Classifying Toric and Semitoric Fans by Lifting Equations from SL₂(Z)
We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group SL₂(Z) to its preimage in the universal cover of SL₂(R) . With this method, we recover the classification of two-dimensional toric fans and obtain a description of their...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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Інститут математики НАН України
2018
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| Цитувати: | Classifying Toric and Semitoric Fans by Lifting Equations from SL₂(Z) / D.M. Kane, J. Palmer, Á. Pelayo // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 48 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859901827555786752 |
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| author | Kane, D.M. Palmer, J. Pelayo, Á. |
| author_facet | Kane, D.M. Palmer, J. Pelayo, Á. |
| citation_txt | Classifying Toric and Semitoric Fans by Lifting Equations from SL₂(Z) / D.M. Kane, J. Palmer, Á. Pelayo // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 48 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group SL₂(Z) to its preimage in the universal cover of SL₂(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.
|
| first_indexed | 2025-12-07T15:57:37Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 016, 43 pages
Classifying Toric and Semitoric Fans
by Lifting Equations from SL2(Z)
Daniel M. KANE, Joseph PALMER and Álvaro PELAYO
University of California, San Diego, Department of Mathematics,
9500 Gilman Drive #0112, La Jolla, CA 92093-0112, USA
E-mail: dakane@ucsd.edu, j.palmer@math.rutgers.edu, alpelayo@ucsd.edu
Received April 17, 2017, in final form February 13, 2018; Published online February 22, 2018
https://doi.org/10.3842/SIGMA.2018.016
Abstract. We present an algebraic method to study four-dimensional toric varieties by
lifting matrix equations from the special linear group SL2(Z) to its preimage in the universal
cover of SL2(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.
Key words: symplectic geometry; integrable system; semitoric integrable systems; toric
integrable systems; focus-focus singularities; SL2(Z)
2010 Mathematics Subject Classification: 52B20; 15B36; 53D05
To Tudor S. Ratiu on his 65th birthday, with admiration.
1 Introduction
Toric varieties [10, 11, 13, 18, 33, 34] have been extensively studied in algebraic and differential
geometry and so have their symplectic analogues, usually called symplectic toric manifolds or
toric integrable systems [6]. The relationship between symplectic toric manifolds and toric
varieties has been understood since the 1980s, see for instance Delzant [14], Guillemin [22, 23].
The article [15] contains a coordinate description of this relation.
In this paper we present an algebraic viewpoint to study non-singular complete four-dimen-
sional toric varieties, based on the study of matrix relations in the special linear group SL2(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
(v0 = vd, v1, . . . , vd−1) ∈
(
Z2
)d
,
where d ∈ Z is the number of faces. A Delzant polygon (or toric polygon) is one for which
det(vi, vi+1) = 1 for each 0 6 i 6 d − 1. This determinant condition forces the vectors to
satisfy the linear equations aivi = vi−1 + vi+1, for i = 0, . . . , d − 1 where v−1 = vd−1, which
are parameterized by integers a0, . . . , ad−1 ∈ Z (see Lemma 3.3 and Fulton [18, p. 43]). These
integers satisfy(
0 −1
1 a0
)(
0 −1
1 a1
)
· · ·
(
0 −1
1 ad−1
)
=
(
1 0
0 1
)
(1.1)
mailto:dakane@ucsd.edu
mailto:j.palmer@math.rutgers.edu
mailto:alpelayo@ucsd.edu
https://doi.org/10.3842/SIGMA.2018.016
2 D.M. Kane, J. Palmer and Á. Pelayo
(this equation appears in [18, p. 44]) but in fact, as can be seen in the exercises on p. 44 of [18],
not all integers satisfying equation (1.1) correspond to a toric fan. To deal with this issue we
take an approach similar to that employed in [42]. In particular, the identity in equation (1.1)
in SL2(Z) can be lifted to the group G presented as
G ∼=
〈
S, T |STS = T−1ST−1
〉
and which, we will see in Lemma 3.2, satisfies SL2(Z) ∼= G/
(
S4
)
. The group G, as is shown
in [42, Section 8.3], is isomorphic to the pre-image of SL2(Z) in the universal cover of SL2(R)
(Proposition 3.7), and thus we can define what we call the winding number of an element of
g ∈ G that evaluates to the identity in SL2(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.
By considering a lift of equation (1.1) to the group G, we obtain an equation with the
property that a collection of integers a0, . . . , ad−1 is a solution if and only if they correspond to
a toric fan. Furthermore, from the integers it is straightforward to recover the fan up to the
appropriate isomorphism. Thus, the collection of toric fans can be studied by instead studying
all a0, . . . , ad−1 which satisfy the equation in G. This allows us to prove some classical results
about toric fans from a new perspective (Section 4), and generalize these results to the semitoric
case (Section 5).
Recall from [37, 38] that a semitoric integrable system is a completely integrable system with
two degrees of freedom and for which one of the Hamiltonians generates a periodic flow (see
Definition 6.1). Semitoric systems are a generalization of toric integrable systems which can
have, in addition to elliptic type singularities, also focus-focus singularities (called nodal singu-
larities in the context of Lefschetz fibrations and algebraic geometry). Focus-focus singularities
appear in algebraic geometry [19], symplectic topology [17, 32, 44, 47], and many simple physical
models such as the spherical pendulum [1] and the Jaynes–Cummings system [12, 26, 41]. Asso-
ciated to a semitoric system there is also a collection of vectors (v0, . . . , vd−1), defined up to the
appropriate notion of isomorphism, which satisfy more complicated equations (given explicitly
in Definition 2.9). Any collection of such vectors is henceforth referred to as a semitoric fan.
A semitoric fan can be thought of as a toric fan for which the relations between some pairs of
adjacent vectors have been changed as a result of the presence of the focus-focus singularities.
Roughly speaking, semitoric fans encode aspects of 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 [5], and mirror symmetry,
see Kontsevich–Soibelman [20, 21, 30].
We present two theorems in this paper which are applications of the algebraic method we
introduce. The first gives a classification of semitoric fans and the second is an application
of the first which describes the path-connected components of the moduli space of semitoric
integrable systems with a fixed number of focus-focus singular points.
Theorem 1.1. Any semitoric fan may be obtained from a standard semitoric fan in a finite
number of steps using four standard transformations.
A more detailed description of Theorem 1.1 is given in Theorem 2.14 and the definitions of
the standard semitoric fans and the four standard transformations used in Theorem 1.1 are given
in Definition 2.11. As an application of Theorem 1.1 to the symplectic geometry of integrable
systems we will prove Theorem 1.2, which is the second main theorem of the paper.
The articles [37, 38] give a classification of semitoric systems in terms of several invariants,
which can be essentially encoded into a polygon (a generalization of its toric counterpart and
very closely related to a semitoric fan) together with finitely many marked interior points, each of
Classifying Toric and Semitoric Fans by Lifting Equations from SL2(Z) 3
which is labeled by a formal power series in two variables and an integer. In contrast to the toric
case, a semitoric system has a family of polygons associated to it, each constructed using a toric
momentum map defined by removing certain closed subsets; the collection of all such polygons
associated to a given system is henceforth referred to as the semitoric polygon associated to
the system. The marked interior points are in fact the images of the focus-focus points of the
integrable system and the formal power series in two variables determines the semiglobal model
around the focus-focus fiber up to a suitable notion of isomorphism (cf. [43, 45]). The collection
of integers labeling the focus-focus points for each polygon is the twisting index invariant. This
invariant encodes how the semiglobal model of the focus-focus fiber “sits” relative to the toric
momentum map used to define each element of the semitoric polygon, so the integer assigned
to a specific focus-focus point depends on the choice of polygon but the difference between the
twisting index of consecutive (ordered according to the x-component) values does not depend
on the choice of polygon and, roughly speaking, this difference measures the “twist” in the
topology of the singular Lagrangian fibration between them. Here we have not described the
group action on the space of possible invariants, which is necessary so that choices made when
constructing the invariants do not effect the outcome. See Section 6.1 for the precise definition
of these invariants and for a discussion of the group action.
For the next result we need to consider the moduli space of semitoric systems as a topological
space so we can study continuous families in this space. The natural topology in this situation
is the one induced by the metric on the moduli space of semitoric systems [35] and which is
described in Section 6.2.3.1 The metric is produced by pulling back a metric from the space of
all invariants of semitoric integrable systems. The role played by the polygon invariant in the
definition of this metric is related to the Duistermaat–Heckman measure.
The twisting index invariant of a system is an equivalence class of a list of integers assigned to
each choice of toric momentum map. The integer associated to a given focus-focus point is not
a well-defined invariant when considered individually, but one can define the “twisting index
class” of a system. Two systems are in the same twisting index class if there exists a choice of
polygon for each system such that the integers for these polygons agree (the precise notion is in
Definition 6.15). Now we can state our second main result.
Theorem 1.2. With respect to the topology induced by the metric defined in [35], any two
semitoric systems with the same number of focus-focus singular points which are in the same
twisting index class may be continuously deformed into one another, up to isomorphism, via a
continuous path of semitoric systems with the same number of focus-focus singular points and
in the same twisting index class.
We state a detailed version of this result later in the paper as Theorem 2.19.
Remark 1.3. The topology from [35] automatically places isomorphism classes of systems with
different numbers of focus-focus points into different components of the space. It would be very
interesting to have a version of the metric, or at least topology, from [35] which would allow us
to study deformations of an isomorphism class of semitoric systems into another with a different
number of focus-focus points but such a comparison gives rise to several nontrivial issues which
are beyond the scope of the present paper. The most pressing issue being if such a deformation
can really occur through semitoric systems, i.e., whether the non-degeneracy condition can be
satisfied for all values of the deformation (under some extra natural hypotheses, degeneracies
must occur, see [25, Proposition 2.8]).
The Jaynes–Cummings system is an important example of a semitoric system with precisely
one focus-focus point and is studied for example in [41] (systems with exactly one focus-focus
1For this paper we only define this metric on certain subsets of the full moduli space because this is all that
is necessary to produce the same topology, see Remark 6.23.
4 D.M. Kane, J. Palmer and Á. Pelayo
point are referred to as systems of Jaynes–Cummings type for this reason). Since there is only
one twisting index class of semitoric systems with exactly one focus-focus point, Theorem 1.2
implies the following.
Corollary 1.4. With respect to the topology induced by the metric defined in [35], any isomor-
phism class of semitoric system with precisely one focus-focus singular point may be continuously
deformed into the isomorphism class of the Jaynes–Cummings system via a continuous path of
isomorphism classes of semitoric systems, each of which have precisely one focus-focus point.
Semitoric systems have been studied by mathematicians and physicists in the past decade, and
there have been contributions to their study from many angles, including mathematical physics
(see, e.g., Babelon–Douçot [3, 4], Dullin [16]). While this paper deals with classical integrable
systems, much of the work on these systems is motivated by inverse spectral problems about
quantum integrable systems as pioneered in the work of Colin de Verdière [8, 9] and others, and
which also has been the subject of recent works [7, 48].
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 struc-
ture 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 about toric fans and in Section 5 we generalize these results to semitoric fans. Indeed,
in Section 6 we use the results of Section 5 to study the connectivity of the moduli space of
semitoric systems.
2 Fans, symplectic geometry, and winding numbers
2.1 Toric fans
A toric variety is a variety which contains an algebraic torus as a dense (in the Zariski topology)
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 [18]. By an algebraic
torus we mean the product C∗ × · · · × C∗, where C∗ = C \ {0}. Under some mild assumptions
(which are automatically satisfied if the variety is smooth), the geometry of a toric variety is
completely determined by the associated fan [18, Section 1.5].
Definition 2.1. A rational strongly convex cone is a convex cone of a vector space with apex
at the origin generated by a finite number of integral vectors which contains no line through the
origin. A fan is a 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.
Definition 2.1 is the general notion of a fan in arbitrary dimension; in this paper we will be
concerned with two-dimensional smooth compact toric varieties and their associated fans, which
for simplicity we will call toric fans. We identify such fans with a sequence of points in Z2.
Definition 2.2. A toric fan is a sequence of lattice points
(v0 = vd, v1, . . . , vd−1) ∈
(
Z2
)d
labeled in counter-clockwise order such that each pair of adjacent vectors generates all of Z2 and
the angle between any two adjacent vectors is less than π radians. That is, det(vi, vi+1) = 1 for
i = 0, . . . , d− 1.
Classifying Toric and Semitoric Fans by Lifting Equations from SL2(Z) 5
Figure 1. (a) The three minimal models from Theorem 2.5. (b) An illustration of a Delzant polygon
produced by corner chopping the Hirzebruch trapezoid.
Definition 2.3. For the purposes of this paper by convex polygon we mean the intersection
in R2 of finitely or infinitely many closed half planes such that there are at most finitely many
corner points in each compact subset of R2.
Definition 2.4. A Delzant polygon (or toric polygon) is a compact convex polygon ∆ in R2
which is simple, rational, and smooth. Recall v ∈ Z2 is primitive if v = kw for some k ∈ Z>0
and w ∈ Z2 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 edge e of ∆ there exists a vector in Z2 which is normal to e;
3) ∆ is smooth if the inwards pointing primitive vectors normal to any pair of adjacent edges
form a basis of Z2.
