text:
\section{Introduction}

The search for the efficient electrical manipulation of magnetic textures has recently received a major boost with the observation of ultrafast current-driven domain wall motion in non-centrosymmetric transition metal ferromagnets \cite{Moore2008,Miron2011,Emori2013,Yang2015}, and very low depinning current threshold of bulk magnetic skyrmions \cite{Jonietz2010,Schulz2012,Yu2012}. The latter are topological magnetic defects \cite{Nagaosa2013} that present some similarities with the more traditional magnetic vortices \cite{He2006,Thomas2006,Eltschka2010,Heyne2010}. In both cases, the magnetic topology induces a Lorentz force on the flowing electrons, resulting in topological Hall effect \cite{Neubauer2009,Nagaosa2013} (see also Ref. \onlinecite{Ndiaye2016}). Both classes of magnetic textures also experience a Hall effect when driven by an electric flow, an effect sometimes referred to as skyrmion Hall effect \cite{Nagaosa2013}. In fact, it is important to emphasize that the current-driven characteristics of skyrmions are mostly similar to that of magnetic vortices. For instance, the universal current-velocity relation \cite{Iwasaki2013a,Iwasaki2013b}, as well as the \textit{colossal} spin transfer torque effect at the edge \cite{Iwasaki2014} have already been predicted in the case of magnetic vortices \cite{He2006}. Notwithstanding, skyrmions display some striking differences with respect to magnetic vortices. As a matter of fact, vortex walls are non-local objects composed of a vortex core and two transverse walls \cite{He2006,Thomas2006,Eltschka2010,Heyne2010}, expanding over several exchange lengths in the wire. In contrast, skyrmions are localized objects with a limited expansion from a few tens to hundred nanometers \cite{Nagaosa2013}. This localization has a dramatic impact on the manner skyrmions behave in the presence of pinning centers (defects, notches etc.). When driven by a current, they can deform their texture, and thereby avoid pinning \cite{Iwasaki2013a,Iwasaki2013b,Iwasaki2014,Sampaio2013,Everschor2012,Lin2013}. In contrast, magnetic vortices are much more sensitive to defects and get easily pinned. \par

Since the robustness of skyrmions with respect to defects might hold the key for efficient ultra-dense data storage \cite{Fert2013}, it is crucial to develop a precise understanding of the nature of the torque exerted on these objects. Recently, Iwasaki \textit{et. al.} have shown that such a robustness is partly due to the presence of a large non-adiabatic torque \cite{Iwasaki2013a}. Indeed, in magnetic textures the spin transfer torque can be generally expressed as
\begin{equation}\label{eq:tor}
{\bf T} = b_{\rm J}({\bf u}\cdot\nabla){\bf M} -\frac{\beta b_{\rm J}}{M_s}{\bf M}\times({\bf u}\cdot\nabla){\bf M},
\end{equation}
where {\bf u} is the direction of current injection, ${\bf m} = {\bf M}/M_s$ is a unit vector in the direction of the magnetization {\bf M} and $M_s$ is the saturation magnetization. The first term in Eq. (\ref{eq:tor}) is the adiabatic torque while the second term is the non-adiabatic torque \cite{Zhang2004,Thiaville2005}. It is well known that while the non-adiabatic parameter in transverse walls is quite small $\beta\approx\alpha$ ($\alpha$ being the magnetic damping of the homogeneous magnet), it is much larger in magnetic vortices, $\beta\approx 8-10\alpha$ \cite{Thomas2006,Eltschka2010,Heyne2010,Bisig2016}. To the best of our knowledge, there is currently no available estimation of the non-adiabaticity parameter in skyrmions, but one can reasonably speculate that it should be of the same order of magnitude as that of magnetic vortices. The recent avalanche of experimental observations of magnetic skyrmions in transition metal multilayers might soon shed light on this question \cite{Jiang2015,Chen2015,Boulle2016,Moreau-Luchaire2016,Woo2016}.\par

That being said, the nature of non-adiabaticity in skyrmions and vortex walls has been scarcely addressed. In transverse domain walls, two major physical mechanisms have been uncovered: spin relaxation \cite{Zhang2004} and spin mistracking \cite{Tatara2004}. Spin relaxation produces a non-adiabatic torque $\beta\sim \tau_{\rm ex}/\tau_{\rm sf}$, where $\tau_{\rm ex}$ is the precession time of the spin about the local magnetization, while $\tau_{\rm sf}$ is the spin relaxation time. Spin mistracking is the quantum misalignement of the flowing electron spin with respect to the local magnetic texture and provides a non-adiabaticity parameter $\beta\sim e^{-\xi \lambda_{\rm dw}}$ that exponentially decreases with the domain wall width $\lambda_{\rm dw}$. It is therefore limited to extremely (atomically) sharp domain walls \cite{Tatara2007,Xiao2006}. We also recently showed that spin diffusion enhances the non-adiabatic torque when the domain wall width is of the order of the spin relaxation length \cite{Akosa2015}. However, none of these effects explains the large difference in non-adiabaticity between transverse walls on the one hand and skyrmions and vortices on the other hand. In a recent work, we proposed that the topological currents induced by the topological Hall effect can enhance the non-adiabatic parameter in magnetic vortices \cite{Bisig2016}. Such an intimate connection between topological Hall effect and non-adiabaticity has also been pointed out by Jonietz \textit{et al.} \cite{Jonietz2010}, but to the best of our knowledge, no theoretical work addresses this issue thoroughly, and all micromagnetic simulations on skyrmions dynamics so far assume a constant $\beta$ parameter \cite{Iwasaki2013a,Iwasaki2013b,Iwasaki2014,Sampaio2013}. \par

As an in-depth follow-up of Ref. \onlinecite{Bisig2016}, the present work investigates the nature of the non-adiabaticity in skyrmions analytically and numerically, and demonstrates that the texture-induced emergent magnetic field inherent in these structures, induces a non-local non-adiabatic torque on the magnetic texture. We provide an explicit expression of the torque and highlight its connection with the spin and charge topological Hall effect.

\section{Emergent Electrodynamics}\label{sec:two}

\subsection{Premises}

It is well-known that when conduction electrons flow in a smooth and slow magnetic texture, ${\bf m}({\bf r} ,t)$, their spin adiabatically changes orientation so that these electrons acquire a Berry phase \cite{Berry1984,Ye1999, Taguchi2001, Onoda2004}. This geometrical phase is attributed to an emergent electromagnetic field $({\bf E}_{\rm em},{\bf B}_{\rm em})$ determined by the magnetic texture gradients \cite{Volovik1987,Bruno2004,Barnes2007,Saslow2007,Tatara2007,Duine2008,Tserkovnyak2008}. Indeed, a time-dependent magnetic texture induces local charge and spin currents through the action of the so-called spin electromotive force \cite{Volovik1987,Barnes2007,Zhang2009,Tanabe2012, Shimada2015,Zang2011,Stern1992}. For the sake of completeness, we derive below this emergent electromagnetic field. Let us considering the simplest Hamiltonian of an \textit{s-d} metal in the presence of a smooth magnetic texture given as
\begin{equation}\label{eq:ham}
\hat{\cal H}=\frac{\hat{\bf p}^2}{2m} + J_{\rm sd}\hat{\bm\sigma}\cdot{\bf m}({\bf r},t).
\end{equation}
The Schr\"odinger equation corresponding to Eq. (\ref{eq:ham}) can be re-written in the rotating frame of the magnetization, using the unitary transformation $\mathcal{U} = e^{-i\frac{\theta}{2}\boldsymbol{\hat{\sigma}}\cdot{\bf n}}$ where ${\bf n}={\bf z}\times{\bf m}/|{\bf z}\times{\bf m}|$ to obtain
\begin{equation}
\tilde{\cal H}=\frac{(\hat{\bf p}- e\boldsymbol{\mathcal{A}})^2}{2m} + J_{\rm sd}\hat{\sigma}_z + e\hat{\mathcal{V}},
\end{equation}
where the vector and scalar potentials are given respectively as $\boldsymbol{\mathcal{ A}}=-\frac{\hbar}{2e} \hat{\bm\sigma}\cdot({\bf m}\times\partial_i{\bf m}){\bf e}_i$ and $\hat{\cal V}=\frac{\hbar}{2e} \hat{\bm\sigma}\cdot({\bf m}\times\partial_t{\bf m})$.
As a consequence, the spin-polarized carriers feel an emerging electromagnetic field on the form \cite{Stern1992,Bruno2004,Barnes2007,Saslow2007,Tatara2007,Duine2008,Tserkovnyak2008,Tanabe2012, Shimada2015,Zhang2009, Zang2011}
\begin{subequations}\label{eq:emf}
\begin{eqnarray}
{\bf E}_{\rm em}^s&=&\frac{s \hbar}{2e} [{\bf m}\cdot(\partial_t{\bf m}\times\partial_i{\bf m})]{\bf e}_i,\\
{\bf B}_{\rm em}^s&=&-\frac{s \hbar}{2e}[{\bf m}\cdot(\partial_x{\bf m}\times\partial_y{\bf m})]{\bf z}.
\end{eqnarray}
 \end{subequations}
The electric field is proportional to the first derivative in time and space and therefore a moving magnetic texture induces a charge current \cite{Barnes2007,Saslow2007,Tatara2007,Duine2008,Tserkovnyak2008} and a self-damping \cite{Zhang2009,Zang2011}. The effect of the magnetic field has so far been overlooked as it requires a second order derivative in space, and is generally considered \textit{small}. Interestingly, this emergent magnetic field turns out to be crucial to understand the spin transport involved in these textures. Indeed,it creates a local ordinary Hall current such that the spin-dependent local charge current driven by the external electric field reads \cite{Bisig2016}
\begin{equation}\label{eq:eff_current}
{\bf j}_e^s =\sigma_0^s {\bf E}+\sigma_0^s{\bf E}_{\rm em}^s+\frac{\sigma_H^s}{B_{\rm H}}{\bf E}\times{\bf B}_{\rm em}^s +\frac{\sigma_H^s}{B_{\rm H}}{\bf E}_{\rm em}^s\times{\bf B}_{\rm em}^s,
 \end{equation}
where $\sigma_0^s$ and $\sigma_H^s$ are respectively the spin-dependent longitudinal and ordinary Hall conductivities, ${\bf E}$ is the external electric field, and $B_{\rm H}=|{\bf B}_{\rm em}^s|$. Inspecting Eq. (\ref{eq:eff_current}), we note that there are two sources of charge or spin currents (i) through the external electric field ${\bf E}$ and (ii) through the emergent electric field driven by the time-variation of the magnetic texture, ${\bf E}_{\rm em}^s$. Therefore, our calculation is able to capture the physics of the motion of the itinerant electrons [Topological (spin) Hall effect] or the magnetic texture itself [skyrmion (vortex) Hall effect]. Let us now assume a rigid magnetic structure for which the time derivative of the magnetization is such that $\partial_t{\bf M} = -({\bf v}\cdot\boldsymbol{\nabla}){\bf M}$, where ${\bf v} = v_x{\bf x} + v_y{\bf y}$ is the velocity of the magnetic structure, and, without loss of generality, an electric field applied along {\bf x} such that ${\bf E} =E{\bf x}$. We can obtain the expressions for the local spin current tensor, ${\mathcal J}_s={\bf M}\otimes({\bf j}_e^\uparrow-{\bf j}_e^\downarrow)$, and charge current vector, ${\bf j}_e={\bf j}_e^\uparrow+{\bf j}_e^\downarrow$, from Eq. (\ref{eq:eff_current}). Explicitly, we obtain
\begin{subequations}\label{eq:currents}
 \begin{eqnarray} \label{eq:currents_a}
\boldsymbol{\mathcal{J}}_s &=&\left[ b_J + \lambda_E^2\left( v_y  + v_x \lambda_H^2\mathcal{N}_{xy}\right)\mathcal{N}_{xy}\right]{\bf M}\otimes{\bf x} \\  \nonumber
&-&\left[\frac{\mathcal{P}_H}{\mathcal{P}_0} \lambda_H^2 b_J + \lambda_E^2  \left(v_x - v_y\lambda_H^2\mathcal{N}_{xy}\right)\right]\mathcal{N}_{xy}{\bf M}\otimes{\bf y},
 \end{eqnarray}
 \begin{eqnarray} \label{eq:currents_b}
{\bf j}_e &=&\sigma_0\left[E +\frac{ \hbar}{2e} \left(\mathcal{P}_0v_y  + \mathcal{P}_H v_x \lambda_H^2\mathcal{N}_{xy}\right)\mathcal{N}_{xy}\right]{\bf x} \\  \nonumber
&-&\sigma_0\left[\lambda_H^2E + \frac{ \hbar}{2e} \left(\mathcal{P}_0 v_x - \mathcal{P}_Hv_y\lambda_H^2\mathcal{N}_{xy}\right)\right]\mathcal{N}_{xy}{\bf y},
 \end{eqnarray}
 \end{subequations}
where $b_J = \hbar \mathcal{P}_{\rm 0}\sigma_0E/2eM_s$, $ \sigma_a = \sigma^{\uparrow}_a + \sigma^{\downarrow}_a$, $\mathcal{P}_a = (\sigma^{\uparrow}_a - \sigma^{\downarrow}_a)/\sigma_a$, (a = 0, {\rm H}), $\lambda_{\rm E}^2 = \hbar^2\sigma_0/4e^2M_s$, $\lambda_{\rm H}^2 = \hbar\sigma_H/2e\sigma_0B_{\rm H}$, and $\mathcal{N}_{\mu\nu}({\bf r}) = {\bf m}\cdot(\partial_\mu{\bf m}\times\partial_\nu{\bf m})$
where $\nu, \mu \in (x, y)$. Finally, $\otimes$ is the direct product between spin space and real space. Eqs. (\ref{eq:currents_a}) and (\ref{eq:currents_b}) constitute a central result in this work. They contain information about the motion of itinerant electrons as they traverse a smooth magnetic texture. Indeed, in its explicit form, one is able to explain the subtle difference between non-adiabatic transport in one- and two-dimensional textures. In particular, in addition to the usual constant adiabatic spin current moving along the direction of the applied current ($\sim{\bf x}$) common in one-dimensional textures, the presence of a non-zero topological charge, $\mathcal{N}_{xy} \ne 0$ (as it is the case for magnetic skyrmions and vortices), leads to a texture-induced emergent magnetic field that induces an additional spatially varying spin current along both the longitudinal ($\sim{\bf x}$) and transverse ($\sim{\bf y}$) directions to the electric field ${\bf E}$. This longitudinal spin current is responsible for (i) topological spin and charge Hall effects already observed in topological textures such as skyrmions \cite{Bruno2004,Neubauer2009,Nagaosa2013} (see also Ref. \onlinecite{Ndiaye2016}) and as we propose, (ii) enhanced non-adiabaticity already observed in vortices \cite{Bisig2016}. To clarify a potential mis-conception, we note here that magnetization variation in more than one direction is a \textit{necessary} but not \textit{sufficient} condition to observe these effects. The sufficient condition is a non-zero \textit{topological charge} ($\mathcal{N}_{xy} \ne 0$), which is the case for magnetic textures such as vortices and skyrmions.\par

