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\title{Twisted Alice Loops as Monopoles}


\author{Katherine M.~Benson}
\email{benson@physics.emory.edu}
\affiliation{
Radcliffe Institute for Advanced Study,
Putnam House, 10 Garden St,
Cambridge, MA \ 02138}
\altaffiliation[Permanent Address: ]{
Department of Physics,
Emory University,  400 Dowman Drive, Suite N202,Atlanta, GA\ 30322}
\author{Tom Imbo}\email{imbo@uic.edu}
\affiliation{Department of Physics,
University of Illinois at Chicago,
845 W. Taylor St,  m/c 273,
Chicago, IL 60607-7059}


%\bigskip
\date{\today}

\begin{abstract}
Symmetry breaking can produce ``Alice'' strings, which alter scattered
charges and carry monopole number and charge when twisted into loops.
We apply recent topological results, fixing Alice strings' stability
and prescribing their twisting into loops with monopole charge, to
several models.  We show that Alice strings of condensed matter
systems (nematic liquid crystals, $^3$He-A, and related systems of
non-chiral Bose condensates and amorphous chiral superconductors) are
topological, and carry fundamental monopole charge when twisted into
loops. Other models yield Alice loops that carry only deposited, and not
fundamental, charge.

\end{abstract}

\maketitle

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%\pacs{11.27.+d, 11.30.Fs, 11.30.Ly,98.80.Cq}
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\section{Introduction}

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Among the defects created in spontaneous symmetry breakdown are Alice
strings.\cite{oldAlice,stringzm,newAlice} Like monopoles
\cite{moncolor}, Alice strings obstruct the global extension of
unbroken symmetries, making them multivalued when parallel transported
around the string. This algebraic obstruction causes nonconservation
of associated charges, when Aharonov-Bohm scattered around the string;
it also induces monopoles, as twisted loops of Alice string.

For gauged theories, these Alice features arise as follows. Gauge flux
on the string's core generates the condensate winding, and acts on
asymptotic particles, through the Wilson line $U(\varphi)$. This
action fixes Aharonov-Bohm scattering around the string, to one
changing both charge and monopole number. Loops of string leave charge
and monopole number well-defined asymptotically, and thus support
nonlocalizable deposited charge (``Cheshire charge'') and deposited
monopole number. \cite{oldAlice,stringzm} Alice loops carry this
deposited monopole number by twisting.\cite{blp,topalice}

Condensed matter Alice strings are global, rather than gauged,
defects. No gauge flux fixes their Aharonov-Bohm scattering,
guaranteeing alteration of charge and monopole number upon string
traversal. However, their Aharonov-Bohm scattering was shown in
\cite{ABrefs, global} to generically induce the Alice behaviors of their
gauged counterparts: they alter charge and monopole number on string
traversal, with twisted loops supporting both Cheshire charge and
deposited monopole number.

The algebraic condition for Alice behavior is noncommutation of a
string's untraced Wilson loop, $U(2\pi)$, with some unbroken symmetry
generator $T_h$.  In \cite{topalice} we established a topological
criterion for Alice behavior, stating when Alice strings {\bf must}
form, and when their Alice features may be deformed away. A
topological string, whose untraced Wilson loop $U(2\pi)$ induces Alice
behavior --- by not commuting with some $T_h$ --- has homotopy
$\pi_o(H)$; that is, it has flux $U(2\pi)$ in a disconnected component
of the unbroken symmetry group $H$.  Similarly, for $G$ and $H$ Lie
groups, monopoles have homotopy $\pi_1(H)$, describing loops
$h(\alpha)$ of different winding in $H$. Our criterion labels strings
with flux $U(2\pi)$ topologically Alice if they alter the topological
charge of monopoles circumnavigating them:
$$h(\alpha) \rightarrow \tilde{h} (\alpha) = U(2\pi)\ h(\alpha)\
U^{-1}(2\pi)\ \not\sim \ \ h(\alpha) \ .$$ This is a topological criterion,  corresponding to a nontrivial action of $\pi_o(H)$ on $\pi_1(H)$, where 
$h_o = U(2\pi) \in \pi_o(H)$ alters the topological winding of loop
$h(\alpha) \in\pi_1(H)$: \beq \tilde{h}
(\alpha) = h_o\ h(\alpha)\ h_o^{-1} \ \ \not\sim \ \ h(\alpha)\ \
.\label{crit}\eeq