Delzant polygons were introduced in the work of Delzant [14] in symplectic geometry, who
built on the work of Atiyah [2], Kostant [31], and Guillemin-Sternberg [24] to give a classification
of symplectic toric 4-manifolds in terms of the Delzant polygon (in fact, their work was in any
dimension and higher dimensional toric integrable systems correspond to higher dimensional
Delzant polytopes which satisfy analogous conditions to those given in this Definition 2.4).
Delzant polygons are similar to Newton polygons, as in [18], 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 by considering the collection of inwards
pointing normal vectors of the polygon. 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.
The natural action of SL2(Z) on R2 induces an action on the set of Delzant polygons. In [18]
Fulton stated the following result for toric fans. We have adapted the statement to relate it to
Delzant polygons.
Theorem 2.5 (Fulton [18, p. 44]). Up to the action of SL2(Z), every Delzant polygon can be
obtained from a Delzant triangle, rectangle, or Hirzebruch trapezoid by a finite number of corner
chops.
The minimal models (the Delzant triangle, rectangle, and Hirzebruch trapezoid with param-
eter k > 1) are defined in Definition 4.9 and depicted in Fig. 1. The corner chop operation
is defined in Definition 2.11 and is a standard operation in algebraic and symplectic geometry
(see, for instance, [28]). It corresponds to an equivariant symplectic blowup.
6 D.M. Kane, J. Palmer and Á. Pelayo
The proof of Theorem 2.5 sketched by Fulton in [18, Section 2.5], which uses only two-
dimensional geometry and basic combinatorial arguments, is relatively long and does not im-
mediately generalize to the case of semitoric polygons we study in this paper. In Section 4
we provide an alternative proof using SL2(Z)-relations which may be easily extended to the
semitoric case.
The moduli space of toric polygons was endowed with a metric given by the Lebesgue measure
of the symmetric difference in [36], which we denote by dT, and in that paper the following
consequence of Theorem 2.5 was proved:
Theorem 2.6 ([36]). The moduli space of toric polygons is path-connected with respect to the
topology induced by dT.
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 2.5. Then one shows, using elementary analysis, that the three minimal models can
be continuously transformed into one another and that the corner chop operation is continuous.
Again, let us emphasize that Theorem 2.5 is not new, but in this paper we will prove it from
a new viewpoint which we believe to be more natural and which generalizes to the case of
semitoric fans.
2.2 Semitoric fans
Motivated by semitoric polygons (originally defined in [38, Definition 2.5]) we define semitoric
fans. A semitoric polygon is a family of polygons which are similar to Delzant polygons except
that the presence of the focus-focus singular points in semitoric systems may cause some corners
of the polygons to not satisfy the smoothness condition of Definition 2.4. Such corners must
satisfy other conditions and are known as fake or hidden corners. They can exist on the top
or bottom boundary of the polygon but there will always be an infinite subfamily of semitoric
polygons related to the system for which the hidden and fake corners are all on the top boundary,
and we will use one of these polygons to produce the associated semitoric fan. The choice of
which polygon with this property is used is reflected in the symmetry group of semitoric fans
which we introduce in Definition 2.10. Let
T =
(
1 1
0 1
)
.
Definition 2.7 ([38, Section 4.1 and Definition 4.1]). Let v, w ∈ Z2. The ordered pair (v, w) of
vectors:
1) is on the top boundary if both vectors are in the open lower half-plane;
2) satisfies the Delzant condition if det(v, w) = 1;
3) satisfies the hidden condition if det(v, Tw) = 1; and
4) satisfies the fake condition if det(v, Tw) = 0.
Remark 2.8. Notice that a pair (v, w) satisfies both the fake and Delzant conditions if and
only if
v =
(
k + ε
ε
)
and w =
(
k
ε
)
for some k ∈ Z and ε ∈ {−1,+1} and in order for such a pair to be in the top boundary we can
only have the case in which ε = −1.
Classifying Toric and Semitoric Fans by Lifting Equations from SL2(Z) 7
Definition 2.9. Let d ∈ Z with d > 2. A semitoric fan is a collection of primitive vectors
(v0 = vd, v1, . . . , vd−1) ∈
(
Z2
)d
labeled in counter-clockwise order such that each pair of adjacent
vectors (vi, vi+1) for i ∈ {0, . . . , d−1} is labeled as a Delzant, fake, or hidden corner. We require
that each labeled pair of vectors satisfies the corresponding condition from Definition 2.7 and
we further require that all fake and hidden corners be on the top boundary. The defect of
a semitoric fan is the number of corners which are either fake or hidden.
Notice that the labeling of the pairs is required only because of the case described in Re-
mark 2.8 in which a pair can satisfy both the fake and Delzant conditions. In all other cases the
corner type of a pair of vectors can be uniquely determined by inspecting the vectors involved.
Definition 2.9 is inspired by the toric case. Theorem 2.5 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 2.10. The symmetry group of semitoric fans is given by
G′ =
{
T k | k ∈ Z
}
,
where G′ acts on a semitoric fan by acting on each vector in the fan in the standard fashion.
Definition 2.11.
1. Let c ∈ Z>0. The standard semitoric fan of defect c is the fan (u0, . . . , uc+3) ∈
(
Z2
)c+4
given by
u0 =
(
0
−1
)
, u1 =
(
1
0
)
, u2 =
(
c
1
)
, u3 =
(
−1
0
)
,
and
u4+n =
(
−c+ n
−1
)
for n = 0, . . . , c−1 in which the first four pairs of vectors are Delzant corners and the rest
are fake corners.
2. Let (v0 = vd, . . . , vd−1) ∈
(
Z2
)d
be a semitoric fan. The following are called the four fan
transformations:
(a) Suppose that (vi, vi+1) is a Delzant corner for some i ∈ {0, . . . , d− 1}. Then
(v0, . . . , vi, vi + vi+1, vi+1, . . . , vd−1) ∈
(
Z2
)d+1
obtained by inserting the sum of two adjacent vectors between them is a semitoric fan
with the new pairs (vi, vi+ vi+1) and (vi+ vi+1, vi+1) both labeled as Delzant corners
(Lemma 5.1). The process of producing this new fan from the original generalizes
corner chopping for toric fans (cf. [28]). For this reason, it is also referred to as
a corner chop(ping) in the context of semitoric fans.
(b) Suppose that (vi−1, vi) and (vi, vi+1) are Delzant corners and that vi = vi−1 + vi+1.
Then
(v0, . . . , vi−1, vi+1, . . . , vd−1) ∈
(
Z2
)d−1
is a semitoric fan with the pair (vi−1, vi+1) being labeled as a Delzant corner (Lem-
ma 5.2). This process of producing this new fan from the original generalizes reverse
corner chopping for toric fans (cf. [28]). For this reason, it is also referred to as
a reverse corner chop(ping) in the context of semitoric fans.
8 D.M. Kane, J. Palmer and Á. Pelayo
(c) Suppose that the pair (vi, vi+1) is a hidden corner. Then
(v0, . . . , vi, T vi+1, vi+1, . . . , vd−1) ∈
(
Z2
)d+1
is a semitoric fan with (vi, T vi+1) a Delzant corner and (Tvi+1, vi+1) a fake corner
(Lemma 5.3). The process of producing this fan is called removing the hidden corner
(vi, vi+1).
(d) Suppose that the pair (vi, vi+1) is a fake corner and the pair (vi+1, vi+2) is a Delzant
corner on the top boundary. Then
(v0, . . . , vi, T vi+2, vi+2, . . . , vd−1) ∈
(
Z2
)d
is a semitoric fan with (vi, T vi+2) a Delzant corner and (Tvi+2, vi+2) a fake corner
(Lemma 5.4). The process of producing this fan is called commuting a fake and
a Delzant corner.
Remark 2.12. The corner chop and reverse corner chop can be obtained as equivariant sym-
plectic blowups and blowdowns, respectively.
Remark 2.13. The standard semitoric fans from Definition 2.11 are, for the purposes of this
paper, analogous to the minimal models from toric geometry (as in Theorem 2.5) because any
semitoric fan can be transformed into a standard semitoric fan using the four fan transformations.
That being said, the set of standard semitoric fans does not correspond to the set of all minimal
semitoric fans, which are those fans which do not admit a reverse corner chop. The problem
of classifying all minimal semitoric fans is being addressed in a future paper (which is now
available, see [27]).
The transformations of removing a hidden corner and commuting a fake and a Delzant corner
can also be reversed, but we will not need the inverses of those operations for the proofs in this
paper, so we do not list them here. Using the algebraic results from Section 3 we show the
following in Section 5.
Theorem 2.14. Let d > 3 be an integer. Any semitoric fan (v0, . . . , vd−1) ∈
(
Z2
)d
of defect
c ∈ Z>0 may be transformed into a semitoric fan G′-equivalent to the standard semitoric fan of
defect c by using the four fan transformations finitely many times.
Remark 2.15. The method we are using to study semitoric manifolds is analogous to the
method we use to study toric manifolds. Theorem 2.5 explains how to generate the toric poly-
gons and is used to prove that the space of toric polygons is path-connected (Theorem 2.6)
which implies that the space of toric manifolds is path-connected (Theorem 2.18). Similarly,
Theorem 2.14 shows how to generate the semitoric polygons, and as an application we prove
Lemma 6.31 which describes the connected components in the space of semitoric ingredients
(Definition 6.13) and this implies Theorem 2.19, which describes the connected components of
the moduli space of semitoric systems.
Remark 2.16. It is likely that there are connections between our work and that of Gross and
Siebert [19, 20, 21]. In particular, it seems to us that one of the objects that Gross and Siebert
use is very closely related to the notion of a semitoric fan. Understanding this would be an
entirely different project which merits further study.
Classifying Toric and Semitoric Fans by Lifting Equations from SL2(Z) 9
2.3 Algebraic tools: the winding number
It is shown in Lemma 3.2 that the special linear group SL2(Z) may be presented as
SL2(Z) =
〈
S, T |T−1ST−1 = STS, S4 = I
〉
,
where
S =
(
0 −1
1 0
)
and T =
(
1 1
0 1
)
.
Thus equation (1.1) becomes
ST a0 · · ·ST ad−1 = I, (2.1)
where a0, . . . , ad−1 ∈ Z and I denotes the 2 × 2 identity matrix. Given v0, v1 ∈ Z2 with
det(v1, v2) = 1 a set of vectors
(v0, v1, . . . , vd−1) ∈
(
Z2
)d
may be produced by
vi+2 = −vi + aivi+1
for i = 0, . . . , d− 1 where we define vd = v0 and vd+1 = v1. In this way associated to each list of
integers satisfying equation (1.1) there is an ordered collection of vectors unique up to SL2(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 counter-clockwise order, then they are a toric fan. The reason that
not all sequences of integers which satisfy equation (1.1) correspond to a toric fan is that the
vectors v0, . . . , vd−1 ∈ Z2 may circle more than once around the origin, and thus not be labeled
in counter-clockwise order (see Fig. 2). Thus, we see that equation (2.1) is merely a necessary
and not sufficient condition for a sequence (a0, . . . , ad−1) to correspond to a toric fan.
Let K = ker(Z ∗ Z→ SL2(Z)) where Z ∗ Z denotes the free group with generators2 S and T
and the map Z ∗ Z → SL2(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 ∈ Z2 one term at a time. We
know that this sequence ends 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 R2 \ {(0, 0)} by
considering the piecewise linear path between the ends of the vectors, see Definition 4.4. 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
gives a precise definition of the number of times an ordered collection of vectors circles the
origin. Let w : Z ∗ Z→ Z be the unique homomorphism satisfying
w(S) = 3 and w(T ) = −1.
We find that if σ ∈ K then w(σ) is a multiple of 12 and w(σ)/12 is the winding number associated
to the word σ, see Lemma 4.2. We present the group
G =
〈
S, T |STS = T−1ST−1
〉
2Starting in Section 3 we will adopt Notation 3.1 to differentiate between the generators S and T in the
different groups.
10 D.M. Kane, J. Palmer and Á. Pelayo
on which w descends to a well-defined homomorphism wG : G → Z. In fact, G is isomorphic
to the pre-image of SL2(Z) in the universal cover of SL2(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 SL2(R). The fundamental group of SL2(R) is Z and the classical winding number of this loop
in SL2(R) coincides with wG(g)/12. Finally, in Corollary 4.6 we show that integers a0, . . . , ad−1 ∈ Z
correspond to a toric fan if and only if the equality
ST a0 · · ·ST ad−1 = S4
is satisfied in G. This correspondence is the basis of our method to study toric and semitoric
fans.
Remark 2.17. We note that [42] also needed to make use of this winding number for similar
reasons. They define the homomorphism w (Φ in their notation) in Section 8.4, and come up
with an interesting interpretation of it. While w(γ) can be interpreted as a winding number for γ
that map to the identity in SL2(Z), for other values of γ, the interpretation is not so obvious.
Firstly, [42] come up with an interpretation for elements of the universal cover of SL2(R).
