Classifying Toric and Semitoric Fans by Lifting Equations from ${\rm SL}_2({\mathbb Z})$

We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group ${\rm SL}_2({\mathbb Z})$ to its preimage in the universal cover of ${\rm SL}_2({\mathbb R})$. With this method we recover the classification of two-dimensional toric fans, and obtain a description of their semitoric analogue. As an application to symplectic geometry of Hamiltonian systems, we give a concise proof of the connectivity of the moduli space of toric integrable systems in dimension four, recovering a known result, and extend it to the case of semitoric integrable systems with a fixed number of focus-focus points and which are in the same twisting index class. In particular, we show that any semitoric system with precisely one focus-focus singular point can be continuously deformed into a system in the same isomorphism class as the Jaynes-Cummings model from optics.

The relationship between symplectic toric manifolds and toric varieties has been understood since the 1980s, see for instance Delzant [10], Guillemin [14,15]. The article [11] contains a coordinate description of this relation.
In this paper we present an algebraic viewpoint to study four-dimensional toric varieties, based on the study of matrix relations in the special linear group SL 2 (Z). Indeed, one can associate to a rational convex polygon ∆ the collection of primitive integer inwards pointing normal vectors to its faces, called a toric fan. This is a d-tuple where d ∈ Z is the number of faces. A Delzant polygon (or toric polygon) is one for which det(v i , v i+1 ) = 1 for each 0 i d − 1. This determinant condition forces the vectors to satisfy certain linear equations which are parameterized by integers a 0 , . . . , a d−1 ∈ Z. These integers satisfy (this equation appears in [12], page 44) but in fact not all integers satisfying Equation (1) correspond to a Delzant polygon. This is an equation in SL 2 (Z) and in this paper we lift it to the group G presented as where SL 2 (Z) ∼ = G/(S 4 ). The group G is the pre-image of SL 2 (Z) in the universal cover of SL 2 (R), and thus we can define what we call the winding number of an element of g ∈ G that evaluates to the identity in SL 2 (Z). Roughly speaking, we view g as a word in S and T and by applying this word to a vector one term at a time we produce a path around the origin. We define the winding number of g to be the winding number of this path in the classical sense. The equation in G analogous to Equation (1) has the property that a collection of integers a 0 , . . . , a d−1 satisfy the equation if and only if they correspond to a toric fan, and from the integers it is straightforward to recover the fan. As such, this is a method to translate problems about toric fans into equivalent problems about the algebraic structure of the group G. This allows us to simplify the proofs of some classical results about toric fans (Section 4), and generalize these results to the semitoric case (Section 5). Associated to a semitoric system there is also a collection of vectors (v 0 , . . . , v d−1 ) which satisfy more complicated equations (given explicitly in Definition 2.5) known as a semitoric fan. A semitoric fan can be thought of as a toric fan which includes additional information coming from the focus-focus singularities (called nodal singularities in the context of Lefschetz fibrations and algebraic geometry) of semitoric integrable systems. Indeed, focus--focus singularities appear in algebraic geometry [13], symplectic topology [20,31,34], and many simple physical models such as the spherical pendulum [1] and the Jaynes-Cummings system [7,17,30]. Roughly speaking, semitoric fans encode the singular affine structure induced by the singular fibration associated to a semitoric integrable system. This affine structure also plays a role in parts of symplectic topology, see for instance Borman-Li-Wu [3], and mirror symmetry, see Kontsevich-Soibelman [18].
We present two theorems in this paper which are applications of the algebraic method we introduce: • Theorem 2.8 gives a classification of semitoric fans; • Theorem 2.11 is an application of Theorem 2.8 which describes the pathconnected components of the moduli space of semitoric integrable systems.
The paper is divided into two blocks. The first one concerns toric and semitoric fans and requires no prior knowledge of symplectic or algebraic geometry, while the second block, which consists only of Section 6, contains applications to symplectic geometry and will probably be most interesting to those working on differential geometry or Hamiltonian systems. The structure of the paper is as follows. In Section 2 we state our main results, and the applications to symplectic geometry. In Section 3 we define the necessary algebraic structures and prove several general algebraic results. In Section 4 we use this new algebraic approach to recover classical results and about toric fans and in Section 5 we generalize these results to semitoric fans. The last part of this paper contains applications to the symplectic geometry of semitoric integrable systems, where we will prove connectivity of certain moduli spaces analogous to the result in [24] that the moduli space of symplectic toric manifolds is path-connected. Indeed, in Section 6 we use the results of Section 5 to study the connectivity of the moduli space of semitoric systems.

