all its geodesics are periodic with least period $l$; that $\{g_t\}$ is a
\emph{$P_l$-deformation} of $g_0$ if $g_t$ is a $P_l$-metric for all $t$; that
the deformation $\{g_t\}$ is \emph{trivial} if $(M,g_t)$ is isometric to
$(M,g_0)$ for all $t$; and that $g$ is \emph{$P_l$-rigid} (or \emph{Zoll
rigid}) if every $P_l$-deformation of $g$ is trivial.  Then Funk's results
showed that $\mathbb{RP}^2$ is Zoll rigid, while Guillemin's convergence
argument showed that every odd function gives rise to a
$P_l$-deformation of $S^2$.} These results were extended to $S^n$ and
$\mathbb{RP}^n$, for arbitrary $n$, by Michel (\cite{mich73}) and Weinstein
(see \cite[p.~120]{besse78}).

Around this time, similar integral transforms arose in several related
contexts.  A smooth family $\{g_t\}$ of Riemannian metrics on a compact
manifold $M$ is said to be \emph{isospectral} if the Laplace spectrum of
$(M,g_t)$ is independent of $t$.  Such families are plentiful: given any local
one-parameter group of diffeomorphisms $\phi_t:M\rightarrow M$ and any metric
$g$ on $M$, one obtains an isospectral family by setting $g_t$ equal to the
pullback of $g$ along $\phi_t$.  Examples of this type are trivial in the
sense that $(M,g_t)$ is isometric to $(M,g)$ for every $t$.  One says that
$(M,g)$ is \emph{spectrally rigid} if every isospectral deformation is trivial
in this sense, and \emph{spectrally rigid to first order} if every isospectral
deformation agrees with a trivial one to first order in the deformation
parameter $t$.

In the case of Riemannian globally symmetric spaces of compact type, results
of Guillemin in \cite{gui79} implied that spectral rigidity is related to a
certain natural ``thickening'' of the Funk transform.  To be specific, let us
refer to any integral transform obtained by integrating over a fixed class of
totally geodesic submanifolds of fixed dimension as a ``Radon transform.''
(Thus the Funk transform is the Radon transform associated with compact
submanifolds of dimension one.)  For the case where $M$ is a Riemannian
globally symmetric space of compact type, Guillemin's results implied that the
initial derivative of any isospectral deformation must lie in the kernel of
the Radon transform associated with maximal totally geodesic flat tori.  One
says that $M$ is \emph{Guillemin rigid} if the Lie derivatives of the metric
(i.e.\ the initial derivatives of trivial deformations) fill the entire kernel
of the maximal flat Radon transform for symmetric two-forms.\footnote{See
\cite{gasqui04} for the definition of the Radon transform on symmetric
$p$-forms.}  Then Guillemin rigidity is a sufficient condition for first-order
spectral rigidity.  Since the kernel of the Funk transform is contained in the
kernel of the maximal flat Radon transform, an observation of Michel
(\cite[Proposition~2.2.4]{mich73}) implies that Guillemin rigidity is also a
sufficient condition for first-order Zoll rigidity of $P_l$-metrics.

Each Riemannian globally symmetric space $M$ of compact type admits a unique
\emph{adjoint form} (or \emph{universal covered space}): another symmetric
space admitting $M$ as a Riemannian cover, and not itself properly covering
any other symmetric space.  (For example, the adjoint form of the sphere $S^n$
is the real projective space $\mathbb{RP}^n$.) In \cite{gri92}, Grinberg
conjectured that the maximal flat Radon transform for functions is injective
if and only if $M$ coincides with its adjoint form.

In a monumental study (\cite{gasqui04}), Gasqui and Goldschmidt determined
precisely which Grassmannians (over $\mathbb{R}$, $\mathbb{C}$, or
$\mathbb{H}$) are Guillemin rigid.  Aware of Grinberg's conjecture, they
observed that their criterion is equivalent to the requirement that $M$
coincide with its adjoint form, and went on to prove that, if Grinberg's
conjecture is true, then the only candidates for Guillemin rigidity are
the irreducible adjoint spaces.

In the present paper, we prove a refinement of Grinberg's conjecture: the
kernel of the maximal flat Radon transform consists precisely of those
functions orthogonal to all functions pulled back from the adjoint form
(\pref{thm:main}).  Applied to the sphere $S^2$, this recovers Funk's result
of 1911: the kernel of the Funk transform consists of those functions
orthogonal to all functions pulled back from $\mathbb{RP}^2$ (i.e.\ the
functions orthogonal to all even functions, namely the odd functions).  Our
proof is based on elementary representation theory of the isometry group of
$M$, and treats all Riemannian globally symmetric spaces of compact type
uniformly; there is no case-by-case analysis.

The organization of the paper is as follows: in \pref{sec:prelim} we fix
notation and gather basic tools; in \pref{sec:reynolds} we show that
the maximal flat Radon transform can be viewed as a Reynolds operator
for a certain group of isometries of $M$; in \pref{sec:support} we
relate this Reynolds operator to highest weight theory; in \pref{sec:adjoint}
we relate the resulting highest weight theory to the extraction of
the adjoint form; and in \pref{sec:main} we use our accumulated results
to prove Grinberg's conjecture.

\section{Preliminaries}
\label{sec:prelim}

In this section, we fix notation and gather some useful results from the
    literature.

\subsection{Symmetric spaces and isometry groups}

Let $M$ be a Riemannian globally symmetric space of compact type, and let $G$ be
    the identity component of the isometry group of $M$.
Then $G$ is a compact semisimple Lie group (\cite[Lemma~IV.3.2, Definitions~V.1,
    and Definition~V.4]{helgason78}).
Denote its Lie algebra by $\lie{g}_0$, and put $\lie{g}=\cplx\otimes_{\real}
    \lie{g}_0$.

Fix a point $p\in M$, let $\sigma$ be the geodesic symmetry of $M$ at $p$,
    let $K$ be the isotropy group of $p$ in $G$, and let $K_0$ be the identity
    component of $K$.
Define $\theta:G\rightarrow G$ by $\theta(g)=\sigma g\sigma^{-1}$; then
    $\theta$ is an involution of $G$.
Letting $K_{\theta}$ be the set of fixed points of $\theta$, we have
    $K_0\subseteq K\subseteq K_{\theta}$ (\cite[Theorem~IV.3.3]{helgason78}).
These three groups share a common Lie algebra $\lie{k}_0$ (namely the set of
    fixed points of $d\theta$), whose complexification we denote by $\lie{k}$.

Since $G$ is transitive on $M$ (\cite[Theorem~IV.3.3]{helgason78}), we have
    $$M\simeq G/K.$$

\subsection{The adjoint form}

Let $Z$ be the center of $G$, and define $$K_Z=\{g\in G\,|\,\theta(g)^{-1}g\in
    Z\}.$$
Evidently this is a subgroup of $G$ containing $K$.
Since $G$ is semisimple, $Z$ is finite, so $K$ has finite index in $K_Z$.

Let $\ad(G)=G/Z$ denote the adjoint group of $G$.
Since $\theta$ stabilizes $Z$, we have an induced involution on $\ad(G)$, whose
    fixed point set we denote by $\ad(G)^{\theta}$.
One shows easily that $K_Z$ is the pre-image of $\ad(G)^{\theta}$ under
    the natural projection $G\rightarrow\ad(G)$; hence the space
    $$\ad(M)=G/K_Z\simeq \ad(G)/\ad(G)^{\theta}$$ is again a Riemannian
    globally symmetric space of compact type; we refer to it as the
    \emph{adjoint form} of $M$.

Since $K$ has finite index in $K_Z$, the natural map $\pi:M\rightarrow\ad(M)$
    is a covering map.
In fact, $\ad(M)$ is minimal in the isogeny class of $M$, by which we mean that
    it is covered by every Riemannian globally symmetric space whose
    universal cover is isometric to that of $M$ (\cite[Corollary~VII.9.3]
    {helgason78}).

\subsection{Representative functions}

Let $H$ be any closed subgroup of $G$.
Since $G$ and $H$ are both compact, they are both unimodular and the
    homogeneous space $G/H$ admits a unique normalized left Haar measure
    $\mu_{G/H}$ (\cite[Theorem~29E, Lemma~30A, and Theorem~33D]{loomis53}).

Define an action of $G$ on $L^2(G/H)$ by the formula $(gf)(x)=f(g^{-1}x)$.
Since the inner product $(f_1,f_2)=\int f_1\overline{f_2}\,d\mu_{G/H}$ is
    invariant under this action, $L^2(G/H)$ is a unitary (hence continuous)
    representation of $G$.

Let $C(G/H)$ denote the algebra of continuous functions on $G/H$.
Let $\repf(G/H)$ denote the set of $G$-finite vectors in $C(G/H)$.
The elements of $\repf(G/H)$ are called \emph{representative functions}
    on $G/H$.
Since $G$ is compact, we have the inclusions $$\repf(G/H)\subseteq
    C(G/H)\subseteq L^2(G/H).$$
By the Peter-Weyl Theorem (\cite[Theorem~III.5.7]{bd85}), the algebra
    $\repf(G/H)$ is dense in $L^2(G/H)$.

In the case where $H$ is the trivial subgroup, we can say more.
Define an action of $G\times G$ on $L^2(G)$ by the formula $((g_1,g_2)f)(x)
    =f(g_1^{-1}xg_2)$.
Then (again since $G$ is unimodular) this action is also unitary.
Let $e$ denote the identity element of $G$, and let $LG$ and $RG$
    denote the groups $G\times\{e\}$ and $\{e\}\times G$, respectively.
Then $f\in L^2(G)$ is $LG$-finite if and only if it is $RG$-finite
    (\cite[Proposition~III.1.2]{bd85}).
Moreover, \cite[Proposition~III.1.5]{bd85} gives the $G\times G$-module
    decomposition $$\repf(G) \simeq\bigoplus_{V\in\Irr(G)} V\otimes V^*$$
    where $\Irr(G)$ denotes the set of isomorphism classes of irreducible
    representations of $G$.
(Since $G$ is compact, it follows from the Peter-Weyl Theorem that all of its
    irreducible representations are finite-dimensional.)

Returning to the case of arbitrary closed $H$, we note that the natural
    projection $G\rightarrow G/H$ induces an injection $\repf(G/H)
    \rightarrow\repf(G)$ intertwining the (left) action of $G$.
Identifying $\repf(G/H)$ with its image under this injection, we shall
    regard $\repf(G/H)$ as a subalgebra of $\repf(G)$.
From \cite[Example~III.6.3]{bd85} we have
    $$\repf(G/H) \simeq\bigoplus_{V\in\Irr(G)}V\otimes\left(V^*\right)^H$$
    where $(V^*)^H$ denotes the space of $H$-invariant vectors in $V^*$.
More generally, whenever $H_1\subseteq H_2$ are both closed subgroups of
    $G$, we regard $\repf(G/H_2)$ as a subalgebra of $\repf(G/H_1)$ (and hence
    also a subalgebra of $\repf(G)$).