In particular, they write elements of the group as pairs (M, [γ]) where M ∈ SL2(R) and [γ]
a homotopy class of paths in R2−{0} from b to Mb for some fixed basepoint b. The composition
rule can then be written as (M, [γ])(M ′, [γ′]) = (MM ′, [γM
′
γ′]) where γM
′
denotes the path
obtained by applying M ′ pointwise to γ, and the multiplication with γ′ denotes concatenation.
In this language, [42] find an interpretation of w relating it to the modular form ∆. In
particular, if M ∈ SL2(Z) is given by M =
(
a b
c d
)
, then treating M as a linear fractional
transformation in the standard way, we have that
∆(Mz) = (cz + d)12∆(z).
Taking logs of this one finds that log(∆(Mz))− log(∆(z)) is some branch of 12 log(cz+d). If M
is given not just as a matrix, but as the endpoint of a path from the identity in SL2(R), then
this defines the appropriate branch. If M corresponds to the element (M, [γ]) in the universal
cover of SL2(R) (where γ uses z as its basepoint), [42] show that
log(∆(Mz))− log(∆(z)) = 12 log(cz + d) + 2πiw((M,γ)), (2.2)
where the branch of log(cz + d) is defined using the path γ. In fact, [42] use equation (2.2)
to define w, and then show in Section 8.5 the action on generators. We note that from this
expression, it is clear that if M = I, that since log(∆) is analytic on the upper half plane, that
w((M,γ)) comes entirely from the branch of γz. In particular, this means that w((I, γ)) is 12
times the winding number of γz.
2.4 Applications to symplectic geometry
While Delzant polygons are in correspondence with closed 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 : M → R2) where (M,ω) is a connected
symplectic 4-manifold and F : M → R2 is an integrable system given by two maps J,H : M → R2
such that J is a proper map which generates a periodic flow (see Definition 6.1 for the precise
definition). In [38] the authors prove a result analogous to Delzant’s, classifying semitoric
systems satisfying a mild assumption via a list of ingredients which includes a family of polygons
(there is an overview of this result in Section 6.1).
In [35] the second author defines a metric space structure for the moduli space of semi-
toric systems using [38], which is related to the Duistermaat–Heckman measure. The metric
Classifying Toric and Semitoric Fans by Lifting Equations from SL2(Z) 11
is designed to induce a topology that respects continuous transformations of the invariants of
semitoric systems (this follows tautologically from the definition) and to be compatible with the
known topology on toric integrable systems from [36], in the sense that the projection map from
isomorphism classes of compact semitoric systems with zero focus-focus points to toric systems
is continuous, as proven in [35, Corollary 4.14] (this map is an projection since the isomorphism
for toric integrable systems is strictly weaker than that for semitoric systems). A natural ques-
tion is whether, with respect to a such structure, the space is path-connected. That is, can two
isomorphism classes of semitoric systems be continuously deformed (with respect to the above
topology) into one another, via a continuous path of isomorphism classes of semitoric systems?
This is preceded by [36] in which the authors construct a natural metric on the moduli space
of symplectic toric manifolds which is related to the Duistermaat–Heckman measure and prove
Theorem 2.6, which is used to conclude the following.
Theorem 2.18 ([36]). With respect to the topology defined in [36], the moduli space of toric
manifolds is path-connected.
Similarly, Theorem 2.14 implies the following statement.
Theorem 2.19. If (M,ω, F ) and (M ′, ω′, F ′) are simple semitoric integrable systems such that:
(i) they have the same number of focus-focus singularities;
(ii) they are in the same twisting index class,
then there exists a continuous (with respect to the topology defined in [35]) path of semitoric
systems with the same number of focus-focus points and in the same twisting index class between
them, up to isomorphism. That is, the space of isomorphism classes of semitoric systems with
fixed number of focus-focus points and in a fixed twisting index class is path-connected.
The number of focus-focus singularities is discussed in Section 6.1.1, the twisting index class
of a semitoric system is defined in Definition 6.15, and the notion of a simple semitoric integrable
system is defined in Definition 6.1.
Remark 2.20. It is important to note that while there is only one twisting index class for
systems with exactly one focus-focus point, this does not mean that the twisting index invariant
is trivial in this case. That is, there exist systems with exactly one focus-focus point for which
all invariants agree except for the twisting index and the systems are not isomorphic.
3 Algebraic set-up: matrices and SL2(Z) relations
Notation 3.1. We will present several different groups on generators S and T because it is
important to be able to easily see the standard homomorphisms between these groups. When
referring to a word in S and T we will use a subscript to indicate which group that word belongs
to. For instance, we write (S4)SL2(Z) to refer to the element of SL2(Z). 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 , so σ =H η is shorthand for
(σ)H = (η)H . For example, S4 =SL2(Z) I but S4 6=G I. Finally, if v ∈ Z2 then Tv will always
mean TSL2(Z)v.
The 2× 2 special linear group over the integers, SL2(Z), is generated by the matrices
SSL2(Z) =
(
0 −1
1 0
)
and TSL2(Z) =
(
1 1
0 1
)
.
12 D.M. Kane, J. Palmer and Á. Pelayo
We will see that to each closed toric (resp. semitoric) integrable system there is an associated
toric (resp. semitoric) fan and we will use the algebraic structure of SL2(Z) to study these fans.
For our purposes, the following presentation of SL2(Z) will be the most natural way to view the
group.
Lemma 3.2. The 2× 2 special linear group over the integers, SL2(Z), may be presented as
SL2(Z) ∼=
〈
S, T |T−1ST−1 = STS, S4
〉
.
Proof. It is well-known that
SL2(Z) ∼=
〈
S, T | (ST )3 = S2, S4
〉
(3.1)
(see for instance [29, equation (A.2)]). The result follows from equation (3.1) and the observation
that
(ST )3 =SL2(Z) S
2 ⇔ STS =SL2(Z) T
−1ST−1. �
For v, w ∈ Z2 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].
Lemma 3.3. Let u, v, w ∈ Z2 and det(u, v) = 1. Then det(v, w) = 1 if and only if there exists
some a ∈ Z such that w = −u+ av.
Proof. In the basis (u, v) we know that v = ( 0
1 ). Write w = ( ba ) for some a, b ∈ Z. Then we
can see that det(v, w) = −b so det(v, w) = 1 if and only if b = −1. That is, w = −u+ av. �
The result of Lemma 3.3 can be easily summarized in a matrix equation, as we will now show.
Let
(v0 = vd, v1 = vd+1, v2, . . . , vd−1) ∈
(
Z2
)d
be a toric fan and define Ai = [vi, vi+1] for i = 0, . . . , d. Note that Ad = A0.
Lemma 3.4. For each i ∈ 0, . . . , d− 1 there exists an integer ai ∈ Z such that Ai+1 = AiST
ai.
Proof. By the definition of a toric fan we know that for each 0 6 i < d− 2 we have that
det(vi, vi+1) = det(vi+1, vi+2) = 1
so by Lemma 3.3 there exists ai ∈ Z such that vi+2 = −vi + aivi+1. Then
AiST
ai = [vi+1,−vi + aivi] = [vi+1, vi+2] = Ai+1,
and this concludes the proof. �
It follows that
Ad = Ad−1
(
ST ad−1
)
SL2(Z)
= Ad−2
(
ST ad−2ST ad−1
)
SL2(Z)
= · · ·
= A0
(
ST a0 · · ·ST ad−1
)
SL2(Z)
,
which means A0 = Ad = A0(ST a0 · · ·ST ad−1)SL2(Z), and so
ST a0 · · ·ST ad−1 =SL2(Z) I. (3.2)
This is a restatement of equation (1.1) which is from [18, p. 44]. So to each toric fan of d vectors
there is an associated d-tuple of integers which satisfy equation (3.2), but having a tuple of
integers which satisfy equation (3.2) is not enough to assure that they correspond to a toric fan.
The determinant of the vectors will be correct but, roughly speaking, if the vectors wind around
the origin more than once then they will not be labeled in the correct order to be a toric fan, as
it occurs in the following example.
Classifying Toric and Semitoric Fans by Lifting Equations from SL2(Z) 13
Figure 2. These vectors do not form a fan because they are not labeled in counter-clockwise order.
Example 3.5. Consider the sequence of integers a0 = −1, a1 = −1, a2 = −2, a3 = −1, a4 = −1,
a5 = 0 and notice that
ST−1ST−1ST−2ST−1ST−1ST 0 =SL2(Z) I,
so equation (3.2) is satisfied but these integers do not correspond to any toric fan. This can be
seen by considering the vectors:
v0 =
(
1
0
)
, v1 =
(
0
1
)
, v2 =
(
−1
−1
)
, v3 =
(
1
0
)
, v4 =
(
−1
1
)
, v5 =
(
0
−1
)
which travel twice around the origin (a formal definition of this is given in Definition 4.4), see
Fig. 2, and form one possible sequence of vectors which can be obtained from the given integers
a0, . . . , a5. Since the action of SL2(Z) preserves the property of being labeled in counter-clockwise
order this means that none of the sequences of vectors which can be obtained from these integers
is a toric fan.
So we need extra information that is not captured by viewing this word in S and T as an
element of SL2(Z). For a more obvious example notice that even though they are equal in SL2(Z)
we can see that S4 corresponds to a toric fan while S8 does not. From [18, p. 44] we know that
integers (a0, . . . , ad) ∈ Zd which satisfy equation (1.1) correspond to a toric fan if and only if
a0 + · · ·+ ad−1 = 3d− 12,
so we would like to prove that
3d−
d−1∑
i=0
ai
12
is the number of times that the vectors corresponding to (a0, . . . , ad−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 lift to a group which projects onto SL2(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 SL2(Z) is a quotient of Z∗Z
by Lemma 3.2 so there exists a natural projection map π1 : Z ∗ Z→ SL2(Z). Also define a map
w : Z ∗ Z→ Z to be the unique homomorphism such that
w(SZ∗Z) = 3 and w(TZ∗Z) = −1.
14 D.M. Kane, J. Palmer and Á. Pelayo
Given any word in the kernel of the projection from Z∗Z to SL2(Z) we will see that w evaluates
to 12 times the image of the map to π1(SL2(R)) that is obtained by lifting the word in the
natural way to the universal cover of SL2(R), and w is the unique homomorphism on all of Z∗Z
with this property. Given a toric fan with associated integers (a0, . . . , ad−1) ∈ Zd we will show
that
w
((
ST a0 · · ·ST ad−1
)
Z∗Z
)
= 12.
Both π1 and w factor over the same group G which is the fiber product of SL2(Z) and Z
over Z/(12). Now we can see that we wanted the particular presentation of SL2(Z) from
Lemma 3.2 so that the relationship between G and SL2(Z) would be clear. This discussion
is made precise in the following proposition.
Proposition 3.6. The following diagram commutes:
Z ∗ Z
π1
zz
π2
��
w
��
SL2(Z)
wSL2(Z)
��
G
π3oo
wG
��
Z/(12) Z.π4
oo
The group G is the fiber product of SL2(Z) and Z over Z/(12) and satisfies
G ∼=
〈
S, T |STS = T−1ST−1
〉
,
each of π1, π2, π3, and π4 is a projection, and w : Z ∗ Z → Z, wG : G → Z, wSL2(Z) : SL2(Z) →
Z/(12) are given by the same formal expression
Sb0T a0 · · ·Sb`T a` 7→ 3
∑̀
i=0
bi −
∑̀
i=0
ai. (3.3)
Proof. It can be seen that the map wSL2(Z) : SL2(Z) → Z/(12) is well-defined by noting that
both relations in SL2(Z) as presented in Lemma 3.2 preserve the value of the formula (equa-
tion (3.3)) up to a multiple of 12. Similarly, since the relation STS =SL2(Z) T
−1ST−1 preserves
the value of the equation we know that wG 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 isomorphic to the fiber product of SL2(Z) and Z
over Z/(12) we must only show that wG restricted to the fibers is bijective. That is, we must
show that
wG �π−1
3 (A) : π−1
3 (A)→ π−1
4 (wSL2(Z)(A))
is a bijection for each A ∈ SL2(Z). To show it is surjective, fix A ∈ SL2(Z) and let g ∈ π−1
3 (A).
For any k ∈ Z notice π3
(
S4kg
)
= π3(g) so S4kg ∈ π−1
3 (A). Since wG
(
S4kg
)
= wG(g) + 12k and
π−1
4 (wSL2(Z)(A)) = wG(g) + 12Z the map is surjective. To show it is injective it is sufficient
to consider only A = I. Since
(
S4
)
G
is in the center of G we know that SL2(Z) ∼= G/
(
S4
)
so π−1
3 (I) =
{
S4k
G | k ∈ Z
}
. Since wG
(
S4k
G
)
= 12k we know for each choice of k this maps to
a distinct element of Z. �
Classifying Toric and Semitoric Fans by Lifting Equations from SL2(Z) 15
There is another useful sense in which G is an unwinding of SL2(Z). While SL2(Z) is discrete,
and thus does not have a natural cover, it sits inside the group SL2(R), which has a universal
cover. We claim that G is the preimage of SL2(Z) inside of the universal cover of SL2(R).