Toric fans
A toric variety is a variety which contains an algebraic torus as a dense open subset such that the standard action of the torus on itself can be extended to the whole variety. That is, a toric variety is the closure of an algebraic torus orbit [21]. By an algebraic torus we mean the product C * × . . . × C * , where C * = C \ {0}. It is well known that the geometry of a toric variety is completely classified by the associated fan. In general, a fan is set of rational strongly convex cones in a real vector space such that the face of each cone is also a cone and the intersection of any two cones is a face of each. In this paper we will be concerned with twodimensional nonsingular complete toric varieties and their associated fans, which we will simply call toric fans. These fans are given by a sequence of lattice points (v 0 = v d , v 1 , . . . , v d−1 ) ∈ (Z 2 ) d labeled in counter-clockwise order such that each pair of adjacent vectors generates all of Z 2 and the angle between any two adjacent vectors is less than π radians. That is, det(v i , v i+1 ) = 1 for i = 0, . . . , d − 1.
Definition 2.1. A Delzant polygon 1 (or toric polygon) is a convex polygon ∆ in R 2 which is simple, rational, and smooth. Recall v ∈ Z 2 is primitive if v = kw for some k ∈ Z >0 and w ∈ Z 2 implies k = 1 and w = v.
1. ∆ is simple if there are exactly two edges meeting at each vertex; 2. ∆ is rational if for each face f of ∆ there exists a vector in Z 2 which is normal to f ; 3. ∆ is smooth if the inwards pointing primitive vectors normal to any pair of adjacent edges form a basis of Z 2 .
Delzant polygons were introduced in the work of Delzant [10] in symplectic geometry, who built on the work of Atiyah [2], Kostant [19], and Guillemin-Sternberg [16] to give a classification of symplectic toric 4-manifolds in terms of the Delzant polygon (in fact, their work was in any dimension). Delzant polygons are similar to Newton polygons except that the vertices of Delzant polygons do not have to have integer coordinates. Just as in the case with Newton polygons, a toric fan may be produced from a Delzant polygon considering the collection of inwards pointing normal vectors of the polygon 2 .
In [12] Fulton stated the following result for toric fans. We have adapted the statement to relate it to Delzant polygons. Theorem 2.2 (Fulton [12], page 44). Up to the action of SL 2 (Z), every Delzant polygon can be obtained from a Delzant triangle, rectangle, or Hirzebruch trapezoid by a finite number of corner chops.
The corner chop operation is defined in Definition 2.7 and the minimal models (the Delzant triangle, rectangle, and Hirzebruch trapezoid) are defined in Definition 4.9 and depicted in Figure 1. The proof of Theorem 2.2 sketched by Fulton in [12], which uses only two-dimensional geometry and basic combinatorial arguments, is relatively long and does not immediately generalize. In Section 4 we provide an alternative proof using SL 2 (Z)-relations. This proof may be easily extended to the semitoric case.
As a consequence of Theorem 2.2 in [24] it was recently proved: 24]). The moduli space of toric polygons is path-connected.
That is, any two toric polygons may be deformed onto each other continuously via a path of toric polygons. One shows this by first knowing how to generate all toric polygons as in Theorem 2.2. Then one shows, using elementary analysis, 1 Higher dimensional toric integrable systems correspond to higher dimensional Delzant polytopes which satisfy analogous conditions to those given in this definition. 2 Notice that since Delzant polygons are required to be convex we automatically have that the angle between adjacent vectors in the fan is less than π. that the four minimal models can be continuously transformed into one another and that the corner chop operation is continuous. Again, let us emphasize that the results about toric fans are not new, we have just used a new viewpoint to arrive at classical results. We will see that this new viewpoint allows us to generalize the known results.

Semitoric fans
In analogy with semitoric polygons (originally defined in [  Definition 2.5 is inspired by the toric case 3 . Theorem 2.2 states that any toric fan can be produced from a minimal model using only corner chops. Similarly, our goal is to use a series of transformations to relate any semitoric fan to a standard form up to the action of the appropriate symmetry group.
Definition 2.6. The symmetry group of semitoric fans is given by where G acts on a semitoric fan by acting on each vector in the fan. Definition 2.7.
1. Let c ∈ Z 0 . The standard semitoric fan of complexity c is the fan (u 0 , . . . , u c+3 ) ∈ (Z 2 ) c+4 given by The following are called the four fan transformations: obtained by inserting the sum of two adjacent vectors between them. The process of producing this new fan from the original is known as corner chopping.
(b) A reverse corner chop is the procedure by which a single vector which is the sum of its adjacent vectors is removed from the fan. This is the inverse of a corner chop.
(c) Suppose that the pair (v i , v i+1 ) is a hidden corner. Then is a semitoric fan with (v i , T v i+1 ) a Delzant corner and (T v i+1 , v i+1 ) a fake corner (Lemma 5.1). The process of producing this fan is known as removing the hidden corner (v i , v i+1 ).
(d) Suppose that the pair (v i , v i+1 ) is a fake corner and the pair (v i+1 , v i+2 ) is a Delzant corner. Then is a semitoric fan with (v i , T v i+2 ) a Delzant corner and (T v i+2 , v i+2 ) a fake corner (Lemma 5.2). The process of producing this fan is known as commuting a fake and a Delzant corner.
Using the algebraic results from Section 3 we show the following.
Theorem 2.8. Let d 1 be an integer. Any semitoric fan (v 0 , . . . , v d−1 ) ∈ (Z 2 ) d of complexity c ∈ Z 0 may be transformed into a semitoric fan G -equivalent to the standard semitoric fan of complexity c by using the four fan transformations.
Remark 2.9. The method we are using to study semitoric manifolds is analogous to the method we use to study toric manifolds. Theorem 2.2 explains how to generate the toric polygons and is used to prove that the space of toric polygons is path-connected (Theorem 2.3) which implies that the space of toric manifolds is connected (Theorem 2.10). Similarly, Theorem 2.8 shows how to generate the semitoric polygons, and as an application we prove Lemma 6.23 which describes the connected components in the space of semitoric ingredients (Definition 6.9) and this implies Theorem 2.11, which describes the connected components of the moduli space of semitoric systems.

Algebraic tools: the winding number
It is shown in Lemma 3.1 that the special linear group SL 2 (Z) may be presented as

Thus Equation (1) becomes
where a 0 , . . . , a d−1 ∈ Z and I denotes the 2 × 2 identity matrix. Given v 0 , v 1 ∈ Z 2 with det(v 1 , v 2 ) = 1 a set of vectors In this way associated to each list of integers satisfying Equation (1) there is an ordered collection of vectors unique up to SL 2 (Z). It can be seen that the determinant between any adjacent pair of these vectors is one and thus if these vectors are labeled in counterclockwise order, then they are a toric fan. The reason that not all sequences of integers which satisfy Equation (1) correspond to a toric fan is that the vectors v 0 , . . . , v d−1 ∈ Z 2 may circle more than once around the origin, and thus not be labeled in counter-clockwise order (see Figure 2). Thus, we see that viewing ST a 0 · · · ST a d−1 as an element of SL 2 (Z) is losing too much information.
Let K = ker(Z * Z → SL 2 (Z)) where Z * Z denotes the free group with generators S and T and the map Z * Z → SL 2 (Z) is the natural projection. For any word in K a sequence of vectors may be produced by letting the word act on a vector v ∈ Z 2 one term at a time. We know we will end back at v, but the sequence of vectors produced contains more information about the word. This sequence can be used to define a path in R 2 \ {(0, 0)}. Of particular interest, especially when studying toric and semitoric fans, is the winding number of such a path. That is, the number of times the path, and hence the collection of vectors, circles the origin. This construction is explained in detail in Section 4, and in particular Definition 4.4 given a precise definition of the number of times an ordered collection of vectors circles the origin. We find that the winding number of a word σ ∈ K is given by where s is the number of appearances of S in σ and t is the number of appearances of T in σ. We present the group on which the winding number descends to a well-defined function W : G → Z. In fact, G is isomorphic to the pre-image of SL 2 (Z) in the universal cover of SL 2 (R) (Proposition 3.7). Thus, if K is the image of K projected to G, then given some g ∈ K there is an associated closed loop in SL 2 (R). The fundamental group of SL 2 (R) is Z and the classical winding number of this loop in SL 2 (R) coincides with W (σ) for any σ ∈ Z * Z which projects to g. Finally, in Corollary 4.6 we show that integers a 0 , . . . , a d−1 ∈ Z correspond to a toric fan if and only if the equality ST a 0 · · · ST a d−1 = S 4 is satisfied in G. This correspondence is the basis of our method to study toric and semitoric fans.