One says that $H$ is a \emph{spherical subgroup} of $G$ if
    $\dim\left((V^*)^H\right) \leq1$ for all $V\in\Irr(G)$.
By passing to the complexified Lie algebra $\lie{k}$ and using the proof of
    \cite[Theorem~12.3.12]{gw09}, one sees that $K_0$ is spherical.
Hence $K$ and $K_Z$ are also spherical.

\subsection{The torus transform}

Let $T$ be a maximal flat totally geodesic torus of $M$ containing $p$.
Let $\lie{a}_0$ be the tangent space to $T$ at $p$, and let $\lie{p}_0
    \subset\lie{g}_0$ be the $-1$-eigenspace of $d\theta$.
By \cite[Proposition~V.6.1]{helgason78}, the differential of the map
    $g\mapsto gp$ identifies $\lie{a}_0$ with a maximal abelian subspace of
    $\lie{p}_0$.

By \cite[Theorem V.6.2]{helgason78}, $G$ is transitive on the set $M'$ of all
    maximal flat totally geodesic tori in $M$.
Let $L$ be the stabilizer of $T$ in $G$.
Then $M'$ is in bijective correspondence with the homogeneous space $G/L$, with
    which we shall now identify it.

Let $A=\exp(\lie{a}_0)$ be the Lie subgroup of $G$ with Lie algebra $\lie{a}_0$,
    and let $\overline{A}$ be the closure of $A$.
Then $\overline{A}$ is a compact abelian subgroup of $G$ whose elements
    satisfy the equation $\theta(a)=a^{-1}$.
It follows that the Lie algebra of $\overline{A}$ is an abelian subspace of
    $\lie{p}_0$ containing $\lie{a}_0$.
But $\lie{a}_0$ is maximal abelian in $\lie{p}_0$, so we have equality, and
    $A$ is in fact a closed subgroup of $G$ (hence a torus).

Since $T$ is totally geodesic, it follows from \cite[Theorem~IV.3.3]
    {helgason78} that the geodesics of $T$ passing through $p$ are precisely
    the orbits of $p$ under one-parameter subgroups of $A$.
But every point of $T$ is joined to $p$ by some such geodesic, so it follows
    that $A$ acts transitively on $T$.
Evidently the full stabilizer $L$ is also transitive on $T$, so we can write
    $$T\simeq L/(L\cap K)\simeq A/(A\cap K).$$

The normalized Haar measure $\mu_{L/(L\cap K)}$ is an $A$-invariant
    measure on $T$, so the uniqueness of such measures forces $\mu_{L/(L\cap K)}
    =\mu_{A/(A\cap K)}$.
For brevity, let us denote this measure by $\mu_T$, and define a map
    $\tau_0:\repf(M)\rightarrow\repf(M')$ by the formula
    $$(\tau_0(f))(gT) = \int_{x\in T} f(gx)\,d\mu_T(x).$$
That $\tau_0(f)$ depends only on the torus $gT$ and not on $g$ itself follows
    from the $L$-invariance of $\mu_T$; that it depends continuously on $gT$
    follows from the compactness of $G$ and $T$; and that it is $G$-finite follows
    from the fact that $\tau_0$ intertwines the left action of $G$.

In the sequel we shall show that $\norm{\tau_0(f)}_2\leq\norm{f}_2$
    (\pref{cor:bounded}), from which it will follow that $\tau_0$ admits a unique
    bounded extension $\tau:L^2(M)\rightarrow L^2(M')$.
This extension is known in the literature as the \emph{maximal flat Radon
    transform} on $L^2(M)$; we shall call it the \emph{torus transform} for
    short.

\subsection{Weights and representations}

Let $\lie{h}_0$ be any maximal abelian subspace of $\lie{g}_0$ containing
    $\lie{a}_0$.
Then by \cite[Lemma~VI.3.2]{helgason78}, $\lie{h}_0$ is a Cartan subalgebra of
    $\lie{g}_0$, and it is stabilized by the action of $\theta$.
Denote by $\lie{t}_0$ be the set of $\theta$-fixed points in $\lie{h}_0$.
Since $\lie{a}_0$ is maximal among abelian subspaces of $\lie{p}_0$, we have
    $\lie{h}_0 = \lie{t}_0 \oplus \lie{a}_0$.
Let $H$ denote the maximal torus of $G$ with Lie algebra $\lie{h}_0$.

Next let $\lie{a}$, $\lie{t}$, and $\lie{h}$ denote the complexifications of
    $\lie{a}_0$, $\lie{t}_0$, and $\lie{h}_0$, respectively.
The complexification of $d\theta$ stabilizes $\lie{a}$, $\lie{t}$,
    and $\lie{h}$.
From now on we shall use the same symbol $\theta$ to denote the involution
    of $G$, its differential, the complexification of its differential,
    and the induced involutions on the duals of all the vector spaces on
    which these act.

Let $\Delta\subset\lie{h}^*$ be the root system of $\lie{g}$ with respect to
    $\lie{h}$, and let $\Sigma\subset\lie{a}^*$ be the restricted root system
    of the pair $(\lie{g},\lie{k})$ (i.e.\ the set of non-zero restrictions
    of members of $\Delta$ to $\lie{a}$).
Choose a positive system $\Delta^+$ for $\Delta$ such that the non-zero
    restrictions of its members form a positive system $\Sigma^+$ for
    $\Sigma$.
Let $\Pi\subseteq\Delta^+$ and $\Pi'\subseteq\Sigma^+$ be the resulting sets
    of simple roots.

Denote by $B$ the Killing form of $\lie{g}$.
Since $B$ has non-degenerate restriction to $\lie{h}$, it defines an isomorphism
    $\lie{h}\rightarrow\lie{h}^*$, by means of which we transfer $B$ to an inner
    product $(\cdot,\cdot)$ on $\lie{h}^*$.
Then $B$ is negative definite on $\lie{h}_0$, but $(\cdot,\cdot)$ is positive
    definite on $\real\Delta$, the real span of $\Delta$
    (\cite[Proposition II.6.6, Corollary~II.6.7, and
    Theorem~III.4.4]{helgason78}).
For any $\alpha,\beta\in\real\Delta$ with $\beta\neq0$, define $$\comps{\alpha,\beta}=
    \frac{2(\alpha,\beta)}{(\beta,\beta)}.$$

Since the restriction of $B$ to $\lie{a}$ is also non-degenerate, we can use
    the same procedure to define an inner product on $\lie{a}^*$.
With this inner product, $\lie{a}^*$ is naturally isometric to $\ann(\lie{t})
    \subseteq\lie{h}^*$, with which we shall now identify it.

Let $W$ denote the normalizer of $\lie{h}_0$ in $G$, modulo its centralizer.
Then $W$ acts faithfully on $\lie{h}_0$, hence also on $\lie{h}$ and its dual.
One finds that $W$ is generated by the root reflections $\{s_{\alpha}\,|\,
    \alpha\in\Delta\}$; hence $W$ stabilizes $\real\Delta$.
We say that a weight $\lambda\in\real\Delta$ is \emph{dominant} if $(\lambda,
    \alpha)\geq0$ for all $\alpha\in\Pi$.
Each element of $\real\Delta$ is $W$-conjugate to a unique dominant weight
    (\cite[Theorem~10.3 and Lemma~10.3B]{hum72}).

Similarly, let $W_{\lie{a}}$ denote the normalizer of $\lie{a}_0$ in $K$,
    modulo its centralizer.
Then $W_{\lie{a}}$ acts faithfully on $\lie{a}_0$ and hence also on $\lie{a}$
    and its dual.
It is generated by the root reflections $\{s_{\alpha}\,|\,\alpha\in\Sigma\}$,
    so it stabilizes $\real\Sigma$.
A restricted weight $\nu\in\real\Sigma$ is \emph{dominant} if $(\nu,\alpha)
    \geq0$ for all $\alpha\in\Pi'$.\
Each element of $\real\Sigma$ is $W_{\lie{a}}$-conjugate to a unique dominant
    restricted weight (\cite[Corollary~VII.2.13 and Theorem~VII.2.22]{helgason78}).

We say that a weight $\lambda\in\real\Delta$ is \emph{algebraically integral}
    if $\comps{\lambda,\alpha}\in\ints$ for all $\alpha\in\Pi$.
A weight $\omega\in\real\Delta$ is dominant and algebraically integral if and
    only if it is the highest weight of some finite-dimensional irreducible
    representation $V(\omega)$ of $\lie{g}$.
In this case every weight of $V(\omega)$ is algebraically integral
    (\cite[Theorem~5.5]{knapp02}).

We say that a weight $\lambda\in\real\Delta$ is \emph{analytically integral}
    if it is the complexified differential of a character of $H$.
In this case it is harmless to use the same symbol $\lambda$ to denote
    the character of which it is the differential, and to say that a subset
    $S\subseteq H$ is \emph{annihilated by $\lambda$} if $\lambda(s)=1$ for all
    $s\in S$.
Every analytically integral weight is algebraically integral.
A weight $\omega\in\real\Delta$ is dominant and analytically integral if and only
    if it is the highest weight of the differential of some irreducible
    representation of $G$, which we shall also denote by $V(\omega)$.
In this case every weight of $V(\omega)$ is analytically integral
    (\cite[Proposition~4.59, Lemma~5.106, and Theorem~5.110]{knapp02}).

\section{Properties of the torus transform}
\label{sec:reynolds}

In this section we prove that $\tau_0$ can be extended to a bounded operator
    on $L^2(M)$, and that it coincides with the restriction of a Reynolds
    operator on $\repf(G)$.