Proposition 3.7. The group G is isomorphic to the preimage of SL2(Z) within the universal
cover of SL2(R).
Proof. Let G′ be the preimage of SL2(Z) in the universal cover of SL2(R). We note that there
exists a homomorphism, φ from G to G′ defined by
φ(SG) =
(
cos
(
πt
2
)
− sin
(
πt
2
)
sin
(
πt
2
)
cos
(
πt
2
) )
06t61
, φ(TG) =
(
1 t
0 1
)
06t61
,
where the paths given are to represent elements of the universal cover of SL2(R). Consider
how φ(SG)φ(TG)φ(SG) and φ(TG)−1φ(SG)φ(TG)−1 act on any basis of Z2. Both paths will take
the basis to the same point without traversing completely around the origin (i.e., there is a ray
starting at the origin in R2 that the basis does not cross as either path acts on it). From this fact
it is easy to see that they are homotopic relative to endpoints, 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 SL2(Z). Furthermore, π = π′ ◦ φ.
Thus, to show that φ is an isomorphism, it suffices to show that φ : ker(π) → ker(π′) is an
isomorphism.
Notice that ker(π) is 〈S4
G〉. On the other hand, ker(π′) ∼= π1(SL2(R)) = Z, and is generated
by φ(SG)4. This completes the proof. �
We will see in Corollary 4.6 that (as shown in [42, equation (3)]) there is a one to one
correspondence between toric fans up to the action of SL2(Z) and lists of integers a0, . . . , ad−1 ∈ Z
satisfying
ST a0 · · ·ST ad−1 =G S
4. (3.4)
Equation (3.4) is a refinement of equation (3.2) which implies both that the successive pairs
of vectors form a basis of Z2 and that the vectors are labeled in counter-clockwise order. In
Proposition 5.6 we produce an analogous equation for semitoric fans.
Now we would like to simplify these toric fans. We will understand which integers a0, . . . , ad−1
are possible in an element (ST a0 · · ·ST ad−1)G ∈ G corresponding to a toric fan by studying
PSL2(Z) ∼= SL2(Z)/(−I). We again use Notation 3.1 and write SPSL2(Z) and TPSL2(Z) as the ele-
ments in PSL2(Z) corresponding to SSL2(Z) and TSL2(Z) in SL2(Z). The following lemma is impor-
tant for this and will also be useful later on when classifying semitoric fans. If (ST a0 · · ·ST ad−1)G
∈ G projects to the identity in SL2(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.
Lemma 3.8. Suppose that
ST a0 · · ·ST ad−1 =PSL2(Z) I (3.5)
for some d ∈ Z, d > 0. Then if d > 3 there exist i, j ∈ Z satisfying 0 6 i < j 6 d− 1 such that
ai, aj ∈ {−1, 0, 1}. Furthermore:
1) if d > 3 then i, j can be chosen such that i 6= j − 1 and (i, j) 6= (0, d− 1);
2) if d = 3 then a0 = a1 = a2 = 1 or a0 = a1 = a2 = −1;
16 D.M. Kane, J. Palmer and Á. Pelayo
3) if d = 2 then a0 = a1 = 0;
4) if d = 1 then equation (3.5) cannot hold.
Note that part (1) is the statement that i and j are not consecutive in the cyclic group Z/(d).
Of course, such i and j may be chosen if three or more elements of the list a0, . . . , ad−1 are in
the set {−1, 0, 1}.
Proof. It is well known that PSL2(Z) acts faithfully on the real projective line RP1 = R∪{∞}
by linear fractional transformations:(
a b
c d
)
(x) =
ax+ b
cx+ d
for x ∈ R and
(
a b
c d
)
(∞) =
a
c
.
Let d > 3. To find a contradiction suppose that at most two of a0, . . . , ad−1 are in {−1, 0, 1},
and also suppose that if there are two in {−1, 0, 1} that they are consecutive or indexed by 0
and d− 1. Notice that
ST a0 · · ·ST ad−1 =PSL2(Z) I implies that ST a1 · · ·ST ad−1ST a0 =PSL2(Z) I
by conjugating each side with (ST a0)PSL2(Z). This conjugation method and renumbering the
integers can be used to assure that ai /∈ {−1, 0, 1} for i = 1, . . . , d − 2. Since this expression is
equal to the identity in PSL2(Z) it acts trivially on R ∪ {∞}. In particular, we have(
ST a0 · · ·ST ad−1
)
PSL2(Z)
(∞) =∞.
Notice that SPSL2(Z)(x) = −1/x and T aPSL2(Z)(x) = x + a for a ∈ Z. Further notice that for any
a ∈ Z \ {−1, 0, 1} and x ∈ (−1, 1) \ {0} we have (ST a)PSL2(Z)(x) ∈ (−1, 1) \ {0}. We see that
(ST ad−1)PSL2(Z)(∞) = 0 and since ad−2 /∈ {−1, 0, 1} we know (ST ad−2)PSL2(Z)(0) ∈ (−1, 1)\{0}.
Putting these facts together we have(
ST a0 · · ·ST ad−1
)
PSL2(Z)
(∞) =
(
ST a0 · · ·ST ad−2
)
PSL2(Z)
(0)
=
(
ST a0 · · ·ST ad−3
)
PSL2(Z)
(x) for some x ∈ (−1, 1) \ {0}
=
(
ST a0
)
PSL2(Z)
(y) for some y ∈ (−1, 1) \ {0}
=
−1
y + a0
6=∞.
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(
ST a0ST a1ST a2
)
PSL2(Z)
(∞) =
(
ST a0ST a1
)
PSL2(Z)
(0) = −
(
−1
a1
+ a0
)−1
.
For this function to be the identity we would need
−1
a1
+ a0 = 0.
This implies that a0a1 = 1 so since they are both integers we have a0 = a1 = ε where ε ∈
{−1, 1}. Conjugating by (ST a0)PSL2(Z), we find symmetrically, that a1a2 = 1, and thus that
a0 = a1 = a2 = ±1.
If d = 2 then we have(
ST a0ST a1
)
PSL2(Z)
(∞) =
(
ST a0
)
PSL2(Z)
(0) =
−1
a0
,
Classifying Toric and Semitoric Fans by Lifting Equations from SL2(Z) 17
so we must have a0 = 0 and then(
ST 0ST a1
)
PSL2(Z)
(x) =
(
S2T a1
)
PSL2(Z)
(x) = x+ a1,
so we are also forced to have that a1 = 0, as stated in the lemma. If d = 1 then (ST a0)PSL2(Z)(∞)
= 0 for any choice of a0 ∈ Z, so there are no solutions. �
Note that ST 0ST 0 6=SL2(Z) I and ST 1ST 1ST 1 6=SL2(Z) I, so we have the following corollary.
Corollary 3.9. If d ∈ Z, d > 0, and a0, . . . , ad−1 ∈ Z are such that
ST a0 · · ·ST ad−1 =SL2(Z) I,
then d > 3. Furthermore, if d = 3 then a0 = a1 = a2 = −1.
Remark 3.10. In terms of toric fans, Corollary 3.9 is saying that every toric fan has at least
three vectors and, up to the action of SL2(Z), there is only one toric fan with exactly three
vectors (the fan of CP2).
4 Toric fans
Let a0, a1, . . . , ad−1 ∈ Z be a collection of integers such that
ST a0 · · ·ST ad−1 =SL2(Z) I.
This means that
ST a0 · · ·ST ad−1 =G S
4k for some k ∈ Z
by Proposition 3.6. We claim that these integers correspond to a toric fan if and only if k = 1.
The idea is that from a0, a1, . . . , ad−1 we can always construct a sequence of vectors for
which every pair of consecutive vectors is a basis of Z2, but k = 1 precisely when these vectors
are labeled in counter-clockwise order. The only relation in the group G, which is STS =G
T−1ST−1, preserves the number of times the vectors circle the origin, so it is natural to study
winding in G. Now we make this idea precise.
Lemma 4.1. Let g ∈ ker(π3). Then wG(g)
12 ∈ Z.
Proof. Since π3(g) = I we know wSL2(Z) ◦ π3(g) = 0, so by Proposition 3.6 π4 ◦ wG(g) = 0.
Thus wG(g) ∈ ker(π4) = {12k | k ∈ Z}. �
Recall that G is isomorphic to the preimage of SL2(Z) in the universal cover of SL2(R) by
Proposition 3.7. Let φ be the isomorphism from G to its image in the universal cover of SL2(R)
with
φ(SG) =
(
cos
(
πt
2
)
− sin
(
πt
2
)
sin
(
πt
2
)
cos
(
πt
2
) )
06t61
, φ(TG) =
(
1 t
0 1
)
06t61
.
This means to each element of the kernel of π3 we can associate a closed loop based at the
identity in SL2(R) denoted φ(g). The fundamental group π1(SL2(R)) is isomorphic to Z and is
generated as
〈
φ
(
S4
G
)〉
, so let ψ : π1(SL2(R))→ Z be the isomorphism with ψ(φ(S4
G)) = 1.
Lemma 4.2. Let g ∈ ker(π3 : G→ SL2(Z)). Then
ψ ◦ φ(g) =
wG(g)
12
.
18 D.M. Kane, J. Palmer and Á. Pelayo
Proof. Since ker(π3) is generated by S4
G, it suffices to check that ψ
(
φ
(
S4
G
))
=
wG(S4
G)
12 = 1, but
this holds by definition. �
Definition 4.3. Define W : ker(π3)→ Z by
W (g) =
wG(g)
12
.
We call W (g) the winding number of g ∈ ker(π3).
Definition 4.4. Let
(v0 = vd, v1, . . . , vd−1) ∈
(
Z2
)d
with det(vi, vi+1) > 0 for i = 0 . . . , d−1. We define the number of times (v0, . . . , vd−1) circles the
origin to be the winding number of the piecewise linear path in
(
R2
)∗
= R2 \ {(0, 0)} produced
by concatenating the linear paths between vi and vi+1 for i = 0, . . . , d− 1.
Lemma 4.5. Let a0, . . . , ad−1 ∈ Z such that ST a0 · · ·ST ad−1 =SL2(Z) I and let v0, v1 ∈ Z2 such
that det(v0, v1) = 1. Then d > 3 so we may define v2, . . . vd−1 by
vi+2 = −vi + aivi+1,
where vd = v0 and vd+1 = v1. The winding number W ((ST a0 · · ·ST ad−1)G) ∈ Z is the number
of times that (v0, . . . , vd−1) ∈
(
Z2
)d
circles the origin.
Proof. Corollary 3.9 states that d > 3 in this situation. For i = 1, . . . , d − 2 let Ai be the
matrix [vi, vi+1], let Ad−1 = [vd−1, v0], and recall that Ai+1 = Ai(ST
ai)SL2(Z). Identify G as
a subgroup of the universal cover of SL2(R) by choosing the identity as a basepoint. Thus, for
each i there is a natural path between I and (ST ai)SL2(Z), and multiplying this path on the left
by Ai we obtain a path from Ai to Ai(ST
ai)SL2(Z) = Ai+1. These paths for i = 0, . . . , d − 1
can be concatenated to form a path from A0 to A0. Projecting this path into the first column
vector of the appropriate matrix gives a path in
(
R2
)∗
. We claim that this path is homotopic
to the path formed by taking line segments between vi and vi+1. This is easily verified because
both paths between vi and vi+1 travel counterclockwise less than a full rotation.
We know that W ((ST a0 · · ·ST ad−1)G) equals ψ of the path in SL2(R) by Lemma 4.2, and
now we need to show that this equals the winding number in
(
R2
)∗
of the first column vectors.
This holds because projection onto the first column vector of a matrix induces an isomorphism
of fundamental groups between SL2(R) and
(
R2
)∗
which sends the generator [φ(SG)4] to the
homotopy class of a loop which circles counter-clockwise once around the origin, a path with
winding number 1. �
Corollary 4.6. There exists a bijection from the set of all sequences (a0, . . . , ad−1) ∈ Zd, d > 0,
which satisfy
ST a0 · · ·ST ad−1 =G S
4
to the collection of all toric fans modulo the action of SL2(Z). This bijection sends (a0, . . . , ad) ∈
Zd to the equivalence class of fans{
(v0 = vd, v1 = vd+1, v2, . . . , vd−1) ∈
(
Z2
)d | v0, v1 ∈ Z2, det(v0, v1) = 1
}
in which
vi+2 = −vi + aivi+1
for i = 0, . . . , d− 1.
Classifying Toric and Semitoric Fans by Lifting Equations from SL2(Z) 19
Proof. Let (v0 = vd, v1, . . . , vd−1) ∈
(
Z2
)d
be a toric fan. That is, det(vi, vi+1) = 1 for each
i = 0, . . . , d− 1 and the vectors are labeled in counter-clockwise order. It is shown in Section 3,
equation (3.2) that associated integers (a0, . . . , ad−1) ∈ (Z)d exist such that
ST a0 · · ·ST ad−1 =SL2(Z) I,
which means
ST a0 · · ·ST ad−1 =G S
4k
for some k ∈ Z with k > 0. By Lemma 4.5 we know
W
((
ST a0 · · ·ST ad−1
)
G
)
= 1,
so that the vectors will be labeled in the correct order for it to be a fan. Thus, W
(
S4k
G
)
= 1 but
W
(
S4k
G
)
= k so k = 1. Notice such a construction is well-defined on equivalence classes of toric
fans because the integers are prescribed via linear equations and fans in a common equivalence
class are related by a linear map.
