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Research Article Symplectic Toric Geometry and the Regular Dodecahedron

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Research Article Symplectic Toric Geometry and the Regular Dodecahedron Hindawi Publishing Corporation Journal of Mathematics Volume 2015, Article ID 967417, 5 pages http://dx.doi.org/10.1155/2015/967417 Research Article Symplectic Toric Geometry and the Regular Dodecahedron Elisa Prato Dipartimento di Matematica e Informatica “U. Dini”, Piazza Ghiberti 27, 50122 Firenze, Italy Correspondence should be addressed to Elisa Prato; [email protected] Received 30 July 2015; Accepted 26 October 2015 Academic Editor: Fred´ eric´ Mynard Copyright © 2015 Elisa Prato. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The regular dodecahedron is the only simple polytope among the platonic solids which is not rational. Therefore, it corresponds neither to a symplectic toric manifold nor to a symplectic toric orbifold. In this paper, we associate to the regular dodecahedron a highly singular space called symplectic toric quasifold. 1. Introduction Definition 1 (simple polytope). A dimension convex poly- ∗ tope Δ⊂(R ) is said to be simple if each of its vertices is According to a well-known theorem by Delzant [1], there contained in exactly facets. exists a one-to-one correspondence between symplectic toric manifolds and simple rational convex polytopes satisfying a Definition 2 (quasilattice). A quasilattice in R is the Z-span special integrality condition. One of the important features of a set of R-spanning vectors, 1,...,,ofR . of Delzant’s theorem is that it provides an explicit procedure Notice that SpanZ{1,...,} is an ordinary lattice if and for computing the symplectic manifold that corresponds to only if it admits a set of generators which is a basis of R . eachgivenpolytope.Asitturnsout,therearemanyimport- ∗ Consider now a dimension convex polytope Δ⊂(R ) ant examples of simple convex polytopes that do not fall into this class, either because they do not satisfy Delzant’sintegral- having facets. Then, there exist elements 1,..., in R ity condition, or, even worse, because they are not rational. and 1,..., in R such that The regular dodecahedron is one of the most remarkable ∗ examples of a simple polytope that is not rational. In this Δ=⋂ { ∈R ( ) |⟨,⟩≥}. (1) paper, we apply a generalization of Delzant’s construction [2] =1 to simple nonrational convex polytopes and we associate to Definition 3 (quasirational polytope). Let be a quasilattice ∗ the regular dodecahedron a symplectic toric quasifold.Quasi- in R . A convex polytope Δ⊂(R ) is said to be quasirational folds are a natural generalization of manifolds and orbifolds with respect to if the vectors 1,..., in (1) can be chosen introduced by the author in [2]; they are not necessarily in . Hausdorff spaces and they are locally modeled by quotients ∗ of manifolds by the action of discrete groups. We remark that each polytope in (R ) is quasirational This paper is structured as follows. In Section 2, we withrespecttosomequasilattice:justtakethequasilattice recall the generalized Delzant construction. In Section 3, we that is generated by the elements 1,..., in (1). Notice that apply this construction to the regular dodecahedron and we if 1,..., canbechoseninsideanordinarylattice,then describe the corresponding symplectic toric quasifold. thepolytopeisrationalintheusualsense. 2. The Generalized Delzant Construction Definition 4 (quasitorus). Let ⊂R be a quasilattice. We call quasitorus of dimension thegroupandquasifold We begin by recalling a few useful definitions. = R /. 2 Journal of Mathematics For the definition and main properties of symplectic quasifolds and of Hamiltonian actions of quasitori on sym- plectic quasifolds, we refer the reader to [2, 3]. On the other hand, the basic facts on polytopes that are needed can be found in Ziegler’s book [4]. We are now ready to recall from [2] the generalized Delzant construction. For the purposes of this paper, we will restrict our attention to the special case =3. 3 3 ∗ Theorem 5. Let be a quasilattice in R and let Δ⊂(R ) be a simple convex polytope that is quasirational with respect to . Then, there exists a 6-dimensional compact connected symplectic quasifold and an effective Hamiltonian action of 3 3 the quasitorus = R / on such that the image of the corresponding moment mapping is Δ. Proof. Let us consider the space C endowed with the standard symplectic form 0 =1/(2)∑=1 ∧ and the action of the torus = R /Z given by : × C → C (2) Figure 1: The regular dodecahedron. 