Recall that $\repf(M)$ can be identified with the algebra of right $K$-invariants
    in $\repf(G)$, and $\repf(M')$ can be identified with the algebra of right
    $L$-invariants.
We have

\begin{thm}
\label{thm:reynolds}
Let $R_A:\repf(G)\rightarrow\repf(G)$ be the operator orthogonally projecting
    $\repf(G)$ onto its space of right $A$-invariants, namely
    $$(R_A(f))(x) = \int_{a\in A}f(xa)\,d\mu_A(a)$$ where $\mu_A$ denotes
    normalized Haar measure on $A$.
Then for any $f\in\repf(M)$, we have $$\tau_0(f) = R_A(f).$$
\end{thm}

\begin{proof}
The torus $T$ is isomorphic to $A/(A\cap K)$.
The positive linear functional $I:C(A)\rightarrow\cplx$ defined by $$I(f) =
    \int_{x(A\cap K)\in A/(A\cap K)}\left(\int_{y\in(A\cap K)}f(xy)\,
    d\mu_{A\cap K}(y)\right)\,d\mu_{A/(A\cap K)}(x)$$ is $A$-invariant,
    preserves monotone limits, and satisfies $I(1)=1$; consequently $I$
    coincides with Haar integration over $A$ (\cite[Definition~12A and Theorem~29D]{loomis53}).
If $f$ is right $K$-invariant, then the inner integral is just $f(x)$
    (this is depends only on the coset $x(A\cap K)$) and we have

\begin{align*}
(\tau_0(f))(g) &= \int_{x(A\cap K)\in A/(A\cap K)} f(gx)\,d\mu_{A/(A\cap K)}(x) \\
&= \int_{a\in A} f(ga)\,d\mu_A(a) \\
&= (R_A(f))(g).
\end{align*}
\end{proof}

\begin{cor}
\label{cor:bounded}
For any $f\in\repf(M)$, we have $\norm{\tau_0(f)}_2\leq\norm{f}_2$.
\end{cor}

\begin{proof}
As above, the positive linear functional $J:C(G)\rightarrow\cplx$ defined by
    $$J(f) = \int_{xK\in G/K}\left(\int_{y\in K}f(xy)\,d\mu_K(y)\right)\,
    d\mu_{G/K}(x)$$ coincides with Haar integration on $G$.
It follows immediately that the injection $\repf(M)\rightarrow\repf(G)$ preserves
    $L^2$-norms.
Similarly, the injection $\repf(M')\rightarrow\repf(G)$ preserves $L^2$-norms.
But the identification of $\repf(M)$ and $\repf(M')$ with their images in $L^2(G)$
    sends $\tau_0$ to the restriction of an orthogonal projection operator.
\end{proof}

\section{Supports of $\lie{k}$-invariants}
\label{sec:support}

Let $\omega$ be an analytically integral dominant weight.
Since $K_0$ is a spherical subgroup of $G$, the irreducible $G$-module
    $V(\omega)$ admits at most a one-dimensional space of $\lie{k}$-invariants.
Let $v^{\omega}$ be a non-zero $\lie{k}$-invariant in $V(\omega)$, if one
    exists, and zero otherwise.
Write $$v^{\omega} = \sum_{\lambda} v^{\omega}_{\lambda}$$ where
    $v^{\omega}_{\lambda}$ is a weight vector of weight $\lambda$.

\begin{defn}
The \emph{support} of $\omega$ is the set $\supp(\omega)=
    \{\lambda\,|\,v^{\omega}_{\lambda}\neq0\}$.
\end{defn}

In this section, we give a necessary and sufficient condition for a weight
    $\lambda$ of $V(\omega)$ to lie in $\supp(\omega)$ (\pref{thm:support}).
We begin with a lemma.

\begin{lem}
\label{lem:helglem}
Let $\alpha$ be a positive root.
If $\theta(\alpha)\neq\alpha$ then $-\theta(\alpha)$ is also a positive
    root.
On the other hand, if $\theta(\alpha)=\alpha$ then the root spaces
    $\lie{g}_{\alpha}$ and $\lie{g}_{-\alpha}$ are both contained in $\lie{k}$.
\end{lem}

\begin{proof}
This is \cite{helgason78} Lemma~VI.3.3.
\end{proof}

We shall also need a few facts concerning a certain partial order on $\lie{a}^*$ (which, recall, we have identified with $\ann(\lie{t}))\subseteq\lie{h}^*$).

\begin{defn}
For $\lambda, \mu \in \lie{a}^*\simeq\ann\lie{t}\subseteq\lie{h}^*$, define $\lambda\preceq\mu$ if and only if $\mu-\lambda$ lies in the non-negative span of $\Pi'$.
\end{defn}

\begin{lem}
\label{lem:preceq}
Suppose $\lambda\in\lie{a}^*$, and let $\mu$ be the unique dominant $W_{\lie{a}}$-conjugate of $\lambda$.
Then $\lambda\preceq\mu$.
\end{lem}
\begin{proof}
The proof is essentially the same as the argument given in \cite[Theorem~VII.2.22]{helgason78}.
Let $\nu$ be any maximal element in the $W_{\lie{a}}$-orbit of $\lambda$.
Then for any $\alpha\in\Pi'$ we have $\nu-s_{\alpha}(\nu) = \comps{\nu,\alpha}\alpha.$
Maximality of $\nu$ now forces $\comps{\nu,\alpha}\geq0$.
Since $\alpha$ was arbitrary in $\Pi'$ this shows that $\nu$ is dominant, and hence $\nu=\mu$.
But now the $W_{\lie{a}}$-orbit is a finite poset with \emph{unique} maximal element $\mu$; hence $\mu$ is also a largest element and the lemma is proved.
\end{proof}

We now need a pair of lattices in $\ann(\lie{t})\subseteq\lie{h}^*$.

\begin{defn}
Let $\Lambda$ denote the set $\{\rho-\theta(\rho)\}$ where $\rho$ runs over the root lattice, and let $\widehat{\Lambda}$ denote the set $\{\mu-\theta(\mu)\}$ where $\mu$ runs over the lattice of algebraically integral weights.
\end{defn}

\begin{lem}
\label{lem:lattice}
Both $\Lambda$ and $\widehat{\Lambda}$ are discrete additive subgroups of $\ann(\lie{t})$ that span $\ann(\lie{t})$.
\end{lem}
\begin{proof}
Both are obviously additive subgroups.
The span of $\Lambda$ (resp.\ $\widehat{\Lambda}$) is all of $\ann(\lie{t})$ because the span of the root lattice (resp.\ the weight lattice) is all of $\lie{h}^*$, and the map $\mu\mapsto(\mu-\theta(\mu))$ is a surjective linear map of $\lie{h}^*$ onto $\ann(\lie{t})$.
Finally, $\Lambda$ (resp.\ $\widehat{\Lambda}$) is discrete because it is a subgroup of the root lattice (resp.\ the weight lattice).
\end{proof}
\begin{lem}
\label{lem:stable}
Both $\Lambda$ and $\widehat{\Lambda}$ are stable under the action of $W_a$.
\end{lem}
\begin{proof}
The group $W_{\lie{a}}$ is a quotient of the centralizer of $\theta$ in $W$ (\cite[Theorem~VII.8.10]{helgason78}).
But this centralizer evidently stabilizes both $\Lambda$ and $\widehat{\Lambda}$.
\end{proof}

\begin{lem}
\label{lem:highest_in_support}
Suppose that $V(\omega)$ admits a non-zero $\lie{k}$-invariant.
Then $\omega\in\supp(\omega)$.
\end{lem}
\begin{proof}
Let $\lambda$ be any element of $\supp(\omega)$, maximal with respect to $\preceq$.
If $\lambda=\omega$ then we are finished; otherwise, since $\lambda$ is not the highest weight of $V(\omega)$, we can find some positive root $\alpha$ such that $X_{\alpha}v^{\omega}_{\lambda}\neq0$ (where $X_{\alpha}$ is some non-zero element of the root space $\lie{g}_{\alpha}$).
If $\alpha$ is fixed by $\theta$, then \pref{lem:helglem} gives $X_{\alpha}
    \in\lie{k}$, contradicting the $\lie{k}$-invariance of $v^{\omega}$.
Consequently $\theta(\alpha)\neq\alpha$.
But $v^{\omega}$ must be annihilated by $X_{\alpha} + \theta(X_{\alpha})
    \in\lie{k}$, so we have $$\theta(X_{\alpha})(v^{\omega}_{\lambda+
    \alpha-\theta(\alpha)}) = -X_{\alpha}v^{\omega}_{\lambda},$$ forcing
    $\lambda+\alpha-\theta(\alpha)\in\supp(\omega)$.
Again by \pref{lem:helglem}, $-\theta(\alpha)$ is also a positive root, so $\lambda\prec\lambda+\alpha-\theta(\alpha)$, contradicting the maximality of $\lambda$.
\end{proof}

\begin{lem}
Suppose that $V(\omega)$ admits a non-zero $\lie{k}$-invariant.
Then $\omega\in\widehat{\Lambda}$.
\end{lem}
\begin{proof}
It suffices to consider the case in which $G$ is simply connected.
Consider the set $$F_0=\{a\in A\,|\,a^2=e\}$$ where $e$ denotes the identity element of $G$.
This is a subset of $K^{\theta}$.
Indeed, let $S$ denote the set of all $a\in\lie{a}$ such that $\mu(a)\in\pi i\ints$ for every algebraically integral weight $\mu$; then $F_0=\exp(S)$, so $F_0$ is actually a subset of $K_0$.
Then, since $\omega\in\supp(\omega)$, it follows that $F_0$ acts trivially on $v^{\omega}_{\omega}$.
Thus, for any $a\in S$ (i.e.\ any $a$ such that $\mu(a)\in2\pi i\ints$ for every  $\mu\in\widehat{\Lambda}$) we must also have $\omega(a)\in2\pi i\ints$.

Next, choose a generating set $\{\mu_1,\dots,\mu_k\}$ for $\widehat{\Lambda}$ such that $\{\mu_1,\dots,\mu_k\}$ is a basis for $\ann(\lie{t})$.
(This is possible by \pref{lem:lattice}.)
Then $\omega\in\supp(\omega)$ forces $\omega\in\ann(\lie{t})$, so we can write $$\omega=c_1\mu_1+\dots+c_k\mu_k$$ for some scalars $c_1,\dots,c_k$.
Let $b_1,\dots,b_k$ be the basis for $\lie{a}$ dual to $\mu_1,\dots,\mu_k$, and put $a_i=2\pi i b_i$, so that $a_1,\dots,a_k\in S$.
Then for each $i$, we have $\omega(a_i) = 2\pi ic_i\in2\pi i\ints$, showing that each $c_i$ is an integer.
\end{proof}

\begin{lem}
\label{lem:lhat_to_l}
Suppose that $\lambda\in\widehat{\Lambda}$, and $w\in W_{\lie{a}}$.
Then $\lambda-w(\lambda)\in\Lambda.$
\end{lem}
\begin{proof}
Choose an algebraically integral weight $\mu$ with $\lambda = \mu-\theta(\mu)$.
Since $\Lambda\subseteq\widehat{\Lambda}$ and $W_{\lie{a}}$ is generated by restricted root reflections, it suffices to prove the result in the case where $w=s_{\alpha|_{\lie{a}}}$ for some root $\alpha$ with $\theta(\alpha)\neq\alpha$.

There are two cases to consider.
Suppose first that $\alpha-\theta(\alpha)$ is proportional to some root $\beta$.
Then
\begin{align*}
\lambda-w(\lambda) &= \comps{\lambda,\beta}\beta \\
&= 2\comps{\mu,\beta}\beta \\
&= \comps{\mu,\beta}(\beta-\theta(\beta))
\end{align*}
and, since $\mu$ is algebraically integral, this lies in $\Lambda$.