Applications to symplectic geometry
While Delzant polygons are in correspondence with toric manifolds, semitoric polygons are associated with the so called semitoric integrable systems. A semitoric integrable system (or semitoric manifold ) is given by a triple (M, ω, F : where M is a connected symplectic 4-manifold and F : M → R 2 is an integrable system given by two maps J, H : M → R 2 such that J is a proper map which generates a periodic flow (see Section 6 for the precise definition). Semitoric integrable systems are of interest in mathematical physics, symplectic geometry and spectral theory, because they exhibit rich features from dynamical, geometric, and topological viewpoints. Examples of semitoric systems permeate the physics literature. For instance the Jaynes-Cummings system [7,17], which is one of the most thoroughly studied examples of semitoric system [30], models simple physical phenomena. It is obtained by coupling a spin and an oscillator, and its phase space is S 2 × R 2 . Many papers have been written in the past ten years about the symplectic geometry of semitoric integrable systems, see for instance [32,33,26,27,28,29,25]. In [27] the authors prove a result analogous to Delzant's, classifying semitoric systems via a list of ingredients which includes a family of polygons (there is an overview of this result in Section 6.1).
One question which had remained open is whether the set of all semitoric systems has a reasonable underlying geometry, and whether as such one can view it as a topological space or a metric space. In [23] the second author defines a metric space structure for the moduli space of semitoric systems using [27], and a natural question is whether, with respect to such structure, the space is pathconnected. That is, can two semitoric systems be continuously deformed onto one another, via a path of semitoric systems? This is preceded by [24] in which the authors construct a natural metric on the moduli space of symplectic toric manifolds and prove Theorem 2.3, which is used to conclude the following. Similarly, Theorem 2.8 implies the following statement. The number of focusfocus singularities and the twisting index of a semitoric system are explained in Section 6. (i) they have the same number of focus-focus singularities; (ii) they have the same sequence of twisting indices, then there exists a continuous (with respect to the topology defined in [23]) path of semitoric systems with the same number of focus-focus points and same twisting indices between them. That is, the space of semitoric systems with fixed number of focus-focus points and twisting index invariant is path-connected. 3 Algebraic set-up: matrices and SL 2 (Z) relations The 2 × 2 special linear group over the integers, SL 2 (Z), is generated by the matrices S = 0 −1 1 0 and T = 1 1 0 1 .
We will see that to each toric (resp. semitoric) integrable system there is an associated toric (resp. semitoric) fan and we will use the algebraic structure of SL 2 (Z) to study these fans. For our purposes, the following presentation of SL 2 (Z) will be the most natural way to view the group. Proof. It is well-known that (see for instance [4]) where (−I) 2 = I and −I is in the center of SL 2 (Z). To obtain the relations in the statement of the lemma from those in Equation (2) notice To obtain the relations of Equation (2) from those in the statement of the lemma we only have to show that S 2 behaves as −I. That is, we must show that S 2 squares to the identity and is in the center of the group. We have that (S 2 ) 2 = I and that S 2 commutes with S. To show that S 2 commutes with T notice This concludes the proof.
For v, w ∈ Z 2 let [v, w] denote the 2 × 2 matrix with v as the first column and w as the second and let det(v, w) denote the determinant of the matrix [v, w]. Proof. In the basis (u, v) we know that v = 0 The result of Lemma 3.2 can be easily summarized in a matrix equation, as we will now show. Let

be a toric fan and define
Proof. By the definition of a toric fan we know that for each 0 and this concludes the proof.