The topological charge and spin Hall effects arising from the magnetic texture induced-emergent electromagnetic field given by Eq. (\ref{eq:emf}) can be quantified by the charge and spin Hall angles defined respectively as $\theta_{\rm TH}={\rm tan}^{-1}\left(\int j_e^y d^2{\bf r}/ \int j_e^x d^2{\bf r}\right)$ and $\theta_{\rm TSH}={\rm tan}^{-1}\left(\frac{2e}{\hbar}\int \mathcal{J}_s^y d^2{\bf r}/ \int j_e^x d^2{\bf r}\right)$ to obtain
\begin{equation}\label{eq:th}
\theta_{\rm TH} \sim -\mathcal{Q} \lambda_{\rm H}^2 \hspace{2 mm} \mbox{and} \hspace{2 mm}
\theta_{\rm TSH} \sim \mathcal{Q} \mathcal{P}_{\rm H}\lambda_{\rm H}^2,
\end{equation}
where $\mathcal{Q}$ is the topological number defined as $\mathcal{Q} =\frac{1}{4\pi}\int \mathcal{N}_{xy} d^2{\bf r}$.
From Eq. (\ref{eq:th}), one can straightforwardly deduce that $\theta_{\rm TSH} \approx - \mathcal{P}_{\rm H}\theta_{\rm TH}$ which, although very simple, turns out to be far reaching as it captures most of the important physics in static magnetic textures as confirmed by our numerical calculations in the following section.

\subsection{Topological spin torque}

In the previous section, we reminded the basics of the topologically-driven spin and charge currents in magnetic textures. Let us now investigate the impact of this spin current on the dynamics of the magnetic texture itself. By virtue of the spin transfer mechanism, this spin current exerts a torque on the local magnetization, ${\bf T}_{\rm t}=- {\bm\nabla} \cdot {\cal J}_s$, which explicitly reads
\begin{eqnarray}\label{eq:tor_em}
{\bf T}_{\rm t} &=& - \left[b_J + \lambda_{\rm E}^2\left( v_y + v_x\lambda_{\rm H}^2 \mathcal{N}_{xy} \right)\mathcal{N}_{xy}\right]\partial_x{\bf M}\\ \nonumber
&& + \left[\frac{\mathcal{P}_{\rm H}}{ \mathcal{P}_{\rm 0}}\lambda_{\rm H}^2b_J + \lambda_{\rm E}^2\left( v_x - v_y\lambda_{\rm H}^2\mathcal{N}_{xy}\right)\right]\mathcal{N}_{xy} \partial_y{\bf M}. \nonumber
\end{eqnarray}
A remarkable consequence of Eq. (\ref{eq:tor_em}) is that, since $\partial_y{\bf m} \sim {\bf m}\times\partial_x{\bf m}$ in magnetic vortices and skyrmions, the finite topological charge ($\mathcal{N}_{xy} \ne 0$) induces an \textit{intrinsic topological} non-adiabatic spin transfer torque. This non-adiabatic torque is \textit{intrinsic} as it does not rely on impurities or defects (in contrast with the non-adiabaticity studied in Refs. \onlinecite{Zhang2004, Akosa2015}), and \textit{topological} since its origin is associated to the topology of the magnetic texture.  \par

To quantify these effects, we study the dynamics of an isolated magnetic skyrmion and vortex under the action of the torque given in Eq. (\ref{eq:tor_em}) in the context of Thiele formalism of generalized forces acting on a rigid magnetic structure \cite{Thiele1973}. The equation of motion governing the dynamics of these structures is given by the extended LLG equation
\begin{eqnarray}
\partial_t{\bf M}&=&-\gamma{\bf M}\times{\bf H}_{\rm eff} + \frac{\alpha}{M_s}{\bf M}\times\partial_t{\bf M} - {\bf T}
\end{eqnarray}
where the torque {\bf T} is given as
\begin{equation}\label{eq:tor_tot}
{\bf T} ={\bf T}_{\rm t} + \frac{\beta b_{\rm J}}{M_s}{\bf M}\times\partial_x{\bf M}
\end{equation}
with $\beta$ being a spatially constant non-adiabatic parameter arising from, e.g., spin relaxation \cite{Zhang2004}. To make our analysis simple without missing any interesting physics, we adopt the magnetization profile of an \textit{isolated} skyrmion and vortex core in spherical coordinates as ${\bf m}=(\sin\theta\cos\Phi,\sin\theta\sin\Phi,\cos\theta)$, where the polar angle $\theta$ is defined for an isolated skyrmion as $\cos\theta = p(r_0^2-r^2)/(r_0^2 + r^2)$, and for an isolated vortex as $\cos\theta = p(r_0^2-r^2)/(r_0^2 + r^2)$ for $r \le r_0$, and $\theta = \pi/2 $ for $r >r_0$. $p = \pm 1$ defines the skyrmion (vortex core) \textit{polarity}, $r_0$ defines the radius of the skyrmion (vortex) core. For both textures, the azimutal angle is defined as $\Phi = q Arg(x + iy) + c \pi/2$, where $q = \pm 1$ is the \textit{vorticity} and the $c =\pm 1$ defines the in-plane curling direction otherwise called the \textit{chirality}. For these magnetic profiles, the \textit{topological charge} $\mathcal{Q} =\frac{1}{4\pi}\int \mathcal{N}_{xy} d^2{\bf r}$
is such that $\mathcal{Q} = \frac{1}{2}pq$ for an isolated vortex core and $\mathcal{Q} = (1 - \mathcal{S})pq$ for an isolated skyrmion, where $\mathcal{S} = r_0^2/(r_0^2 +R^2) \to 0$ for $R \gg r_0$. Using these profiles, we obtain the analytical expressions of the velocity components as
\begin{subequations}\label{eq:velocities}
\begin{eqnarray}\label{eq:vx}
v_x &=& -\frac{\eta_{\rm eff} + \alpha_{\rm eff}\beta_{\rm eff}}{\eta_{\rm eff}^2 + \alpha_{\rm eff}^2}b_J,
\end{eqnarray}
\begin{eqnarray}\label{eq:vy}
v_y &=& pq\frac{\eta_{\rm eff}\beta_{\rm eff} - \alpha_{\rm eff}}{\eta_{\rm eff}^2 + \alpha_{\rm eff}^2}b_J,
\end{eqnarray}
\end{subequations}
where the effective paremeters $\eta_{\rm eff}$, $\beta_{\rm eff}$ and $\alpha_{\rm eff}$ depend on the magnetic texture. For the sake of completeness only, we consider the effect of the renormalization of the gyromagnetic ratio represented by $\eta_{\rm eff}$ which is of order of unity and equals $1 + \frac{16\mathcal{S}_4}{5}\frac{\lambda_{\rm E}^2}{r_0^2}\frac{\lambda_{\rm H}^2}{r_0^2}$ and $1 + \frac{31}{5}\frac{\lambda_{\rm E}^2}{r_0^2}\frac{\lambda_{\rm H}^2}{r_0^2}$ for an isolated skyrmion and vortex respectively, where $\mathcal{S}_k = \sum_{i=0}^{i = k}(\mathcal{S})^i$. However, this is not the focus of this study as this effect is very small and can be neglected.