On the other hand, suppose that $\alpha-\theta(\alpha)$ is not proportional to any root.
Then certainly $\theta(\alpha)$ is not proportional to $\alpha$.
But since exactly one of $\{\alpha,\theta(\alpha)\}$ is positive, we see that neither $\alpha-\theta(\alpha)$ nor $\alpha+\theta(\alpha)$ is a root, so \cite[Lemma~9.4]{hum72} forces $(\alpha,\theta(\alpha))=0.$
Then
\begin{align*}
\lambda-w(\lambda) &= 2\frac{(\lambda, \alpha-\theta(\alpha))}{(\alpha-\theta(\alpha),\alpha-\theta(\alpha))}(\alpha-\theta(\alpha)) \\
&= \frac{(\lambda,\alpha-\theta(\alpha))}{(\alpha,\alpha)}(\alpha-\theta(\alpha)) \\
&= 2\frac{(\lambda,\alpha)}{(\alpha,\alpha)}(\alpha-\theta(\alpha))
\end{align*}
and, since $\lambda$ itself is algebraically integral, this again belongs to $\Lambda$.
\end{proof}

At last we have

\begin{thm}
\label{thm:support}
Suppose that $V(\omega)$ admits a non-zero $\lie{k}$-invariant, and let
    $\lambda$ be any weight of $V(\omega)$.
Then $\lambda\in\supp(\omega)$ if and only if $\theta(\lambda)=-\lambda$
    and $\omega-\lambda$ lies in the lattice $\Lambda$.
\end{thm}

\begin{proof}
First suppose that $\lambda\in\supp(\omega)$.
Since $\lie{t}\subseteq \lie{k}$ and $v^{\omega}$ is $\lie{k}$-invariant, we have for any $t\in\lie{t}$ $$0 = tv^{\omega} = \sum_{\mu}\mu(t)v^{\omega}_{\mu}$$ from which it follows that $\lambda(t)=0$.
Then, for any $x\in\lie{h}$, write $x = x_+ + x_-$ with $x^+\in\lie{t}$
    and $x_-\in\lie{a}$.
We have

\begin{align*}
(\theta(\lambda))(x) &= \lambda(\theta(x_+ + x_-)) \\
&= \lambda(x_+-x_-) \\
&= \lambda(x_-) \\
&= \lambda(-x_+-x_-) \\
&= (-\lambda)(x)
\end{align*}

so that $\theta(\lambda)=-\lambda.$
The proof that $\omega-\lambda$ belongs to the lattice $\Lambda$ is by downward induction with respect to the partial order $\preceq$.
In the base case, $\omega-\lambda=0$ certainly belongs to $\Lambda$.
Otherwise $\lambda$ is not a highest weight of $V(\omega)$, so by the argument given in the proof of \pref{lem:highest_in_support}, there is some positive root $\alpha$ with $\lambda+\alpha-\theta(\alpha)\in\supp(\omega)$, and the result follows immediately by induction.

On the other hand, suppose that $\theta(\lambda)=-\lambda$ and $\omega-\lambda$
    belongs to $\Lambda$.
We shall prove that $\lambda\in\supp(\omega)$, again by downward induction with respect to $\preceq$.
The base case is \pref{lem:highest_in_support}.
For the general case, recall that $W_{\lie{a}}$ denotes the Weyl group of the pair $(G,K)$; in particular, each of its elements is a coset of an element of $K$.
It follows that $W_{\lie{a}}$ stabilizes the set
    $\{\mu|_{\lie{a}}\}_{\mu\in\supp{\omega}}$, so
    we may assume without loss of generality that $\lambda|_{\lie{a}}$ is dominant.
But this implies that $\lambda$ itself is a dominant weight of $\lie{g}$.
To see this, begin by noting that $\lambda\in\ann(\lie{t})$ since
    $\theta(\lambda)=-\lambda$.
Then for any positive root $\alpha$ we can write $\alpha = \alpha_+ + \alpha_-$
    with $\alpha_+\in\ann(\lie{a})$ and $\alpha_-\in\ann(\lie{t})$, giving
    $$(\lambda,\alpha) = (\lambda, \alpha_-) = (\lambda|_{\lie{a}}, \alpha|_{\lie{a}}) \geq 0$$
    and proving that $\lambda$ is dominant.
Now write $$\omega-\lambda = \sum_{\alpha\in\Delta^+}
    k_{\alpha}(\alpha-\theta(\alpha))$$ with $k_{\alpha}\in\ints$.
Since $\omega$ is the \emph{highest} weight of $V(\omega)$ we may take all
    $k_{\alpha}$ non-negative.
Evidently we may also take $k_{\alpha}=0$ whenever $\alpha$ is $\theta$-
    fixed.
Now choose $\alpha\in\Delta^+$ with $k_{\alpha}>0$.
Let $\lambda'$ be the unique dominant $W_a$-conjugate of $\lambda+\alpha-\theta(\alpha)$.
By \pref{lem:preceq}, we have $\lambda\preceq\lambda'$.
Evidently $\theta(\lambda')=-\lambda'$.
Choose $w\in W_{\lie{a}}$ with $$w(\lambda+\alpha-\theta(\alpha)) = \lambda'.$$
We have
\begin{align*}
\omega-\lambda' &= \omega - w(\lambda-\omega+\omega+\alpha-\theta(\alpha))\\
&= \omega - w(\omega) + w(\omega-\lambda) - w(\alpha-\theta(\alpha))
\end{align*}
which, by Lemmas~\ref{lem:stable} and \ref{lem:lhat_to_l}, lies in $\Lambda$.
Then by induction, $\lambda'\in\supp(\omega)$, so also $\lambda+\alpha-\theta(\alpha)\in\supp(\omega).$

Since $\theta(\alpha)\neq\alpha$, it follows from the Cauchy-Schwartz inequality that $(\theta(\alpha),\alpha)<\abs{\alpha}^2$.
Hence
\begin{align*}
(\lambda + \alpha - \theta(\alpha), \alpha) &= (\lambda,\alpha) + \abs{\alpha}^2 - (\theta(\alpha),\alpha) \\
&> (\lambda,\alpha) \\
&\geq0.
\end{align*}
from which it follows that $$X_{-\alpha}v^{\omega}_{\lambda+\alpha-\theta(\alpha)} \neq 0.$$
Now since $v^{\omega}$ is $\lie{k}$-invariant, we must have $$X_{-\alpha}v^{\omega} = -\theta(X_{-\alpha})v^{\omega}.$$
Extracting the component of each side of weight $\lambda-\theta(\alpha)$ gives $$X_{-\alpha}v^{\omega}_{\lambda+\alpha-\theta(\alpha)} = -\theta(X_{-\alpha})v^{\omega}_{\lambda}$$ forcing $v^{\omega}_{\lambda}\neq0$ and hence $\lambda\in\supp(\omega)$.

\end{proof}

\section{Representative functions on the adjoint form}
\label{sec:adjoint}

In this section we prove \pref{cor:inclusion} characterizing the image
    of the injection $\repf(\ad(M))\rightarrow\repf(M)$.
We begin with a trivial modification of \cite{kr71}, Proposition~1:

\begin{lem}
Let $F$ be the group $\{a\in A|a^2\in Z\}$.
Then $K_Z=FK_0$.
\end{lem}

\begin{proof}
Suppose first that $a\in F$.
We have $\theta(a)^{-1}a=a^2\in Z$, so $F\subseteq K_Z$.
Now $K_0$ is also the connected component of $K_Z$, so $F$ normalizes it
    and hence $FK_0$ is a group with $FK_0 \subseteq K_Z$.
On the other hand, suppose $g\in K_Z$.
Using the $K_0AK_0$ decomposition for $G$ (\cite[Theorem~V.6.7]{helgason78}),
    we can write $g=k_1ak_2$ for $k_i\in K_0$ and $a\in A$.
Then $z=\theta(g)^{-1}g=k_2^{-1}ak_1^{-1}k_1ak_2=k_2^{-1}a^2k_2$
    lies in the center of $G$.
But then $a^2=k_2zk_2^{-1}=z$, showing that $a\in F$.
Thus $g=k_1ak_2$ lies in $FK_0$.
\end{proof}

\begin{lem}
The group $F$ coincides with the set $\{\exp(a)\,|\,a\in\lie{a}\text{ and }
    (\alpha-\theta(\alpha))(a)\in2\pi i\ints,\,\forall\alpha\in\Pi\}$.
\end{lem}

\begin{proof}
Evidently $F$ coincides with $\{\exp(a)\,|\,a\in\lie{a}\text{ and }
    \alpha(2a)\in2\pi i\ints,\,\forall \alpha\in\Pi\}$.
But $(\theta(\alpha))(a)=-\alpha(a)$.
\end{proof}

This leads immediately to

\begin{cor}
\label{cor:gens}
Let $x_1,\dots,x_{\dim(\lie{a})}$ be the basis for $\lie{a}$ dual to $\Pi'$
    and put $S=\{\exp(\pi ix_1),\dots,\exp(\pi i x_{\dim(\lie{a})})\}$.
Then $F$ is generated by $S$.
\end{cor}

\begin{lem}
\label{lem:weightann}
Let $\lambda$ be an analytically integral weight annihilating $\lie{t}$.
Then $\lambda$ annihilates $F$ if and only if $\lambda$ belongs to the lattice
    $\Lambda$.
\end{lem}

\begin{proof}
Suppose first that $\lambda$ annihilates $F$.
In the notation of \pref{cor:gens}, we must have $\lambda(x_i)\in2\ints$.
Let $\{\alpha_i\}$ be a set of positive roots restricting to $\Pi'$.
Put $\mu = \sum_i(\frac{\lambda(x_i)}{2} (\alpha_i-\theta(\alpha_i))$.
Then $\lambda$ agrees with $\mu$ on both $\lie{t}$ and $\lie{a}$, so in
    fact $\lambda=\mu$.