It follows that
This is a restatement of Equation (1) which is from [12]. So to each toric fan of d vectors there is an associated d-tuple of integers which satisfy Equation (3), but having a tuple of integers which satisfy Equation (3) is not enough to assure that they correspond to an toric fan. The determinant of the vectors will be correct but, roughly speaking, if the vectors wind around the origin more then once then they will not be labeled in the correct order to be a toric fan, as it occurs in the following example.  (3) is satisfied for c = 0 but these integers do not correspond to a toric fan. This is because the vectors they produce: travel twice around the origin 4 , see Figure 2.
So we need extra information that is not captured by viewing this word in S and T as an element of SL 2 (Z). For a more obvious example notice that even though they are equal in SL 2 (Z) we can see that S 4 corresponds to a toric fan while S 8 does not. From [12] we know that integers (a 0 , . . . , a d ) ∈ Z d which satisfy Equation (1) correspond to a toric fan if and only if is the number of times that the vectors corresponding to (a 0 , . . . , a d−1 ) circle the origin. In order to prove this we will need some more terminology, and in order to keep track of the extra information about circling the origin we will need to consider a group which is larger than SL 2 (Z).
Consider instead the free group with generators S and T . This group is isomorphic to the free product of Z with itself so we will denote it by Z * Z. We know SL 2 (Z) is a quotient of Z * Z by Lemma 3.1 so there exists a natural projection map π 1 : Z * Z → SL 2 (Z). Also define a map w : Z * Z → Z by Given a toric fan with associated integers (a 0 , . . . , a d−1 ) ∈ Z d we will show that w(ST a 0 · · · ST a d−1 ) = 12. Both π 1 and w factor over the same group G which is the fiber product of SL 2 (Z) and Z over Z/ (12). Now we can see that we wanted the particular presentation of SL 2 (Z) from Lemma 3.1 so that the relationship between G and SL 2 (Z) would be clear. This discussion made precise in the following proposition.
Proposition 3.5. The following diagram commutes.
The group G is the fiber product of SL 2 (Z) and Z over Z/ (12) and is given by each of π 1 , π 2 , π 3 , and π 4 is a projection, and w : (12) are given by the same formal expression Proof. It can be seen that the map w SL 2 (Z) : SL 2 (Z) → Z/(12) is well-defined by noting that both relations in SL 2 (Z) as presented in Lemma 3.1 preserve the value of the formula up to a multiple of 12. Similarly, since the relation ST −1 S = ST S preserves the value of the equation we know that w G is well-defined. Since each of these functions to Z or Z/(12) is given by the same formal expression and since each π is a quotient map, the diagram commutes.
To show that G with the associated maps is the fiber product of SL 2 (Z) and Z over Z/ (12) we must only show that w G restricted to the fibers is bijective. That is, we must show that is a bijection for each A ∈ SL 2 (Z). To show it is surjective, notice that for any g ∈ G π 3 (S 4k g) = π 3 (g) and w G (S 4k g) = w G (g) + 12k for any k ∈ Z. To show it is injective it is sufficient to consider only A = I. Since Since w G (S 4k ) = 12k we know for each choice of k this maps to a distinct element of Z. Notation 3.6. We have several groups with generators S and T . To denote the different equalities in these groups we will use an equal sign with the group in question as a subscript. That is, if an equality holds in the group H we will write = H . For example, S 4 = SL 2 (Z) I but S 4 = G I.
There is another useful sense in which G is an unwinding of SL 2 (Z). While SL 2 (Z) is discrete, and thus does not have a natural cover, it sits inside the group SL 2 (R), which has a universal cover. We claim that G is the preimage of SL 2 (Z) inside of the universal cover of SL 2 (R).
Proposition 3.7. The group G is isomorphic to the preimage of SL 2 (Z) within the universal cover of SL 2 (R).
Proof. Let G be the preimage of SL 2 (Z) in the universal cover of SL 2 (R). We note that there exists a homomorphism, φ from G to G defined by where the paths given are to represent elements of the universal cover of SL 2 (R).
and thus φ does actually define a homomorphism. We have left to show that φ defines an isomorphism. To show this, we note that each of G and G have obvious surjections π and π to SL 2 (Z). Furthermore it is clear that π = π • φ. Thus, to show that φ is an isomorphism, it suffices to show that φ : ker(π) → ker(π ) is an isomorphism. However, it is clear that ker(π) is S 4 . On the other hand, ker(π ) ∼ = π 1 (SL 2 (R)) = Z, and is generated by φ(S) 4 . This completes the proof.
We will see that there is a one to one correspondence between toric fans up to the action of SL 2 (Z) and lists of integers a 0 , . . . , a d−1 ∈ Z satisfying (Corollary 4.6). Equation (5) is a refinement of Equation (3) which implies both that the successive pairs of vectors form a basis of Z 2 and that the vectors are labeled in counter-clockwise order. In Proposition 5.4 we produce an analogous equation for semitoric fans. Now we would like to simplify these toric fans. We will understand which integers a 0 , . . . , a d−1 are possible in an element ST a 0 · · · ST a d−1 ∈ G corresponding to toric fan by studying PSL 2 (Z) = SL 2 (Z)/(−I). The following lemma is important for this and will also be useful later on when classifying semitoric fans. If ST a 0 · · · ST a d−1 ∈ G projects to the identity in SL 2 (Z) then by Lemma 3.8 we can see when one of the exponents must be in the set {−1, 0, 1}. In any of these cases, we will be able to use relations in G to help simplify the expression.
Proof. It is well known that PSL 2 (Z) acts on the real projective line R ∪ {∞} by linear fractional transformations: Let d > 3. Suppose that at most two of a 0 , . . . , a d−1 are in {−1, 0, 1} and if there are two in {−1, 0, 1} that they are consecutive or indexed by 0 and d − 1. Notice that by conjugating each side with ST a 0 . This conjugation method and renumbering the integers can be used to assure that Since this expression is equal to the identity in PSL 2 (Z) it acts trivially on R ∪ {∞}.
In particular, we have Putting these facts together we have This contradiction finishes the d > 3 case. If d = 3 essentially the same result holds except that it is not possible to choose two elements that are non-consecutive. Notice For this function to be the identity we would need This implies that a 0 a 1 = 1 so since they are both integers we have a 0 = a 1 = where ∈ {−1, 1}. Conjugating by ST a 0 , we find symmetrically, that a 1 a 2 = 1, and thus that a 0 = a 1 = a 2 = ±1.
If d = 2 then we have so we must have a 0 = 0 and then so we are also forced to have that a 1 = 0, as stated in the lemma. If d = 1 then ST a 0 (∞) = 0 for any choice of a 0 ∈ Z, so there are no solutions.