\subsection{Non-adiabaticity parameter}

The dynamics given by Eqs. (\ref{eq:vx}) and (\ref{eq:vy}) correctly describes the motion of both an isolated skyrmion and vortex core. The dynamics of these two structures is very similar, the major difference being contained in the effective parameters $\eta_{\rm eff}$, $\beta_{\rm eff}$ and $\alpha_{\rm eff}$. The first effective parameter of interest in this study is $\beta_{\rm eff}$, which provides a direct connection between emergent field-induced topological Hall effect and non-adiabaticity. Indeed, while there has been much discussion about the mechanisms responsible for non-adiabatic spin transfer in the literature \cite{Zhang2004, Tatara2004}, the proposed mechanisms although very successful in describing non-adiabatic effects in one-dimensional domain walls, have failed to address the large non-adiabaticity measured in vortex walls. Here, we show that emergent-field induced topological torque gives rise to an additional non-adiabatic torque and a resulting effective non-adiabaticity parameter given as
\begin{equation}\label{eq:beta}
	 \beta_{\rm eff} = \left\{ \begin{array}{lll}
	  \beta + \frac{4\mathcal{S}_2}{3}\frac{\mathcal{P}_{\rm H}}{ \mathcal{P}_{\rm 0}}\frac{\lambda_{\rm H}^2}{r_0^2} , & \mbox{for skyrmion} \\
      \beta \mathcal{C} + \frac{7}{3}\frac{\mathcal{P}_{\rm H}}{ \mathcal{P}_{\rm 0}}\frac{\lambda_{\rm H}^2}{r_0^2}, & \mbox{for vortex core} ,
	 \end{array}
	 \right.\,
\end{equation}
where the geometric factor $\mathcal{C} = 1 + \ln\sqrt{R/r_0}$. Eq. (\ref{eq:beta}) reveals that associated with these textures is an intrinsic non-adiabaticity parameter which results in an overall enhancement of their one-dimensional and/or non-topological counterpart, $\beta$. This enhancement of the non-adiabaticity parameter is a direct consequence of the topology-induced {\em transverse} spin current ${\cal J}_s^y$, as shown in Eq. (\ref{eq:currents_a}).

\subsection{Damping parameter}

In the context of magnetic textures dynamics, it is practically impossible to discuss the non-adiabaticity parameter $\beta_{\rm eff}$ without a mention of the damping parameter
$\alpha_{\rm eff}$ as both parameters govern the motion of these structures, see Eq. (\ref{eq:vy}). Different mechanisms for magnetic dissipation have been proposed \cite{Mills2003, Kambersky1976, Kambersky2007,Gilmore2008, Akosa2016a,Zhang2009,Zang2011}. Our calculation reveals that a non-steady state magnetization induces an emergent electric field that results in an intrinsic damping solely due to the topological nature of these textures and thus an overall enhancement of the damping given as
\begin{equation}\label{eq:alpha}
	 \alpha_{\rm eff} = \left\{ \begin{array}{lll}
	 \alpha +\frac{4\mathcal{S}_2}{3} \frac{\lambda_{\rm E}^2}{r_0^2} , & \mbox{for skyrmion}\\
     \alpha \mathcal{C} + \frac{7}{3}\frac{\lambda_{\rm E}^2}{r_0^2}, & \mbox{for vortex},
	 \end{array}
	 \right.\,
\end{equation}
which is consistent with Ref. \onlinecite{Zang2011}.

These result are far reaching as they provide a very transparent mechanism which explains the subtle difference in the measured non-adiabaticity in vortices compared one-dimensional domain walls \cite{Bisig2016}. However, as we have shown, two dimensional magnetization variation is a necessary but far from a sufficient condition for this effect, with the sufficient condition being a non-zero topological charge $(\mathcal{N}_{\rm xy} \ne 0)$ as discussed in preceding section.

\section{Numerical Results}

\subsection{Tight-binding model}

The derivation proposed in the previous section relied on the adiabatic transport of conduction electrons in a smooth and slowly varying magnetic textures. In other words, the spin of the conduction electrons remains aligned on the local magnetization direction and no spin mistracking is considered \cite{Tatara2004}. In principle, these formulas do not hold when the magnetic texture becomes sharp (i.e. when the skyrmion size is of the order of the spin precession length). In this section, we verify our analytical results numerically using a tight-binding model to test the validity of the adiabatic model discussed in the previous section. The local spin/charge densities and currents as well as the corresponding spin transfer torque are computed numerically using the non-equilibrium wave function formalism \cite{Groth2014}. The system is composed of a scattering region containing an isolated skyrmion or vortex core, attached to two ferromagnetic leads as shown in Fig. \ref{fig:sys_a}. We model this system as a two dimensional square lattice with lattice constant $a_0$, described by the Hamiltonian
\begin{equation} \nonumber
\mathcal{H} = \sum_i \epsilon_i \hat{c}_i^+ \hat{c}_i - t \sum_{<ij>} \hat{c}_i^+ \hat{c}_j - J_{\rm sd} \sum_i \hat{c}_i^+{ \bf m}_i \cdot \boldsymbol{\hat{\sigma}} \hat{c}_i,
\end{equation}
where $\epsilon_i$ is the onsite energy, $t$ is the hopping parameter, the sum $<ij>$ is restricted to nearest neighbors, ${\bf m}_i$ is a unit vector along the local moment at site $i$ coupled by exchange energy of strength $J_{\rm sd}$ to the itinerant electrons with spin represented by the Pauli matrices $\boldsymbol{\hat{\sigma}}$. The label $i $ and $ j$ represent the lattice site and $\hat{c}_i^+=(c_i^\uparrow,c_i^\downarrow)^+$ is the usual fermionic creation operator in the spinor form.
\begin{figure}[t!]
\includegraphics[width=8.0cm]{two_terminal-eps-converted-to.pdf}
\caption{Schematic diagram of tight-binding system made up of a central scattering region (isolated magnetic skyrmion/vortex) attached to two ferromagnetic leads (red boxed) L and R at chemical potentials $\mu_L$ and $\mu_R$ respectively. To ensure smooth magnetization variation from the leads to the scattering region, we consider an optimal system size compared to the radius ($R \gg r_0$).}\label{fig:sys_a}
\end{figure}
The $z$-axis is chosen as the quantization axis while the $x$-axis is the direction of current flow. For all our numerical calculations except stated otherwise, we used the parameters: hopping constant $t = 1$, exchange energy $J_{\rm sd} = 2t/3$, onsite energy $\epsilon_i = 0$, transport energy $\epsilon_F = -4.8J_{\rm sd}$, bias $eV = 0.1J_{\rm sd}$, and a large system size of $401\times401a_0^2$ to ensure smooth magnetization variation from system to leads to avoid unphysical oscillations of torques close to the leads. The local spin density ${\bf S}_n^\alpha$ at site $n$ and local spin current density ${\bf J}_{n-1}^\alpha$ between site $n-1$ and $n$ at a particular transport energy for electrons from the $\alpha$-lead can be calculated from their respective operators
\begin{equation}\label{eq:sden}
{\bf S}_n^{\alpha} = \frac{\hbar}{2}\sum_{\nu} \begin{pmatrix} {\Psi^{\alpha \uparrow}}^{*}_{n\nu}\\ {\Psi^{\alpha \downarrow}}^{*}_{n\nu}\end{pmatrix}^T\boldsymbol{\sigma}\begin{pmatrix}\Psi_{n\nu}^{\alpha \uparrow}\\
\Psi_{n\nu}^{\alpha \downarrow}
\end{pmatrix}
\end{equation}
and
\begin{equation}\label{eq:curr}
{\bf J}_{n-1}^\alpha = \frac{1}{2i}\sum_{\nu} t_{n\nu, n-1\nu}\begin{pmatrix}{\Psi^{\alpha \uparrow}}^{*}_{n\nu}\\ {\Psi^{\alpha \downarrow}}^{*}_{n\nu}\end{pmatrix}^T\boldsymbol{\sigma}\begin{pmatrix}\Psi_{n-1\nu}^{\alpha \uparrow}\\
\Psi_{n-1\nu}^{\alpha \downarrow}
\end{pmatrix} + c.c,
\end{equation}
where $\Psi_{n\nu}^{\alpha \sigma}$ are $\nu$ propagating mode of the spin-$\sigma$ wave functions from the $\alpha$-lead at site $n$. The quantum mechanical average is calculated by integrating over the small energy window $\epsilon_F - \frac{_{eV}}{^2}$ and $\epsilon_F + \frac{_{eV}}{^2}$ as
\begin{equation}
\langle{\bf S}_n\rangle = \sum_\alpha \int\frac{d\epsilon}{2\pi}f_\alpha {\bf S}_n^{\alpha},
\end{equation}
where $f_\alpha$ is the Fermi-Dirac function for lead-$\alpha$. A similar formula apply for $\langle{\bf J}_{n-1}\rangle$. The corresponding charge density and current density can be obtained from Eqs. (\ref{eq:sden}) and (\ref{eq:curr}) by replacing $\frac{\hbar}{2}\boldsymbol{\sigma}$ by e${\bf I}$ where ${\bf I}$ is the $2\times2$ identity matrix.

\begin{figure}[t!]
\includegraphics[width=9.0cm]{Fig_XY-eps-converted-to.pdf}
\caption{Two-dimensional profile of torque components showing contributions from different spin current sources. The adiabatic torque is dominated by the contribution from the longitudinal spin current (a) compared to the transverse spin current (b). The converse is true for the non-adiabatic torque which is dominated by contribution from the transverse spin current (d) compared to the longitudinal spin current (c).}\label{fig:tor2d}
\end{figure}

\subsection{Results and discussion}
\begin{figure}[t!]
\includegraphics[width=6.0cm]{Fig_X-eps-converted-to.pdf}
\caption{One-dimensional profile of torque components showing contribution from different spin current sources and the spin density. The adiabatic torque (a) is dominated by the contribution from $J_s^{x}$ while the non-adiabatic torque (b) is dominated by contribution from $J_s^{y}$. Results shows a very good match between the torque calculated from the spin density ($S$) and that calculated from the spin current $J_s = J_s^{x} + J_s^{y}$.}\label{fig:tor1d}
\end{figure}

Our objective in this section is to ascertain the source of the different contributions to the adiabatic and non-adiabatic torques to uncover their respective microscopic origin and establish a direct correspondence with our theoretical predictions in Section \ref{sec:two}. The most natural and reliable method for calculating the torque {\bf T}$_n$ at site {\bf n} on the local moment {{\bf m}$_n$ is by using the local spin densities as
\begin{equation}\label{eq:torsdn}
{\bf T}_n = \frac{2J_{\rm sd}}{\hbar}\langle{\bf S}_n\rangle \times{\bf m}_n,
\end{equation}
 since this is a conserved quantity, and therefore well defined. However, this method gives little or no information about the microscopic origin of the torque. Therefore, we also calculate the torque from the local spin current as
 \begin{equation}\label{eq:torcur}
{\bf T}_n = \langle{\bf J}_n\rangle - \langle{\bf J}_{n-1}\rangle.
\end{equation}
This way, we are able to unambiguously separate the different contributions to the torque arising from the spin current along ${\bf y}$ and ${\bf x}$, quantify its origin, and compare our numerical results with our analytical predictions. A caveat to this method however is that the spin current is a non-conserved quantity especially in systems with spin-orbit coupling or sharp magnetic textures (i.e. sizable spin mistracking). This notwithstanding, since our considerations are based on smooth magnetic textures we can argue that the spin current is well defined and to make sure of this, we used both definitions of the local torque and ensured that the calculated torque using both methods are the same. We deduce the local adiabatic $T_n^{\rm ad}$ and non-adiabatic $T_n^{\rm na}$ torque contribution from the calculated local torque {\bf T}$_n$ by recasting the local torque in the form ${\bf T}_n = T_n^{\rm ad}\partial_x{\bf m}_n - T_n^{\rm na}{\bf m}_n\times\partial_x{\bf m}_n$. Throughout this study except otherwise stated, the torques are reported in units of $2J_{\rm sd}/\hbar$.