On the other hand, suppose that $\lambda$ belongs to $\Lambda$.
It follows immediately that $\lambda(x_i)\in2\ints$ for all $i$, and hence
    that the corresponding weight of $G$ annihilates $F$.
\end{proof}

\begin{thm}
Let $V(\omega)$ be an irreducible representation of $G$ containing a non-zero
    $\lie{k}$-invariant.
Then $V(\omega)$ contains a non-zero $K_Z$-invariant if and only if $\omega
    \in\Lambda$.
\end{thm}

\begin{proof}
Since both $K_Z$ and $K_0$ are spherical subgroups of $G$, we need only
    determine when $v^{\omega}$ is also $K_Z$-invariant.
This happens if and only if $v^{\omega}$ is $F$-invariant, which happens
    if and only if all the weights in $\supp(\omega)$ annihilate $F$.
By \pref{lem:weightann}, this happens if and only if every member of
    $\supp(\omega)$ belongs to $\Lambda$, and by \pref{thm:support}
    this happens if and only if $\omega$ itself belongs to $\Lambda$.
\end{proof}

\begin{cor}
\label{cor:inclusion}
The image of the injection $\repf(\ad(M))\rightarrow\repf(M)$ induced by the
    covering map consists precisely of those $V(\omega)$ contained in $\repf(M)$
    such that $\omega^*$, the highest weight of the dual of $V(\omega)$, belongs
    to $\Lambda$.
\end{cor}

\begin{proof}
The ring $\repf(M)$ consists of those $V(\omega)$ such that $V(\omega)^*$ contains
    a non-zero $K$-invariant, and the ring $\repf(\ad M)$ consists of those
    $V(\omega)$ such that $V(\omega)^*$ contains a non-zero $K_Z$-invariant.
\end{proof}

\section{The main result}
\label{sec:main}

\begin{thm}
\label{thm:main}
The kernel of the torus transform $\tau$ is the orthogonal complement of the
    image of the embedding $L^2(\ad(M))\rightarrow L^2(M)$ induced by the
    covering map.
\end{thm}

\begin{proof}
Since $\repf(M)$ is dense in $L^2(M)$, the kernel of $\tau$ is the closure
    of the kernel of $\tau_o$.
Since $A$ is connected, \pref{thm:reynolds} implies that $\ker\tau_0$ consists of
    all those $V(\omega)$ such that $\supp(\omega^*)$ contains no weight
    restricting to zero on $\lie{a}$.
But by \pref{thm:support}, these are precisely the $V(\omega)$ such that
    $\omega^*\not\in\Lambda$, and by \pref{cor:inclusion} these are precisely
    the $V(\omega)$ not in the image of the injection $\repf(\ad(M))
    \rightarrow\repf(M)$ induced by the covering map.
The result follows by taking closures.
\end{proof}

\section*{Acknowledgement}

The authors wish to thank Todor Milev and Alfred No\"el for helpful
    conversations.

\bibliographystyle{amsalpha}
\bibliography{torus}

\end{document}


TITLE: Operator norm of $\mathcal{A}:\ell^2 \rightarrow \mathbb{R}$
QUESTION [0 upvotes]: If $\ell^2$ denotes the sequence space, containing all sequences such that
$$\sum |x_n|^2 < \infty$$
and $\mathcal{A}$ is a (pressumingly bounded) linear operator, mapping such sequences to some real number $\in \mathbb{R}$. How would I go on defining an operator norm of such mapping? Is it even defined? I have searched through my literature and lecture notes but am unable to find something. Especially, if we have a sequence of $\mathcal{A}_n \in \ell^2$ mapping some sequence to an indexed number $r_n \in \mathbb{R}$?

REPLY [3 votes]: Given any linear map $T: V \to W$ between normed vector spaces, you can define
$$\|T\| := \sup_{v\in V, \|v\|_V\le 1}\|Tv\|_W.$$
Apply this with $V= \ell^2, W = \mathbb{R}.$


TITLE: Normal subgroup in $G=G_1 \rtimes G_2$
QUESTION [3 upvotes]: Given $G=G_1 \rtimes G_2$, so that $\rtimes$ is a semi direct product.
(1) Is it always true that $G/G_1=G_2$ as a quotient group?
(2) Are there occasions that $G/G_2$ makes sense?

REPLY [0 votes]: To expand, $G/G_2$ always makes sense. It is the set of (left or right, it's a little ambiguous) cosets of $G_2$ in $G$. However, unless $G_2$ is normal it won't be a group, but it will at least have the same size as $G_1$.


TITLE: Simpler ways to show that $n^2$ divides a polynomial?
QUESTION [4 upvotes]: I want to show that $n^2 \mid P(n)$, where $$P(n) = \frac{n^2(n+1)^2(n+2)(n+3)}{48}$$ for every odd positive integer $n$. The approach I took involved showing that $\cfrac{P(n)}{n^2}$ is always an integer (for such $n$), but then I had to create a polynomial even more complex and then prove nine different cases. While it did provide a valid proof (as far as I know), I have a feeling it was more work than I needed.
So my question is: are there "simpler" proofs to this problem, and what are their approaches/methods? By simpler I roughly mean: prove less cases, reduce the problem to a simpler form, etc; basically a solution that takes up less "space" on paper. (I know that's not the best explanation, sorry!)
Thank you very much!

REPLY [2 votes]: Hint $\, 3\,$ divides a product of $3$ consecutive integers (proof), and even $\,n\!+\!1 = \color{#0a0}{4k}\,$ or $\,\color{#90f}{4k\!+\!2}\,$ and
$$\color{#c00}{16}\mid \underbrace{(\color{#0a0}{4k})^2}_{\Large\color{#c00}{ 4^2}}(4k\!+\!1)(4k\!+\!2)\ \ \ {\rm and}\ \ \ \color{#c00}{16}\mid \underbrace{(\color{#90f}{4k\!+\!2})^2}_{\Large \color{#c00}{2^2}}(4k\!+\!3)\underbrace{(4k\!+\!4)}_{\Large\color{#c00} 4}$$


TITLE: Scrabble Probability
QUESTION [0 upvotes]: Scrabble player 1 and Scrabble player 2 each draw $7$ letters from a bag of $100$ at the start of a Scrabble game.  If there are $6$ "T"s in a bag of Scrabble letters, what is the probability player 1 will draw $3$ of them?

REPLY [1 votes]: Since Scrabble tiles are taken without replacement, you would use the hypergeometric distribution to find the probability of 3 successes in 7 draws, given that there are 6 potential successes in the entire population of 100 tiles.
The hypergeometric distribution is defined as $\dfrac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}$, where $N$ is the total population (here, the 100 tiles), $n$ is the number of draws you will make (in this case, 7 draws), $K$ is the total number of successes in the population (the 6 T tiles), and $k$ is the number of successful draws you will make (in this case, 3).
Plugging in the numbers will get you the result:
$P(X = 3) = \dfrac{\binom{6}{3}\binom{94}{4}}{\binom{100}{7}} \approx 0.00381. $
You can think about this intuitively as follows: we're choosing 3 "successes" out of 6 possible successes, and 4 "failures" out of 94 possible failures, and there are 100 choose 7 possible ways of choosing any combination of tiles at all.


TITLE: Why Do Hurricane Balls Spin So Fast?
QUESTION [8 upvotes]: I was wondering if anyone could offer an explanation as to why the balls described in this video spin so fast.
Here's the setup: Two metal balls  are wielded together. When spun with air, they acquire a massive amount of rpm.

REPLY [1 votes]: All above answers hold some validity.  Additionally, the high speed stream of air is directed initially at the outside of the spinning balls, obviously.  However, once the spinning is stable, the steam of air is gradually directed closer and closer to the center of the spinning mass.  That is the real reason the very high RPM is achieved.  Think of the technique as a transmission shifting into a higher and higher gear as the speed increases.


text:
\begin{document}

\title{Electric-magnetic duality and topological order on the lattice}

\author{Oliver Buerschaper}

\affiliation{
  Max-Planck-Institut f\"ur Quantenoptik, Hans-Kopfermann-Stra\ss{}e
  1, D-85748 Garching, Germany}

\author{Matthias Christandl}

\affiliation{
  Fakult\"at f\"ur Physik, Ludwig-Maximilian-Universit\"at,
  Theresienstra\ss{}e 37, D-80333 Munich, Germany}

\affiliation{
  Institute f\"ur Theoretische Physik, Eidgen\"ossische Technische
  Hochschule Z\"urich, Wolfgang-Pauli-Stra\ss{}e 27, CH-8093 Z\"urich,
  Switzerland}

\author{Liang Kong}

\affiliation{
  California Institute of Technology, Center for the Physics of
  Information, Pasadena, CA 91125, USA}

\affiliation{
  Institute for Advanced Study (Science Hall), Tsinghua University,
  Beijing 100084, China}

\author{Miguel Aguado}

\affiliation{
  Max-Planck-Institut f\"ur Quantenoptik, Hans-Kopfermann-Stra\ss{}e
  1, D-85748 Garching, Germany}

\date{\today}

\pacs{}

\begin{abstract}
  Dualities are deep conceptual tools in many areas of physics and
  mathematics; in particular, electric-magnetic (EM) duality has a
  long tradition ranging from Dirac's magnetic poles to $S$-duality in
  string and field theory.  Here we investigate the duality structure
  in quantum lattice systems with topological order, a collective
  order also appearing in fractional quantum Hall systems.  We define
  EM duality for all of Kitaev's quantum double models based on
  discrete gauge theories with Abelian and non-Abelian groups, and
  identify its natural habitat as a new class of topological models
  based on Hopf algebras.  We interpret these as extended string-net
  models, whereupon Levin and Wen's string-nets, which describe all
  topological orders on the lattice with parity and time-reversal
  invariance, arise as magnetic and electric projections of the
  extended models.  We conjecture that all string-net models can be
  extended in an analogous way, using a more general algebraic
  structure, such that EM duality continues to hold.  Physical
  applications include topology probes in the form of pairs of dual
  tensor networks.
\end{abstract}

\maketitle

\emph{Duality} ranks among the deepest ideas in physics.  Dualities
are powerful probes into the internal structure of mathematical and
physical theories, including quantum many-body systems and field
theories~\cite{Savit}.  Frequently a duality relates weakly and
strongly coupled regimes of physical systems, providing precious
information beyond the reach of perturbative methods.  Many physical
instances of duality involve the exchange of electric and magnetic
degrees of freedom.  Such a symmetry for the Maxwell equations
\emph{in vacuo} led Dirac~\cite{Dirac} to the introduction of
pointlike sources of magnetic field to extend this
\emph{electric-magnetic} (EM) duality to matter, providing a unique
argument for the quantisation of electric charge.  Dirac's insight has
had profound influence in field theory and beyond (see,
e.~g.,~\cite{Seiberg}).

\emph{Topological order}~\cite{WenOrder} appears in a fascinating
class of condensed matter phases, notably the fractional quantum Hall
effect.  Topological phases are sensitive only to global properties of
the underlying space: their effective quantum field theories have only
non-local degrees of freedom~\cite{TQFT}.  This non-local order has
deep connections to high-energy physics and a rich mathematical
structure.  In $2D$, it is associated with localised objects with
fractional statistics, that is to say, \emph{anyons}~\cite{Wilczek}.
In topological lattice models, these anyons appear as excitations.