Using this criterion we constructed a prescription for twisting
Alice strings into loops carrying monopole charge. We showed that the
twisted Wilson line
\beq U(\varphi, \alpha) = h^{-1}(\alpha)\ U(\varphi)\ h(\alpha)\
h_o^{-1}\ ,\label{twistWil}\eeq
with $h(\alpha)$ and $h_o$ as in our criterion above,
generates a single-valued twisted Alice loop.  $U(\varphi,\alpha)$
interpolates between the point $h_o^{-1}$ at $\varphi = 0$ and the
loop $h^{-1}(\alpha)\ \tilde{h}(\alpha)$ at $\varphi = 2\pi$. The
twisted Alice loop thus carries a nontrivial monopole charge $
h^{-1}(\alpha)\ \tilde{h}(\alpha)$ if and only if the Alice string
obeys our topological criterion. This monopole charge corresponds to
that deposited on the Alice loop in the monopole circumnavigation
$h^{-1}(\alpha) \ \rightarrow \
\tilde{h}^{-1}(\alpha)$. \cite{topalice}

Two points are key. First, our topological arguments above only indicate
that {\bf deposited} monopole charge can be carried by a twisted Alice
loop. Monopole charge is typically not deposited in single
fundamental units, leaving open the question of whether twisted
Alice loops can carry {\bf fundamental} monopole charge. Second, the
function $h(\alpha)$ in our construction of twisted Wilson line
$U(\varphi, \alpha)$ need not be the loop in $H$ representing a fundamental
monopole: it can be any function in $H$ that renders $U(\varphi, \alpha)$
singlevalued in the angle $\alpha$.

We exploit both points here, in examining the monopole charge carried
by model twisted Alice loops. We find that whether a twisted Alice
loop can support fundamental monopole charge depends closely on the
symmetry-breaking pattern.  Specifically, it depends on the initial
symmetry group $G$, through the identification of loops in $H$ with
monopoles via the exact sequence for $\pi_2(G/H)$; and it depends on
algebraic commutations in the model, which may allow an accidental
choice of twisted Wilson line $U(\varphi, \alpha)$, which unwinds in
$G$ the fundamental monopole's associated loop in $H$.


We explore these possibilities in an intuition-building gallery of
Alice string models.  These include the original Alice string of
Schwarz, coinciding with the Alice string of liquid crystals and of
non-chiral Bose condensates \cite{oldAlice, leonvol}; the Alice string
of $^3$He-A \cite{volmin}, arising anew for unconventional
spin-triplet superconductors \cite{spintrips}; a nontopological Alice
string introduced in \cite{stringzm}; and a new topological Alice
string which forms in the symmetry breakdown $SU(3) \rightarrow
O(2)$. For the Schwarz and $^3$He-A Alice strings, we show that
twisted Alice loops can support fundamental monopole charge, even
though monopole charge is deposited on string loops in even
increments.  Thus, for the Alice strings of interest to condensed
matter, even the most fundamental singular point defect, or monopole,
can take the form of a twisted Alice loop. In contrast, in the $SU(3)
\rightarrow O(2)$ model, twisted Alice loops support only the even
deposited monopole charge, and cannot form fundamental monopoles.




\section{\label{model:canon}The Schwarz, or Nematic,  Alice String}

We start with the simplest example, the canonical Schwarz Alice
string, \cite{oldAlice} whose symmetry-breaking pattern coincides with
Alice strings in nematic liquid crystals and in non-chiral Bose
condensates. \cite{leonvol}

Here $G$ is $SO(3)$, with 
Higgs $\phi$ transforming in the adjoint representation. 
When $\phi$ develops the vev
$\vev = {\rm diag}\ (1,1,-2)\ \ ,
$
$SO(3)$ breaks to the residual symmetry $H = O(2)$, containing  z-rotations $R_z\,(\alpha)$ and the discrete symmetry element
$ h_o = R_x\, (\pi) = {\rm diag}\ (1, -1, -1)\ .$
Here $\pi_o (H) = Z_2$ and $\pi_1(H) = Z$ so we have topological strings and monopoles. The canonical Alice string has Wilson line
$ U(\varphi) = R_x\, (\varphi/2)$
with $U(2\pi) = h_o$. This string is Alice, as $U(2\pi)$ fails to commute
with the unbroken symmetry generator $T_z$; in fact, on parallel
transport around the string,
\beq T_z \rightarrow U\ (2\pi) \ \ T_z \ \ U^{-1}\ (2\pi) = -T_z \ \ .
\label{canAliceT}\eeq

This canonical Alice string meets our topological criterion, of changing
topological monopole charge upon circumnavigation. By the exact
sequence for $\pi_2 (G/H)$, topological monopoles are associated with
nontrivial loops in $O(2)$ which can be unwound in $SO(3)$. Since only
even winding loops in $O(2)$ can be unwound in $SO(3)$, the
fundamental monopole in this canonical Alice model has a loop in
$O(2)$ of winding 2.