Our numerical results for the two-dimensional profile of spin transfer torque components [calculated using the spin current definition in Eq. (\ref{eq:torcur})] for an isolated skyrmion of core radius $10 a_0$ is shown in Fig. \ref{fig:tor2d}. As shown in Figs. \ref{fig:tor2d} (a) and (b), the adiabatic torque is dominated by the contribution of the {\em longitudinal} spin current ($J_s^{x}$), which is at least an order of magnitude larger than the contribution from the transverse spin current ($J_s^{y}$). The non-adiabatic torque [Figs. \ref{fig:tor2d} (c) and (d)], is largely dominated by contribution of the {\em transverse} spin current ($J_s^{y}$). These results confirm the analysis based on the analytical derivations of the previous section, Eqs. (\ref{eq:currents_a}) and (\ref{eq:tor_em}). \par

Fig. \ref{fig:tor1d} displays the one-dimensional profile of the torque components obtained by summing over the transverse direction for a skyrmion radius $r_0 = 10a_0$. Consistently with Fig. \ref{fig:tor2d}, these results show that the non-adiabatic torque is dominated by the texture-induced {\em transverse} spin current while its adiabatic counterpart is dominated by the {\em longitudinal} spin current. In addition, we also performed these calculations using Eq. ({\ref{eq:torsdn}) (open circles). The results obtained by this method overlap with the results obtained using Eq. (\ref{eq:torcur}). This indicates that for skyrmions with radius $r_0 \ge 10 a_0$ the spin mistracking is negligible.\par

\begin{figure}[t!]
\includegraphics[width=9.5cm]{Fig_r0-eps-converted-to.pdf}
\caption{Effective adiabatic and non-adiabatic torque dependence on the skyrmion radius. The adiabatic torque (a) is almost non-dependent on the radius of the sykrmion while the non-adiabatic torque (b) shows substantial dependence on the skyrmion radius$r_0$. S$^{\rm fit}$ represents fit of our numerical data to analytical result.}\label{fig:radius}
\end{figure}

Finally, we investigated the scaling laws governing the different contributions of the topological torque components with respect to the skyrmion radius. To achieve this, we calculated the normalized adiabatic and non-adiabatic torque components by integrating the projections of the local torque on ${\bf m}\times\partial_y{\bf m}$ and ${\bf m}\times\partial_x{\bf m}$ respectively and normalize accordingly i.e.
\begin{subequations}
\begin{equation}
\tilde{{\rm T}}_{\rm ad} = \left[\int {\bf T}\cdot({\bf m}\times\partial_y{\bf m}) d^2{\bf r}\right] /\int M_s \mathcal{N}_{\rm xy} d^2{\bf r}
\end{equation}
\begin{equation}
\tilde{{\rm T}}_{\rm na} = \left[\int{\bf T}\cdot({\bf m}\times\partial_x{\bf m})d^2{\bf r} \right]/\int pqM_s\mathcal{N}_{\rm xy}d^2{\bf r}.
\end{equation}
\end{subequations}
As shown in Figs. \ref{fig:radius}(a) and (b), while the adiabatic torque is almost independent on the skyrmion radius, the non-adiabatic torque shows a substantial dependence on the skyrmion radius $r_0$ which is in accordance with our analytical results. As a matter of fact, a simple fit of our numerical data to our analytical result given by Eq. (\ref{eq:tor_em}), yields an effective non-adiabatic parameter $\beta_{\rm eff} = 3.5\beta$ for a skyrmion radius of $10a_0$, where $\beta$ is the constant non-adiabaticity parameter obtained in the limit of very large skyrmion radius. For all the range of skyrmion sizes investigated, we also find that the adiabatic assumption exploited to derive the torque expression, Eq. (\ref{eq:tor_em}), is valid. In other words, it confirms that the large non-adiabatic torque in topologically non-trivial magnetic textures can not be explained by spin mistracking.

\section{Conclusion}
We investigated the nature of adiabatic and non-adiabatic spin transfer torque in topologically non-trivial magnetic textures, such as skyrmions and magnetic vortex cores. We showed that the topological spin current flowing through such textures induce an {\em intrinsic topological} non-adiabatic torque, ${\bf T}_t\sim [(\partial_x{\bf m}\times\partial_y{\bf m})\cdot{\bf m}]\partial_y{\bf m}$. Our numerical calculations confirm the physics highlighted by our analytical derivations and confirm that spin transport is mostly adiabatic up to a very good accuracy in these structures, thereby ruling out spin mistracking. Besides providing a reasonable explanation for the enhanced non-adiabaticity in skyrmions and magnetic vortices, our theory opens interesting perspectives for the investigation of current-driven skyrmion dynamics. As a matter of fact, it has been recently proposed that the peculiar robustness of magnetic skyrmions against defects is related to the presence of non-adiabaticity \cite{Iwasaki2013a}. Therefore, understanding the role of this topological non-adiabatic torque when magnetic skyrmions interact with defects is of crucial importance to control current-driven skyrmion motion and achieve fast velocities \cite{Jonietz2010,Schulz2012,Yu2012,Jiang2015,Woo2016}. \par

CA and AM acknowledges financial support from the King Abdullah University of Science and Technology (KAUST) through the Award No OSR-CRG URF/1/1693-01 from the Office of Sponsored Research (OSR). The authors thank M. Kl\"aui, Kyung-Jin Lee, Gen Tatara, A. Bisig and A. Abbout for inspiring discussions.


text:
\section{#1}\setcounter{equation}{0}}
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\defJ.\ Fr\"ohlich} \def\DB{D.\ Buchholz} \def\RH{R.\ Haag{J.\ Fr\"ohlich} \def\DB{D.\ Buchholz} \def\RH{R.\ Haag}
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\parindent0mm \parskip2mm \rcd{\thefootnote}{\fnsymbol{footnote}}
\rightline{DESY 93-116}
{\Large \bf Subfactors and Coset Models\footnote{Talk at the Workshop on
``Generalized Symmetries in Physics'', Clausthal (FRG), July 1993, \\
to appear in the proceedings, eds.\ H.-D.\ Doebner {\it et al.}}}
\\[9mm]
{\large Karl-Henning Rehren }\\[3mm]
II.\ Inst.\ Theor.\ Physik, Univ.\ Hamburg, D-22761 Hamburg (FR Germany)
\\ {\small e-mail: i02reh@dsyibm.desy.de} \\[5mm]
\rcd{\thefootnote}{\arabic{footnote}} \addtolength{\baselineskip}{-1pt}
\bq
 {\bf Abstract:} Some facts about von Neumann algebras and finite
 index inclusions of factors are viewed in the context of local \qft.
 The possibility of local fields intertwining \sps\ sectors with \bgs\
 is explored. Conformal embeddings and coset models serve as examples.
 The associated \sy\ concept is pointed out.
\eq
\addtolength{\baselineskip}{1.5pt}
\section{Introduction}
\parskip3mm \rcd{\theequation}{\thesection.\arabic{equation}}
The present article is a pedestrian's view of some basic mathematical
facts about \vna s and the structure theory of subfactors relevant for
the physical problem of implementability of \sps\ sectors by charged
fields. Its intention is to facilitate the access to abstract results by
formulating them in terms of algebraic relations of rather direct
physical significance, and to illustrate them by a large class of models
(coset models of \cf\ \qft) well known to a broad audience. The present
work is a precursory study related to a joint program in progress with
\KF, \RL, and \JR\ about local extensions of local quantum field
theories.

The general analysis we shall present below was incited by a recent
result of a model study of chiral \ca s \cite{Bul}, which at first sight
quite puzzled the author. The authors of \cite{Bul} were searching for
{\it local} extensions of chiral $SU(2)$ \ca s which are identified by
their local conformal block functions satisfying the $SU(2)$ Ward
identities. They found a solution at level 10 for the isospin 3 sector,
clearly different from the standard non-local vertex operator (Coulomb
gas) solution.

The ``fusion rules'' which are read off the local solution are
\be [3] [3] = [0] + [3] \ee
in contrast to the standard fusion rules
\be [3] [3] = [0] + [1] + [2] + [3] + [4] .\ee
This seems to contradict the message from the general theory of \sps\
sectors \cite{DHR} that the fusion rules are {\it intrinsic} to a given
local \qft. Moreover, the isospin 3 sector is known to have non-trivial
\bgs, so there is the surprising fact that one can associate it with
local correlation functions. In fact, the local solution found in
\cite{Bul} has been identified as the conformal embedding of the $SU(2)$
\ca\ at level 10 into the $SO(5)$ (or $Sp(4)$) \ca\ at level 1. It is
well known that the \vrep\ of the latter splits into the \rep s $[0]$
and $[3]$ of the former, where the primary field for the \rep\ $[3]$ is
the isospin 3 multiplet of $SO(5)$ currents orthogonal to the embedded
$SU(2)$ currents. The \cf\ blocks and the fusion rules (1.1) are just
the correlation functions and the operator product expansions of these
$SO(5)$ currents.

The second solution found in \cite{Bul} is much less surprising. It is
the \cf\ embedding \cite{BBSW} of $SU(2)$ at level 4 into $SU(3)$ at
level 1, generating the isospin 2 sector. This sector has \pgs, and the
fusion rule $[2][2] = [0]$ coincides with the standard one. In fact, the
detailed analysis shows that the $SU(2)_4$ theory is exactly the
subtheory of $\ZZ_2$-invariants of the $SU(3)_1$ theory, where the
global \sy\ group $\ZZ_2$ is the involutive Lie group \amor\ of $SU(3)$
associated with the symmetric space $SU(3)/SU(2)$, lifted to the \ca.

The previous solution is an example of a model, where a sector of a
local \qft, which has genuine \bgs, is generated by local fields of some
larger local \qft. In fact, \cf\ coset models generically exhibit this
feature, and the $SU(2)_4 \subset SU(3)_1$ case is rather the exception.
The aim of the present article is a general analysis of this scheme, as
an alternative to the widespread conviction that charged fields
generating sectors with \bgs\ should exhibit braid-like \comrel s
at \spl\ distance.

The apparent conflict between locality and \bgs\ dissolves due to some
intrinsic ``defect''. As we shall see, one can only find {\it
intertwining} fields
\be \psi a = \rho(a) \psi \ee
for the \emor\ corresponding to the sector in question \cite{DHR},
but not {\it implement} the \emor\ itself:
\be \rho(a) \stackrel!= \sum_i \psi^i \, a \, \psi^{i*} \ee
as it is the case with \pgs\ \cite{DR2}. The defect is responsible for
a mismatch between the abstract fusion rules of the theory of \sps\
sectors \cite{DHR} and the multiplicative structure (operator product
expansions) of the generating fields. This then solves the previously
mentioned puzzle. Namely, the fusion rules (1.1) are not the intrinsic
ones but refer to the embedding into a larger theory.

In order to place the above example in a general context, and to
elucidate the point where the standard reasoning fails, we resort to the
theory of \vna s and subfactors. By the comparison of the general case
of a subfactor $\sub AB$ with the case when the subfactor is given by
the invariants under the action of a \sy\ group, we shall very
explicitly see the role of the gauge group (of the first kind) to
exclude the occurrence of the defect, thus leading to the standard
duality theory \cite{DR2}.