Kitaev's proposal~\cite{Kitaev} to use topologically ordered quantum
lattice systems to store and process quantum information linked
topological order and quantum computation.  His pioneering
\emph{quantum double} $\mathrm{D} (G)$ models are built in
analogy to gauge theories with finite gauge groups.  In this context,
the topological degrees of freedom are Wilson loops; violations of
gauge invariance and magnetic fluxes are described analogously,
hinting at EM duality.  Particularly simple are the models based on
Abelian groups, including the toric code, the first laboratory for
many ideas in topological order.  Although there are deeper objects at
play, $\mathrm{D} (G)$ models can be conveniently studied using group
theory; there is an explicit \emph{local} understanding of anyonic
excitations and their interplay in terms of group representations.
These models provide us with the first emergence of duality in
topologically ordered systems, namely a self-duality of the toric code
and Abelian $\mathrm{D} (G)$ models, involving the direct and dual
lattices; an EM duality, in gauge theory language.  Its importance has
been stressed by Fendley~\cite{Fendley}.

Levin and Wen defined their landmark string-net (SN)
models~\cite{LevinWen} from the physical intuition that topological
phases are renormalisation group fixed points. SNs are built on ideas
from category theory to describe \emph{all} topological orders on the
lattice with parity and time-reversal invariance.  Technically, they
use data from unitary tensor categories; in this spirit, dualities in
SN models, in the sense of equivalences of unitary modular tensor
categories, can be identified with invertible domain walls~\cite{KK}.
While SN models are broader in scope than Kitaev's quantum double
models, their excitations are, however, not understood as thoroughly
as those of the $\mathrm{D} (G)$ models, especially locally.

So far the extension of the EM duality of the toric code to
non-Abelian models has remained an open problem, and thus a key source
of insight into the structure of topological order has been only
available in the very restricted Abelian setting.  The natural place
to look for a well defined EM duality are Kitaev's quantum double
models, since the excitations there are clearly understood and can be
assigned electric and magnetic quantum numbers, while this
characterisation is absent in SN models.

Here we uncover EM duality for all of Kitaev's $\mathrm{D} (G)$'s,
both Abelian and non-Abelian, and beyond.  We do this by identifying
its natural setting as the quantum double models based on Hopf
algebras, a class of models anticipated in~\cite{Kitaev} and defined
and studied in~\cite{BMCA}.  Then we use Fourier techniques~\cite{BA}
to write these as \emph{extended} string-net models (ESNs), a variant
of Levin and Wen's SNs with a canonically enlarged Hilbert space, such
that anyons can be understood in representation theoretic
language~\cite{footnotea}.  ESNs can be defined independently, in
analogy to Levin and Wen's SNs, by data from unitary tensor
categories, equipped in this case with a so-called fibre
functor~\cite{forthcoming}.  By EM duality, each ESN maps to two SNs,
its electric and magnetic projections (as depicted in
figure~\ref{fig:schememodels}).  We conjecture that \emph{all} SNs are
obtained in such a way from parent ESNs with a generalised algebraic
structure, in which EM duality continues to hold.  These results lead
us to propose the duality landscape of figure~\ref{fig:dualities}.

\begin{figure}
\begin{center}

  \includegraphics{fig1.eps}
\end{center}
\caption{\label{fig:dualities} \textbf{A duality landscape}.  We
  represent: Quantum doubles $\mathrm{D} (A)$ based on Abelian groups,
  including the toric code, all of which are self-dual; general group
  QD models $\mathrm{D} (G)$, related by duality to quantum double
  models $\mathrm{D} (G^\ast)$ based on algebras of \emph{functions}
  on groups, whose intersection with $\mathrm{D} (G)$ is $\mathrm{D}
  (A)$; these are all instances of quantum double models $\mathrm{D}
  (H)$ based on Hopf algebras, a larger class closed under duality,
  with cases of self-duality beyond groups~\cite{BMCA}.  We conjecture
  that EM duality can be defined for extended string-net models
  $\mathrm{ESN}$, and that these can be identified as quantum double
  models $\mathrm{D} (W)$ based on weak Hopf $C^\ast$-algebras.
  Arrows denote model identifications upon lattice dualisation.}
\end{figure}

\section{\label{sec:toric} Kitaev's models based on groups}

Let us begin by reviewing the well-known self-duality in the toric
code~\cite{Kitaev}, the quantum double model based on the
$\mathbb{Z}_2$ group.  This is a model of qubits along the bonds of a
$2D$ lattice $\Lambda$ (the dimension of the local Hilbert spaces
being the size of the group), with a Hamiltonian of the form

\begin{equation}\label{generic_hamiltonian}
  \mathcal{H}
=
  - \, \sum_v A_v - \, \sum_p B_p
\; ,
\end{equation}
where $v$ runs over vertices and $p$ runs over plaquettes of
$\Lambda$.  Mutually commuting vertex and plaquette operators are
projectors involving Pauli matrices:

\begin{equation}\label{toric_code_projectors}
  A_v
=
  \frac{
    1 + \bigotimes_{ i \in v } \sigma^x_i
  }{
    2
  }
\; , \quad
B_p
=
  \frac{
    1 + \bigotimes_{ j \in p } \sigma^z_j
  }{
    2
  }
\; ,
\end{equation}
with support on the bonds around the corresponding vertex or
plaquette.  Self-duality means that, upon a global unitary

\begin{equation}\label{}
  U_\Lambda
=
  \bigotimes_{ i \in \Lambda } U_i
\; , \quad
  U_i
=
  \frac{ ( \sigma^x + \sigma^z )_i }{ \sqrt{2} } \,
\end{equation}
built from Hadamard maps on each edge $i$, the toric code on $\Lambda$
gets mapped to a toric code on the dual lattice $\Lambda^\ast$:

\begin{equation}\label{hadamard_for_toric_code}
  U_\Lambda \, A_v \, U_\Lambda^\dagger
=
  \widetilde{B}_v
\; , \quad
  U_\Lambda \, B_p \, U_\Lambda^\dagger
=
  \widetilde{A}_p
\; ,
\end{equation}
where now $\widetilde{B}_v$ corresponds to dual plaquette $v$, and
$\widetilde{A}_p$ to dual vertex $p$ in $\Lambda^\ast$.  Ground states
of the frustration-free Hamiltonian~(\ref{generic_hamiltonian})
minimise each term in the sum, that is, they satisfy all vertex and
plaquette constraints $A_v \lvert \psi \rangle = \lvert \psi \rangle$,
$B_p \lvert \psi \rangle = \lvert \psi \rangle$.  Breakdown of any one
such constraint means the state is in an excited level, and can be
interpreted as the presence of a localised excitation, or particle,
sitting at the vertex or plaquette that is broken; from the gauge
theory interpretation, these are called electric and magnetic
excitations.  The electric-magnetic nature of
duality~(\ref{hadamard_for_toric_code}) comes then from the
interchange of vertices and plaquettes in going from the original to
the dual lattice.

Hidden in the action of the global Hadamard map $U_\Lambda$ are,
first, the mapping from $\Lambda$ to the dual lattice and the
corresponding reinterpretation of vertex and plaquette operators, and
second, a mapping of elements of the group $\mathbb{Z}_2$ to the
functions from this group to the scalars.  If we obtain again a toric
code in the dual lattice it is because, for an \emph{Abelian} group,
the space of functions has the same structure as the group algebra,
which is the space of linear combinations of group elements.  One can
exploit this algebraic fact to extend the self-duality of the toric
code to all $\mathrm{D} (G)$ models for Abelian groups in a
straightforward way: this is the $\mathrm{D} (A)$ region of
figure~\ref{fig:dualities}.

For general $\mathrm{D} (G)$ models, we start with a lattice with
oriented bonds, whose Hilbert spaces have an orthonormal basis $\{
\lvert g \rangle \}$ labelled by elements of a finite group $G$.  The
Hamiltonian of the $\mathrm{D} (G)$ model is still of the
form~(\ref{generic_hamiltonian}), but now the mutually commuting
vertex and plaquette operators are the projectors defined in
figure~\ref{fig:vertexplaquettedg}.

\begin{figure}
\begin{center}

  \includegraphics{fig2.eps}
\end{center}
\caption{\label{fig:vertexplaquettedg} \textbf{Projectors in the
    $\mathrm{D} (G)$ model.}  Vertex ($A_v$) and plaquette ($B_p$)
  projectors in the Hamiltonian of Kitaev's $\mathrm{D} (G)$ model are
  defined by their action on a basis of the corresponding multi-edge
  Hilbert space.  Oriented edges are labelled by group elements
  denoting kets in the orthonormal computational basis $\{ g \}_{ g
    \in G }$; change of orientation is implemented by applying the
  inversion map $g \mapsto g^{-1}$.  The vertex operator projects onto
  gauge-invariant configurations (=symmetrises over simultaneous
  multiplication with group elements).  The plaquette operator
  involves a Kronecker delta on the unit element $e \in G$ and
  projects onto trivial magnetic flux across $p$.}
\end{figure}
The obvious parallelism of plaquettes and vertices in the Abelian case
is lost if $G$ is non-Abelian.  Upon switching to the dual lattice, we
cannot reinterpret the operators $A_v$ and $B_p$ as plaquette and
vertex operators in a quantum double model based on a group, since the
functions on a non-Abelian group cease to have the same structure as
the group algebra itself.  Hence, the extension of the EM duality to
non-Abelian groups is not possible within the class of quantum doubles
based on groups.  Yet the construction of the Abelian duality can be
generalised to the non-Abelian case, resulting in dual models which
are no longer quantum double models based on groups, but rather
quantum double models based on algebras of functions; this is shown as
the arrow between the regions $\mathrm{D} (G)$ and $\mathrm{D}
(G^\ast)$ in figure~\ref{fig:dualities}.  To make sense of this, one
has to widen the construction of quantum double lattice models beyond
the group case.

As we will show, the natural habitat for the EM duality of $\mathrm{D}
(G)$ models is the class of quantum double models based on \emph{Hopf
  algebras}~\cite{BMCA}; this is the region~$\mathrm{D} (H)$ in
figure~\ref{fig:dualities}.  This is the smallest setting containing
all $\mathrm{D} (G)$ models and closed under tensor products and EM
duality; the latter, moreover, takes a remarkably simple form in the
language of Hopf algebras.

\section{\label{sec:hopf} The $\mathrm{D} (H)$ lattice models}

When studying quantum many-body systems, more general notions of
symmetry emerge than those furnished by group theory.  In particular,
linear transformations acting on tensor products of vector spaces lead
naturally to Hopf algebras.  For a detailed account of these in our
context we refer to~\cite{BMCA}.  In the following we give an
intuitive grasp of their structure, which is necessary to understand
the $\mathrm{D} (H)$ lattice models.