Toric fans
Let a 0 , a 1 , . . . , a d−1 ∈ Z be a collection of integers such that This means that by Proposition 3.5. We claim that these integers correspond to a toric fan if and only if k = 1. The idea is that k = 1 precisely when the vectors in the corresponding fan are labeled in counter-clockwise order, and the only relation in the group G, which is ST S = T −1 ST −1 , preserves the number of times the vectors circle the origin. Now we make this idea precise.
Recall that G is isomorphic to the preimage of SL 2 (Z) in the universal cover of SL 2 (R) by Proposition 3.7. Let φ be the isomorphism from G to the universal cover of SL 2 (R) with This means to each element of the kernel of π 3 we can associate a closed loop based at the identity in SL 2 (R) denoted φ(g). The fundamental group π 1 (SL 2 (R)) is isomorphic to Z and is generated as φ(S 4 ) , so let ψ : π 1 (SL 2 (R)) → Z be the homomorphism with ψ(φ(S 4 )) = 1.
We call W (g) the winding number of g ∈ ker(π 3 ).
We define the number of times (v 0 , . . . , v d−1 ) circles the origin to be the winding number of the piecewise linear path in (R 2 ) * = R 2 \ {(0, 0)} produced by concatenating the linear paths between v i and v i+1 for i = 0, . . . , d − 1.
Thinking of the elements ST a i as elements of G, and thus as elements of the universal cover of SL 2 (R), this gives a path from A i to A i+1 , and concatenating these paths gives a path from A 0 to itself in SL 2 (R). Projecting this path into the first column vector of the appropriate matrix gives a path in (R 2 ) * . We claim that this path is homotopic to the path formed by taking line segments between v i and v i+1 . This is easily verified because both paths between v i and v i+1 travel counterclockwise less than a full rotation. We know that W (ST a 0 · · · ST a d−1 ) equals ψ of the path in SL 2 (R), and now we need to show that this equals the winding number in (R 2 ) * of the first column vectors. To show this we note that the element of π 1 ((R 2 ) * ) (the group is abelian, so we may ignore basepoint) given by the image of the element of π 1 (SL 2 (R)) under the natural projection map. Thus, we merely need to show that the this projection acts correctly on a generator of π 1 (SL 2 (R)), but it is easy to see that the image of φ(S 4 ) yields a path of winding number 1.
to the collection of all toric fans modulo the action of SL 2 (Z). This bijection sends (a 0 , . . . , a d ) ∈ Z d to the equivalence class of fans for each i = 0, . . . , d − 1 and the vectors are labeled in counter-clockwise order. It is shown in Section 3 Equation (3) that associated integers (a 0 , . . . , for some k ∈ Z with k 0. By Lemma 4.5 we know so that the vectors will be labeled in the correct order for it to be a fan. Thus, where v 0 , v 1 ∈ Z 2 are any two vectors for which det(v 0 , v 1 ) = 1. Then for each i = 0, . . . , d − 1 we have so by induction all of these determinants are 1. By Lemma 4.5, the path connecting adjacent vectors wraps around the origin only once, and since each v i+1 is located counterclockwise of v i , we have that the v i 's must be sorted in counterclockwise order.
Now that we have set up the algebraic framework the following results are straightforward to prove. First we prove that any fan with more than four vectors can be reduced to a fan with fewer vectors.
Proof. By Corollary 4.6 we know that to the fan (v 0 = v d , . . . , v d−1 ) there is an associated list of integers a 0 , . . . , We must only show that for some i ∈ Z we have a i = 1. Since S 4 = PSL 2 (Z) I we can use Lemma 3.8 to conclude that there exist i, j ∈ Z satisfying 0 i < j −1 d−2 such that a i , a j ∈ {−1, 0, 1} and (i, j) = (0, d−1). By way of contradiction assume that a i , a j ∈ {−1, 0}. Conjugate Equation (7) by ST an for varying n ∈ Z to assure that i = 0 and j = d − 1. Then at each of these values we may use either where > 2 because we started with at least five ST -pairs and have only reduced by two. This means This implies that W (ST b 0 · · · ST b −1 ) = 0 and thus, by Lemma 4.5, that the corresponding collection of vectors winds no times about the origin. However, this is impossible since for such a sequence of vectors v i+1 is always counterclockwise of v i .
The case in which a vector in the fan is the sum of the adjacent vectors is important because this means the fan is the result of corner chopping a fan with fewer vectors in it. Now that we have the proper algebraic tools, we will be clear about the specifics of the corner chopping and reverse corner chopping operations. Suppose is also a fan. Next notice so this new fan has associated to it the tuple of integers (a 0 , . . . , a i−1 − 1, a i+1 − 1, . . . , a d−1 ) ∈ Z d−1 . An occurrence of 1 from the original tuple of integers has been removed and the adjacent integers have been reduced by 1. Algebraically, this move corresponds to the relation ST S = G T −1 ST −1 . Geometrically this move corresponds to the inverse of chopping a corner from the associated polygon (as is shown in Figure 1). The corner chopping of a toric polygon is done such that the new face of the polygon produced has inwards pointing normal vector given by the sum of the adjacent inwards pointing primitive integer normal vectors. Now we can see that Lemma 4.7 tells us that fans with five or more vectors are the result of corner chopping a fan with fewer vectors. We will next classify all possible fans with fewer than five vectors.
for some d ∈ Z, d 0.
1. If d = 4 then up to a cyclic reordering the set of integer quadruples which satisfy this equation is exactly a 0 = 0, a 1 = k, a 2 = 0, a 3 = −k for each k ∈ Z.

If d < 3 then there do not exist integers satisfying Equation (8).
Proof. Notice ST a 0 · · · ST a d−1 = G S 4 implies that ST a 0 · · · ST a d−1 = PSL 2 (Z) I. Next notice that so that is the only possibility for d = 3. Now suppose that d = 4. Lemma 3.8 tells us that at least one of the a i is in the set {−1, 0, 1}. By conjugation (which cyclically permutes the order of the integers) we may assume that a 0 ∈ {−1, 0, 1}. If a 0 = 1 then so for this to equal S 4 in G we must have a 1 − 1 = a 2 = a 3 − 1 = −1 by the d = 3 argument above. It is straightforward to check that ST SST −1 S = G S 4 so we have found a solution.
If a 0 = −1 then notice so by Lemma 3.8 we must have a 1 + 1 = a 2 = a 3 + 1 = ±1. This time, if a 1 + 1 = a 2 = a 3 + 1 = −1 then Equation (8) does not hold, since the left side will equal S 6 , but if a 1 + 1 = a 2 = a 3 + 1 = 1 then the equation holds. So we have found another solution which is ST −1 SST S = G S 4 .
Finally suppose that a 0 = 0. Notice so we can use Lemma 3.8 to conclude that we need a 2 = a 1 + a 3 = 0. Let a 1 = k ∈ Z. Now we have that for any k ∈ Z. Finally, observe that the other two possibilities we derived in the d = 4 case are just reorderings of this one with k = 1.

