---
author:
- 'Ben McCanna, Hannah M. Price'
title: 'Supplemental material for “Superfluid Vortices in Four Spatial Dimensions"'
---

Radial calculations
===================

Firstly, in the main text, we presented numerical calculations performed by imaginary time evolution of the radial equation \[Eq (11)\]. We used a forward Euler method for time-discretization and second order finite differences in space. In these calculations, we separated out the two polar angles $\theta_j$ and the phase $S$, leaving us with an effectively two-dimensional calculation involving the real function $f(r_1,r_2)$, in terms of the two radial coordinates. These calculations were performed over a system of size $R=100\xi$ with hard-wall boundary conditions at $r_j=0$ and $r_j = R$. The former condition is required due to the centrifugal term diverging at the vortex cores; consequently the precise location of the vortex cores was an assumption in these calculations. The hard walls were placed at a large radius (much greater than several healing lengths) so that we always examined vortex cores within a homogeneous region. (Future studies could include the effect of additional trapping potentials, such as harmonic traps along some or all directions.) We were able to achieve a resolution of $0.05\xi$, having removed two coordinates from the problem. The calculations were converged to an accuracy threshold of $10^{-14}$.

Cartesian 4D calculations {#sec:4DCart}
=========================

Secondly, in the main text, we discussed results for imaginary time evolution on the full 4D GPE under simple and double rotations, using again the forward Euler method for time-discretization and second order finite differences in space. These calculations made fewer assumptions than the radial simulations described above, but at the cost of smaller system sizes and lower resolutions as these were now fully four-dimensional calculations.

In these full 4D calculations, we discretized space using a Cartesian grid within a 4D hyper-sphere of radius $R=8.25\xi$ with hard-wall boundary conditions at the edge. (A hyper-sphere rather than a hyper-cube was chosen as the majority of the 4D volume of a hyper-cube is taken up by regions “in the corners”, that is, outside of the hyper-sphere that just fits inside.) We were able to obtain resolutions of up to $0.2\xi$, and by repeating simulations at different resolutions, we checked that our main conclusions were qualitatively insensitive to the coarse-graining of the numerics. At the system sizes and resolutions we have been able to reach, the homogeneous region extends over a few healing lengths. The calculations were converged to an accuracy threshold of $10^{-12}$.

A benefit of performing the full 4D calculations was that we were able to test our ansatz by allowing the phase to evolve, and by removing the boundary condition at $r_j=0$ mentioned above. More precisely, we used an initial state with homogeneous density away from the edge of the ball, and a phase profile given by $\operatorname{arctan2}(y,x)+\operatorname{arctan2}(w,z)$, for the doubly rotating case, and $\operatorname{arctan2}(y,x)$ for the singly rotating case. We tested the robustness of our results to noise (up to 20% of the background value) added to the real and imaginary parts of the initial $\psi$. Note that we measure the applied frequency in units of the critical frequency of a single vortex in a homogeneous 2D disk of the same radius as our 4D ball; this is given (in our units) by [@pethick2002] $$\Omega_{\text{crit}}^{2D} = \mu \log(2.07 R/\xi) \left( \frac{R}{\xi}\right)^2.$$ For the results shown in the main text, both the frequencies of rotation used were roughly $2.5\Omega_{\text{crit}}^{2D}$. Further work could investigate the effect of double rotation with unequal frequencies.

Isoclinic Rotations
===================

As discussed in the main text, generic rotations in 4D are double rotations, which have one fixed point and two planes of rotation with two corresponding rotation angles $\alpha, \beta \in [-\pi,\pi]$. Note that a double rotation displaces every vector in $\mathbb{R}^4$ through an angle in the interval $[\alpha,\beta],$ with the endpoints given by the two planes of rotation [@lounesto2001]. Simple rotations are a special case corresponding to either $\alpha=0$ or $\beta=0$. Another important special case is when $\alpha = \pm\beta$; these are called isoclinic rotations [@lounesto2001], and are the case that we study in this paper. Isoclinic rotations still have one fixed point, but no longer have two unique planes of rotation. Since any vector is displaced under a double rotation by an angle between $\alpha$ and $\beta$, isoclinic rotations displace every vector by the same angle. This means that for any given isoclinic rotation there is a continuum of pairs of planes that can each be though of as the two completely orthogonal rotation planes. Numerically, in this paper, we break this degeneracy since the phase winding of our initial state picks out the $xy$ and $zw$ planes in particular.