In applying our topological criterion, we choose $h_o$ as our
representative of the string, a nontrivial element of
$\pi_o (H)$, and $h(\alpha) = R_z(2\alpha)$ as our representative of
the fundamental Alice monopole, a winding 2 element of $\pi_1(H)$. This gives
$$\tilde{h} (\alpha) = h_o\ h(\alpha)\ h_o^{-1}  = h^{-1} \, (\alpha)\ \ ,$$
from equation (\ref{canAliceT}). Note that $h^{-1} \, (\alpha)$ has $O(2)$ winding -2, topologically distinct from $h(\alpha)$ of $O(2)$ winding 2. Thus 
$ \tilde{h} (\alpha)\ \not\sim \ 
h(\alpha)$ and our topological criterion is fulfilled.

We now construct a monopole as a twisted Alice loop. From Eq.~(\ref{twistWil}), the twisted Wilson line 
$$ U(\varphi, \alpha) = h^{-1}(\alpha/2)\ U(\varphi)\ h(\alpha/2)\ h_o^{-1}\ \ 
$$ generates an Alice loop with single-valued condensate. (We take
$h(\alpha/2)$ because we need only for $h$ to be single-valued in
$\alpha$, and $h(\alpha/2)$, the winding 1 loop in $O(2)$, first
fulfills that requirement.)  $U(\varphi, \alpha)$ interpolates between
$ h_o^{-1}$ at $\varphi = 0$ and $ h^{-1}(\alpha)$ at $\varphi =
2\pi$. It is thus the fundamental antimonopole in the model. Note that
the inverse twisted Alice loop, with twisted Wilson line $
U^{-1}(\varphi, \alpha)$, generates the fundamental monopole.

\section{\label{model:helium3a}The  Alice String of $^3$He-A}

A more complicated global symmetry-breaking pattern describes the
Alice string expected in $^3$He-A \cite{volmin,volbook}, and more recently
predicted in amorphous chiral superconductors with p-wave pairing,
such as Sr$_2$RuO$_4$.\cite{spintrips}

Here $G$ is $SO(3)_L\times SO(3)_S\times U(1)_N$, describing spatial
rotations, spin rotations, and a $U(1)$ phase symmetry associated with
number conservation of helium atoms. (The U(1) symmetry is
approximate, as is independence of spin and orbital rotations due to
minimal spin-orbit coupling, but both describe $^3$He-A well.)  The
matrix order parameter $A$ transforms under symmetry transformations
as
$A \ \rightarrow e^{2i\theta}\ R_S \ A\ R_L^{-1}\ ,$
where $R_S$ and $R_L$ are spin and orbital rotations, respectively.

The order parameter develops the form
$$A_{ij} = \Delta_A\ \hat{d}_i\ (\hat{m}_j + i\,\hat{n}_j)\ \ ,$$ where
$\hat{m}$ and $\hat{n}$ are perpendicular, determining $\hat{l} =
\hat{m} \times\hat{n},$ the direction of the
condensate's angular momentum vector. This breaks $G$ to the residual symmetry
$H= U(1)_{S_{\hat{d}}} \ \times\ U(1)_{L_{\hat{l}} - N/2}\ \times Z_2,$
consisting of spin rotations about the $\hat{d}$ axis; spatial
rotations about the $\hat{l}$ axis when compensated by a matching
$U(1)_N$ phase rotation, and the discrete $Z_2$ transformation $h_o$, with
$h_o: \hat{d}, \ \ \hat{m} + i\,\hat{n}\ \rightarrow \   - \hat{d}, \ \ 
 -\ (\hat{m} + i\,\hat{n}) \ .$