Now suppose that (a0, . . . , ad−1) ∈ Zd satisfy ST a0 · · ·ST ad−1 =G S
4 and define (v0, . . . , vd−1)
∈
(
Z2
)d
by
vi+2 = −vi + aivi+1,
where v0, v1 ∈ Z2 are any two vectors for which det(v0, v1) = 1. Then for each i = 0, . . . , d − 1
we have
det(vi+1, vi+2) = det
(
0 −1
1 ai
)
det(vi, vi+1) = det(vi, vi+1),
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 vi+1 is located counterclockwise of vi,
we have that the vi’s must be sorted in counterclockwise order.
It is straightforward to see that these constructions are inverses of one another. �
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. This result is well-known but the following proof is new.
Lemma 4.7 ([18, Claim, p. 43]). If (v0 = vd, . . . , vd−1) ∈
(
Z2
)d
is a toric fan with d > 4 then
there exists some i ∈ {0, . . . , d− 1} such that vi = vi−1 + vi+1.
Proof. By Corollary 4.6 we know that to the fan (v0 = vd, . . . , vd−1) there is an associated list
of integers a0, . . . , ad−1 ∈ Z such that vi+2 = −vi + aivi+1 and
ST a0 · · ·ST ad−1 =G S
4. (4.1)
We must only show that for some i ∈ Z we have ai = 1. Since S4 =PSL2(Z) I we can use
Lemma 3.8 to conclude that there exist i, j ∈ Z satisfying 0 6 i < j − 1 6 d − 2 such that
ai, aj ∈ {−1, 0, 1} and (i, j) 6= (0, d− 1). By way of contradiction assume that ai, aj ∈ {−1, 0}.
Conjugate equation (4.1) by ST an for varying n ∈ Z to assure that i 6= 0 and j 6= d− 1. Then
at each of these values we may use either ST 0S =G S2 or ST−1S =G S2TST to reduce the
number of ST -pairs by one or two and produce a factor of S2
G, which we can move to the front
of the word because S2
G is in the center of G. These reductions do not interfere with one another
because the values in question are not adjacent. So we end up with
S4ST b0 · · ·ST b`−1 =G S
4
20 D.M. Kane, J. Palmer and Á. Pelayo
and thus
ST b0 · · ·ST b`−1 =G I,
where ` > 3 by Corollary 3.9. This implies that W ((ST b0 · · ·ST b`−1)G) = 0 and thus, by
Lemma 4.5, that any corresponding collection of vectors winds no times about the origin. How-
ever, this is impossible since for such a sequence of vectors each vector is always counterclockwise
from the previous vector and ` > 1. �
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 (v0 = vd, v2, . . . , vd−1) ∈
(
Z2
)d
is a toric fan with associated integers (a0, . . . , ad−1) ∈
Zd. Then
vi+2 = −vi + aivi+1
so if ai = 1 then we have that vi+1 = vi + vi+2. Now we see that in this case
det(vi, vi+2) = det(vi,−vi) + det(vi, vi+1) = 1,
so
(w0 = v0, . . . , wi = vi, wi+1 = vi+2, . . . , wi−2 = vd−1) ∈
(
Z2
)d−1
is also a fan. Next notice
−wi + (ai+1 − 1)wi+1 = −(vi + vi+2) + ai+1vi+2 = −vi+1 + ai+1vi+2 = wi+2
and
−wi−1 + (ai−1 − 1)wi = (−vi−1 + ai−1vi)− vi = vi+1 − vi = wi+1,
so this new fan has associated to it the tuple of integers
(a0, . . . , ai−1 − 1, ai+1 − 1, . . . , ad−1) ∈ Zd−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 STS =G
T−1ST−1. Geometrically this move corresponds to the inverse of chopping a corner from the
associated polygon (as is shown in Fig. 1) and from the point of view of symplectic geometry
this corresponds to an equivariant symplectic blowup. 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.
Lemma 4.8. Suppose that integers a0, . . . , ad−1 ∈ Z satisfy
ST a0 · · ·ST ad−1 =G S
4 (4.2)
for some d ∈ Z, d > 0.
Classifying Toric and Semitoric Fans by Lifting Equations from SL2(Z) 21
1. If d = 4 then up to a cyclic reordering the set of integer quadruples which satisfy this
equation is exactly a0 = 0, a1 = k, a2 = 0, a3 = −k for each k ∈ Z.
2. If d = 3 then a0 = a1 = a2 = −1.
3. If d < 3 then there do not exist integers satisfying equation (4.2).
Proof. Notice ST a0 · · ·ST ad−1 =G S4 implies that ST a0 · · ·ST ad−1 =SL2(Z) I so Corollary 3.9
implies items (2) and (3). Moreover, we also have that
ST a0 · · ·ST ad−1 =PSL2(Z) I,
so we can apply Lemma 3.8. Suppose that d = 4. Lemma 3.8 tells us that at least one of the ai
is in the set {−1, 0, 1}. By conjugation (which cyclically permutes the order of the integers) we
may assume that a0 ∈ {−1, 0, 1}. If a0 = 1 then
STST a1ST a2ST a3 =G ST
a1−1ST a2ST a3−1,
so for this to equal S4
G in G we must have a1 − 1 = a2 = a3 − 1 = −1 by the case of d = 3. It is
straightforward to check that STSST−1S =G S
4 so we have found the required solution.
If a0 = −1 then notice
ST−1ST a1ST a2ST a3 =G S
4 implies ST a1+1ST a2ST a3+1 =G S
2 =PSL2(Z) I,
so by Lemma 3.8 we must have a1+1 = a2 = a3+1 = ±1. This time, if a1+1 = a2 = a3+1 = −1
then equation (4.2) does not hold, since the left side will equal S6, but if a1 +1 = a2 = a3 +1 = 1
then the equation holds. So we have found another solution, ST−1SSTS =G S
4, which has the
form described in the statement of the lemma.
Finally, suppose that a0 = 0. Notice
ST 0ST a1ST a2ST a3 =G S
4 implies ST a2ST a1+a3 =G S
2 =PSL2(Z) I,
so we can use Lemma 3.8 to conclude that we need a2 = a1 + a3 = 0. Let a1 = k ∈ Z. Now we
have that
ST 0ST kST 0ST−k =G S
4
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. �
Definition 4.9. A Delzant triangle is the convex hull of the points (0, 0), (0, λ), (λ, 0) in R2
for any λ > 0. A Hirzebruch trapezoid with parameter k ∈ Z>0 is the convex hull of (0, 0),
(0, a), (b, a), and (b+ ak, 0) in R2 where a, b > 0. A Hirzebruch trapezoid with parameter zero
is a rectangle.
These are shown in Fig. 1. So we see that the fan corresponding to any Delzant triangle is((
1
0
)
,
(
0
1
)
,
(
−1
−1
))
with associated integers (−1,−1,−1) and the fan corresponding to a Hirzebruch trapezoid with
parameter k is((
0
1
)
,
(
−1
−k
)
,
(
0
−1
)
,
(
1
0
))
with associated integers (0, k, 0,−k). The following Theorem is immediate from Lemmas 4.7
and 4.8.
22 D.M. Kane, J. Palmer and Á. Pelayo
Theorem 4.10 ([18]). Every Delzant polygon can be obtained from a polygon SL2(Z)-equivalent
to a Delzant triangle, a rectangle, or a Hirzebruch trapezoid by a finite number of corner chops.
Proof. Let ∆ be any Delzant polygon with d edges, let (v0, . . . , vd−1) ∈
(
Z2
)d
be the associated
fan of inwards pointing primitive normal vectors, and let (a0, . . . , ad−1) ∈ Zd be the integers
associated to this fan. By Lemma 4.7 if d > 4 then ai = 1 for some i ∈ {0, . . . , d− 1} so the fan
is the result of a corner chop for some fan with d− 1 vectors. That is, ∆ is the result of a corner
chop of some Delzant polygon with d−1 edges. If d < 5 then Lemma 4.8 lists each possibility. If
d = 4 and a0 = a1 = a2 = a3 = 0 then ∆ is SL2(Z)-equivalent to a rectangle, if a0 = 0, a1 = k,
a2 = 0, and a3 = −k for k ∈ Z \ {0} then ∆ is SL2(Z)-equivalent to a Hirzebruch trapezoid, and
if d = 3 with a0 = a1 = a2 = −1 then ∆ is SL2(Z)-equivalent to a Delzant triangle. �
5 Semitoric fans
Now we will apply the method from Section 4 to classify semitoric fans (Definition 2.9).
The first step in the classification is given by a series of lemmas which we will use to ma-
nipulate the semitoric fans in a standard form. In Lemma 5.1 we describe the process of corner
chopping, in Lemma 5.2 we describe the process of reverse corner chopping, in Lemma 5.3 we
describe the process of removing a hidden corner, and in Lemma 5.4 we describe the process
of commuting a fake and Delzant corner. In each of these lemmas we prove the result of the
corresponding process is still a semitoric fan. These four processes are defined in Definition 2.11.
Lemma 5.1. If (v0, . . . , vd−1) ∈
(
Z2
)d
is a semitoric fan and (vi, vi+1) is a Delzant corner, then
(w0 = v0, . . . , wi = vi, wi+1 = vi + vi+1, wi+2 = vi+1, . . . , wd = vd−1) ∈
(
Z2
)d+1
is a semitoric fan with the pairs (wi, wi+1) and (wi+1, wi+2) being labeled as Delzant corners.
Proof. It is immediate that det(wi, wi+1) = det(wi+1, wi+2) = 1 and the vectors are still in
counter-clockwise order because the new vector was inserted between two adjacent vectors. �
Lemma 5.2. If (v0, . . . , vd−1) ∈
(
Z2
)d
is a semitoric fan, (vi−1, vi) and (vi, vi+1) are Delzant
corners, and vi = vi−1 + vi+1, then
(w0 = v0, . . . , wi−1 = vi−1, wi = vi+1, . . . , wd−2 = vd−1) ∈
(
Z2
)d−1
is a semitoric fan with the pair (wi−1, wi) being labeled as a Delzant corner.
Proof. Notice det(wi−1, wi) = det(vi−1, vi − vi−1) = 1 so we must only show that the vectors
in the new fan are labeled in counter-clockwise order, but this is the case because the vectors in
the original fan were and to get the new fan we have only removed a vector. �
Lemma 5.3. If (v0, . . . , vd−1) ∈
(
Z2
)d
is a semitoric fan and (vi, vi+1) is a hidden corner, then
(w0 = v0, . . . , wi = vi, wi+1 = Tvi+1, wi+2 = vi+1, . . . , wd = vd−1) ∈
(
Z2
)d+1
is a semitoric fan in which (wi, wi+1) is a Delzant corner and (wi+1, wi+2) is a fake corner.
Proof. We know that det(vi, T vi+1) = 1 because that pair of vectors forms a hidden corner.
Notice that
det(wi, wi+1) = det(vi, T vi+1) = 1,
Classifying Toric and Semitoric Fans by Lifting Equations from SL2(Z) 23
so that corner is Delzant and
det(wi+1, Twi+2) = det(Tvi+1,Tvi+1) = 0,
so if (wi+1, wi+2) is on the top boundary then it is a fake corner. It is on the top boundary
because vi+1 is in the lower half plane, TSL2(Z) sends the lower half plane to the lower half plane,
and (wi+1, wi+2) = (Tvi+1, vi+1). The vectors in the new fan are still in counter-clockwise
order because wi+1 is in the lower half plane so wi+2 = T−1wi+1 is counter-clockwise to it and
since wi and wi+1 are both in the lower half plane and det(wi, wi+1) > 0 we see they are in
counter-clockwise order. �
Lemma 5.4. If (v0, . . . , vd−1) ∈
(
Z2
)d
is a semitoric fan and (vi, vi+1) is a fake corner and
(vi+1, vi+2) is a Delzant corner on the top boundary, then
(w0 = v0, . . . , wi = vi, wi+1 = Tvi+2, wi+2 = vi+2, . . . , wd−1 = vd−1) ∈
(
Z2
)d
is a semitoric fan in which (wi, wi+1) is a Delzant corner and (wi+1, wi+2) is a fake corner.
Proof. We know det(vi, T vi+1) = 0 so vi = Tvi+1 since they are both on the top boundary and
we also know det(vi+1, vi+2) = 1. Now we can check that
det(wi, wi+1) = det(vi, T vi+2) = det(Tvi+1, T vi+2) = det(vi+1, vi+2) = 1
and
det(wi+1, Twi+2) = det(Tvi+2, T vi+2) = 0.