The general case is described by a faithful normal \cex\
\be \mu : B \arr A \ee
i.e.\ a positive unit preserving map of $B$ onto $A$ satisfying
\be \mu(abc) = a\mu(b)c \qquad (a,c \in A,\; b \in B). \ee
A \cex\ generalizes the abstract properties of a group average
\be \mu(b) := |G|^{-1}\sum_{g \in G} \a_g(b), \ee
which special case we shall always compare to the general situation.

Associated with the \cex\ is a ``canonical'' \emor\ $\g : B \arr A$, the
structure theory of which is treated in Sect.\ 2. It is of direct
physical significance: the sector decomposition of $\gamma$ as an \emor\
of $B$ allows to recover the \sy\ group in the case (1.7), and naturally
leads to Ocneanu's generalized \sy\ concept (``paragroups'' \cite{Oc})
in the general case, see Sect.\ 4. The sector decomposition of $\gamma$
as an \emor\ of $A$ corresponds to the decomposition of a \rep\ of $B$
upon restriction to $A$, provided the former is obtained from an
invariant state $\omega_0 = \omega_0 \circ \mu$. In the quantum field
theoretical context (i.e.\ $A$ and $B$ are a pair of local \vna s from
local quantum field theories $\aa$ and $\bb$ and $\omega_0$ is the
vacuum state in which the \sy\ is unbroken), this is the ``branching
rule'' for the vacuum sector of the larger theory. The sectors of the
subtheory thus obtained are interpolated by fields from the larger
theory.

The canonical \emor\ uniquely characterizes the underlying inclusion
$\sub AB$. We show how to recover detailed structure information about
the latter in terms of intertwining operators and their algebraic
properties. In Haag's spirit that ``fields are just coordinates on local
algebras'', we are putting coordinates on an inclusion.

Our conclusion in the quantum field theoretical context is that indeed
some sectors with \bgs\ can be generated by a local field algebra. The
exact relation to the \st\ is discussed in Sect.\ 3. In fact, this
possibility is realized in (almost) all coset models of \cf\ \qft.

The models we have in mind include the previously mentioned \cf\
embedding $SU(2)_{10} \subset SO(5)_1$, but also coset models \cite{GKO}
of the form
\be Diag(G \times G)_{k+l} \otimes W \subset G_k \otimes G_l \ee
where $G_k$ stands for the level $k$ chiral \ca\ of the compact
simple Lie group $G$, or still more generally
\be H_l \otimes W \subset G_k \ee
where $\sub HG$ is a pair of compact semi-simple Lie groups (not
necessarily simple such that the levels may have several components for
the simple components of $H$ and $G$). In all cases, $W$ stands for a
local \qft\ decoupled from the subgroup currents. It is trivial for \cf\
embeddings and otherwise contains the coset \emt\ given by
\be T^{(G)} = T^{(H)} + T^{\rm coset} \ee
in terms of the Sugawara expressions for $T^{(G)}$, $T^{(H)}$. If the
central charge of $T^{\rm coset}$ is $c>1$, then $W$ may contain further
primary fields of higher scaling \dmn s.

We consider a coset model as an extension of the local theory of \obs s
$\aa$ into a local theory of ``charged'' fields $\bb$. The detailed
knowledge of these models precisely confirms and exemplifies our general
findings. In Sect.\ 4, we briefly sketch the generalized \sy\ associated
with such an extension.

According to Wassermann \cite{Wa}, the local \vna s of \ca\ models are
defined -- in purely group theoretical terms -- as $\pi_0(L_IG)''$ where
$LG$ is the group of smooth maps (loops) $: S^1 \arr G$, the local
subgroup $L_IG$ consists of those maps with support in the interval $I$
of the circle, and $\pi_0$ is the projective \vrep\ of level $k$.
The term ``vacuum'' refers to the implementation of the group of \dmor s
of the circle which act as \amor s of the loop group, and in particular
to the spectrum of the ``rigid'' rotations, while the level refers to
the cocycle associated with a projective \rep. Similar definitions can
be given for the algebra of the (coset) \emt\ in terms of the \dmor\
group of $\s$ and its local subgroups.

Following \cite{Wa}, we shall argue in Sect.\ 5 that the index of the
inclusion of the corresponding local \vna s can be finite only when the
\vrep\ of $\bb$ decomposes finitely into \rep s of $\aa$, and give
a formula to compute the index. The branching rules for coset models
being known, this formula is very explicit.

The contents of this article are not really new, although we sometimes
adopt an unconventional point of view. By viewing a class of physical
models in the light of the abstract structure theory for the canonical
\emor, some model findings are systematically related to the general
algebraic structures which underly local extensions of local quantum
field theories. Moreover, the latter are formulated in physically
evident terms. Some novel insight concerns the properties of
intertwining fields and the relation between their local commutativity
and \bgs, the precise identification of the points where more general
subtheories depart from the rather special gauge \sy\ inclusions, as
well as some algebraic control over the defect going along with
non-integer statistical \dmn s.

\sectreset{Inclusions of type {\it \bf III} factors}
We collect some well known results about finite index inclusions of
\typethree\ factors \cite{PP,Ko,Hi,Lo2,Lo3}, which we shall need for the
subsequent discussion, and qualify the abstract statements in the
special case when the inclusion is given by the action of a \sy\ group.
The generalization to Hopf algebra actions is also known \cite{Lo4}.

We consider an irreducible inclusion $\sub AB$ of \typethree\ factors.
With this pair, we have in mind a local \vna\ $\bb(\oo)$ from some \qft\
and a local \vna\ $\aa(\oo)$ from some subtheory. However, the purely
mathematical statements in this section do not refer to the physical
context. The local aspects will be treated in Sect.\ 3.
\bq
 {\bf Proposition 1:} Let $\sub AB$ be an irreducible inclusion of
 \typethree\ factors with finite index $\lambda$. There are a unique
 normal \cex\ $\mu: B \arr A$ which satisfies the operator estimate
 \be \mu(bb^*) \geq \lambda^{-1} \cdot bb^* \qquad (b \in B), \ee
 and an isometry $V \in B$ such that
 \be \barr{lc} (i) & \mu(bV^*)V = V^*\mu(Vb) = \lambda^{-1}b \qquad
 (b \in B) \\ (ii) & \mu(VV^*) = \lambda^{-1}. \earr \ee
 The map $\g : B\arr \sub AB$
 \be \g(b) := \lambda \cdot \mu(VbV^*) \ee
 is an \emor, called the canonical \emor, and $V$ is an \itw\ $: id
 \arr \g$, i.e.
 \be V \, b = \g(b) \, V. \ee
\eq
The canonical \emor\ was originally introduced in \cite{Lo1} in terms of
modular conjugations. It is very useful in \rep\ theory, since it allows
the restriction of an \emor\ $\sigma$ of $B$ to an \emor\ $\g \circ
\sigma\vert_A$ of $A$, and the induction of an \emor\ $\sigma$ of $A$ to
an \emor\ $\sigma\circ\gamma$ of $B$. Considering \emor s as \rep s of
spatial \vna s, these prescriptions correspond to the restriction and
(Mackey) induction of \rep s.

We specify the statement of Prop.\ 1 in the case when $A$ are the
``gauge invariants'' of $B$ with respect to some \sy\ group.
\bq
 {\bf Proposition $\bf 1'$:} Let in Prop.\ 1 $A = B^\a$ be the fixpoint
 subalgebra under an outer action $\a : G \arr Aut(B)$ of a \sy\
 group $G$. Then $\lambda = |G|$ (in particular, $G$ is finite), and
 $\mu$ is the group average:
 \be \mu = |G|^{-1} \sum_{g \in G} \a_g\,. \ee
 The operators
 \be V_g := \a_g(V) \ee
 form a complete system of \on\ isometries:
 \be V_g^*V_h = \delta_{gh} \qquad {\rm and} \qquad
 \sum_{g \in G} V_gV_g^* = 1, \ee
 carrying the left regular \rep\ of $G$:
 \be \a_g(V_h) = V_{gh}. \ee
 The canonical \emor\ is
 \be \g(b) = \sum_{g \in G} V_g \a_g(b) V_g^*. \ee
\eq
The completeness of $V_g$ ist just (2.2($ii$)): $|G| \cdot \mu(VV^*) =
1$. The \on ity is verified by summing $V^*\a_g(VV^*)V$ over the group,
which implies $\sum_{g \neq e} (V^*V_g)(V^*V_g)^* = 0$ and therefore
$V^*V_g = \delta_{ge}$. (2.9) follows from the definitions.

Next, we discuss the reducibility of \emor s. Let $\sigma : M
\arr M$ be an \emor\ of a \typethree\ factor $M$. It is reducible iff
the relative commutant $\sigma\comm M$ is non-trivial. Every projection
$e \in \sigma\comm M$ corresponds to a sub-\emor\ $\sigma_e$ defined as
follows. Pick an isometry $w \in M$ such that $ww^* = e$. Then
\[ \sigma_e(\cdot) := w^* \sigma(\cdot) w. \]
By construction, $w$ is an \itw\ $: \sigma_e \arr \sigma$. If
\[ 1 = \sum_s e_s \qquad {\rm and} \qquad  e_s = \sum_i e_s^i \]
are the central partition of unity in $\sigma\comm M$, and partitions of
$e_s$ into minimal projections in $\sigma\comm M$, respectively, then
the corresponding sub-\emor s $\sigma_s^i = w_s^{i*} \sigma(\cdot)
w_s^i$ fall into inner equivalence classes (sectors) labelled by $s$,
while $i = 1,\ldots n_s$ counts the multiplicity of the sector $[s]$
within $\sigma$. It is possible and convenient to choose $\sigma_s^i =
\sigma_s$ independent of $i$, and to write
\[ \sigma \simeq \bigoplus_s n_s \; \sigma_s. \]
For reducible inclusions one defines $Index(\sub NM)$ to be the
{\it minimal}\ index \cite{Hi}, and calls $d(\sigma) :=
\sqrt{Index[\sub {\sigma(M)}M]}$ the {\it \dmn} of an \emor. The \dmn\
is additive and multiplicative:
\[ d(\sigma) = \sum_s n_s d(\sigma_s) \qquad {\rm and} \qquad
d(\sigma\rho) = d(\sigma) d(\rho).\]
\bq
 {\bf Proposition 2:} Let $\sub AB$ be as in Prop.\ 1. Denote by
 $\rho : A \arr A$ the restriction of $\g$ to $A$. Then
 \be \rho(A) \subset \g(B) \subset A \subset B \ee
 is (the beginning of) a Jones tower. In particular, all inclusions are
 irreducible with index $\lambda$ and $d(\g) = d(\rho) = \lambda$. Both
 $\sub {\g(B)}B$ and $\sub {\rho(A)}A$ are reducible (unless $\lambda =
 1$):
 \be (a) \quad \rho \simeq \bigoplus_s N_s \; \rho_s \qquad {\rm and}
 \qquad (b) \quad \g \simeq \bigoplus_t M_t \; \g_t \,, \ee
 either decomposition containing the identity as a sub-\emor\ with
 multiplicity 1.

