Toric Quasifolds

Let us write the $2$ and $3$–dimensional unit spheres as follows

$$S^{2}=\{\,(z,x)\in\mbox{\bbb{C}}\times\mbox{\bbb{R}}\,|\,|z|^{2}+x^{2}=1\,\},$$

$$S^{3}=\{\,(z,w)\in\mbox{\bbb{C}}^{2}\,|\,|z|^{2}+|w|^{2}=1\,\}.$$

The surjective mapping

	$\displaystyle f\,\colon$	$\displaystyle S^{3}$	$\displaystyle\longrightarrow S^{2}$
		$\displaystyle\left(z,w\right)$	$\displaystyle\longmapsto\left(2z\overline{w},|z|^{2}-|w|^{2}\right)$

is known as the Hopf fibration. It is easily seen that the fibers of this mapping are given by the orbits of the circle group

$$S^{1}=\{\,e^{2\pi i\theta}\,|\,\theta\in\mbox{\bbb{R}}\,\}$$

acting on $S^{3}$ by complex multiplication as follows:

$$e^{2\pi i\theta}\cdot(z,w)=\left(e^{2\pi i\theta}z,e^{2\pi i\theta}w\right).$$

Therefore $S^{2}$ can be identified with the space of orbits $S^{3}/S^{1}$. Notice that the $S^{1}$–orbits through the points $(0,1)$ and $(1,0)$ of $S^{3}$ correspond, respectively, to the south pole, $S=(0,-1)$, and north pole, $N=(0,1)$, of $S^{2}$.

For each $(z,w)\in S^{3}$, we denote by $[z:w]\in S^{3}/S^{1}\simeq S^{2}$ the corresponding orbit. Let us describe the standard atlas of $S^{2}$. Consider the covering given by the open subsets

$$U_{S}=\{\,[z:w]\in S^{2}\;|\;w\neq 0\,\}$$

$$U_{N}=\{\,[z:w]\in S^{2}\;|\;z\neq 0\,\}.$$

As the notation suggests, the first is a neighborhood of the south pole $S=[0:1]$, while the second is a neighborhood of the north pole $N=[1:0]$. Finally, we have homeomorphisms:

$$\begin{array}[]{ccc}B(1)&\longrightarrow&U_{S}\\ z&\longmapsto&\left[z:\sqrt{1-|z|^{2}}\right]\end{array}$$

$$\begin{array}[]{ccc}B(1)&\longrightarrow&U_{N}\\ w&\longmapsto&\left[\sqrt{1-|w|^{2}}:w\right].\end{array}$$

This simple quotient construction can be extended to the orbifold setting as follows. Let $p,q$ be two relatively prime positive integers and consider the $3$–dimensional ellipsoid

$$S^{3}_{p,q}=\{\,(z,w)\in\mbox{\bbb{C}}^{2}\,|\,p|z|^{2}+q|w|^{2}=pq\,\}.$$

The circle group $S^{1}$ acts on $S^{3}_{p,q}$ as follows:

$$e^{2\pi i\theta}\cdot(z,w)=\left(e^{2\pi ip\theta}z,e^{2\pi iq\theta}w\right).$$

(3)

Taking the space of orbits in this case yields the $2$–dimensional orbifold $S^{2}_{p,q}=S^{3}_{p,q}/S^{1}$, called orbisphere. It admits the two singular points $S=[0:\sqrt{p}]$ and $N=[\sqrt{q}:0]$.

Similarly to what we have done for the sphere, for each $(z,w)\in S^{3}_{p,q}$, we denote by $[z:w]\in S^{2}_{p,q}$ the corresponding orbit. We then consider the covering given by the two open subsets

$$U_{S}=\{\,[z:w]\in S^{2}_{p,q}\;|\;w\neq 0\,\}$$

$$U_{N}=\{\,[z:w]\in S^{2}_{p,q}\;|\;z\neq 0\,\}.$$

The first is a neighborhood of the point $S=[0:\sqrt{p}]$, while the second is a neighborhood of the point $N=[\sqrt{q}:0]$. The mappings

$$\begin{array}[]{ccc}B(q)/\Gamma_{\frac{1}{q}}&\longrightarrow&U_{S}\\ \left[z\right]&\longmapsto&\left[z:\sqrt{p-\frac{p}{q}|z|^{2}}\right];\end{array}$$

$$\begin{array}[]{ccc}B(p)/\Gamma_{\frac{1}{p}}&\longrightarrow&U_{N}\\ \left[w\right]&\longmapsto&\left[\sqrt{q-\frac{q}{p}|w|^{2}}:w\right]\end{array}$$

are homeomorphisms, turning $U_{S}$ and $U_{N}$ into orbifold charts.