Cuts of the radial profile
==========================

As discussed in the main text and shown in Fig. 2(b), we have numerically verified for the solution of the radial equation \[Eq (11) in the main text\] that far from the intersection point of the vortex planes the corresponding denstity profile is well approximated by a product state of the 2D vortex profiles. To visualise this in an alternative way, we have plotted in Fig S2(a) cuts of Fig 2(a) for specific values of $r_2$, and then rescaled these by $f_1(r_2)$ in Fig S2(b). As shown the rescaled curves approach $f_1(r_1)$ for large values of $r_2$, verifying the approximation as expected.

(picA) at (-0.25,0) [![ (a) Cuts of Fig 2 (a) in the main text, given by fixed values of $r_2$. (b) As in (a) but rescaled by the 2D vortex density profile $f_1(r_2)$; note the convergence to $f_1(r_1)$ for large values of $r_2$, showing that the order parameter can be well approximated as a product of 2D vortex profiles away from the intersection point. Close to the intersection point, this approximation breaks down, as can be seen from the deviation between these rescaled cuts in this region.[]{data-label="fig:RadialCuts"}](4DCuts1.pdf "fig:"){width="\picwidth"}]{}; (a) at ($(picA)+(-0.5\picwidth,0.25\picwidth)$) [(a)]{}; (picB) at (0.25,0) [![ (a) Cuts of Fig 2 (a) in the main text, given by fixed values of $r_2$. (b) As in (a) but rescaled by the 2D vortex density profile $f_1(r_2)$; note the convergence to $f_1(r_1)$ for large values of $r_2$, showing that the order parameter can be well approximated as a product of 2D vortex profiles away from the intersection point. Close to the intersection point, this approximation breaks down, as can be seen from the deviation between these rescaled cuts in this region.[]{data-label="fig:RadialCuts"}](4DCuts2.pdf "fig:"){width="\picwidth"}]{}; (b) at ($(picB)+(-0.5\picwidth,0.25\picwidth)$) [(b)]{};

Energy calculation for two intersecting vortex planes in a 4D superfluid
========================================================================

Here, we numerically verify Eq (12) in the main text, which predicts that the energy cost of two intersecting and completely orthogonal vortex planes in a 4D superfluid can be decomposed as a sum of the individual kinetic energies associated with each vortex plane in isolation.

Firstly, we used the numerical solution of the 4D radial density profile presented in Fig 2 in the main text to calculate the energy of the intersecting vortex planes as a function of system size in each plane. We then produced a fit of this energy to the functional form of Eq (12) in the main text, with the coefficient of $R_j/\xi$ inside the logarithm as the fitting parameter. From this we obtained $2.06$ which is very close to the known coefficient of $2.07$ (in our units) within the logarithmic form of the vortex energy in $2D$ and $3D$ [@pethick2002]. This shows that the energy of our numerical solution for the radial equation is consistent with being a sum of two individual vortex energies.

Secondly, we performed further simulations on a Cartesian 4D grid, with the same parameters as Fig 3 in the main text, except for the convergence accuracy which was chosen to be $10^{-10}$ to speed up calculations. We repeated these calculations for different values of $\Omega\equiv\Omega_{1}=\Omega_{2}$, ranging between two and three times $\Omega_{\text{crit}}^{2D}$, in order to numerically verify the expected dependence of the energy on the rotation frequency. Here we used three different initial states: one with no phase winding, one with “simple” winding in one plane, and one with “double” winding in two planes. The resulting values for $E$ and $\mu$ as a function of $\Omega$ are shown in Fig \[fig:energies\], given in units of $\mu_0$ (the chemical potential of a homogeneous state with no vortices or hard-walls but the same number of particles). We obtain straight lines for each of these data series, showing that each state has well defined angular momentum.