 Let $W \in A$ be the \itw s $: \rho_s \arr \rho$ according to the above
 decomposition theory for $\rho$, and $V \in B$ as in Prop.\ 1. The map
 $W \mapsto \psi := W^*V$ is bijective onto the linear space $\hh_s
 \subset B$ of operators satisfying
 \be \psi a = \rho_s(a) \psi \qquad (a \in A). \ee
 $\hh_s$ is an $N_s$-\dmn al Hilbert space of isometries with scalar
 product $\psi^*\psi' \in A' \cap B = \CC$.
\eq
In the quantum field theoretical context, $A$ are local \obs s
$\aa(\oo)$, $B$ are local fields $\bb(\oo)$, and $\rho_s$ are
(localized) \emor s of $\aa$ representing the \sps\ sectors \cite{DHR},
see Sect.\ 3. We have thus established the existence of intertwining
fields for all sectors contained in $\rho$, and conversely all sectors
which have intertwining fields in $\bb$ are contained in $\rho$.

{\it Proof of the last statements of Prop.\ 2}: The map $W \mapsto
\psi$ is injective since $\mu(\psi\psi^*) = \lambda^{-1} W^*W > 0$. It
is inverted by $\psi \mapsto W:= \lambda \mu(V\psi^*) \in A$. The stated
intertwining properties of $\psi$ resp.\ $W$ are readily checked.

The following specification of Prop.\ 2 in the group symmetric case is
easily obtained with group theoretical arguments.
\bq
 {\bf Proposition $\bf 2'$:} Let $A = B^\a$ be as in Prop.\ $1'$.
 Then
 \be (a) \quad \rho \simeq \bigoplus_r dim(r) \rho_r \qquad {\rm and}
 \qquad (b) \quad \g \simeq \bigoplus_{g \in G} \a_g \ee
 where in $(a)$ the sum extends over the unitary \irrep s
 $\tau^r$ (of \dmn\ $dim(r)$) of the \sy\ group $G$. For every sector
 $[r]$ and an \on\ basis of \itw s $W_r^i : \rho_r \arr \rho$ in $A$,
 the corresponding charged fields $\psi^i = \sqrt{|G|/dim(r)} \cdot
 (W_r^i)^*V \in \hh_r \subset B$ are a complete set of \on\ isometries:
 \be (i) \quad \psi^{i*}\psi^j = \delta_{ij} \qquad \quad
 (ii) \quad \sum_{i=1}^{dim(r)} \psi^i\psi^{i*} = 1 \ee
 which transform linearly in the \rep\ $\tau^r$:
 \be \a_g(\psi^i) = \sum_{j=1}^{dim(r)} \psi^j\cdot\tau^r_{ji}(g) \ee
 and implement the sectors:
 \be \rho_r(a) = \sum_{i=1}^{dim(r)} \psi^i \, a \, \psi^{i*}. \ee
\eq
The reader will find the sketch of a constructive proof in the appendix.

With Prop.\ $1'$ we have recovered the Cuntz algebra (= $C^*$ algebra
generated by a complete set of \on\ isometries) of intertwining fields
which implement the sectors. In the context of \sps\ sectors in local
\qft, it was first derived in \cite{DR1}, and is the basic tool for the
reconstruction \cite{DR2} of the gauge symmetric field algebra
from the sector structure of the \obs s.

If one looks through the argument for the completeness relation
(2.14($ii$)) (see the appendix) one finds that it owes its specific
form to the linear transformation law (2.15). Namely, by our general
construction, for every $\psi \in\hh_s$ the ``average'' of $\psi\psi^*$
is a multiple of unity:
\be \mu(\psi\psi^*) = \lambda^{-1} \cdot W^*W \propto 1 .\ee
On the other hand, introduce the joint range projection of the Hilbert
space of isometries $\hh_s$ in terms of any \on\ basis $\psi^i \in
\hh_s$
\be E_s := \sum_{i=1}^{N_s} \psi^i \psi^{i*}.\ee
Now, under the {\it linear} group action (2.15), the average
$\mu(\psi\psi^*)$ is a multiple of $\sum_i \psi^i\psi^{i*}$, hence
$E_s = 1$. In the general case, however, (2.17) is just an abstract
relation in $B$ rather than an algebraic relation among the operators
$\psi \in \hh_s$, while $E_s$ will be a non-trivial projection in
$\rho_s(A)' \cap B$. This ``defect'' $E_s < 1$ is related to the
mismatch between the multiplicity $N_s$ and the dimension $d(\rho_s)$
in (2.11(a)):
\bq
 {\bf Proposition 3:} If $\sub AB$ has finite index, then the
 multiplicities $N_s$ in (2.11($a$)) are not larger than the \dmn s
 $d(\rho_s)$:
 \be N_s \leq d(\rho_s). \ee
 Equality holds iff the defect projection $E_s$ is unity.
\eq
{\it Proof:} For $\psi^i \in \hh_s$ as above, the operators $X_i :=
\g(\psi^i)$ are a system of orthogonal \itw s in $A$
\[  X_i \rho(a) = \rho\rho_s(a) X_i\,. \]
Hence the $N_s$ projections $X_iX_i^* \in \rho\rho_s\comm A$ correspond
to $N_s$ disjoint sub-endo\-mor\-phisms equivalent to $\rho$ contained
in $\rho\rho_s$. The additivity and multiplicativity of \dmn s implies
\[ N_s \cdot d(\rho) \leq d(\rho\rho_s) = d(\rho)d(\rho_s) \]
where equality holds iff $\sum_i X_iX_i^*$ is a partition of unity.
Since $d(\rho) = \lambda$ is finite, the inequality (2.19) follows,
and since $\sum_i X_iX_i^* = \g(E_s)$, equality holds iff $E_s = 1$.

In the group \sy\ case, we conclude $d(\rho_r) = N_r = dim(r)$ and
$\rho\rho_s \simeq d(\rho_s) \cdot \rho$. The \dmn\ formula for finite
groups, $|G| = \sum_r dim(r)^2$, is recovered from (2.13). It becomes
also clear from Prop.\ 3 that sectors with non-integer \dmn\
$d(\rho_s)$ must suffer the defect. Finally, the defect is related to
the depth \cite{Oc} of the inclusion $\sub AB$: suppose the defect to be
absent in all sectors $\rho_s$. Then together with $\rho_s$ all
sub-\emor s of $\rho_s\rho_r$ are implemented, and by Prop.\ 2 are
already contained in $\rho$. Then $\g(B) \subset A$ and $\sub AB$ have
depth 2 and $\sub AB$ is given by the action of a Hopf algebra
\cite{Lo4}.

The failure of Prop.\ 3 at infinite index will be exemplified in Sect.\
6.

\sectreset{The relation to statistics}
We have so far considered a single pair of \vna s $\sub AB$. A local
\qft\ is given by a local net, i.e.\ an assignment of an algebra of
local \obs s $\aa(\oo)$ to every \spt\ region $\oo$ such that local
algebras at \spl\ distance commute. These algebras are \typethree\
factors for typical bounded (``double cone'') regions \cite{BDF}. In the
case of chiral \ca s, the local algebras $\aa_I$ are assigned to the
intervals $I$ of the circle. $\aa_I$ are also \typethree\ factors
\cite{Wa,FG}. The total algebra $\aa$ of \obs s is the $C^*$ algebra
generated by all its local subalgebras.

A theory is covariant with respect to the \spt\ \sy\  group (the \pc\
group or, on the circle, the \mb) if the latter acts by \amor s
on the local net, i.e.\
\be \a_x(\aa(\oo)) = \aa(x\oo) .\ee
A \rep\ is covariant if the \amor s $\a_x$ are implemented by unitary
operators in the Hilbert space. The spectrum condition refers to the
positivity of the generators of the translations in a given \rep.

Let us now turn to a subtheory. By this we mean a pair of local nets
$\aa$ and $\bb$ such that for every region
\be \aa(\oo) \subset \bb(\oo), \ee
and the covariance \amor s $\a_x^{(\bb)}$ of $\bb$ coincide on $\aa$
with $\a_x^{(\aa)}$ (we shall thus drop the distinction).

The analysis of Sect.\ 2 then applies to every single inclusion (3.2).
It is important to note that the canonical \emor\ $\g$ certainly, and
the \cex\ $\mu$ possibly depends on $\oo$. There arise thus consistency
problems when extending the maps $\mu, \g, \rho$ globally. These
problems are not subject of the present article. Let us therefore --
rather than deriving it from some more general principles -- make the
certainly very reasonable assumption that $\mu$ is consistently defined
for all local algebras (i.e.\ $\mu^{(\oo_1)}\vert_{\bb(\oo_2)} =
\mu^{(\oo_2)}$ if $\oo_2 \subset \oo_1$) and commutes with $\a_x$.
Furthermore, we shall assume that the vacuum state $\omega_0$ of $\bb$
is the unique \pc\ invariant state. Then $\omega_0$ is also invariant
under the \cex:
\be \omega_0 \circ \a_x = \omega_0 = \omega_0 \circ \mu \;. \ee
We fix $\oo_0$ and apply Prop.\ 1 and 2 to the inclusion $\aa(\oo_0)
\subset \bb(\oo_0)$. In particular, $V$ is an isometry in $\bb(\oo_0)$.
Then we extend $\gamma$ globally by (2.3):
\be \g(b) := \lambda\cdot\mu(VbV^*) \qquad\quad (b \in \bb). \ee
By locality, (1.6), and (2.2), $\rho := \g\vert_\aa$ is an \emor\ of
$\aa$ localized in $\oo_0$, namely it acts trivially on $a \in
\aa(\oo')$ if $\oo'$ is \spl\ separated from $\oo_0$. The same is not
true for $\g \in End(\bb)$, nor does $\g$ map $\bb(\oo)$ into
$\aa(\oo)$ unless $\oo$ contains $\oo_0$.

The operators $\psi \in \hh_s \subset \bb(\oo_0)$ are local \itw s for
the sub-\emor s $\rho_s$ of $\rho$ which are also localized in $\oo_0$.

Next, let $\rho$ correspond to a covariant \rep\ of $\aa$. Then
\cite{DHR} there is a unitary cocycle $x \mapsto U_x \in \aa$
\be U_{xy} = \a_x(U_y)U_x \ee
such that $\rho_x := Ad_{U_x} \circ \rho = \a_x \rho \a_x^{-1}$ is
equivalent to $\rho$ and localized in $x\oo_0$. Then also the sub-\emor
s $\rho_s$ are covariant (with cocycle $U^s_x$). We claim:
\bq
 {\bf Lemma:} \hfill $\a_x(\psi) = U^s_x \cdot \psi \qquad
 (\psi \in \hh_s)$. \hfill (3.6)
\eq \setcounter{equation}{6}
{\it Proof:} It is easily checked that $(U^s_x)^*\a_x(\psi)$ are again
\itw s for $\rho_s$ and therefore belong to $\hh_s$. Thus the map
$(x,\psi) \mapsto (U^s_x)^*\a_x(\psi)$ is a finite \dmn al unitary
\rep\ of the \pc\ or \mb\ on $\hh_s$. Since every such \rep\ is trivial,
one gets (3.6).