We consider Hopf algebras $H$ as spaces of transformations on a
many-body Hilbert space.  First of all, the linear nature of the
target is naturally extended to its transformations, so Hopf algebras
are vector spaces.  We must be able to compose transformations and to
include the identity transformation, so $H$ has a multiplication of
vectors, and a unit.  Most importantly, we must have a rule to
\emph{distribute} the action of an element of $H$ into a tensor
product of target spaces; this is the so-called comultiplication.
Additionally $H$ has a trivial representation $\varepsilon$, called
counit, making precise the notion of spaces invariant under the action
of $H$.  Like for groups, the representation theory of Hopf algebras
includes a notion of conjugate representation, implemented via an
antipode mapping, which for groups is just the inversion $g \mapsto
g^{-1}$.

In order to construct Hilbert spaces, we use Hopf $C^\ast$-algebras,
which in addition have an inner product.  We will only consider
finite-dimensional Hopf $C^\ast$-algebras, which come equipped with a
canonical, highly symmetric element, the \emph{Haar integral} $h \in
H$, invariant under multiplication in the sense $a \cdot h =
\varepsilon(a) \, h$ for all elements $a$ of $H$.  This canonical
element is crucial for the construction of the lattice model and its
ground states.

\begin{figure}
\begin{center}

  \includegraphics{fig3.eps}
\end{center}
\caption{\label{fig:hopf} \textbf{Structure and dualities in Hopf
    algebras.}  The \emph{first row} represents the structure maps of
  a Hopf algebra $H$.  Solid lines represent elements of $H$, and
  dotted lines represent scalars in $\mathbb{C}$.  The \emph{second
    and third rows} illustrate the algebraic duality within the class
  of Hopf algebras, taking as examples a group algebra $H = \mathbb{C}
  G$ and its dual $H^\ast$, whose structure is defined implicitly in
  the figure.  For each diagram, the upper expression is equal to the
  lower expression; the structure maps of $H$ and $H^\ast$ are
  represented by solid and dashed lines, respectively, according to
  the conventions of the first row.  Hence, the multiplication in
  $H^\ast$ is determined by the comultiplication in $H$, the unit in
  $H^\ast$ is determined by the counit in $H^\ast$, etc.; this duality
  holds for general finite-dimensional Hopf algebras.  We use the form
  of the structure maps in the basis $\{ g; \, g \in G \}$ of the
  group algebra, and the basis $\{ \delta_\ell; \, \ell \in G \}$ of
  its dual, where $\delta_\ell : g \mapsto \delta_{ \ell, \, g }$ are
  Kronecker delta functions.  As for the extra structure needed to
  define our lattice models, the $C^\ast$-algebra structures of $H$
  and $H^\ast$ are defined by declaring $\{ g \}$ to be an orthonormal
  basis, and $\{ \delta_\ell \}$ to be orthonormal up to a factor, $(
  \delta_\ell, \, \delta_m ) = \lvert G \rvert^{-1} \, \delta_{ \ell,
    \, m }$.  The Haar integrals $h$ and $\phi$ in the group algebra
  and its dual are defined in equation~(\ref{haargroups}).}
\end{figure}

The root of the EM duality to be unveiled in the following is an
algebraic fact: \emph{The class of Hopf $C^\ast$-algebras is closed
  under dualisation}.  That is, given a Hopf $C^\ast$-algebra $H$, its
dual space $H^\ast$ of \emph{functions} from $H$ to the scalars is
again a Hopf $C^\ast$-algebra, whose structure is, moreover,
determined by the structure of $H$, as illustrated by
figure~\ref{fig:hopf}.  This closure property is shared by the class
of Abelian group algebras (which are all self-dual), but not for the
whole class of group algebras.  The landscape of
figure~\ref{fig:dualities} reflects these statements in the world of
lattice models.

\begin{figure}
\begin{center}

  \includegraphics{fig4.eps}
\end{center}
\caption{\label{fig:vertexplaquettedh} \textbf{The core of $\mathrm{D}
    (H)$ models}.  Topological models based on Hopf algebras are
  defined via these vertex and plaquette representations of $u \in H$
  and $f \in H^\ast$~\cite{footnoteb}.  Oriented edges carry, in
  general, elements of a finite-dimensional Hopf $C^\ast$-algebra $H$;
  for Kitaev's models, take $H$ to be the group algebra $\mathbb{C}
  G$.  Orientation can be reversed by using the antipode map $S$
  referred to in the text.  The sums denote the appropriate
  \emph{comultiplication} or splitting of algebra elements into
  several tensor factors.  Vertex operators $A_{v, \, p}$ now depend
  also on a plaquette $p$ marking the beginning of a loop around the
  vertex $v$, and correspondingly for the plaquettes.  These operators
  probe the algebraic structure of the models: Given $v$ and $p$,
  $A_{v, \, p} (u)$ and $B_{p, \, v} (f)$ interact nontrivially,
  building representations of Drinfel'd's \emph{quantum double}
  algebra $\mathrm{D} (H)$~\cite{drinfeld}.  The irreducible
  representations of $\mathrm{D} (H)$ classify the superselection
  sectors, and thus the anyonic structure of excitations, of the
  quantum double lattice models~\cite{Kitaev}.}
\end{figure}
Figure~\ref{fig:vertexplaquettedh} defines a set of operators on a
lattice whose bonds carry elements of~$H$, a finite-dimensional Hopf
$C^\ast$-algebra.  These operators depend on elements of the algebra
and its dual, respectively.  When these are taken as the Haar
integrals $h$ of $H$ and $\phi$ of $H^\ast$, the resulting operators

\begin{equation}\label{ophamiltdh}
  A_v = A_{ v, \, p } ( h )
\; , \quad
  B_p = B_{ p, \, v } ( \phi )
\end{equation}
define via~(\ref{generic_hamiltonian}) the Hamiltonian of a
topological lattice model: the quantum double model based on $H$, or
$\mathrm{D} (H)$ model for short~\cite{BMCA}.

Kitaev's models are a particular subclass of $\mathrm{D} (H)$ models,
recovered when $H$ is the group algebra of a finite group $G$.  Both
this algebra $\mathbb{C} G$ and its dual $G^\ast$ have particularly
simple Hopf $C^\ast$-algebra structures, summarised in
figure~\ref{fig:hopf}.  In particular, their Haar integrals read

\begin{equation}\label{haargroups}
  h
=
  \frac{ 1 }{ \lvert G \rvert } \,
  \sum_{ g \in G } g
\in
  \mathbb{C} G
\; , \quad
  \phi = \delta_e \in G^\ast
\; .
\end{equation}
Taking into account this equation, operators in
figure~\ref{fig:vertexplaquettedg}, defining the Hamiltonian of the
$\mathrm{D} (G)$ model, can be checked to coincide with the general
expression~(\ref{ophamiltdh}).

\section{\label{sec:em} Electric-magnetic duality}

We now define EM duality for general $\mathrm{D} (H)$ models.
Consider the following unitary map $\mathcal{U}$ from $H$ to its dual:

\begin{equation}\label{emduality}
  a \stackrel{\mathcal{U}}{\longmapsto} f_a \; ,
\quad
  f_a (b)
=
  \sqrt{ \lvert H \rvert } \,
  \phi ( a \cdot b )
\; ,
\end{equation}
where $\lvert H \rvert$ is the dimension of $H$, and the function
$f_a$ is constructed via the dual Haar integral $\phi \in H^\ast$ (for
instance, for groups, $f_g = \sqrt{ \lvert G \rvert } \, \delta_{
  g^{-1} }$).  We associate this mapping with the transformation of
$\Lambda$ into its dual lattice $\Lambda^\ast$ as shown in
figure~\ref{fig:dualbondvar}.

\begin{figure}
\begin{center}

  \includegraphics{fig5.eps}
\end{center}
\caption{\label{fig:dualbondvar} \textbf{Going over to the dual
    lattice}. The mapping $\mathcal{U}: H \rightarrow H^\ast$ is here
  interpreted as a lattice dualisation.  The solid bond is in
  $\Lambda$, while the dashed bond is its dual in $\Lambda^\ast$.  The
  orientation of the dual edge is chosen by convention so that it
  points to the left of the direct edge.  The degree of freedom $a \in
  H$ in the Hilbert space $H$ of the bond in $\Lambda$ becomes a
  degree of freedom $f_a$ of the Hilbert space $H^\ast$ of the dual
  bond.  When applied simultaneously on all edges of $\Lambda$, this
  mapping implements EM duality.}
\end{figure}

The vertex and plaquette representations in
figure~\ref{fig:vertexplaquettedh}, associated with the $\mathrm{D}
(H)$ model on $\Lambda$, are mapped by $\mathcal{U}_\Lambda =
\bigotimes_{ i \in \Lambda } \mathcal{U}_i$ into precisely the
representations associated with the $\mathrm{D} ( H^\ast )$ model on
the dual lattice $\Lambda^\ast$~\cite{footnotec}:

\begin{equation}\label{maprepresentationstodual}
  \mathcal{U}_\Lambda \,
  B_{ p, \, v } ( f ) \,
  \mathcal{U}_\Lambda^\dagger
=
  { \widetilde A }_{ p, \, v } ( f )
\; , \quad
  \mathcal{U}_\Lambda \,
  A_{ v, \, p } ( u ) \,
  \mathcal{U}_\Lambda^\dagger
=
  { \widetilde B }_{ v, \, p } ( u )
,
\end{equation}
where $p$ is a plaquette of $\Lambda$ and a vertex of $\Lambda^\ast$;
this expression generalises equation~(\ref{hadamard_for_toric_code})
and identifies $\mathcal{U}_\Lambda$ as the EM duality mapping.

Thus, $\mathcal{U}_\Lambda$ \emph{identifies} the $\mathrm{D} (H)$
model on $\Lambda$ with the $\mathrm{D} ( H^\ast )$ model on
$\Lambda^\ast$, as it transforms the Hilbert spaces and Hamiltonian of
the former into those of the latter.  We write this symbolically as $[
  \mathrm{D} (H) ]_\Lambda = [ \mathrm{D} ( H^\ast ) ]_{ \Lambda^\ast
}$.

But EM duality is deeper than a mere identification of energy levels.
The Hamiltonian involves just the operators $A_v ( h )$ and $B_p (
\phi )$, but, according to~ (\ref{maprepresentationstodual}),
$\mathcal{U}_\Lambda$ induces a transformation of all operators of
figure~\ref{fig:vertexplaquettedh} that respects their algebra
structure, and relates the anyonic excitations of both
models~\cite{forthcoming}.