For the case with no phase winding, we find that $E/\mu_0 N = 0.931$ and $\mu /\mu_0= 1.622$ are constants which do not depend on frequency, as expected; this data series is therefore plotted with a straight line joining the dots as a guide to the eye. For the double winding case, we have performed a linear fit, obtaining $E/\mu_0 N = 1.119 - 0.083\Omega/\Omega_{\text{crit}}^{2D}$ and $\mu/\mu_0 = 1.822 - 0.083\Omega/\Omega_{\text{crit}}^{2D}$. The gradient, $-0.083$, is equal to $-2\Omega_{\text{crit}}^{2D}/\mu_0$, meaning that this is the expected gradient of $-2$ corresponding to particles having one unit of angular momentum in each plane of rotation. For the simple winding case, we fix the gradient to be half that of the double winding line, since this state has angular momentum in only one of the two planes, and perform a linear fit with only the y intercept as a free parameter. We then obtain $E/\mu_0 N = 1.023$ and $\mu/\mu_0=1.722$ when $\Omega=0$. This gives an energy cost of $0.188=1.119-0.931$ for the intersecting vortex planes and $0.092=1.023-0.931$ for the single plane, as compared to the state with no vortices. We expect from Eq (12) of the main text that these energy costs should be related by a simple factor of two for this geometry, and indeed we find numerically that $0.188 - 2\times0.092 \simeq 0$.

(picA) at (-0.25,0) [![(a) Energy and (b) Chemical potential of numerical steady states of the 4D doubly rotating GPE \[Eq (9) in the main text\] with different initial phase profiles. The lines correspond to fits and guides to the eye, respectively, as detailed in the text. The gradient and intercept of these lines give the angular momentum and energy at zero frequency, respectively, of each state, which agree with expected behaviour.[]{data-label="fig:energies"}](energies.pdf "fig:"){width="\picwidth"}]{}; (a) at ($(picA)+(-0.5\picwidth,0.25\picwidth)$) [(a)]{}; (picB) at (0.25,0) [![(a) Energy and (b) Chemical potential of numerical steady states of the 4D doubly rotating GPE \[Eq (9) in the main text\] with different initial phase profiles. The lines correspond to fits and guides to the eye, respectively, as detailed in the text. The gradient and intercept of these lines give the angular momentum and energy at zero frequency, respectively, of each state, which agree with expected behaviour.[]{data-label="fig:energies"}](mus.pdf "fig:"){width="\picwidth"}]{}; (b) at ($(picB)+(-0.5\picwidth,0.25\picwidth)$) [(b)]{};

Simple Rotations
================

As described in the main text, we expect that a simple rotation should be able to stablise a single vortex plane, extending the concept of 2D point vortices and 3D line vortices straightfowardly to four-dimensional systems. Assuming the rotation is in plane 1 \[as defined in the main text\], this would correspond to a condensate wavefunction of the form: $$\begin{aligned}
\psi=f(r_1, r_2) e^{i k_1 \theta_1} \label{eq:single}\end{aligned}$$ with $f(0,r_2)=0$, and such that this wave-function approximately takes the form $\psi\propto (x+iy)$ near the vortex core.

We have verified this minimal vortex structure numerically by performing imaginary time evolution on the full 4D GPE under simple rotation in the plane orthogonal to the expected vortex core (i.e. Eq (9) in the main text with $\Omega_1\neq0, \Omega_2=0$). The corresponding density and phase profiles for the numerical stationary state are shown for selected 2D cuts in Fig. \[fig:Simple2DCuts\]. Here, the initial state was chosen as detailed in Section II and the rotation frequency was chosen as $2\Omega_{\text{crit}}^{2D}$. These numerical calculations were performed within a discretized 4D hyper-sphere of radius $8.25 \xi$, and with resolution $0.5\xi$.

As can be seen in Fig. \[fig:Simple2DCuts\], the observed density and phase profiles are in good agreement with the single vortex plane \[Eq \[eq:single\]\]. In particular, the density is depleted for the plane defined by $z=0$ and $w=0$, as is expected for a single vortex plane that approximately takes the form $\psi\propto (x+iy)$ near the vortex core. Depending on the 2D cut, this vortex plane either appears as a point \[see (e) and (g)\], as a line \[see (a) and (c)\] or as a plane \[not shown\]. Furthermore, around the vortex plane, the superfluid rotates, as can be seen from the winding of the phase in panels (f) and (h) and from the phase jumps in (b) and (d).