Locality of $\bb$ implies that $\a_x(\psi)$ and $\psi$ commute when
$x\oo_0$ is \spl\ to $\oo_0$. Therefore ($i,j = 1,\ldots N_s$)
\[ U^s_x\psi^i\psi^j = \psi^jU^s_x\psi^i = \rho_s(U^s_x)\psi^j\psi^i .\]
Now, $U_x^{s*}\rho_s(U^s_x)$ is the \st\ operator $\eps_s =
\eps(\rho_s,\rho_s)$ \cite{DHR}, so locality reads
\be \eps_s\,\psi^i\psi^j = \psi^j\psi^i .\ee
When there is no defect as discussed in Prop.\ 3, this can be solved for
$\eps_s$:
\be  \eps_s = \sum_{ij} \psi^j\psi^i\psi^{j*}\psi^{i*} \ee
implying $\eps_s^2 = 1$, i.e.\ \pgs\ as expected for sectors
implemented by local fields. In contrast, when there is a defect, one
only gets
\be \eps_s^2\psi^i\psi^j = \psi^i\psi^j ,\ee
i.e.\ the non-trivial joint range projection of $\psi^i\psi^j$ is an
eigenprojection of the monodromy operator with eigenvalue 1. On the
other hand, the monodromy operator is diagonalized by the projections
for the decomposition of $\rho_s^2 \simeq \bigoplus \rho_k$ into its
irreducible components \cite{DHR}. Thus, although $\psi^i\psi^j$ are
\itw s for $\rho_s^2$, they do not project to \itw s for irreducible
sub-\emor s $\rho_k$ unless the monodromy eigenvalue
\be \frac{\kappa(\rho_k)}{\kappa(\rho_s)^2} \stackrel != 1. \ee
Here $\kappa$ is the \st\ phase \cite{DHR} related to the (fractional)
spin of a sector. Namely, every candidate \itw\ $T^*\psi^i\psi^j \in
\bb$ (with $\aa \ni T : \rho_k \arr \rho_s^2$) for a subsector $\rho_k$
violating (3.10) must vanish. This immediately explains the
``truncation'' of the fusion rules (1.1) as compared to the intrinsic
fusion rules (1.2).

Actually, one can prove more, thereby restricting the branching
rules for the \vrep\ in the first place. (3.10) is then a
trivial consequence.
\bq
 {\bf Proposition 4:} All subsectors of $\rho$ have \st\ phase
 $\kappa(\rho_s) = 1$.
\eq
{\it Proof:} The ``master'' equation underlying (3.7):
\be \eps_\rho VV = VV, \ee
where $\eps_\rho = \eps(\rho,\rho)$, obtains in the same way as (3.7).
Introduce isometric \itw s in $A$: $W := \lambda^{1/2}\mu(V) : id \arr
\rho$ and $R := \g(V)W : id \arr \rho^2$, and observe $V^*R =
\lambda^{-1/2}V$. Define the left-inverse $\phi_\rho(a) := R^*\rho(a)R$
and compute $d(\rho)\phi_\rho(\eps_\rho) = \lambda\mu(V^*\eps_\rho V)$.
Next, compute ($\hat R \equiv \lambda^{1/2}R$)
\[ V^*\eps_\rho V = \hat R^*V\eps_\rho V = \hat R^*\rho(\eps_\rho)VV =
\rho(\hat R^*)\eps_\rho^* VV = \rho(\hat R^*)VV = V \hat R^*V = VV^*. \]
The equality in the middle comes from the theory of \st\ \cite{DHR}.
Therefore,
\be \kk_\rho := d(\rho)\phi_\rho(\eps_\rho) = 1. \ee
Thus $\phi_\rho$ is the standard left-inverse of $\rho$, and since the
spectrum of $\kk_\rho$ is given by the \st\ phases $\kappa(\rho_s)$ of
all subsectors of $\rho$ \cite{DHR}, the claim follows.

The lemma and the constraint (3.10) derived from it, as well as
Prop.\ 4 fail if $\sub AB$ has infinite index (see Sect.\ 6).

\sectreset{The generalized \sy}
An alternative way to show the completeness (2.14($ii$)) of the
charged isometries in the group symmetric case relies on the gauge
principle: since the defect projection $E_s \in \rho_s(A)' \cap B$ is
gauge invariant by the linear transformation law (2.15), it is actually
contained in $A$ and therefore must be a scalar.

In the general case, an explicit formula like (2.15) is lacking. The
gauge transformations $\a_g$ -- which by (2.13($b$)) are the irreducible
components in the decomposition of the canonical \emor\ -- are replaced
by the \emor s $\g_t$ (2.11($b$)). The latter will not act linearly on
the \itw\ spaces $\hh_s$ of charged field operators. Yet, it seems
natural to consider the action of $\g_t$ on $B$ as the generalized \sy.

The questions arise what sort of a generalized group $\g_t$ form, and
in which sense $A$ consists of invariant quantities. We shall not
elaborate on this program here, but only include three remarks to
illustrate our conception of (gauge) \sy: everything that is apt to
characterize the position of a subalgebra in an algebra (the gauge
invariant quantities among all fields) is a good candidate for a
generalized symmetry.

1.) The underlying structure to $\g_t$ and all sub-\emor s of their
products is described by the even part of the bipartite graph associated
with the inclusion $\sub AB$ as discussed by Ocneanu \cite{Oc}. The odd
part of the graph is obtained if one writes the canonical \emor\ in the
form
\be \g = \sigma\bar\sigma  \qquad {\rm where} \quad \sigma(B) = A. \ee
The odd part of the graph then comprises all subsectors of $\gamma^n
\sigma$. In (4.1), $\sigma$ is an irreducible \emor\ of $B$ (since $\sub
AB$ is irreducible), and $\bar\sigma$ is a conjugate \emor\ uniquely
determined up to inner equivalence by the requirements that $\bar\sigma$
is irreducible and $\sigma\bar\sigma$ contains the identity.

2.) If it is only known that all irreducible sub-\emor s $\g_t$ of $\g$
in (2.11($b$)) have \dmn\ 1, i.e.\ are \amor s, then one can show
\cite{Lo3,KHR} that the latter may be (uniquely) chosen within their
inner equivalence classes such that they form a group of order $\lambda$
pointwise preserving $A$. In other words, it is possible to recover the
\sy\ group $G$ of $A = B^\a$ from its canonical \emor\ $\g : B \arr A$.
The argument is simple: since for every $t$, $\g_t^{-1}\g = \g_t^{-1}
\sigma \circ \bar\sigma$ again contains the identity, one concludes
that $\g_t^{-1}\sigma$ is conjugate to $\bar\sigma$ and therefore
equivalent to $\sigma$, and so is $\g_t\sigma$. It follows that
$\g_t^{-1}\g$ and $\g_t\g$ contain the same sectors as $\g$, so the
sector multiplication of $\g_t$ must form a group. Now, if $\g_t\sigma
= Ad_{U_t}\circ \sigma$, then $\a_t := Ad_{U_t^*}\circ \g_t$ preserve $A
= \sigma(B)$ pointwise and satisfy the composition law of the group up
to inner conjugation with some cocycle in $B$. Since the cocycle
commutes with $A$ it is scalar and the inner conjugation is trivial.

3.) There is a generalized ``gauge principle'' which allows to recover
(up to inner conjugation) the observables $A$ from the knowledge
of the \emor s $\g_t \in End(B)$ together with their multiplicities.
Namely, according to (2.11($b$)) one may construct $\g \in End(B)$ which
has the abstract properties characterizing canonical \emor s. E.g.,
there is a pair of isometric \itw s $V : id \arr \g$ and $W : \g \arr
\g^2$ satisfying the identities \cite{Lo4}
\be \barr{c} V^*W = \lambda^{-1/2} = \g(V^*)W \\ \g(W)W = WW, \qquad
\g(W^*)W = WW^*. \earr \ee
 From this one recovers the \cex\ $\mu(\cdot) := W^*\g(\cdot)W$ and the
observables $A := \mu(B) =$ fixpoints of $\mu$. Conversely, $\mu, V,
\g$ are the data of Prop.\ 1 associated with $\sub AB$, and $W =
\lambda^{1/2}\mu(V)$.

These remarks justify our proposal for the generalized \sy\ underlying
the inclusion $\sub AB$.

\sectreset{The index of coset models}
Let us now turn to the specific coset models of \cf\ \qft. It is known
\cite{Wa} that for \ca s as well as for the algebras of the \emt\ on the
circle, Haag duality holds in the \vrep, i.e.\ the local \vna s
associated with complementary intervals $I$ and $I^c = \s \setminus I$
are each other's commutants:
\be \po(\aa_I) = \po(\aa_{I^c})'. \ee
In other \rep s, Haag duality will be violated in general. Consider
the inclusion (locality!)
\be \pi(\aa_I) \subset \pi(\aa_{I^c})'. \ee
If $\pi = \po \circ\rho$ is given by an \emor\ of $\aa$ which is
localized in $I$, i.e.\ $\rho(a') = a'$ for $a' \in \aa_{I^c}$,
then due to (5.1), (5.2) turns into
\be \rho(\aa_I) \subset \aa_I\, . \ee
The index of the inclusions (5.2), (5.3) is given by the square of the
statistical \dmn\ $d(\pi)^2 \equiv d(\rho)^2$ of the \emor\ $\rho$
\cite{Lo2}, i.e., the notion of statistical \dmn\ for localized
\emor s of the $C^*$ algebra $\aa$ and the notion
                                       of \dmn\ for \emor s of a local
\vna\ $\aa_I$ as introduced in Sect.\ 2 coincide.

Now, denote the pairs of algebras of the coset models (1.8), (1.9)
by $\sub \aa\bb$. Clearly, the inclusion holds also for the local
algebras: $\aa_I \subset \bb_I$. Let $\pi$ be a \per\ of $\bb$ and
$\pi_|$ the restriction of $\pi$ to $\aa$. The following idea is due
to Wassermann \cite{Wa}. Consider the chain of inclusions
\be \pi_|(\aa_I)\;\stackrel\a\subset\; \pi(\bb_I) \;\stackrel\b\subset\;
\pi(\bb_{I^c})' \;\stackrel{\a'}\subset\; \pi_|(\aa_{I^c})' .\ee
The index of $(\a)$ is both independent of the interval $I$ by \cf\
covariance and independent of the \rep\ $\pi$ since all \rep s are
locally equivalent. We may therefore define $Index(\a)$ as the ``index
of the subtheory'' $\sub \aa\bb$.

Since the index of a subfactor equals the index of its commutant,
$(\a)$ and $(\a')$ have the same index. The index of $(\b)$ is given by
the statistical \dmn\ of $\pi$, $d(\pi)^2$, and the index of the total
inclusion $(\a'\circ\b\circ\a)$ is given by the statistical \dmn\ of the
reducible \rep\ $\pi_| \simeq \bigoplus_s N_s \pi_s$ of $\aa$,
$d(\pi_|)^2$. One can solve for the index of $\sub\aa\bb$:
\be Index[\sub\aa\bb] = \frac {d(\pi_|)}{d(\pi)} =
\frac{\sum_s N_s d(\pi_s)}{d(\pi)}. \ee
Clearly, finite index requires finite reducibility of $\pi_|$.

In coset models, the branching rules for the decomposition of a \rep\ of
$\bb$ upon restriction to $\aa$ are well known, and the statistical \dmn
s of the involved \rep s can be computed from the chiral partition
functions $\chi_\pi(\b) = Tr_\pi\exp(-2\pi\b L_0)$ as the ``asymptotic
\dmn s'' $d_{\rm as}(\pi) := \lim \chi_\pi(\b)/\chi_0(\b)$ in the
high-temperature limit $\b \searrow 0$. (Actually, the coincidence of
the asymptotic and statistical \dmn s has been established only in a few
special cases (e.g., see \cite{Wa}), but is widely believed to hold in
general. Note that the independence of (5.5) of the \rep\ $\pi$ is
manifest with the asymptotic dimension.)