\section{\label{sec:qdsn} From quantum doubles to string-nets}

We now make the connection with Levin and Wen's string-net
models~\cite{LevinWen}.  In~\cite{BA} the $\mathrm{D} (G)$ model was
identified, via a Fourier mapping, with an \emph{extended} string-net
model on the same lattice, with bond degrees of freedom given by
irreducible representations (irreps) of $G$ and auxiliary matrix
indices depending on each irrep (see also~\cite{Kadar}).  String-net
vertex conditions amount to knitting the matrix indices together by
means of $3j$ symbols, effectively mapping the local Hilbert spaces
isometrically to those of a Levin-Wen SN model, whose degrees of
freedom and underlying category theoretical structure are given solely
by irreps of $G$ and their fusion properties.  The ground levels of SN
and their extended versions are identical.

Now the representation theory of finite Hopf $C^\ast$-algebras is
essentially identical to that of finite groups (the only real
difference being that the fusion of representations need not be
commutative).  Therefore the construction of~\cite{BA} can be
generalised immediately to any $\mathrm{D} (H)$ model.  That is, if $H$
has matrix irreps $D^\mu$ of dimension $\lvert \mu \rvert$, we define
a basis in $H$ as

\begin{equation}\label{fourierbasish}
  [ \mu a b ]
=
  \sqrt{ \lvert H \rvert \lvert \mu \rvert } \,
  \sum_{ (h) }
  D^\mu_{ a b } ( h^{(1)} ) \,
  h^{(2)}
\; ,
\end{equation}
where $a, \, b = 1, \, \ldots, \, \lvert \mu \rvert$ are matrix
indices for irrep $D^\mu$.  This is an orthonormal basis for each
bond, and the analysis of~\cite{BA} carries through intact to show
that the $[ \mathrm{D} (H) ]_\Lambda$ model can be written as an
extended string-net model $[ \mathrm{ESN}_H ]_\Lambda$.  The latter
reduces to a string-net model $[ \mathrm{SN}_H^{ \mathrm{mag} }
]_\Lambda$, whose degrees of freedom are the irreps of $H$, and whose
ground level is identical to that of the extended model.

The same construction, as applied to the $[ \mathrm{D} ( H^\ast ) ]_{
  \Lambda^\ast }$ model using the Fourier basis for $H^\ast$, yields
an equivalent extended string-net $[ \mathrm{ESN}_{ H^\ast } ]_{
  \Lambda^\ast }$ on the dual lattice.  This, in turn, reduces to
another string-net model $[ \mathrm{SN}_H^{ \mathrm{el} } ]_{
  \Lambda^\ast }$ on the dual lattice, whose bond degrees of freedom
are irreps of $H^\ast$.  We regard the two string-net models $[
  \mathrm{SN}_H^{ \mathrm{mag} } ]_\Lambda$ and $[ \mathrm{SN}_H^{
    \mathrm{el} } ]_{ \Lambda^\ast }$ as the \emph{magnetic} and
\emph{electric} projections of the original quantum double model (see
figure~\ref{fig:schememodels}).  By construction, they satisfy $[
  \mathrm{SN}^{ \mathrm{el} }_H ]_{ \Lambda^\ast } = [ \mathrm{SN}^{
    \mathrm{mag} }_{ H^\ast } ]_{ \Lambda^\ast }$.

\begin{figure}
\begin{center}

  \includegraphics{fig6.eps}
\end{center}
\caption{\label{fig:schememodels} \textbf{From quantum doubles to
    string-nets.}  Quantum double models and string-net models are
  interconnected by EM duality, the Fourier mapping, and a projection
  to SN degrees of freedom.  The direct lattice is drawn solid, the
  dual lattice dashed.  The first layer represents the quantum double
  models in both their EM-dual formulations.  Via the Fourier mapping
  we write the quantum doubles as extended string-net models, depicted
  in the second layer, which are entirely equivalent.  The degrees of
  freedom of the ESN model on $\Lambda$ are labelled by irreducible
  representations $\mu$ of $H$ and auxiliary indices $a$, $b$; for the
  dual model on $\Lambda^\ast$ the labels are irreducible
  representations $\widetilde{\mu}$ of $H^\ast$, and auxiliary indices
  $\widetilde{a}$, $\widetilde{b}$.  These models support excitations
  both at vertices (electric) and at plaquettes (magnetic), which we
  represent by blobs and lozenges.  An electric excitation $e$ becomes
  a magnetic excitation $\widetilde{m}$ in the dual, and viceversa.
  The third layer represents the SN models obtained, respectively, by
  projecting to degrees of freedom $\mu$ on the direct lattice, or
  $\mu^\ast$ on the dual lattice; these we refer to as the magnetic
  and electric projections of the ESN.  Excitations in SN models are
  less symmetric than in the ESN: for instance, violation of a vertex
  constraint implies the violation of all surrounding plaquette
  constraints (see~\cite{BA}).}
\end{figure}

For instance, in~\cite{BA} the magnetic SN projection of the
$\mathrm{D} (G)$ model was analysed.  Its \emph{electric} SN projection,
on the other hand, has the same local degrees of freedom as the
$\mathrm{D} (G)$ model, since group elements are the (one-dimensional)
irreps of the dual of the group algebra~\cite{footnoted}.

Thus, EM duality can be given a meaning in the subclass of string-nets
obtained by reduction of ESNs.  This leads us to the following

\emph{Conjecture}: All SN models can be extended into quantum double
models based on a sufficiently general algebraic structure, the class
of weak Hopf $C^\ast$-algebras \cite{hayashi}\cite{footnotee}.  EM
duality can be defined for all extended SN models so that the pattern
of figure~\ref{fig:schememodels} holds.

\section{\label{sec:phys} Probing topology with tensor networks}

Some physical consequences of EM duality are immediate.  First of all,
EM duality allows us to probe the topology of the surface underlying
the lattice models by using only locally defined states.

As shown in~\cite{BMCA}, each $[ \mathrm{D} (H) ]_\Lambda$ model has one
canonical tensor network ground state $\lvert \psi ( H, \, \Lambda )
\rangle$ constructed from \emph{identical} tensors at each site, which
are defined solely by the structure of $H$.

{}From the corresponding canonical state $\lvert \psi ( H^\ast, \,
\Lambda^\ast ) \rangle$ of the dual $[ \mathrm{D} (H^\ast) ]_{
  \Lambda^\ast }$ model on the dual lattice we can obtain another
ground state of the original $[ \mathrm{D} (H) ]_\Lambda$ model by
using the duality map~(\ref{emduality}):

\begin{equation}\label{dualgroundstate}
  \lvert \widetilde{\psi} ( H, \, \Lambda ) \rangle
=
  \mathcal{U}_\Lambda^\dagger \,
  \lvert \psi ( H^\ast, \, \Lambda^\ast ) \rangle
\; .
\end{equation}

The relation between $\lvert \psi ( H, \, \Lambda ) \rangle$ and
$\lvert \widetilde{\psi} ( H, \, \Lambda ) \rangle$ depends on the
topology of the surface underlying $\Lambda$.  On the sphere, for
instance, the ground level is nondegenerate, so both ground states are
the same.  On a topologically nontrivial surface such as the torus,
these ground states can be shown to always be linearly independent.

For instance, in the toric code the tensor network construction of the
canonical state $\lvert \psi ( H, \, \Lambda) \rangle$ coincides with
the \emph{projected entangled-pair state} (PEPS) ansatz
in~\cite{VerstraetePower}.  On the torus this yields the logical
$\lvert ++ \rangle$ state; the dual ground state $\lvert
\widetilde{\psi} ( H, \, \Lambda ) \rangle$ is the logical $\lvert 00
\rangle$ state.  See~\cite{forthcoming} for a detailed construction,
including the non-Abelian cases.

On the other hand, the correspondence between string-net and quantum
double models can be used to relate these PEPS to the tensor network
descriptions of string-net ground states put forward in~\cite{BAV}
and~\cite{GuTopo}.  These come from the $\lvert \psi ( H, \, \Lambda )
\rangle$ construction of the corresponding $[ \mathrm{D} (H)
]_\Lambda$ model, while the $\lvert \widetilde{\psi} ( H, \, \Lambda )
\rangle$ can be seen to yield a \emph{new} TN given simply by the
knitting together of $3j$ symbols at vertices in the ESN degrees of
freedom.  More details will be given in~\cite{forthcoming}.  The
Fourier construction is also expected to relate the \emph{entanglement
  renormalization} analyses of $\mathrm{D} (H)$~\cite{AguadoVidal,
  BMCA} and SN models~\cite{Koenig}.

\section{\label{sec:discussion} Discussion and outlook}

We have defined EM duality for non-Abelian $\mathrm{D} (G)$ models,
showing how it arises naturally in the context of $\mathrm{D} (H)$
models.  The connection to SN models comes from the Fourier
construction of extended string-nets and their electric and magnetic
projections.  We conjecture that all SN models have extended versions,
and that EM duality can be defined for all the extended models.  EM
duality offers nonlocal information about the systems, unveiling
global characteristics of space, from tensor networks defined locally.
Beyond this, it serves as an \emph{organising principle} for
topological models; indeed, we envisage a net of dualities for
topological phases (not unlike that connecting theories of strings and
branes).  This then reflects on the field theories underlying
topological systems and on their mathematical structures, such as
category theory.

We wish to stress the importance of the \emph{whole class} of models
based on Hopf algebras.  This is indeed \emph{the} natural class of
models to study to understand electric-magnetic duality for lattice
models based on non-Abelian groups.  We thus shift the focus away from
lattice gauge theories based on groups, which are a primary concern in
high energy theory, even when the Hopf algebra language is
applied~\cite{Oeckl}.  Ours is, moreover, the first proposal for a
general EM duality governing topologically ordered systems.

The Hopf algebraic language makes the EM duality transparent and easy
to grasp, instead of obscuring it; in addition, this language can be
learnt here hands-on from the standpoint of quantum many-body systems,
which can even be built in a laboratory.  We also provide, with the
Fourier construction, a way to understand some aspects of category
theory in terms of representations, more familiar to many physicists.

Our next goal is defining the extended string-nets for all SN
models~\cite{forthcoming}, casting ESNs and their dualities in the
language of categories.  The nature of the anyonic models controlling
the excitations of the $\mathrm{D} (H)$ models has a bearing on the
experimental realisation of string-net models, which we contend would
be most natural in their~\emph{extended} incarnations (protocols for
anyonic manipulation in the spirit of~\cite{brennenetal}
and~\cite{brennenetaltwo} would then be possible).  A key question to
be explored is the meaning of EM upon perturbation of the models
considered, that is, away from topological fixed points.

\textbf{Acknowledgements:} We would like to thank J.~M.~Mombelli,
J.~Slingerland, and G.~Vidal for discussions.  LK is supported by the
Gordon and Betty Moore Foundation through CPI in Caltech and NSF under
Grant No.~PHY-0803371 and the Basic Research Young Scholars Programs
of Tsinghua University.