; (letter) at (-1.5\*,.75\*); at (letter) [(a)]{}; at (0,0) [![Density (a,c,e,g), and phase (b,d,f,h) profiles for 2D cuts of the numerical stationary state under simple rotation. These cuts are given by (a,b) $x=y$ and $z=w$, (c,d) $x=y$ and $z=-w$, (e,f) $x=z$ and $y=w$, (g,h) $x=z$ and $y=-w$. Numerical calculations were performed for a superfluid confined within a discretized 4D hyper-sphere of radius $8.25 \xi$ and resolution $0.5\xi$; this discretization is reflected in the pixellation, particularly at the boundaries of the plots. The observed density and phase profiles are in good agreement with a single vortex plane \[Eq \[eq:single\]\]. []{data-label="fig:Simple2DCuts"}](simplexyzw.pdf "fig:"){width="\picwidth"}]{};

at ($ (letter) + (2*\dx,0) $) [(b)]{};

at ($ (letter) + (4*\dx,0) $) [(c)]{}; at (4\*,0) [![Density (a,c,e,g), and phase (b,d,f,h) profiles for 2D cuts of the numerical stationary state under simple rotation. These cuts are given by (a,b) $x=y$ and $z=w$, (c,d) $x=y$ and $z=-w$, (e,f) $x=z$ and $y=w$, (g,h) $x=z$ and $y=-w$. Numerical calculations were performed for a superfluid confined within a discretized 4D hyper-sphere of radius $8.25 \xi$ and resolution $0.5\xi$; this discretization is reflected in the pixellation, particularly at the boundaries of the plots. The observed density and phase profiles are in good agreement with a single vortex plane \[Eq \[eq:single\]\]. []{data-label="fig:Simple2DCuts"}](simplexyz-w.pdf "fig:"){width="\picwidth"}]{};

at ($ (letter) + (6*\dx,0) $) [(d)]{};

at ($ (letter) + (0,-2*\dy) $) [(e)]{}; at (0,-2\*) [![Density (a,c,e,g), and phase (b,d,f,h) profiles for 2D cuts of the numerical stationary state under simple rotation. These cuts are given by (a,b) $x=y$ and $z=w$, (c,d) $x=y$ and $z=-w$, (e,f) $x=z$ and $y=w$, (g,h) $x=z$ and $y=-w$. Numerical calculations were performed for a superfluid confined within a discretized 4D hyper-sphere of radius $8.25 \xi$ and resolution $0.5\xi$; this discretization is reflected in the pixellation, particularly at the boundaries of the plots. The observed density and phase profiles are in good agreement with a single vortex plane \[Eq \[eq:single\]\]. []{data-label="fig:Simple2DCuts"}](simplexzyw.pdf "fig:"){width="\picwidth"}]{};

at ($ (letter) + (2*\dx,-2*\dy) $) [(f)]{};

at ($ (letter) + (4*\dx,-2*\dy) $) [(g)]{}; at (4\*,-2\*) [![Density (a,c,e,g), and phase (b,d,f,h) profiles for 2D cuts of the numerical stationary state under simple rotation. These cuts are given by (a,b) $x=y$ and $z=w$, (c,d) $x=y$ and $z=-w$, (e,f) $x=z$ and $y=w$, (g,h) $x=z$ and $y=-w$. Numerical calculations were performed for a superfluid confined within a discretized 4D hyper-sphere of radius $8.25 \xi$ and resolution $0.5\xi$; this discretization is reflected in the pixellation, particularly at the boundaries of the plots. The observed density and phase profiles are in good agreement with a single vortex plane \[Eq \[eq:single\]\]. []{data-label="fig:Simple2DCuts"}](simplexzy-w.pdf "fig:"){width="\picwidth"}]{};

at ($ (letter) + (6*\dx,-2*\dy) $) [(h)]{};

Double Rotations
================

As shown in the main text, the double rotation of a 4D superfluid can stabilise a new type of vortex configuration consisting of two vortex planes intersecting at a point. In Fig. \[fig:2DCuts\], we plot the density and phase profiles for additional 2D cuts of the numerical stationary state shown in Fig 3 in the main text. As can be seen, these profiles have a much richer structure as compared to the case of a single vortex plane shown in Fig. \[fig:Simple2DCuts\], as the phase winds simultaneously around both vortex cores with two independent winding numbers. This is also in contrast to 3D systems where two vortex lines may intersect and reconnect over time, but a pair of intersecting vortices is not stabilised by rotation as a stationary state of the system.