Since (vi+1, vi+2) is on the top boundary, vi+1 and Tvi+1 are in the lower half plane, so
(wi+1, wi+2) = (Tvi+2, vi+2) is in the top boundary which means it is a fake corner, as de-
sired. The argument for why the new vectors are still in counter-clockwise order is similar to
the argument in the proof of Lemma 5.3. �
Recall that G′ =
{
T kSL2(Z) | k ∈ Z
}
is the symmetry group of semitoric fans, given in Defini-
tion 2.10.
Lemma 5.5. Suppose that (v0, . . . , vd−1) ∈
(
Z2
)d
is a semitoric fan. Then after a finite number
of corner choppings the fan will be G′-equivalent to one in which two adjacent vectors are
(
0
−1
)
and ( 1
0 ).
Proof. Let vd = v0. If
vi =
(
1
0
)
for some i ∈ {0, . . . , d− 1} then notice
vi−1 =
(
a
−1
)
for some a ∈ Z. This is because (vi−1, vi) is not on the top boundary so it must be a Delzant
corner. Then by the action of T−aSL2(Z) ∈ G
′, which does not change vi, we can attain the required
pair of vectors.
Otherwise, renumber so that v0 is in the lower half plane and v1 is in the upper half plane.
Then insert the vector v0 + v1 between them. This new vector will have a second component
with a strictly smaller magnitude than that of v0 or of v1. Since the magnitude is an integer,
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 ). �
24 D.M. Kane, J. Palmer and Á. Pelayo
Figure 3. Any semitoric fan with defect c ∈ Z>0 can be transformed into the standard fan of defect c.
This image has c = 3.
Now we can put Lemmas 5.3, 5.4, and 5.5 together to produce a standard form for semitoric
fans (see Fig. 3). This standard form will be important to us in Section 6 because it can be
obtained from any semitoric fan of defect c by only using transformations which are continuous
in the space of semitoric polygons.
Proposition 5.6. Let (v0, . . . , vd−1) ∈
(
Z2
)d
be a semitoric fan of defect c ∈ Z>0.
1. By only corner chopping, reverse corner chopping, removing hidden corners, commuting
fake and Delzant corners, and acting by the symmetry group G′ we can obtain a new
semitoric fan (w0 = w`+c, . . . , w`+c−1) ∈
(
Z2
)`+c
with `+ c > d such that
• w0 =
(
0
−1
)
and w1 = ( 1
0 );
• each corner (wi, wi+1) for i = 0, . . . , `− 1 is Delzant;
• each corner (wi, wi+1) for i = `, . . . , `+ c− 1 is fake; and
• w` = T cw0 so det(w`−1, T
cw0) = 1.
2. The collection of vectors (w`, w1, . . . , w`−1) is a toric fan;
3. The semitoric fan (w0, . . . , wc+`−1) can be obtained via a finite number of corner chops
and reverse corner chops from a fan (u0, . . . , uc+3) ∈
(
Z2
)c+4
where
u0 =
(
0
−1
)
, u1 =
(
1
0
)
, u2 =
(
c
1
)
, u3 =
(
−1
0
)
,
and
u4+n =
(
−c+ n
−1
)
for n = 0, . . . , c − 1. In this fan the first four pairs are Delzant corners and the rest are
fake corners.
Proof. The first part is immediate from Lemmas 5.3, 5.4, and 5.5. By Lemma 5.5 we know
after a finite number of corner choppings and renumbering it can be arranged that the first two
vectors in the fan are(
0
−1
)
and
(
1
0
)
.
Classifying Toric and Semitoric Fans by Lifting Equations from SL2(Z) 25
Then we invoke Lemma 5.3 to remove all of the hidden corners, and finally use Lemma 5.4
commute all of the fake corners to be adjacent and arrive at the fan (w0, . . . , w`+c−1) ∈
(
Z2
)`+c
in the statement of the proposition. Notice that (wi, wi+1) being fake for i = `, . . . , ` + c − 1
implies that
det(w`, Tw`+1) = · · · = det(w`+c−1, Tw0) = 0.
Now, we know both vectors in a fake corner must have negative second component by definition,
so this implies that
w` = Tw`+1 = T 2w`+2 = · · · = T cw0.
Now (T cw0 = w`, w1, . . . , w`−1) ∈
(
Z2
)`
is a toric fan (we know the vectors are in counter-
clockwise order because we started with a semitoric fan) so we have established part (2) of the
theorem. Denote the associated integers for this toric fan by b0, . . . , b`−1 ∈ Z so that
ST b0 · · ·ST b`−1 =G S
4. (5.1)
To prove part (3) we will use corner chops and reverse corner chops on this toric fan to remove
vectors in the set (w2, . . . , w`−1) until the remaining vectors are in the required form. The idea
here is that any corner chop or reverse corner chop on the toric fan (w`, w1, . . . , w`−1) which
does not remove w` or w1 and does not add a vector between w` and w1 can also be performed
on the semitoric fan (w0, . . . , w`+c−1).
Here we proceed by cases on the values of ` and b0, . . . , b`−1. We will see in Case I that
the semitoric fan is already of the required form and we will show any that other case can
be transformed into Case I by corner chops and reverse corner chops on the semitoric fan
(w0, . . . , w`+c−1). Here it is important to recall that w` and w1 are already fixed, so the integers
b0, . . . , b`−1 completely determine the toric fan (w`, w1, . . . , w`−1).
Case I. ` = 4 and b0 = b1 = b2 = b3 = 0.
In this case the toric fan (w`, w1, w2, w3) must be((
−c
−1
)
,
(
1
0
)
,
(
c
1
)
,
(
−1
0
))
,
which means that the semitoric fan (w0, . . . , w`+c−1) is in the required form for part (3) of the
theorem.
Case II. ` = 4 and b0, b1, b2, b3 not all zero.
In this case, by Lemma 4.8, we must either have b0 = b2 = 0 and b1 = −b3, or b0 = −b2 and
b1 = b3 = 0. First assume b0 = b2 = 0 and b1 = −b3. In this case equation (5.1) becomes
S2T b1S2T−b1 =G S
4.
If b1 > 0 then a corner chop will produce
(
S2T b1STSTST−b1+1
)
G
and a reverse corner chop
will produce from this
(
S2T b1−1S2T−b1+1
)
G
. This can be repeated b1 times to recover Case I. If
b1 < 0 then a corner chop can produce
(
S2T b1+1STSTST−b1
)
G
and from this a reverse corner
chop can produce
(
S2T b1+1S2T−b1−1
)
G
. Again, this can be repeated b1 times to recover Case I.
Observe that the above corner chops and reverse corner chops do not affect the first and last
element of the toric fan; thus the above operations can be realized as corner chops and reverse
corner chops on the semitoric fan (w0, . . . , w`+c−1) as required.
The case of b0 = −b2 and b1 = b3 = 0 is similar.
26 D.M. Kane, J. Palmer and Á. Pelayo
Case III. ` < 4.
By Lemma 4.8 we must have ` = 3. Perform a corner chop between the vectors w1 and w2
to add the vector w1 + w2 and reduce to either Case I or Case II.
Case IV. ` > 4.
In this case we will show that the semitoric fan can always be transformed into another
semitoric fan with one less length, so repeating this process will eventually yield ` = 4, thereby
reducing to Case I or Case II. There are two subcases in this case.
Case IV (a). bi = 1 for some i such that 0 < i < `− 1.
Here a reverse corner chop may be performed on wi+1, removing it from the semitoric fan
and reducing the length by 1.
Case IV (b). bi 6= 1 for all i = 1, . . . , `− 2.
Any substitution using STS =G T
−1ST−1 in equation (5.1) can be realized as a corner chop
or reverse corner chop on the toric fan (w`, w1, . . . , w`−1) and no such substitution corresponds to
the corner chop on (w`−1, w`) so these transformations can also be realized a corner chops and re-
verse corner chops on the semitoric fan (w0, . . . , w`+c−1). In particular, if
(
ST b0 · · ·ST b`−1
)
G
con-
tains the subword
(
T a+1SST b
)
G
, we can perform a corner chop to obtain
(
T a+1STSTST b+1
)
G
,
and then a reverse corner chop to reduce to
(
T aSST b+1
)
G
. Note that this can be done even
if the T aG was in the
(
ST b0
)
G
term and also notice that a similar argument can be used to
transform
(
T a−1SST b
)
G
into
(
T aSST b−1
)
G
if a < 0. This means that every occurrence of S2
G
may be commuted to the front of the word, and thus we may assume that there is an integer
m > 0 for which bi = 0 if and only if i < m. If m > 4 then(
ST 0
)m−4
ST bm · · ·ST b`−1 =G I,
so
wG
((
ST 0
)m−4
ST bm · · ·ST b`−1
)
= wG(I) = 0,
but the integers (0, . . . , 0, bm, bm+1, . . . , b`−1) have associated vectors which travel some integer
k > 0 times around the origin, since each vector is counterclockwise from the previous one,
which means that
wG
(((
ST 0
)m−4
ST bm · · ·ST b`−1
)
G
)
= 12k
forming a contradiction. Thus, m < 4 and this also implies `−m > 1 because ` > 4.
Let e = 0 if m is even and e = 1 if m is odd, then equation (5.1) implies the following equality
in PSL2(Z):
SeST bm · · ·ST b`−1 =PSL2(Z) I. (5.2)
We proceed by cases on the value of `−m+ e.
We already know that `−m+ e > `−m > 1, and also that `−m+ e = 2 is impossible by
applying Lemma 3.8 to equation (5.2). If e = 1 then `−m+ e = 3 is impossible by Lemma 3.8
and if e = 0 then `−m+ e = 3 is impossible by acting with wG on both sides of equation (5.1).
By Lemma 3.8 if e = 0 and `−m+ e = 3 then equation (5.1) must be
SmST−1ST−1ST−1 =G S
4,
Classifying Toric and Semitoric Fans by Lifting Equations from SL2(Z) 27
which implies
wG
((
SmST−1ST−1ST−1
)
G
)
= wG
(
S4
G
)
,
which can only hold if m = 0 in which case `−m+ e = 3 implies that ` = 3, which is false. If
`−m+ e = 4 then either b`−1 = 0 or b`−2 = 0, both of which are false by assumption.
The only remaining case is ` − m + e > 4, and in this case Lemma 3.8 implies that there
exists some i with m < i < ` − 1 and bi = {−1, 0, 1}. Thus, bi = ±1. If bi = 1, then we may
perform a reverse corner chop on wi+1, as in Case IV (a), to reduce `. Otherwise, bi = −1 and
we use the relation ST−1S =G SSTST . Since
ST−1S =G S
(
T−1ST−1
)
T =G S(STS)T =G SSTST
this relation can actually be realized by a corner chop, which we know corresponds to a legal
move at the level of fans. Here we have introduced an S2
G term, and we next return to the
beginning of Case IV (b). That is, the S2
G term is moved to the front of the word and now the
first m′ powers of TG are zero, where m′ > m, and the new toric fan has length `′ > `. If the
new word contains (STS)G then a reverse corner chop may be performed, as in Case IV (a),
before returning to the beginning of Case IV (b) (with a larger m and `) and otherwise the word
contains
(
ST−1S
)
G
, which we can again remove. This process must terminate by finding an
occurrence of (STS)G and performing a reverse corner chop after at most 3 iterations because
at each iteration the integer m increases and we have already argued that m > 4 is impossible.
Once m = 3 then no iteration of this process can include
(
ST−1S
)
G
because in such a case m can
be increased by one. This is important because on those iterations during which the replacement(
ST−1S
)
G
to
(
S2TST
)
G
is used the total number ` of vectors in the toric fan increases by one,
but we have shown this can happen at most three times.
Thus, the algorithm described in Case IV (b) must eventually allow for enough reverse corner
chops to be performed to reducing the number of vectors in the toric fan to four. This completes
Case IV (b).
Therefore, in any case with ` 6= 4 the semitoric fan can be transformed into one with ` = 4
by Cases III and IV, and any fan with ` = 4 is either in the correct form already (Case I) or can
be transformed into the correct form (Case II). �
Theorem 2.14 is immediate from Proposition 5.6.
Remark 5.7. Notice that Theorem 4.10 is different from Theorem 2.14 because in Theorem 4.10
the minimal models of the Delzant polygons may be achieved through only corner chops. In
Theorem 2.14 we use instead a variety of transformations and in Section 6 we show all of those
transformations are continuous with respect to the topology on the moduli space of semitoric
systems (described in Section 6.2).
6 Application to symplectic geometry
The results of Section 5 have an interpretation in the symplectic geometry of toric manifolds and
semitoric integrable systems. In Section 6.1 we will review the classification theorem of Pelayo–
Vũ Ngo.c [38], in Section 6.2 we define the metric on the moduli space of semitoric systems given
by the second author in [35], and in Section 6.3 we prove the connectivity result for the moduli
space of semitoric systems (with respect to the aforementioned metric) using Theorem 2.14.
Definition 6.1 ([37]). A semitoric integrable system is a 4-dimensional connected symplectic
manifold (M,ω) with an integrable Hamiltonian system F = (J,H) : M → R2 such that J is
a proper momentum map for an effective Hamiltonian S1-action on (M,ω) and F has only
28 D.M. Kane, J. Palmer and Á. Pelayo
non-degenerate singularities which have no real-hyperbolic blocks. Such a system is said to be
a simple semitoric integrable system if there is at most one focus-focus critical point in J−1(x)
for any x ∈ R.