Identifying the defect topology requires care in this setting, as the
exact sequences relating $\pi_2(G/H)$ and $\pi_1(G/H)$, the monopole
and string homotopy groups, to $\pi_1(H)$ and $\pi_0(H)$ are highly
nontrivial.  Note that $\pi_2(G/H) = Z$, corresponding to the loops
$\pi_1( U(1)_{S_{\hat{d}}} )$, which can be unwound in $G$. (Loops of
the other $U(1)$ factor cannot be unwound in $G$, as they contain
unshrinkable $U(1)_N$ loops.) $\pi_1(G/H) = Z_4$, which describes
strings of two different origins. First, the Alice strings, called
half-quantum vortices, have Wilson lines ending in a disconnected
component of $H$, getting topological stability from
$\pi_0(H)$. Second, a $Z_2$ winding one vortex, nontrivial in
$SO(3)_L$ in $G$, induces as its image a $Z_2$ winding one vortex in
$G/H$, with topological stability inherited from $\pi_1 (G)$. These
two classes of vortices are not independent: instead winding twice
about a half-quantum vortex is equivalent to once around a winding
one vortex, and the full string homotopy is $\pi_1(G/H) = Z_4$, or
windings $0, \pm 1/2$, and $1$ modulo 2, with Alice strings corresponding to
windings $\pm 1/2$.

The Volovik-Mineev Alice string, of winding $\pm 1/2$, has order
parameter $A_{ij}$ with $\hat{d} = \hat{x}$ in spin space, and
$\{\hat{l}, \hat{m},\hat{n}\} = \{\hat{x}, \hat{y}, \hat{z}\}$ in ordinary
space. This is acted on by Wilson line
$ U(\varphi) =  e^{\pm i\varphi/2}\ R_{S_{\hat{z}}}(\varphi/2)$
to give, asymptotically in $r$, 
$$A_{ij} (\varphi) = \Delta_A\  e^{\pm i\varphi/2}\ (\cos (\varphi/2)\ \hat{x}_j + \sin  (\varphi/2)\ \hat{y}_j)_S\ (\hat{x}_j + i\hat{y}_j)_L\ \ ,$$ 
single-valued in $\varphi$. Note that  $U(2\pi) = - R_{S_{\hat{z}}}(\pi) $ lies in the same homotopy class as $h_o$. This string is Alice, making 
unbroken symmetry generator $T_{S_{\hat{x}}}$ double-valued
Physically, this means that a particle flips its spin, and hence its magnetization, on circumnavigating the Alice string.

This long-studied Alice string meets our topological criterion, of changing
topological monopole charge upon circumnavigation. By the exact
sequence for $\pi_2 (G/H)$, topological monopoles are associated with
nontrivial loops in $ U(1)_{S_{\hat{x}}}$ which can be unwound in $SO(3)_S$. As in the nematic case,  only
even winding loops in $ U(1)_{S_{\hat{x}}}$ can be unwound in $SO(3)_S$. Thus the
fundamental monopole in $^3$He-A corresponds to  a loop in
$ U(1)_{S_{\hat{x}}}$ of winding 2.

In applying our topological criterion, we choose $U(2\pi)$ as our
representative of the string, a nontrivial element of
$\pi_o (H)$, and $h(\alpha) = R_{S_{\hat{x}}}(2\alpha)$ as our representative of
the fundamental monopole, a winding 2 element of $\pi_1( U(1)_{S_{\hat{x}}})$. This gives
$$\tilde{h} (\alpha) = h_o\ h(\alpha)\ h_o^{-1}  = h^{-1} \, (\alpha)\ \ ,$$
since $T_{S_{\hat{x}}}\rightarrow T_{S_{\hat{x}}}$. Note that $h^{-1} \, (\alpha)$ has $U(1)_{S_{\hat{x}}}$ winding -2, topologically distinct from $h(\alpha)$ of $U(1)_{S_{\hat{x}}}$ with winding 2. Thus 
$ \tilde{h} (\alpha)\ \not\sim \ 
h(\alpha)$, meeting our topological criterion.

As in the nematic case, we construct a monopole as a twisted Alice loop. From Eq.~(\ref{twistWil}), the twisted Wilson line 
$$ U(\varphi, \alpha) = h^{-1}(\alpha/2)\ U(\varphi)\ h(\alpha/2)\ h_o^{-1}\ \ 
$$
generates an Alice loop with single-valued condensate. (Again 
$h(\alpha/2)$ appears,  the winding 1 loop in $ U(1)_{S_{\hat{x}}}$, as our minimal single-valued choice in  constructing  $U(\varphi, \alpha)$.)  $U(\varphi, \alpha)$ interpolates between
$ h_o^{-1}$ at $\varphi = 0$ and $ h^{-1}(\alpha)$ at $\varphi =
2\pi$. It is thus the fundamental antimonopole in the model, which
contains monopoles and antimonopoles of even winding in $ U(1)_{S_{\hat{x}}}$
only. Note that the inverse twisted Alice loop, with twisted Wilson
line $ U^{-1}(\varphi, \alpha)$, again generates the fundamental monopole.