; (letter) at (-1.5\*,.75\*); at (letter) [(a)]{}; at (0,0) [![Additional density (a,c,e,g), and phase (b,d,f,h) profiles along 2D cuts of the numerical stationary state studied in Fig 3 in the main text. These cuts are given by (a,b) $x=y$ and $z=w$, (c,d) $x=y$ and $z=-w$, (e,f) $x=z$ and $y=w$, (g,h) $x=z$ and $y=-w$. The parameters and discretization are detailed in section \[sec:4DCart\]. This discretization is reflected in the pixellation, particularly at the boundaries of the plots. The observed density and phase profiles are in good agreement with our numerical ansatz \[Eq (10) in the main text\], which approximately takes the form $\psi\propto (x+iy)(z+iw)$ near the vortex cores. []{data-label="fig:2DCuts"}](xyzw.pdf "fig:"){width="\picwidth"}]{};

at ($ (letter) + (2*\dx,0) $) [(b)]{};

at ($ (letter) + (4*\dx,0) $) [(c)]{}; at (4\*,0) [![Additional density (a,c,e,g), and phase (b,d,f,h) profiles along 2D cuts of the numerical stationary state studied in Fig 3 in the main text. These cuts are given by (a,b) $x=y$ and $z=w$, (c,d) $x=y$ and $z=-w$, (e,f) $x=z$ and $y=w$, (g,h) $x=z$ and $y=-w$. The parameters and discretization are detailed in section \[sec:4DCart\]. This discretization is reflected in the pixellation, particularly at the boundaries of the plots. The observed density and phase profiles are in good agreement with our numerical ansatz \[Eq (10) in the main text\], which approximately takes the form $\psi\propto (x+iy)(z+iw)$ near the vortex cores. []{data-label="fig:2DCuts"}](xyz-w.pdf "fig:"){width="\picwidth"}]{};

at ($ (letter) + (6*\dx,0) $) [(d)]{};

at ($ (letter) + (0,-2*\dy) $) [(e)]{}; at (0,-2\*) [![Additional density (a,c,e,g), and phase (b,d,f,h) profiles along 2D cuts of the numerical stationary state studied in Fig 3 in the main text. These cuts are given by (a,b) $x=y$ and $z=w$, (c,d) $x=y$ and $z=-w$, (e,f) $x=z$ and $y=w$, (g,h) $x=z$ and $y=-w$. The parameters and discretization are detailed in section \[sec:4DCart\]. This discretization is reflected in the pixellation, particularly at the boundaries of the plots. The observed density and phase profiles are in good agreement with our numerical ansatz \[Eq (10) in the main text\], which approximately takes the form $\psi\propto (x+iy)(z+iw)$ near the vortex cores. []{data-label="fig:2DCuts"}](xzyw.pdf "fig:"){width="\picwidth"}]{};

at ($ (letter) + (2*\dx,-2*\dy) $) [(f)]{};

at ($ (letter) + (4*\dx,-2*\dy) $) [(g)]{}; at (4\*,-2\*) [![Additional density (a,c,e,g), and phase (b,d,f,h) profiles along 2D cuts of the numerical stationary state studied in Fig 3 in the main text. These cuts are given by (a,b) $x=y$ and $z=w$, (c,d) $x=y$ and $z=-w$, (e,f) $x=z$ and $y=w$, (g,h) $x=z$ and $y=-w$. The parameters and discretization are detailed in section \[sec:4DCart\]. This discretization is reflected in the pixellation, particularly at the boundaries of the plots. The observed density and phase profiles are in good agreement with our numerical ansatz \[Eq (10) in the main text\], which approximately takes the form $\psi\propto (x+iy)(z+iw)$ near the vortex cores. []{data-label="fig:2DCuts"}](xzy-w.pdf "fig:"){width="\picwidth"}]{};

at ($ (letter) + (6*\dx,-2*\dy) $) [(h)]{};