21 2 21 2 (( ,..., ),) → ( 1,..., ). This is an effective Hamiltonian action with moment map- Remark 6. We will say that is a symplectic toric quasifold ping given by associated to the polytope Δ.Thequasifold depends on our ∗ choice of quasilattice with respect to which the polytope is C → ( R) : quasirational and on our choice of vectors 1,...,.Note 2 ∗ (3) that the case where the polytope is simple and rational, but → ∑ ∗ +, ∈(R) . constant does not necessarily satisfy Delzant’s integrality condition, =1 was treated by Lerman and Tolman in [5]. They allowed orbifold singularities and introduced the notion of symplectic The mapping is proper and its image is given by the cone toric orbifold. C =+C,whereC denotes the positive orthant of ∗ (R ) .Takenowvectors1,..., ∈and real numbers ,..., 1 as in (1). Consider the surjective linear mapping 3. The Regular Dodecahedron from 3 : R → R , a Symplectic Viewpoint (4) Δ → . Let be the regular dodecahedron centered at the origin and having vertices 3 3 = R3/ Consider the dimension quasitorus .Then, (±1, ±1, ±1) the linear mapping induces a quasitorus epimorphism 3 Π: →. Define now to be the kernel of the mapping 1 ∗ (0, ±, ± ) Π and choose =∑=1 .Denoteby the Lie algebra ∗ inclusion Lie() → R and notice that Ψ= ∘ is a 1 (5) (± ,0,±) moment mapping for the induced action of on C .Then, 3.1 =Ψ−1(0)/ according to [2, Theorem ], the quotient 1 isasymplecticquasifoldwhichisendowedwiththeHamil- (±, ± ,0), ∗ tonian action of the quasitorus /.Since is injective and is proper, the quasifold is compact. If we identify = (1 + √5)/2 3 where is the golden ratio and satisfies the quasitori and / via the epimorphism Π,weget =1+1/ 3 (see Figure 1). It is a well-known fact that the a Hamiltonian action of the quasitorus whose moment polytope Δ is simple but not rational. However, consider the ∗ −1 ∗ 3 mapping Φ has image equal to ( ) (C ∩ ker )= quasilattice that is generated by the following vectors in R : ∗ −1 ∗ ∗ −1 ∗ ( ) (C ∩ im )=( ) ( (Δ)) which is exactly Δ.This −1 1 action is effective since the level set Ψ (0) contains points of 1 =( ,1,0) the form ∈ C , =0̸ , = 1,...,,wherethe -action is =2−2 =2−2(− 1 free. Notice finally that dim dim =(0, ,1) 3) = 6. 2 Journal of Mathematics 3 1 = (1, 0, ) 3 1 =(− ,1,0) 4 1 =(0,− ,1) 5 1 = (1, 0, − ). 6 (6) We remark that these six vectors and their opposites point to thetwelveverticesofaregularicosahedronthatisinscribed in the sphere of radius √3− (see Figures 2 and 3). The quasilattice is known in physics as the simple icosahedral lattice [6]. Now, an easy computation shows that 12 3 ∗ Δ=⋂ {∈(R ) | ⟨, ⟩ ≥−} , (7) =1 where = and 6+ =−,for = 1,...,6. Therefore, Δ is quasirational with respect to .Letusperformthe ± ,...,± generalized Delzant construction with respect to and the Figure 2: The vectors 1 6. vectors 1,...,12. Following the proof of Theorem 5, we consider the surjective linear mapping Moreover, the moment mapping for the induced -action 12 3 12 ∗ : R → R Ψ: C →(n) has the 9 components below: (8) → . 2 2 + −2 It is easy to see that the following relations 1 7 2 2 1 1 + −2 −1 2 8 4 1 2 2 1 1 + −2 ( )=(−1 )( ) 3 9 5 2 (9) 2 2 1 1 + −2 6 −1 3 4 10 2 2 5 + 11 −2 (11) imply that the kernel of , n,isthe9-dimensional subspace 12 2 2 of R that is spanned by the vectors 6 + 12 −2 2 2 2 2 1 +7 1 + 2 −(3 + 4 )+2 + 2 8 2 2 2 2 2 + 3 −(1 + 5 )+2 3 +9 2 2 2 2 1 + 3 −(2 + 6 )+2. 4 +10 −1 The level set Ψ (0) is described by the 9 equations that 5 +11 (10) are obtained by setting these components to 0.Finally,the 6 6 +12 symplectic toric quasifold is given by the compact - −1 3 3 dimensional quasifold Ψ (0)/.Thequasitorus = R / + −( + ) 1 2 3 4 acts on in a Hamiltonian fashion, with image of the corresponding moment mapping given exactly by the dodec- + −( + ) 3 2 3 1 5 ahedron Δ.Theactionof has 20 fixed points, and the 1 +3 −(2 +6). moment mapping sends each of them to a different vertex of the dodecahedron. The quasifold has an atlas made of Since the vectors , = 1,...,12, generate the quasilattice 20 charts, each of which is centered around a different fixed ,thegroup is connected and given by the group exp(n). point. To give an idea of the local behavior of ,wewill 4 Journal of Mathematics √ 2 7 ()= 2 − 1 √ 2 8 ()= 2 − 2 √ 2 9 ()= 2 − 3 2 1 2 2 ()=√ − ( + )+2 10 3 1 2 2 1 2 2 ()=√ − ( + )+2 11 1 2 3 2 1 2 2 ()=√ − ( + )+2. 12 2 1 3 (14) Figure 3: The regular icosahedron. The mapping ̃ induces a homeomorphism ̃ → describe the chart around the fixed point that maps to the Γ (15) vertex (−1, −1, −1). Consider the open neighborhood ̃ of 0 3 [( , , )] → [ (̃ , , )] , in C defined by the inequalities below: 1 2 3 1 2 3 2 where the open subset of is the quotient 1 <2 −1 2 { ∈Ψ (0) | =0,̸ =4,...,12} 2 <2 (16) 2 <2 3 Γ (12) and the discrete group is given by 2 2 2 −2 < 1 + 2 −3 <2 2(ℎ+) 2(ℎ+) 2(+) 3 {( , , )∈ |ℎ,,∈Z}.
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