Semitoric fans
Now we will apply the method from Section 4 to classify semitoric fans (Definition 2.5). The first step in the classification is given by a series of lemmas which we will use to manipulate the semitoric fans in a standard form. a semitoric fan and (v i , v i+1 ) is a hidden corner, then is a semitoric fan in which (w i , w i+1 ) is a Delzant corner and (w i+1 , w i+2 ) is a fake corner.
Proof. We know that det(v i , T v i+1 ) = 1 because that pair of vectors forms a hidden corner. Notice that which concludes the proof.
is a semitoric fan in which (w i , w i+1 ) is a Delzant corner and (w i+1 , w i+2 ) is a fake corner.
Proof. We know det(v i , T v i+1 ) = 0 so v i = T v i+1 since they are both on the top boundary and we also know det(v i+1 , v i+2 ) = 1. Now we can check that which concludes the proof.
In Lemma 5.1 we have described the process of removing a hidden corner and in Lemma 5.2 we have described the process of commuting a fake and Delzant corner. Both of these processes are defined in Definition 2.7.

Proof.
Let for some a ∈ Z. This is because (v i−1 , v i ) is not on the upper boundary so it must be a Delzant corner. Then by the action of T −a ∈ G , which does not change v i , we can attain the required pair of vectors.
Otherwise, renumber so that v 0 is in the lower half plane and v 1 is in the upper half plane. Then insert the vector v 0 + v 1 between them. This new vector will have a second component with a smaller magnitude than that of v 0 or of v 1 . Repeat this process until the new vector lies on the x-axis. Since it is a primitive vector it must be ±1 0 . However since it is the sum of two vectors of opposite sides of the x-axis with the one above being counterclockwise about the origin of the one on bottom, it must be 1 0 .
Proof. The first part is immediate from Lemmas 5.1, 5.2, and 5.3. By Lemma 5.3 we know after a finite number of cuts and renumbering it can be arranged that the first two vectors in the fan are 0 −1 and 1 0 .
Then we invoke Lemma 5.1 to remove all of the hidden corners, and finally use Lemma 5.2 commute all of the fake corners to be adjacent and arrive at the fan (w 0 , . . . , w +c−1 ) ∈ (Z 2 ) +c in the statement of the proposition. Notice that (w i , w i+1 ) being fake for i = , . . . , + c − 1 implies that det(w , T w +1 ) = · · · = det(w +c−1 , T w 0 ) = 0. Now, we know both vectors in a fake corner must have negative second component by definition, so this implies that is a toric fan (we know the vectors are in counter-clockwise order because we started with a semitoric fan) so there must exist b 0 , . . . , b −1 ∈ Z as in Part 2 of the statement of this theorem. Since

Application to symplectic geometry
The results of Section 5 have an interpretation in symplectic geometry of toric manifolds and semitoric integrable systems. In Section 6.1 we will review the Classification Theorem of Pelayo-Vũ Ngo . c [27], in Section 6.2 we define the metric on the moduli space of semitoric systems given by the second author in [23], and in Section 6.3 we prove the connectivity result for semitoric systems using Theorem 2.8.

Invariants of semitoric systems
A single polytope is enough to classify toric integrable systems [10]. Semitoric systems are classified in terms of a list of five invariants [27]. Roughly speaking, the complete invariant is a semitoric polygon together with a set of interior points each labeled with extra information (encoding singularities of so called semitoric type, which semitoric systems posses, but toric systems do not) modulo an equivalence relation. Even without the extra information semitoric polygons are more complicated than toric polygons because they have fake corners and hidden corners (as defined in Definition 6.4). In this section we will define each of invariants of semitoric systems and state the Pelayo-Vũ Ngo . c Classification Theorem. In this section we give the necessary definitions to state the classification theorem and define the metric in [23]. Readers interested in further details may consult [32,33,26,27,28,29].

The number of focus-focus points invariant
While a toric integrable system can only have transversally-elliptic and ellipticelliptic singularities a semitoric integrable system can also have focus-focus singularities (for a definition of these types of singularities see for instance [27]). In [33, Theorem 1] Vũ Ngo . c proves that any semitoric system has at most finitely many focus-focus singular points. The first invariant is a nonnegative integer m f known as the number of singular points invariant.

The Taylor series invariant
In [32] Vũ Ngo . c proves that the semi-global 5 structure of a focus-focus singular point is completely determined by a Taylor series.

The affine invariant and the twisting index invariant
The affine invariant is the polygon which is directly analogous to the Delzant polygon in the toric case and the twisting index represents how the different focusfocus singular points relate to one another. The twisting index will be represented by an integer label on each focus-focus point, but up to the appropriate group action this sequence of integers will only be defined up to the addition of a common integer. These two invariants are described together because the choice of common integer added to the twisting index effects the polygon.
In general, the momentum map image of a semitoric system does not have to be a polygon and does not even have to be convex, but in [33] the author was able to recover a family of convex polygons which take the place of the Delzant polygon in the semitoric case. Definition 6.2. A convex polygonal set, which we will refer to as a polygon, is the intersection in R 2 of finitely or infinitely many closed half planes such that there are at most finitely many corner points in each compact subset of the intersection. A polygon is rational if each edge is directed along a vector with integer coefficients. Let Polyg(R 2 ) denote the set of all rational convex polygons.
Notice that a convex polygonal set may be noncompact. For any λ ∈ R we will use the notation where π 1 : R 2 → R is projection onto the first coordinate. We denote the space of labeled weighted polygons of complexity m f by LWPolyg(R 2 ). This is not yet the affine invariant. The affine invariant will be a subset of a quotient of LWPolyg(R 2 ). Recall Definition 6.4.
2. Let ∆ ∈ Polyg(R 2 ). A vertex of ∆ is a point p ∈ ∂∆ such that the edges meeting at p are not co-linear. For a vertex p ∈ ∆ let u, v ∈ R 2 be primitive inwards pointing normal vectors to the edges of ∆ adjacent to p in the order of positive orientation. Then p is In the case that (∆, ( λ j , j , k j ) m f j=1 ) is a primitive semitoric polygon and ∆ is compact then there exists an associated semitoric fan.