\section{\label{model:nontop}A Nontopological Alice string}

A nontopological Alice string arises when a Higgs $\phi$, transforming
in the adjoint representation under $G=SO(6)$, acquires the vev $\vev
= {\rm diag}(1^3,-1^3)$. As discussed in \cite{stringzm}, this
condensate leaves unbroken an $SO(3)\times SO(3)$ subgroup of $SO(6)$
and a discrete $Z_2$ transformation $h_1=-1\mkern-6mu 1_6$, so
$H=SO(3)\times SO(3)\times Z_2$.  Here $\pi_o(H)=Z_2$ and $\pi_1(H) =
Z_2 \times Z_2$, so topological strings and monopoles form, with
monopoles and antimonopoles topologically identified.  As noted in
\cite{topalice}, strings in this model can have algebraic Alice
behavior, for $U(2\pi) = h_o = {\rm diag} (1^2, (-1)^4) = -
R_{12}(\pi) $, which makes generators $T_{13}$ and $T_{23}$ of
rotations in the $13$- and $23$-planes double-valued. Yet that Alice
behavior fails our topological criterion.  Taking as our nontrivial
monopole loop $h(\alpha) = R_{13}\ (\alpha)$, with monopole charge
(1,0), we find $\tilde{h} (\alpha) = h_o\ h(\alpha)\ h_o^{-1} = h^{-1}
\, (\alpha)\ $, since $T_{13} \rightarrow -\ T_{13}$. Thus the
(1,0) monopole transforms into its antimonopole on traversing the
string. {\bf However,} as monopoles and antimonopoles are identified,
that transformation is nontopological. This string's Alice behavior is
thus nontopological; it can be deformed away by deforming $U(2\pi)$ to the
topologically equivalent value $h_1$, with no algebraic Alice behavior.

We might still hope to construct a (1,0) monopole as a twisted string
loop, taking for our string the algebraic, but nontopological Alice
string $U(\varphi) = R_{34}(\varphi/2) \ R_{56}(\varphi/2)$, with
algebraic Alice flux $U(2\pi) = h_o = - R_{12}(\pi) $ as above. From
Eq.~(\ref{twistWil}),the twisted Wilson line
$$ U(\varphi, \alpha) = h^{-1}(\alpha)\ U(\varphi)\ h(\alpha)\ h_o^{-1}\ \ 
$$
generates an Alice loop with single-valued condensate.  $U(\varphi,
\alpha)$ interpolates between $ h_o^{-1}$ at $\varphi = 0$ and $
h^{-2}(\alpha)$ at $\varphi = 2\pi$. This is a loop in $H$ of winding
(2,0); however, winding (2,0) loops are deformable to the identity in $H$, so
this twisted nontopological Alice loop fails to carry topological
monopole charge.

Recall that, in building singlevalued twisted Alice loops, we required
$ U(\varphi, \alpha)$ to be singlevalued in $\alpha$; we thus
identified $h(\alpha)$ as a loop in $H$. Strictly, we do not need
$h(\alpha)$ to be a loop; all we need is
\beq h^{-1}(2\pi)\ U(\varphi)\ h(2\pi) = U(\varphi)\  .\label{singval}\eeq
We might still hope to build the  fundamental (1,0) monopole as a twisted Alice loop, exploiting this freedom in $h(\alpha)$. Were the twisted loop
$$ U(\varphi, \alpha) = h^{-1}(\alpha/2)\ U(\varphi)\ h(\alpha/2)\ h_o^{-1}\ \ 
$$
single-valued, it would carry fundamental (-1,0) monopole charge, as it  interpolates between
$ h_o^{-1}$ at $\varphi = 0$ and the nontrivial (-1,0) antimonopole $ h^{-1}(\alpha)$ at $\varphi =
2\pi$. However, this twisted loop candidate is not single-valued; it obeys instead
$ h^{-1}(2\pi)\ U(\varphi)\ h(2\pi) = U^{-1}(\varphi)\ .$ We 
thus cannot build a fundamental (1,0) monopole as a twisted Alice loop in
this model, where Alice behavior is nontopological and monopole
charge is $Z_2 \times Z_2$.