We now extend the construction even further. Let $s,t$ be two positive real numbers with $s/t\notin\mbox{\bbb{Q}}$ and consider the $3$–dimensional ellipsoid

$$S^{3}_{s,t}=\{\,(z,w)\in\mbox{\bbb{C}}^{2}\,|\,s|z|^{2}+t|w|^{2}=st\,\}.$$

Simply substituting $p,q$ with $s,t$ in (3), does not define an $S^{1}$–action on $S^{3}_{s,t}$: in fact, if you replace $\theta$ by $\theta+h$, where $h$ is a non–zero integer, we have $e^{2\pi i(\theta+h)}=e^{2\pi i\theta}$ but $(e^{2\pi is(\theta+h)},e^{2\pi it(\theta+h)})\neq(e^{2\pi is\theta},e^{2\pi it\theta})$. The idea is to consider the irrational wrap on the standard two–torus instead:

$$N=\{\,(e^{2\pi is\theta},e^{2\pi it\theta})\in\mbox{\bbb{R}}^{2}/\mbox{\bbb{Z}}^{2}\,|\,\theta\in\mbox{\bbb{R}}\,\}.$$

The standard action of $N$ on $S^{3}_{s,t}$ is now well defined and we take our quasisphere to be the space of orbits $S^{2}_{s,t}=S^{3}_{s,t}/N$. This quotient is the simplest example of quasifold. It is wilder then the sphere and orbisphere, in that it is not a Hausdorff topological space. However, quasisphere charts are a straightforward and very natural generalization of the standard sphere and orbisphere charts.

Exactly as done above, for each $(z,w)\in S^{3}_{s,t}$, we denote by $[z:w]$ the corresponding orbit. We then consider the covering of $S^{2}_{s,t}$ given by the opens subsets

$$U_{S}=\{\,[z:w]\in S^{3}_{s,t}/\mbox{\bbb{R}}\;|\;w\neq 0\,\}$$

$$U_{N}=\{\,[z:w]\in S^{3}_{s,t}/\mbox{\bbb{R}}\;|\;z\neq 0\,\}.$$

The first is a neighborhood of the point $S=[0:\sqrt{s}]$, while the second is a neighborhood of the point $N=[\sqrt{t}:0]$. They are each homeomorphic to the quotient of an open subset of ℂ modulo the action of a countable group. In fact, the mappings

$$\begin{array}[]{ccc}B(t)/\Gamma_{\frac{s}{t}}&\longrightarrow&U_{S}\\ \left[z\right]&\longmapsto&\left[z:\sqrt{s-\frac{s}{t}|z|^{2}}\right]\end{array}$$

$$\begin{array}[]{ccc}B(s)/\Gamma_{\frac{t}{s}}&\longrightarrow&U_{N}\\ \left[w\right]&\longmapsto&\left[\sqrt{t-\frac{t}{s}|w|^{2}}:w\right]\end{array}$$

are homeomorphims.

The fact that quasilattices appear naturally in nonperiodic tilings lead us to explore, jointly with Battaglia, the connection between toric quasifolds and Penrose and Amman tilings.

Penrose rhombus tilings are nonperiodic tilings that are composed by two different types of rhombuses, thick and thin [24]. These rhombuses are simple convex polytopes and it is natural to want to compute the corresponding toric quasifolds. The normals of each rhombus taken separately actually span a lattice, so each of them is rational in its own right. However, if we want to treat simultaneously all of the rhombuses of a given tiling, we need to consider a quasilattice: the natural choice here is the pentagonal quasilattice that we introduced earlier, with normals the relevant fifth roots of unity (see Figure 2).

The generalized toric construction then yields a pair of four–dimensional toric quasifolds, one for each different type of rhombus. They are both given by a quotient of the type $(S^{2}(r)\times S^{2}(r))/\Gamma$, where $S^{2}(r)$ denotes the $2$–sphere of radius $r$ and $\Gamma$ is a countable subgroup of the standard 2–torus. The radius $r$ is $(\frac{1}{2}\sqrt{2+\phi})^{1/2}$ for the thick rhombus and $(\frac{1}{2\phi}\sqrt{2+\phi})^{1/2}$ for the thin one, where $\phi=\frac{1+\sqrt{5}}{2}$ is the golden ratio. The two quasifolds are diffeomorphic but not symplectomorphic.