Interaction of tilted vortex planes
===================================

The phase profile for the orthogonal pair of vortex planes considered in the main text is given by $$S = \arctan\left( \frac{y}{x} \right) + \arctan\left( \frac{w}{z} \right),$$ where the first (second) term will be denoted $S_{1(2)}$ respectively. For this profile the energy density of interaction between the vortices, given by $\nabla S_1 \cdot \nabla S_2$, vanishes identically due to orthogonality, as discussed in the main text. Here we consider a generalisation of the above where the two vortex planes are tilted away from orthogonality, showing that this hydrodynamic interaction energy vanishes for vortex planes intersecting at the centre of a symmetric superfluid. To describe how the planes tilt towards each other we use the following double rotation $$R(\phi_1,\phi_2) = 
        \begin{pmatrix}
            \cos{\phi_1} & 0 & -\sin{\phi_1} & 0 \\
            0 & \cos{\phi_2} & 0 & -\sin{\phi_2} \\
            \sin{\phi_1} & 0 & \cos{\phi_1} & 0 \\
            0 & \sin{\phi_2} & 0 & \cos{\phi_2}
        \end{pmatrix}.
        \label{eq:tiltR}$$ To obtain this form we have used that we are free to choose bases within the $xy$ and $zw$ planes, and that we only care about how the rotation $R$ couples these two planes. We can also without loss of generality consider only angles $\phi_j$ within $(-\pi/4,\pi/4]$. Our phase profile for the tilted planes is given by $$S = \arctan\left( \frac{y_-}{x_-} \right) + \arctan\left( \frac{w_+}{z_+} \right),$$ where we denote rotated coordinates by $\mathbf{r}_\pm = R(\pm\phi_1/2,\pm\phi_2/2) \mathbf{r}$. One plane is rotated by $R(\phi_1/2,\phi_2/2)$ and the other by its inverse so that the total rotation between the two is given by the matrix in Eq . Intuitively, the angles $\phi_j$ describe how far away from orthogonality the two planes are tilted, and in what direction. The velocity field is given by $\nabla S = \nabla S_1 + \nabla S_2$. We compute the gradient of $S_1$ in the $r_-$ frame and of $S_2$ in the $r_+$ frame, resulting in $$\nabla S_1 = \frac{x_-\hat{y}_- - y_-\hat{x}_-}{x_-^2 + y_-^2}, \ \text{and} \ \nabla S_2 = \frac{z_+\hat{w}_+ - w_+\hat{z}_+}{z_+^2 + w_+^2}.$$ Using that $\mathbf{r}_+ = R(\phi_1,\phi_2)\mathbf{r}_-$ we find that the overlap between velocity fields is given by $$\nabla S_1 \cdot \nabla S_2 = \frac{x_-z_+ \sin{\phi_2} + y_-w_+ \sin{\phi_1}}{(x_-^2 + y_-^2)(z_+^2 + w_+^2)},$$ Now we must compute $\frac{1}{2}\int_{D} \rho \nabla S_1 \cdot \nabla S_2 {\mathrm{d}V}$, where $D$ is the domain of the condensate minus the vortex cores. We will make the approximation $\rho = 1$ outside the cores. Therefore, we have $$E_{int} = \frac{1}{2} \int_D \frac{\sin{\phi_1}y_-w_+ + \sin{\phi_2}x_-z_+}{(x_-^2 + y_-^2)(z_+^2 + w_+^2)} {\mathrm{d}x}{\mathrm{d}y}{\mathrm{d}z}{\mathrm{d}w}.$$ Since the integrand is in terms of $x_-,y_-,z_+,$ and $w_+$, we make the transformation to these non-orthogonal variables. This has a Jacobian determinant of $1/\cos{\phi_1}\cos{\phi_2}$, so that $$E_{int} = \frac{1}{2} \int_{D^{'}} \left[
        \frac{\tan{\phi_1}}{\cos{\phi_2}} \frac{y_-w_+}{(x_-^2 + y_-^2)(z_+^2 + w_+^2)}
        + \frac{\tan{\phi_2}}{\cos{\phi_1}} \frac{x_-z_+}{(x_-^2 + y_-^2)(z_+^2 + w_+^2)}
        \right] {\mathrm{d}x_-}{\mathrm{d}y_-}{\mathrm{d}z_+}{\mathrm{d}w_+}.$$ The first term is an odd function of $y_-$ and $w_+$, while the second is odd in both $x_-$ and $z_+$. This means the total interaction between the planes will cancel unless the superfluid is asymmetric about the intersection point in both of the tilted planes simultaneously. Additionally, for small angles the interaction is $O(\theta_{1,2})$. Note that this analysis says nothing about vortex surfaces with curvature.

[2]{} ifxundefined \[1\][ ifx[\#1]{} ]{} ifnum \[1\][ \#1firstoftwo secondoftwo ]{} ifx \[1\][ \#1firstoftwo secondoftwo ]{} ““\#1”” @noop \[0\][secondoftwo]{} sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{} @startlink\[1\] @endlink\[0\] @bib@innerbibempty @noop [**]{} (, ) [**](https://doi.org/10.1017/CBO9780511526022),  ed., London Mathematical Society Lecture Note Series (, )