This possibility to construct $U(\varphi,\alpha)$ single-valued in
$\alpha$, {\bf without} forcing $h(\alpha)$ to be a loop, always
merits investigating. Indeed, in \cite{skyrme, global}, one of us
constructed what is essentially the fundamental monopole in this
model, by exploiting exactly such an accidental algebraic
singlevaluedness. That construction (most clearly in section IIIA of
\cite{global}, taking $F(r), \varphi$ as the spherical coordinates
$\theta, \varphi$ at spatial infinity),is quite similar to the
twisting constructions here. However, it describes a fundamentally
point-like defect, and cannot be interpreted as a twisted loop. 

\section{\label{model:su3}A Topological Alice Loop Carrying only Deposited Monopole Charge }

We consider a slightly modified canonical Alice string. Take $G$ to be
$SU(3)$, with Higgs $\phi$ transforming according to $\phi \rightarrow
g\, \phi\, g^T$ (giving fermions in this model a Majorana mass).
When $\phi$ develops the vev
$\vev = {\rm diag}\ (1,1,-2)\ \ ,
$
$SU(3)$ breaks to the residual symmetry $H = O(2)$, 
identical to that of the canonical Schwarz Alice string.
Again we have $\pi_o (H) = Z_2$ and $\pi_1(H) = Z$,
with topological strings and monopoles. We have the same Alice string
as in the canonical case, making the $O(2)$ generator $T_z$
double-valued. This Alice behavior is again topological, as our topological
criterion, that $\pi_o(H)$ acts nontrivially on $\pi_1(H)$, depends
only on the unbroken symmetry group $H$.

Where we deviate from the canonical Alice string model is in the
identification of twisted Alice loops as monopoles. Here, by the exact
sequence for $\pi_2 (G/H)$, topological monopoles are associated with
nontrivial loops in $O(2)$ which can be unwound in $G$, here
$SU(3)$. {\bf All} nontrivial loops in $O(2)$ can be unwound in
$SU(3)$; thus the fundamental monopole in this model  has a loop
in $O(2)$ of winding 1.

We now construct a monopole as a twisted Alice loop. We take
$U(\varphi) = R_x\, (\varphi/2)$, as in the canonical Alice case, and
$h(\alpha) = R_z(\alpha)$, a loop of winding 1. From
Eq.~(\ref{twistWil}), the twisted Wilson line
$$ U(\varphi, \alpha) = h^{-1}(\alpha)\ U(\varphi)\ h(\alpha)\ h_o^{-1}\ \ 
$$ generates a twisted Alice loop with single-valued condensate.
$U(\varphi, \alpha)$ interpolates between $ h_o^{-1}$ at $\varphi = 0$
and $ h^{-2}(\alpha)$ at $\varphi = 2\pi$. This twisted Alice loop
carries monopole charge of $-2$, which while nontrivial is {\bf not}
the fundamental antimonopole in this model. (Similarly, the inverse
twisted Alice loop, with Wilson line $ U^{-1}(\varphi, \alpha)$,
carries monopole charge +2).

Again, we might still hope to build a fundamental monopole as a
twisted Alice loop, by allowing $h(\alpha)$ above to be not a loop,
but a curve obeying Eq.~(\ref{singval})
This looser
constraint still guarantees singlevaluedness in $\alpha$ of $
U(\varphi, \alpha)$.  Indeed, were the twisted loop
$$ U(\varphi, \alpha) = h^{-1}(\alpha/2)\ U(\varphi)\ h(\alpha/2)\ h_o^{-1}\ \ 
$$ single-valued in $\alpha$, with $h(\alpha) = R_z(\alpha)$ as above,
it would carry fundamental antimonopole charge. This is because it
interpolates between $ h_o^{-1}$ at $\varphi = 0$ and the winding $-1$
loop $ h^{-1}(\alpha)$ at $\varphi = 2\pi$. However, this twisted loop
candidate is not single-valued in $\alpha$; it obeys instead
$ U(\varphi, 2\pi) = R_x^{-1} (\varphi) \ U(\varphi, 0)\  .
$ We thus cannot build a fundamental monopole as a twisted Alice loop
in this model. Instead twisted Alice loops carry only the monopole
charge which topological arguments ensure they must carry: because
monopoles scatter into antimonopoles on transiting Alice loops, Alice
loops {\bf must} support deposited monopole charge, which arises in
units of 2. Our twisting construction creates twisted Alice loops
supporting exactly that deposited charge.


\begin{acknowledgments}
Early stages of this work were supported by NSF grant PHY-9631182 and
by the University Research Committee of Emory University. 
\end{acknowledgments}

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