Something analogous happens for the three–dimensional generalization of this tiling due to Ammann, which is composed by two different types of rhombohedrons, prolate and oblate [29]. Again, each rhombohedron is rational, but to treat them all of them simultaneously we need to consider a quasilattice, known in crystallography as the face–centered icosahedral lattice. As normals we choose the relevant generators. One then obtains a pair of six–dimensional symplectic toric quasifolds, one for each type of rhombohedron. Similarly to what happens for the rhombus tiling, they are given by $(S^{2}(r)\times S^{2}(r)\times S^{2}(r))/\Gamma$, where $\Gamma$ is a countable subgroup of the standard 3–torus. The radius $r$ here is $[2\phi^{2}(3-\phi)]^{-\frac{1}{4}}$ for the oblate rhombohedron and $[2(3-\phi)]^{-\frac{1}{4}}$ for the prolate one. Again, the two spaces here are diffeomorphic but not symplectomorphic.

As we have seen, the quasifolds for both Penrose rhombus tilings and Ammann tilings are global, namely the quotient of a manifold modulo the action of a countable group.

Something entirely different happens for the kite and dart tiling [24]. First of all, the only tile here that is convex, and therefore relevant to our discussion, is the kite. Moreover, the kite, unlike the rhombuses and rhombohedrons, is actually nonrational. So there is no choice but to consider a quasilattice, and the natural one happens to be, again, the pentagonal quasilattice; the normals are, up to sign, the relevant fifth roots of unity (see Figure 3).

Then the resulting toric quasifold is not global. It is the four–dimensional quasifold given by

$$M=\frac{{\left\{\,(z_{1},z_{2},z_{3},z_{4})\in\mbox{\bbb{C}}^{4}\,|\,\phi|z_{1}|^{2}+|z_{2}|^{2}+\phi|z_{3}|^{2}=\phi|z_{2}|^{2}+|z_{3}|^{2}+\phi|z_{4}|^{2}=\phi\,\right\}}}{{\left\{\,\exp\left(-s+\phi t,s,t,-t+\phi s\right)\in\mbox{\bbb{R}}^{4}/\mbox{\bbb{Z}}^{4}\,|\,s,t\in\mbox{\bbb{R}}\,\right\}}}.$$

Let us describe one of its charts. Consider the open subset

$$\tilde{U}=\left\{\,(z_{2},z_{3})\in\mbox{\bbb{C}}^{2}\,|\,|z_{2}|^{2}+\phi|z_{3}|^{2}<\phi,\,\phi|z_{2}|^{2}+|z_{3}|^{2}<\phi\,\right\}$$

and the countable group

$$\Gamma=\left\{\,(e^{2\pi i\phi h},e^{2\pi i\phi k})\in\mbox{\bbb{R}}^{2}/\mbox{\bbb{Z}}^{2}\;|\;h,k\in\mbox{\bbb{Z}}\right\}.$$

Then the mapping

$$\begin{array}[]{ccc}\tilde{U}/\Gamma&\longrightarrow&\left\{[z_{1}:z_{2}:z_{3}:z_{4}]\in M\;|\;z_{1}\neq 0,z_{4}\neq 0\right\}\\ \left[z_{2}:z_{3}\right]&\longmapsto&\left[\sqrt{\phi-|z_{2}|^{2}-\phi|z_{3}|^{2}}:z_{2}:z_{3}:\sqrt{\phi-\phi|z_{2}|^{2}-|z_{3}|^{2}}\right]\end{array}$$

is a homeomorphism.

Decomposing Penrose tiles in half, yielding isosceles triangles as in Figure 4, is a very simple geometrical operation that has important repercussions.

First of all, it is the first step of both the inflation and deflation procedures. In the case of inflation, the triangles are appropriately combined to form a new tiling, whose tiles are rescaled by a factor $\phi$. In the case of deflation, the triangles are further decomposed into smaller ones to yield the half–tiles of another tiling that is rescaled by a factor $1/\phi$. It is easy to see that these operations are inverses of one another. We refer the reader to [2] for a detailed description in the case of rhombus tilings.

Cutting kites in half can also be used to transform a kite and dart tiling into a rhombus tiling.

The triangles are appropriately combined with each other and possibly a dart, in order to form thick and thin rhombuses (see [28] and Figure 5).