An isomorphism of semitoric systems is a symplectomorphism φ : (M1, ω1) → (M2, ω2),
where (M1, ω1, F1 = (J1, H1)) and (M2, ω2, F2 = (J2, H2)) are semitoric systems, such that
φ∗(J2, H2) = (J1, f(J1, H1)) where f : F1(M1) → R is a smooth function such that ∂f
∂H1
is eve-
rywhere positive. We denote the moduli space of simple semitoric systems modulo semitoric
isomorphism by T .
6.1 Invariants of semitoric systems
The momentum map image of a compact toric integrable system, which is a polytope, is sufficient
to classify such systems up to a suitable notion of isomorphism (cf. [14]). Semitoric systems are
classified in terms of a list of invariants [38]. Roughly speaking, the complete invariant is
a polygon together with a set of interior points each labeled with extra information (encoding
singularities of so called focus-focus type, which semitoric systems may possess, 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 6.5). In this section we will define each of the invariants of semitoric
systems and give the necessary definitions to state the Pelayo–Vũ Ngo.c classification theorem.
Readers interested in further details may consult [37, 38, 39, 40, 45, 46].
6.1.1 The number of focus-focus points invariant
While a toric integrable system can only have transversally-elliptic and elliptic-elliptic singu-
larities a semitoric integrable system can also have focus-focus singularities (for a definition of
these types of singularities see for instance [38]). In [46, 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 mf known as the number of focus-focus singular points invariant.
6.1.2 The Taylor series invariant
The semi-global3 structure of a focus-focus singular point is determined by a formal power series
in two variables up to the suitable notion of isomorphism (see [43, 45]).
Definition 6.2. Let R[[X,Y ]] denote the algebra of real formal power series in two variables
and let R[[X,Y ]]0 ⊂ R[[X,Y ]] be the subspace of series
∑
i,j>0
σi,jX
iY j which have σ0,0 = 0 and
σ0,1 ∈ [0, 2π).
The Taylor series invariant is one element of R[[X,Y ]]0 for each of the mf focus-focus
points. Thanks to [43] we understand why the isomorphism need not be taken into account
when there is a global S1-action, that is, as in the case of semitoric systems [37], provided one
assumes everywhere that the Eliasson isomorphisms preserve the global S1-action and the R2-
orientation, in which case the Taylor series is unique (for the general case uniqueness is up to
a (Z2 × Z2)-action, see [43]).
6.1.3 The polygon invariant and the twisting index invariant
The polygon invariant is the invariant which is directly analogous to the Delzant polygon in the
toric case and the twisting index is an integer label on each focus-focus fiber which encodes how
3I.e., in a neighborhood of the fiber over the critical point.
Classifying Toric and Semitoric Fans by Lifting Equations from SL2(Z) 29
the semiglobal (in a neighborhood of a fiber) model of the focus-focus point relates to the global
toric momentum map used to generate the polygon. The different choices of toric momentum
map produce a family of polygons and thus the component of the twisting index associated to
a single focus-focus fiber may change depending on this choice, but if the system has more than
one focus-focus point then the difference of twisting index components of focus-focus fibers
does not depend on the choice of polygon. This is because different choices of polygon can
only change the sequence of integers which is the twisting index by the addition of a common
integer, and for this reason we are able to also define the twisting index class (first introduced
in [35, Definition 3.7], and defined in the present paper in Definition 6.15). The polygon and
twisting index invariants are described together because the choice of common integer added to
the twisting index is related to 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 [46] the author was able to recover a family of
convex polygons which take the place of the Delzant polygon in the semitoric case.
Definition 6.3. Let Polyg
(
R2
)
denote the set of all convex polygons (as in Definition 2.3) which
are rational (i.e., each edge is directed along a vector with integer coefficients), and observe that
all elements of Polyg
(
R2
)
are automatically simple (two edges meet at each vertex).
For brevity, for the duration of this paper by “polygon” we will always mean “rational, convex
polygon”. Notice that a polygon may be noncompact. For any λ ∈ R we will use the notation
`λ =
{
(x, y) ∈ R2 |x = λ
}
and Vert
(
R2
)
= {`λ |λ ∈ R}.
Definition 6.4. A labeled weighted polygon of defect4 mf ∈ Z>0 is defined to be(
∆, (`λj , εj , kj)
mf
j=1
)
∈ Polyg
(
R2
)
×
(
Vert
(
R2
)
× {−1,+1} × Z
)mf
with
inf
s∈∆
π1(s) < λ1 < · · · < λmf < sup
s∈∆
π1(s),
where π1 : R2 → R is projection onto the first coordinate. We denote the space of labeled
weighted polygons of any defect by LWPolyg
(
R2
)
.
The labels kj , j = 1, . . . ,mf , will become the twisting index invariant after the appropriate
quotient. The polygon invariant will be the orbit of a specific type of element of LWPolyg
(
R2
)
.
Recall5
TSL2(Z) =
(
1 1
0 1
)
∈ SL2(Z) so
(
T tSL2(Z)
)k
=
(
1 0
k 1
)
for k ∈ Z. Recall by Notation 3.1 that Tv always means TSL2(Z)v for any v ∈ Z2.
Definition 6.5.
1. For ∆ ∈ Polyg
(
R2
)
a point (x0, y0) ∈ R2 is said to be in the top boundary of ∆ if
y0 = sup{y ∈ R | (x0, y) ∈ ∆}. We denote this by (x0, y0) ∈ ∂top∆.
4In this context this is sometimes called the complexity, but seeing as complexity sometimes refers to half the
dimension of the manifold minus the dimension of the acting torus (so all semitoric systems would be complexity 1)
we have chosen to use the word defect.
5The authors of [37, 38] denote by T the transpose of this matrix. Since in this paper we have discovered the
connection to SL2(Z) we have chosen to instead use the notation standard to SL2(Z)-presentations.
30 D.M. Kane, J. Palmer and Á. Pelayo
2. Let ∆ ∈ Polyg
(
R2
)
. A corner of ∆ is a point p ∈ ∂∆ such that the edges meeting at p
are not co-linear. For a corner p ∈ ∆ let u, v ∈ R2 be primitive inwards pointing normal
vectors to the edges of ∆ adjacent to p in the order of positive orientation. We say that p
satisfies
(a) the Delzant condition if det(u, v) = 1;
(b) the hidden condition if p ∈ ∂top∆ and det(u, Tv) = 1; and
(c) the fake condition if p ∈ ∂top∆ and det(u, Tv) = 0.
3. We say ∆ ∈ Polyg
(
R2
)
has everywhere finite height if the intersection of ∆ with any
vertical line is either compact or empty.
Definition 6.6. An element
(
∆, (`λj , εj , kj)
mf
j=1
)
∈ LWPolyg
(
R2
)
is a primitive semitoric poly-
gon if
1) ∆ has everywhere finite height;
2) εj = +1 for j = 1, . . . ,mf ;
3) any point in ∂top∆ ∩ `λj for some j ∈ {1, . . . ,mf} is a corner which satisfies either the
hidden or fake condition; and
4) all other corners satisfy the Delzant condition.
As is done in [38] we say that those corners of a primitive semitoric polygon which are in the
top boundary and also in some line `λj are either a hidden corner or fake corner corresponding
to which condition they satisfy. Also, all other corners of a primitive semitoric polygon must
by definition satisfy the Delzant condition and thus are referred to as Delzant corners. Notice
here that the corners of a primitive semitoric polygon do not need to be labeled by their type
(hidden, fake, or Delzant) because the type of a corner can by deduced by examining the primitive
semitoric polygon (compare to semitoric fans, see Definition 2.9 and the following discussion).
In the case that
(
∆, (`λj , εj , kj)
mf
j=1
)
is a primitive semitoric polygon and ∆ is compact then
there exists an associated semitoric fan.
6.1.4 The action of Gmf × G
Now we will define a group and the way that its elements act on a labeled weighted polygon
of defect mf , for some fixed mf . For any ` ∈ Vert
(
R2
)
and k ∈ Z fix an origin in ` and let
tk` : R2 → R2 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 = (u1, . . . , umf ) ∈ {−1, 0, 1}mf and
~λ = (λ1, . . . , λmf ) ∈ Rmf let
t~u~λ = tu1`λ1
◦ · · · ◦ t
umf
`λmf
.
Definition 6.7. For any nonnegative mf ∈ Z let Gmf = {−1, 1}mf and let G =
{(
T tSL2(Z)
)k | k ∈
Z
}
. We define the action of ((ε′j)
mf
j=1, (T
t
SL2(Z))
k) ∈ Gmf × G on an element of LWPolyg
(
R2
)
by
(
(ε′j)
mf
j=1,
(
T tSL2(Z)
)k) · (∆, (`λj , εj , kj)mfj=1
)
=
(
t~u~λ
((
T tSL2(Z)
)k
∆
)
, (`λj , ε
′
jεj , k + kj)
mf
j=1
)
,
where ~λ = (λ1, . . . , λmf ) and ~u =
( εj−εjε′j
2
)mf
j=1
.
Classifying Toric and Semitoric Fans by Lifting Equations from SL2(Z) 31
Definition 6.8. A semitoric polygon is the orbit under Gmf×G of a primitive semitoric polygon.
That is, given a primitive semitoric polygon ∆w =
(
∆, (`λj ,+1, kj)
mf
j=1
)
∈ LWPolyg
(
R2
)
the
associated semitoric polygon is the subset of LWPolyg
(
R2
)
given by
[∆w] =
{(
t~u~λ
((
T tSL2(Z)
)k
(∆)
)
, (`λj , 1− 2uj , kj + k)
mf
j=1
)
| ~u ∈ {0, 1}mf , k ∈ Z
}
.
The collection of semitoric polygons is denoted DPolyg
(
R2
)
.
In general the action of Gmf × G may not preserve the convexity of the polygons but it is
shown in [38, Lemma 4.2] that [∆w] ⊂ LWPolyg
(
R2
)
for any primitive semitoric polygon ∆w.
Remark 6.9. A semitoric polygon is a family of polygons which is determined by the choice of
a single primitive semitoric polygon, though this choice is not unique since infinitely many prim-
itive semitoric polygons generate any given semitoric polygon. For instance, [∆, (`0,+1, 0)] and
[∆′, (`0,+1, 1)] are the same semitoric polygon where ∆ is the convex hull of (−1, 0),(0, 1),(1, 0),
and (2, 1), and ∆′ is the convex hull of (−1,−1), (0, 1), (1, 1), and (2, 3), since T t(∆) = ∆′.
Definition 6.10. Let ∆w = (∆, (`λj , εj , kj)
mf
j=1) be a primitive semitoric polygon. Then the
twisting index of ∆w is the vector ~k = (k1, . . . , kmf ) ∈ Zmf .
The twisting index of a semitoric system is the coherent assignment of a twisting index to each
element of the associated semitoric polygon, which is to say that the twisting index invariant is
the set[(
∆, (`λj , εj , kj)
mf
j=1
)]
∈ LWPolyg
(
R2
)
/(Gmf × G).
Remark 6.11. In [37] the authors define the polygon invariant as this set without the integer
labeling (without the kj) and the twisting index invariant as this set including the labeling. In
this paper we define these together because this allows us to give a more transparent formulation
of Theorem 1.2.
Definition 6.12. Two semitoric polygons [∆w] and [∆′w] are in the same twisting index class
if and only if the primitive semitoric polygons
∆w =
(
∆, (`λj , εj , kj)
mf
j=1
)
and ∆w
′ =
(
∆′, (`λj ′ , εj
′, kj
′)
mf
′
j=1
)
,
representing them can be chosen such that mf = mf
′ and kj = k′j for j = 1, . . . ,mf .
6.1.5 The volume invariant
Suppose that ∆w is a primitive semitoric polygon. For each j = 1, . . . ,mf we define a real
number hj ∈ (0, length(π2(∆ ∩ `λj ))) by measuring the Liouville volume of a specific subset
of M related to each focus-focus point. Specifically, if pj ∈ M is a focus-focus point then we
define
hj = vol{x ∈M | J(x) = J(pj) and H(x) < H(pj)},
where vol is the Liouville volume on M . The real numbers h1, . . . , hmf are independent of the
choice of primitive semitoric polygon representing the semitoric polygon associated to a given
semitoric system. The details are in [37, 38].
32 D.M. Kane, J. Palmer and Á. Pelayo
Figure 4. The complete invariant of semitoric systems is the orbit under Gmf
× G of this object.
6.1.6 The complete invariant
The collection of these five invariants forms the complete invariant of a semitoric system, which
can be represented by a polygon ∆ (satisfying the conditions of Definition 6.6) with mf vertical
lines `λ1 , . . . , `λmf and for each j = 1, . . . ,mf there is a distinguished point on `λj a distance hj
from the bottom of ∆. Each of these points is also labeled with an integer kj and a Taylor
series (Sj)
∞. This is shown in Fig. 4. It is important to note that in this rough description we
have omitted the action of group Gmf×G which is a key part to understanding the classification.