The following table lists a few examples with the branching rules for
the \vrep\ (the sectors of \ca s are denoted by the corresponding
isospin, while the sectors of $W = Vir(c)$ are given in the minimal
model nomenclature) and the evaluation of (5.5):
\be \barr{ll} SU(2)_4 \subset SU(3)_1: & [0] \arr [0] + [2] \\ & Index =
1+1 = 2 \\ SU(2)_{10} \subset SO(5)_1: & [0] \arr [0] + [3] \\ & Index =
1 + \sin\frac{7\pi}{12}/\sin\frac{\pi}{12} = 3 + \sqrt 3 \\ SU(2)_2
\otimes Vir(\frac12) \subset SU(2)_1 \otimes SU(2)_1: & [0] \otimes [0]
\arr [0] \otimes [(1,1)] + [1] \otimes [(1,3)] \\ & Index = 1 \cdot 1 +
1 \cdot 1 = 2 \\ SU(2)_3 \otimes Vir(\frac7{10}) \subset SU(2)_2 \otimes
SU(2)_1: & [0] \otimes [0] \arr [0] \otimes [(1,1)] + [1] \otimes
[(1,3)] \\ & \barr{r} \!\! Index = 1 \cdot 1 + 2\cos\frac\pi 5 \cdot
2\cos\frac\pi 5 \\ = 4\cos^2\frac\pi{10}. \earr \earr \ee

As a by-product, we draw an interesting conclusion from the first and
third entries of this list. In both cases, the subtheory $\aa$ is
contained in the subtheory of $\ZZ_2$-invariants of $\bb$, where the
action of $\ZZ_2$ is the involutive Lie group \amor\ of $SU(3)$
associated with the symmetric space $SU(3)/SU(2)$, lifted to the \ca,
and the ``flip'' of the two $SU(2)_1$ tensor factors, respectively. Now,
since the index of $\sub \aa\bb$ is 2, $\aa$ must actually coincide with
the invariant subtheory. Thus, these two examples fall into the class
described by a global gauge group. One would not have expected this
result in terms of the respective Kac-Moody modes $j_n^a$. Another
example of this type, although of infinite index, is the \emt\ theory
$Vir(c=1)$ contained in the $SU(2)_1$ \ca, which in fact coincides with
the subtheory of invariants under the global $SU(2)$ \sy\ \cite{Vir}.

The last entry in the list (5.6) is an inclusion with index of the
``rigid'' Jones form $4\cos^2\frac\pi n$.

\sectreset{Discussion and outlook}
We have seen that some non-trivial sectors of a subtheory $H_l \otimes
W$ and {\it a forteriori} of $H_l$ alone are in fact generated by local
fields of the theory $G_k$. How would one have guessed the extension
$G_k$, if the physical \obs s are chosen to be a given $H_l$ theory? In
particular, what is the role of the coset theory which completely
decouples from the given $H_l$ theory? Let us begin with a simpler
question: Why have models like the last two entries in our list (5.6)
escaped the systematic search in \cite{Bul} for local extensions of
$SU(2)$ \ca s?

The answer is of course that the currents of the extending $G_k$ theory
are not primary with respect to the \emt\ $T^{(H)}$ of the original
theory, but only with respect to the \emt\ $T^{(G)}$ which according to
(1.10) is obtained by including the decoupling coset degrees of freedom.
The proper \spt\ covariance is only implemented by the latter.

For the general problem of the existence of local charged fields
generating sectors of the \obs s $H_l$, however, an extension
\be H_l \subset G_k \ee
rather than (1.9) is perfectly admissible. This incites us to take a
glimpse at infinite index. Namely, the subtheory inclusion (6.1) is
reducible with infinite index whenever $W$ in (1.9) is non-trivial,
i.e.\ $T^{\rm coset} \neq 0$. The branching of the \vrep\ comes
with infinite multiplicities, clearly falsifying Prop.\ 3. One also
readily checks in models that Prop.\ 4 and the phase condition (3.10)
are violated. This is possible since the the argument leading to (3.6)
fails. The infinite-\dmn al multiplicity space $\hh_s$ carries an {\it
infinite}-\dmn al ``internal'' \rep\ of the \mb\ by unitaries in $W$
which modifies (3.6) without affecting the intertwining property of
$\psi$ for the sectors of the observables $H_l$. This fact is the
abstract version of the coset Sugawara formula (1.10). The internal
part of the covariant transformation law for $\psi$ accounts for
$\kappa(\rho_s) \neq 1$ and for the violation of (3.10).

Thus, the reason for the introduction of the coset $W$-algebra is the
demand (principle?) of {\it local} and {\it covariant} charged fields
in the sector-generating algebra $\bb$. The latter then automatically
also contains the decoupled coset algebra.

We suggest that this sort of ``principle'' could be an alternative to
the search for a generalized gauge principle which imposes some linear
transformation law and braided \comrel s on definitely non-local and
therefore {\it a priori} un\obs\ objects. In contrast, our approach
attempts to find {\it local}\ fields which are by definition un\obs\
($\notin \aa$) with the measuring apparatus we have equipped our
observer with, but which admit the option of becoming \obs\ with an
upgraded apparatus (``breaking the \sy'' by switching on ``magnetic
fields'').

Actually, there {\it is} a way to observe the decoupled coset degrees of
freedom. Since gravity couples to the total energy-momentum density, an
observer in the $H_l$ world could detect the excess $T^{\rm coset}$ of
the total \emt\ $T^{(G)}$ beyond the \obs\ \emt\ $T^{(H)}$ through a
gravitational effect (``dark matter''). Clearly, this argument is
highly speculative. It seems not to apply in four \dmn s, where it is
known that all sectors of the algebra of \obs s are generated (actually
implemented) by charged fields (e.g.\ fermions) \cite{DR2} which are
accounted for in the standard \emt. Yet, ignoring this objection, one
could pursue a dark matter scenario like the above. It would entail that
gravitational effects belong to a different concept of observability
than the one given in terms of local quantum measurements. Thus quantum
gravity would have to stand on a very different footing than ordinary
\qft.

We observe that infinite index inclusions like (6.1) can give rise to
``anyonization'' of \bgs. Namely, a given level $k$ theory is always
diagonally embedded into the tensor product of $k$ level 1 theories
(cf.\ (1.8)), and all sectors of the former are obtained by restriction
of the sectors of the latter. But all sectors of level 1 theories for
Lie groups of Dynkin type $A,D,E$ are simple (anyonic) sectors. They
are described by (explicitly known) localized \amor s and can be
implemented by unitary operators which satisfy anyonic \comrel s.
Thus, these are models where one can further extend the local field
algebra $\bb$ into an anyonic field algebra $\ff$ such that all sectors
of $\aa$ arise in the \vrep\ of $\ff$ and the corresponding
intertwining isometries in $\ff$ satisfy anyonic \comrel s.

Finally, we include a remark answering a question raised at this
conference by S.\ Majid.
For the mathematical theory of subfactors, there is a duality between
the inclusions $\sub AB$ and $\g(B) \subset A$ and between the
associated ``paragroups'' \cite{Oc}. In particular, the same structure
theory applies for $\rho = \g\vert_A$ which is the canonical \emor\ for
the latter inclusion and $\g$ which is equivalent to the restriction of
$\rho$ to $\g(B)$. However, when extended to nets of local algebras
$\sub \aa\bb$ this balance will fail: namely in order to guarantee
$\rho$ to be a localized \emor\ of the subtheory we had to require the
compatibility of the \cex\ with the space-time symmetries $\a_x$. Now,
on one hand this implies the existence of \st, and therefore the
semigroup generated by the subsectors of $\rho$ is abelian. On the other
hand, the above compatibility is not preserved by duality: descending
the Jones tower, the \cex\ $\nu: A \arr \g(B)$ is given by $\nu(a) =
\g(V^*aV) = \lambda\mu(VV^*aVV^*)$. So the semigroup generated by $\g_t$
need not be commutative, and it definitely isn't in the case of a
non-abelian symmetry group. If one wants to maintain the mathematical
balance between the ``\sy'' paragroup (generated by $\g_t$) and the
``\sps'' paragroup (generated by $\rho_s$) in physics, either one has to
abandon localization, or the \sy\ has to be commutative (in the weak
sense of composition of equivalence classes). Is this a hint at some
unknown ``principle'' which could explain the phenomenological fact that
all exact \sps\ rules observed in Nature are due to abelian symmetries?
\\[9mm]
{\Large \bf Appendix}\\[5mm]
We sketch how Prop.\ $2'$ follows from Prop.\ 2 by entirely group
theoretical arguments.

First, the decomposition of $\g$ is just (2.9). In order to determine
the decomposition of $\rho$, we must compute the commutant $\rho\comm
A$. The map $g \mapsto |G| \cdot \mu(V\a_g(V^*))$ maps $G$ into the
commutant $\rho\comm A$:
\[ \barr{r} \mu(V\a_g(V^*)) \cdot \rho(a) = \mu(V\a_g(V^*)\rho(a)) =
 \mu(V\a_g(V^*\rho(a))) = \mu(V\a_g(aV^*)) = \quad
 \\ = \mu(Va\a_g(V^*)) = \mu(\rho(a)V\a_g(V^*)) = \rho(a)
 \cdot \mu(V\a_g(V^*)).\earr \]
By invariance of $\mu$: $\mu \circ \a_g = \mu$, and (2.2($i$)), this map
extends linearly to a $*$-\hmor\ from the group algebra $\CC G$ into
$\rho\comm A$, which is actually an \imor. Now, $\CC G$ is a direct sum
of matrix rings corresponding to the \rep s $\tau^r$ of $G$ of \dmn\
$dim(r)$. There are therefore matrix units $X^r_{ik} \in \CC G$ which
reduce the left regular \rep\ of $G$:
\[ g \; X^r_{ik} = \sum_j \ovl{\tau^r_{ji}(g)} \cdot X^r_{jk}\, . \]
Consequently, the operators
\[ a^r_{ik} := |G| \cdot \mu(V\a_{X^r_{ik}}(V^*)) \]
are algebraically matrix units in $\rho\comm A$ and can be written
in the form
\[ a^r_{ik} = W_r^i(W_r^k)^*\]
with \on\ isometries $W_r^i \in A$ yielding $\rho_r$ and the
multiplicities as in (2.13($a$)). Next, (2.2($i$)) yields
\[ \phi_r^{i*} := V^*W_r^i = V^*a^r_{ik}W_r^k =
  \a_{X^r_{ik}}(V^*) W_r^k\]
for every $k$, implying the linear transformation law (2.15) for
$\psi \propto \phi$. By (2.15), the numerical matrix
$(\phi_r^{i*}\phi_r^j)_{ij}$ commutes with the \irrep\
$\tau^r(g)$ and therefore is a multiple of $\d_{ij}$. Also by (2.15),
the average on the left-hand-side of
\[ |G|\mu(\phi_r^i\phi_r^{j*}) = (W_r^i)^* |G|\mu(VV^*) W_r^j =
(W_r^i)^* W_r^j = \delta_{ij} \]
can be computed:
\[ \sum_{g \in G} \a_g(\phi_r^i\phi_r^{j*}) = \delta_{ij}
\frac{|G|}{dim(r)} \sum_{k=1}^{dim(r)} \phi_r^k\phi_r^{k*}.\]
This establishes the completeness relations (together with the missing
normalization of $\psi \propto \phi$), such that finally (2.12) entails
(2.16).

\small \addtolength{\baselineskip}{-1pt}

redpajama_set_name: RedPajamaArXiv