Now, the process of subdividing a simple convex polytope into two smaller ones corresponds, at the (smooth) symplectic level, to the symplectic cutting operation, which was introduced by Lerman [20]. In the toric setting, the original manifold decomposes into two new ones, each corresponding to one of the subdivided polytopes. The decomposition of Penrose tiles motivated us to extend this operation to the simple nonrational toric case. We find, for example, that the toric quasifold corresponding to each half–kite is given by

$$\frac{\left\{\,(z_{1},z_{2},z_{3})\in\mbox{\bbb{C}}^{3}\,|\,|z_{1}|^{2}+\phi|z_{2}|^{2}+\phi|z_{3}|^{2}=1\,\right\}}{\left\{\,(e^{2\pi is},e^{2\pi i\phi s},e^{2\pi i\phi(s+k)})\in\mbox{\bbb{R}}^{3}/\mbox{\bbb{Z}}^{3}\,|\,s\in\mbox{\bbb{R}},k\in\mbox{\bbb{Z}}\,\right\}}.$$

As we have shown, a number of interesting examples of toric quasifolds arise in connection with nonperiodic tilings. There also appears to be a correspondence between some of the fundamental operations in the two theories. We have seen, in fact, that decomposing convex Penrose tiles into half corresponds to cutting the associated symplectic toric quasifolds. We expect, moreover, that recombining these half–tiles, as needed for the inflation and deflation procedures, will correspond to a nonrational generalization of the inverse operation of symplectic cutting, which is given, in the smooth case, by the symplectic sum [14]. We believe that it would be interesting to pursue the study of these connections even further. As a matter of fact, certain nonperiodic tilings have been used as mathematical models for the theory of quasicrystals [28]; these are special materials that were experimentally discovered by Shechtman et al. [27] that have discrete nonperiodic diffraction patterns. Actually, the very existence of these materials had been predicted by Mackay in connection with his studies of Penrose and Ammann tilings [22, 23]. Ultimately, it is quite possible that toric quasifolds might contribute to their theoretical understanding. A first step would consist in analyzing from the toric viewpoint other tilings (and their operations) that are relevant in this respect. Significant (though not the only) examples would be Socolar’s octagonal and dodecagonal tilings, which are used as a basis for a treatment of the elasticity of octagonal and dodecagonal quasicrystals [30].

By slightly perturbing the hyperplanes in (1), it can be shown that every simple or simplicial polytope can be perturbed to a rational one that is combinatorially equivalent ([31, Proposition 2.17]). In the simple case, these perturbations yield, at the toric level, interesting families of quasifolds. For example, jointly with Battaglia and Zaffran, we have used one such perturbation to construct a one–parameter family of toric quasifolds that generalize and contain Hirzebruch surfaces. This perturbation property also holds true for any three–dimensional polytope, not necessarily simple or simplicial [31, Corollary 4.8]. In greater dimensions, there are examples of polytopes for which this does not happen. The first was found by Perles in the sixties and has dimension $8$ (see [31, Example 6.21] and [32]). As we have seen, from the toric viewpoint, these polytopes, being necessarily nonsimple, yield spaces that are stratified by quasifolds. We believe it would be interesting to study these stratified spaces and understand how their geometry is affected by the fact that the corresponding polytopes cannot be deformed to rational ones within their combinatorial class.

In recent years, there has been a surge of interest in nonrational toric geometry, and several alternate approaches to this subject have been introduced. It should be said, first of all, that toric quasifolds can be thought of both as examples of stacks and of diffeological spaces. The stack approach to nonrational toric geometry was espoused first by Hoffman–Sjamaar [16, 15] and then by Katzarkov et al. [19]. Diffeological quasifolds, on the other hand, were studied jointly with Iglesias–Zemmour in [17], providing an explicit link with non–commutative geometry [9]; applications of this viewpoint to the toric setting are work in progress. Other recent points of view, due to Battaglia–Zaffran [7, 8], Lin–Sjamaar [21], Ratiu–Zung [26], and Ishida et al. [18], involve foliations of smooth manifolds, either in the complex or presymplectic setting. We would like to point out that most of the alternate nonrational toric viewpoints are founded on variations on the theme of the fundamental triple (2), beginning, first and foremost, with the quasilattice $Q$. In joint work with Battaglia [6], we describe in detail how many of these different perspectives connect with each other and with ours; a dictionary is provided, in the hope that it will provide clarity and facilitate future interaction in the field.