The action of G m f × G
Now we will define a group and the way that its elements act on a labeled weighted polygon. For any ∈ Vert(R 2 ) and k ∈ Z fix an origin in and let t k : R 2 → R 2 act as the identity on the half-space to the left of and as (T t ) k , with respect to the origin in , on the half-space to the right of . For u = (u 1 , . . . , We define the action of (( j ) where λ = (λ 1 , . . . , λ m f ) and u = ( j=1 . Definition 6.7. A semitoric polygon is the orbit under G m f × G of a primitive semitoric polygon. That is, given a primitive semitoric polygon ∆ w = (∆, ( λ j , +1, k j ) m f j=1 ) ∈ LWPolyg(R 2 ) the associated semitoric polygon is the subset of LWPolyg(R 2 ) given by The collection of semitoric polygons is denoted DPolyg(R 2 ).
In general the action of G m f ×G may not preserve the convexity of the polygons but it is shown in [27,Lemma 4.2] that [∆ w ] ⊂ LWPolyg(R 2 ) for any primitive semitoric polygon ∆ w .
Remark 6.8. The affine invariant is a family of polygons. These polygons are determined by choosing a single primitive semitoric polygon.

The volume invariant
The next invariant is also not independent of the polygon. Suppose that ∆ is a primitive semitoric polygon. For each j = 1, . . . , m f we define a real number h j ∈ (0, length(π 2 (∆ ∩ λ j ))). This is called the volume invariant because it is the Liouville volume of a submanifold that has to be removed from M during the procedure which produces these polygons (the details are in [26,27]).
The complete invariant can be represented by a primitive semitoric polygon ∆ with m f vertical lines λ 1 , . . . , λm f and for each j = 1, . . . , m f there is a distinguished point on λ j a distance h j from the bottom of ∆. Each of these points is also labeled with an integer k j and a Taylor series (S j ) ∞ . This is shown in Figure 4. It is important to note that in this rough description we have omitted the action of group G m f × G which is key to understanding the classification.  A semitoric integrable system is a 4-dimensional connected symplectic manifold (M, ω) with an integrable Hamiltonian system F = (J, H) : M → R 2 such that J is a proper momentum map for a Hamiltonian S 1 -action on M and F has only non-degenerate singularities which have no real-hyperbolic blocks. Such a system is said to be a simple semitoric integrable system is one for which there is at most one focus-focus critical point in J −1 (x) for any x ∈ R.
Semitoric systems and the associated isomorphisms are defined in [26]. We use the following result to study the moduli space of semitoric systems by instead studying the set of semitoric lists of ingredients. Theorem 6.10 ( [27]). There exists a bijection between the set of simple semitoric integrable systems modulo semitoric isomorphism and M, the set of semitoric lists of ingredients. In particular, where the invariants m f , [∆ w ], h j , and (S j ) ∞ are as defined above.

Metric and topology on the moduli space
In [23] the second author defines a metric space structure for T using Theorem 6.10. This metric is constructed by combining metrics on each ingredient. We reproduce this construction briefly in this section. Remark 6.11. It is important to notice that the metric we are using on T in this article is not the same as the metric defined in [23]. Since these two metrics induce the same topology on T [23, Section 2.6] it is suggested in [23,Remark 1.31(3)] that the metric in the present article be used when studying the topological properties of T , such as connectedness.
The metric defined in [23] gives T the appropriate metric space structure, which can be seen when the completion is computed in that article. That metric is defined as the minimum of a collection of functions, one of which is the metric used in this article, so it is immediate that the distance between two systems using Definition 6.18 will never be smaller than the distance between those two systems using the metric studied in [23].

The metric on Taylor series
Recall that the Taylor series in R[[X, Y ]] 0 have the coefficient to X 0 Y 1 in the interval [0, 2π). From the construction in [32] we can see that when defining the topology the endpoints of this interval should be associated. We define a topology on R[[X, Y ]] 0 such that a sequence of Taylor series converges if and only if each term converges in the appropriate space. This is the topology induced by the following metric.
In order for a series to converge with respect to this metric each term except for the coefficient to X 0 Y 1 must converge in R, and the coefficient to X 0 Y 1 must converge in R/2πZ.

The metric on the affine invariant
In [24] the authors use the Lebesgue measure of the symmetric difference of the moment polytopes to define a metric on the space of toric systems. For the portion of the semitoric metric related to the affine invariant something similar is done in [23]. Definition 6.13. Let ν be a measure on R 2 . We say that it is admissible if: 1. it is in the same measure class as the Lebesgue measure on R 2 (that is, µ ν and ν µ where µ is the Lebesgue measure); 2. the Radon-Nikodym derivative of ν with respect to Lebesgue measure depends only on the x-coordinate, that is, there exists a g : R → R such that dν /dµ(x, y) = g(x) for all (x, y) ∈ R 2 ; 3. the function g from Part (2) satisfies xg ∈ L 1 (µ, R) and g is bounded and bounded away from zero on any compact interval.
Since the polygons may be noncompact an admissible measure on R 2 is used in place of the Lebesgue measure. Notice that if ν is an admissible measure and ∆ is a convex noncompact polygon with everywhere finite height then ν(∆) < ∞. This is essentially because the convex polygon can grow at most linearly and the admissible measure is designed to shrink faster than this by Part (3). Since there is a family of polygons they must all be compared. Let * denote the symmetric difference of two sets. That is, for A, B ⊂ R 2 we have Definition 6.14. Let DPolyg m f , k (R 2 ) ⊂ DPolyg(R 2 ) denote orbits under G m f × G of primitive semitoric polygons with twisting index given by k ∈ Z m f . Then Remark 6.15. Let m f be a nonnegative integer and k ∈ Z m f . Suppose that a primitive semitoric polygon [∆ w ] has twisting index k ∈ Z m f where there exists some c ∈ Z such that k j = k j + c for j = 1, . . . , m f . Then [∆ w ] ∈ DPolyg m f , k (R 2 ) because the set [∆ w ] is also the orbit of a primitive semitoric polygon with twisting index k. .
If m f = 0 then this sum will be empty so we instead use To find the distance between two families of polygons we take the sum of the symmetric differences of all of them and in the case that m f = 0 there is a unique polygon.

The metric on the moduli space of semitoric systems
The total metric will be formed by combining the metrics from Section 6.2.1 and Section 6.2.2. The following is from [23].
Definition 6.18. Let ν be an admissible measure on R 2 and let {b n } ∞ n=0 be a linear summable sequence. Let m f , m f be nonnegative integers and k ∈ Z m f . 1. We define the metric on M m f , k to be given by 2. We define the metric on M to be given by where m, m ∈ M.
3. Let Φ : T → M be the correspondence from Theorem 6.10. Then we define the metric on T by D ν, The metric on T depends on the choice of admissible measure and linear summable sequence, but the topology it induces does not. Let T m f , k = Φ −1 (M m f , k ). From Definition 6.18 it can be seen that each T m f , k is a separate component of T , so it is natural to wonder if T m f , k is connected for each m f ∈ Z 0 and k ∈ Z m f . From Theorem 2.11 we know that these are in fact path-connected. (see Definition 2.7). Then there exists a continuous family of (compact) primitive semitoric polygons ∆ t , t ∈ [0, 1], such that the fan related to ∆ 0 is F 0 and the fan related to ∆ 1 is F 1 .

The connectivity of the moduli space of semitoric integrable systems
Proof. Suppose that ∆ is a polygon with associated fan F = (v 0 , . . . , v d−1 ) ∈ (Z 2 ) d , let i ∈ {0, 1, . . . , d − 1}, and let (x 0 , y 0 ) ∈ R 2 be the vertex between the edge normal to v i and the edge normal to v i+1 . Let ε > 0 be smaller than the length of either of these edges. Now let H t (v) denote the half-space given by where · denotes the usual vector dot product First we consider the corner chop operation.
Since H t (v i +v i+1 ) is deforming continuously we see that ∆ t is a continuous family and since the edges of ∆ t are parallel to the edges of ∆ except for the new edge with inwards pointing normal vector given by v i + v i+1 we see that the fan of ∆ t is the corner chop of the fan for ∆ if t ∈ (0, 1]. Since a reverse corner chop is the inverse of this operation, we can use the same path backwards. The continuity of H t (T v i+1 ) implies that ∆ t is a continuous family and by construction it has the desired fan. Thus ∆ t is the required family for the operation of removing hidden corners. Finally, suppose that (v i , v i+1 ) is a fake corner and (v i+1 , v i+2 ) is a Delzant corner. Here we can see that this fan is the result of removing the hidden corner (v i , v i+2 ) from the fan We know (v i , v i+2 ) is a hidden corner because it is given that Thus there is a continuous path from any polygon with fan F to any polygon with fan F . Let ∆ have fan F . But now, making ε smaller if needed, we can consider where (x 1 , y 1 ) is the vertex between the edges corresponding to v i and v i+2 and t ∈ [0, 1]. This is a continuous path to a polygon with the required fan. This completes the proof.
Recall that toric polygons are precisely the compact primitive semitoric polygons with complexity zero. Thus, by Proposition 5.4 and Lemma 6.20 we have recovered Theorem 2.3.
In light of Lemma 5.4 and Proposition 6.20 the only difficulty remaining to prove the following lemma is incorporating the case of semitoric systems which have noncompact polygons as invariants.
Lemma 6.21. Let m f ∈ Z 0 and k ∈ Z m f . Then DPolyg m f , k (R 2 ) is pathconnected.
Proof. We must only consider the primitive semitoric polygons, because if the primitive polygons converge so do all of the polygons in the family. Any two compact primitive semitoric polygons with the same fan can be connected by a continuous path. This path is made by continuously changing the lengths of the edges because the angles of the two polygons must all be the same since they have the same fan. So, by Proposition 6.20 given two elements of DPolyg m f , k (R 2 ) which are compact we know that the corresponding fans are related by the moves listed in that proposition and then by Lemma 5.4 we know these moves correspond to continuous paths of polygons. So we have established that any two compact elements of DPolyg m f , k (R 2 ) are connected by a continuous path.
Next assume that is such that ∆ is noncompact but has only finitely many vertices. Choose N ∈ R such that all of the vertices of ∆ are in the region {(x, y) ∈ R | −N < x < N }. and which has all of the same edge lengths as ∆ with the exception of the two or four edges which intersect N ∪ −N . These edges are extended horizontally by a length of 1 t − 1. By this we mean that if an edge of ∆ which intersected {(x, y) ∈ R 2 | x = N } had as one of its endpoints (x 0 , y 0 ) with x 0 > N then the corresponding edge of ∆(t) would have as its endpoint (x 0 + 1 t − 1, y 0 ). Then define ∆(1) = ∆ and we can see that ∆(t) for t ∈ [0, 1] is a path from ∆ , which is compact, to ∆ so [(∆(t), (λ j , +1, k j ) m f j=1 )] is a continuous path which connects a compact semitoric polygon to [∆ w ]. This process is shown in Figure 6. Now we have connected all of the elements except for those with an infinite amount of vertices. Suppose that is such that ∆ is noncompact and has infinitely many vertices. We will connect [∆ w ] to a polygon which has only finitely many vertices to finish the proof. Since ∆ has everywhere finite height and is the intersection of infinitely many half-planes we can choose two of these planes which are not horizontal and are not parallel to one another. Denote the intersection of these two half-planes by A and notice ∆ ⊂ A. Since the boundaries of these two half-planes must intersect we can see Figure 6: The continuous path from a compact primitive semitoric polygon to a noncompact primitive semitoric polygon with finitely many vertices.
that A can only be unbounded in either the positive or negative x-direction, but not both. Without loss of generality assume that A is unbounded in the positive x-direction.
Let ν be any admissible measure. For any n ∈ Z 0 since ν(A) < ∞ we know there exists some x n ∈ R such that ν(A ∩ [x n , ∞)) < 1 /n, ∆ does not have a vertex on the line xn , and x n > |λ j | for all j = 1, . . . , m f . Let ∆ n denote the polygon which satisfies and has no vertices with x-coordinate greater than x n .
Now we have that 0 < h j (t) < len j (t) and t → h j (t) is a continuous function from [0, 1] to R because it is impossible for a semitoric polygon to have a vertical boundary at λ j for any j ∈ {1, . . . , m f }. Now define for t ∈ [0, 1] which is a continuous path from m to m .
Thus we have established the following result.
Theorem 6.24. The set of connected components of T is and each T m f , k is path-connected.