6.1.7 The classif ication theorem
Definition 6.13 ([38]). A semitoric list of ingredients is
1) a nonnegative integer mf ;
2) a semitoric polygon [∆w] = [
(
∆, (`λj ,+1, kj)
mf
j=1
)
] of defect mf ;
3) a collection of mf real numbers h1, . . . , hmf ∈ R such that 0 < hj < length(π2(∆ ∩ `λj ))
for each j = 1, . . . ,mf ; and
4) a collection of mf Taylor series (S1)∞, . . . , (Smf )∞ ∈ R[[X,Y ]]0.
LetM denote the collection of all semitoric lists of ingredients and letMmf be lists of ingredients
with ingredient (1) equal to the nonnegative integer mf .
We use the following result to study the moduli space of semitoric systems by instead studying
the set of semitoric lists of ingredients.
Theorem 6.14 ([38]). 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,
T ∼=M,
[(M,ω, (J,H))]↔
(
[∆w], (hj)
mf
j=1, ((Sj)
∞)
mf
j=1
)
,
where the invariants mf , [∆w], hj, and (Sj)
∞ are as defined above, and the bijection is as given
in [38].
Classifying Toric and Semitoric Fans by Lifting Equations from SL2(Z) 33
Given a semitoric system, the invariants can be obtained as outlined above, and given semi-
toric list of ingredients a semitoric system can be constructed by a gluing procedure. Explicitly
constructing the invariants from the system and explicitly constructing the system from a given
set of invariants is described in detail in [37, 38].
Definition 6.15. Two semitoric systems (M,ω, F ) and (M,ω, F ′) are said to be in the same
twisting index class if the semitoric polygon associated to (M,ω, F ) and the semitoric polygon
associated to (M ′, ω′, F ′) are in the same twisting index class, as in Definition 6.12.
6.2 Metric and topology on the moduli space
In [35] the second author defines a metric space structure for T using Theorem 6.14. This metric
is constructed by combining metrics on each ingredient. We reproduce this construction briefly
in this section.
6.2.1 The metric on Taylor series
Recall that the Taylor series in R[[X,Y ]]0 have the coefficient to X0Y 1 in the interval [0, 2π).
From the construction in [45] we can see that when defining the topology the endpoints of
the closure of this interval should be identified. 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 any of the following metrics.
Definition 6.16. A sequence {bn}∞n=0 with bn ∈ (0,∞) for each n ∈ Z>0 is called linearly
summable if
∞∑
n=0
nbn <∞. Let {bn}∞n=0 be such a sequence and define dR[[X,Y ]]0: (R[[X,Y ]]0)2→R
by
dR[[X,Y ]]0((S)∞, (S′)∞) =
∞∑
i,j>0,(i,j)6=(0,1)
min{
∣∣σi,j − σ′i,j∣∣ , bi+j}
+ min{
∣∣σ0,1 − σ′0,1
∣∣ , 2π − ∣∣σ0,1 − σ′0,1
∣∣ , b1},
where (S)∞, (S′)∞ ∈ R[[X,Y ]]0 with
(S)∞ =
∑
i,j>0
σi,jX
iY j and (S′)∞ =
∑
i,j>0
σ′i,jX
iY j .
In order for a sequence of series to converge with respect to this metric each term except
for the coefficient to X0Y 1 must converge in R, and the coefficient to X0Y 1 must converge
in R/2πZ.
6.2.2 The metric on the polygon invariant
In [36] the authors use the Lebesgue measure of the symmetric difference of the moment poly-
topes to define a metric on the space of toric systems. For the portion of the semitoric metric
related to the polygon invariant something similar is done in [35].
Definition 6.17. Let ν be a measure on R2. We say that it is admissible if:
1) it is in the same measure class as the Lebesgue measure on R2 (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) ∈ R2;
34 D.M. Kane, J. Palmer and Á. Pelayo
3) the function g from part (2) satisfies xg ∈ L1(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 R2 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 ⊂ R2 we have
A ∗B = (A \B) ∪ (B \A).
Definition 6.18. Let DPolyg
mf ,~k
(
R2
)
⊂ DPolyg
(
R2
)
denote the set whose elements are orbits
under Gmf × G of primitive semitoric polygons with twisting index ~k in Zmf . Then
DPolyg
(
R2
)
=
⋃
mf∈Z>0
~k∈Zmf
DPolyg
mf ,~k
(
R2
)
.
Remark 6.19. Let mf be a nonnegative integer and ~k ∈ Zmf . Suppose that a primitive
semitoric polygon ∆w has twisting index ~k′ ∈ Zmf where there exists some c ∈ Z such that
kj = k′j + c for j = 1, . . . ,mf . Then [∆w] ∈ DPolyg
mf ,~k
(
R2
)
because the set [∆w] is also the
orbit of a primitive semitoric polygon with twisting index ~k.
Definition 6.20. Let mf ∈ Z>0 and ~k ∈ Zmf . Let
∆w =
(
∆, (`λj ,+1, kj)
mf
j=1
)
and ∆′w =
(
∆′, (`λ′j ,+1, kj)
mf
j=1
)
∈ LWPolyg
(
R2
)
be primitive semitoric polygons so [∆w], [∆′w] ∈ DPolyg
mf ,~k
(
R2
)
are semitoric polygons in the
same twisting index class. Then, if mf > 0 we define the distance between them to be
dνP([∆w], [∆′w]) =
∑
~u∈{0,1}mf
ν
(
t~u~λ(∆) ∗ t~u~λ′(∆
′)
)
.
If mf = 0 then this sum will be empty so we instead use
dνP([∆w], [∆′w]) = inf
k∈Z
{
ν(∆ ∗ T k(∆′))
}
.
For any mf ∈ Z>0 and for any ~k ∈ Zmf , the function dνP : DPolyg
mf ,~k
(
R2
)
×DPolyg
mf ,~k
(
R2
)
→
R is a metric [35]; let τD denote the topology that it induces on DPolyg
mf ,~k
(
R2
)
.
The distance defined in Definition 6.20 does not depend on the choice of primitive semitoric
polygon by [35, Proposition 4.5], because we have specifically chosen representatives which have
the same twisting index ~k. 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 mf = 0 there in no twisting
index ~k to align so we take whichever alignment is best6.
6This is slightly different from the definition in [35] to correct for a small mistake (which doesn’t affect the
main results of [35]).
Classifying Toric and Semitoric Fans by Lifting Equations from SL2(Z) 35
6.2.3 The metric on the moduli space of semitoric systems
The metric on semitoric systems will be formed by combining the metrics from Section 6.2.1
and Section 6.2.2.
Definition 6.21. LetM
mf ,~k
⊂Mmf denote those lists of ingredients with the polygon invariant
in the set DPolyg
mf ,~k
(
R2
)
.
We will only define the metric on each separate T
mf ,~k
because that is all we will need for this
paper, see Remark 6.23.
Definition 6.22 ([35]). Let ν be an admissible measure on R2 and let {bn}∞n=0 be a linearly
summable sequence. Let mf be a nonnegative integer and ~k ∈ Zmf .
1. We define the metric on M
mf ,~k
relative to ν and {bn}∞n=0 to be given by
d
ν,{bn}∞n=0
mf ,~k
(m,m′) = dνP([∆w], [∆′w]) +
mf∑
j=1
(
dR[[X,Y ]]0((Sj)
∞, (S′j)
∞) +
∣∣hj − h′j∣∣ ),
where m,m′ ∈M
mf ,~k
are given by
m =
(
[∆w], (hj)
mf
j=1, ((Sj)
∞)
mf
j=1
)
, m′ =
(
[∆′w], (h′j)
mf
j=1, ((S
′
j)
∞)
mf
j=1
)
.
2. Let Φ: T →M be the correspondence from Theorem 6.14, and let
T
mf ,~k
= Φ−1(M
mf ,~k
).
The metric on T
mf ,~k
is given by Dν,{bn}
∞
n=0
mf ,~k
= Φ∗d
ν,{bn}∞n=0
mf ,~k
.
Remark 6.23. The metrics on each T
mf ,~k
can be combined to form a metric on the whole
space T such that each T
mf ,~k
is an entire connected component (so the different T
mf ,~k
either
coincide or are in separate components) in a number of ways, see [35]. For instance, we could
take the distance between any two elements of T
mf ,~k
to be min
{
1,Dν,{bn}
∞
n=0
mf ,~k
}
and the distance
between any two elements not in the same T
mf ,~k
to be 1. For the purposes of this paper we are
only interested in the topology of T so we will not define a metric on all of T . Though it is
extended to all of T it is best to think of the metric from [35] as a metric and topology on each
space T
mf ,~k
and not the total space.
Remark 6.24. It is important to notice that the metric we are using in this article is not
the same as the metric defined in [35]. Since these two metrics induce the same topology [35,
Section 4.6] it is suggested in [35, Remark 3.15(4)] that the metric in the present article be used
when studying the topological properties, such as connectedness.
The metric defined in [35] produces 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 6.22 will never be smaller than the distance
between those two systems using the metric studied in [35].
The metric on each component T
mf ,~k
depends on the choice of admissible measure and linearly
summable sequence, but the topology it induces does not.
36 D.M. Kane, J. Palmer and Á. Pelayo
Proposition 6.25 ([35, Theorem A]). Let ν be an admissible measure on R2 and {bn}∞n=0
a linearly summable sequence. Then the space
(
T
mf ,~k
,Dν,{bn}
∞
n=0
mf ,~k
)
is a metric space for any
choice of mf ∈ Z>0 and ~k ∈ Zmf . Furthermore, the topology induced by Dν,{bn}
∞
n=0
mf ,~k
on T
mf ,~k
does not depend on the choice of ν or the choice of {bn}∞n=0.
It follows from Proposition 6.25 that the following notion is well defined. Observe that for
each mf ∈ Z>0,~k ∈ Zmf there exists a unique ~k′ ∈ Zmf with k′1 = 0 such that T
mf ,~k
= T
mf ,~k′
.
Moreover, if T
mf ,~k′
= T
mf ,~k′′
and k′1 = k′′1 = 0 then k′ = k′′, so the union ∪~k∈Zmf ,k1=0
(T
mf ,~k
)
is the set of isomorphism classes of semitoric systems with mf focus-focus points and any two
distinct sets in the above union are disjoint.
Definition 6.26. The topology on T is the disjoint union topology on
T =
⊔
mf∈Z>0,
~k∈Zmf with k1=0
T
mf ,~k
,
where the topology on each T
mf ,~k
is induced by the metric Dν,{bn}
∞
n=0
mf ,~k
for a choice of admissible
metric ν and linear summable sequence {bn}∞n=0.
This topology is defined so that each T
mf ,~k
is a separate component of T , so it is natural to
wonder if T
mf ,~k
is connected for each fixed mf ∈ Z>0 and ~k ∈ Zmf . The remainder of the paper
is devoted to proving Theorem 2.19, which states that these are in fact path-connected.
6.3 The connectivity of the moduli space of semitoric integrable systems
Definition 6.27. Let ∆w ∈ LWPolyg
(
R2
)
be a compact primitive semitoric polygon with
∆w =
(
∆, (`λj ,+1, kj)
mf
j=1
)
.
Then the associated semitoric fan is the semitoric fan F formed by the inwards pointing primitive
integer normal vectors to the edges of ∆ in which the pair of vectors in F are labeled as fake,
hidden, or Delzant to correspond with the labeling of the corners of ∆.
Let ν be an admissible measure (cf. Definition 6.17), endow the set Polyg
(
R2
)
with the
topology induced by the metric given by the ν-measure of the symmetric difference, and endow
Polyg
(
R2
)
× Rmf × Zmf with the product topology. The topology induced on Polyg
(
R2
)
does
not depend on the choice of admissible measure and the topology induced on the set of compact
rational convex polygons by the Lebesgue measure of the symmetric difference agrees with the
subset topology induced by the topology on Polyg
(
R2
)
, see [35].
Definition 6.28. The set of primitive semitoric polygons of defect mf ∈ Z>0 inherits a topology
from the bijection
(
∆, (`λj ,+1, kj)
mf
j=1
)
7→
(
∆, (λj)
mf
j=1, (kj)
mf
j=1
)
, which we denote by τ0.
Lemma 6.29. Every semitoric fan is the associated fan of some compact primitive semitoric
polygon, and each relation in Theorem 2.14 corresponds to some continuous transformation
of compact primitive semitoric polygons. More specifically, suppose that two semitoric fans
F0,F1 ∈
(
Z2
)d
are related by
1) performing corner chops;
2) performing reverse corner chops;
Classifying Toric and Semitoric Fans by Lifting Equations from SL2(Z) 37
3) removing hidden corners; or
4) commuting fake and Delzant corners;
(see Definition 2.11). Then there exists a continuous (with respect to τ0, Definition 6.28) family
of (compact) primitive semitoric polygons
∆t
w =
(
∆t, (`λtj ,+1, kj)
mf
j=1
)
,
t ∈ [0, 1] with fixed k1, . . . , kmf , such that the semitoric fan associated to ∆0
w is F0 and the fan
associated to ∆1
w is F1.
Proof. Given any semitoric fan it is possible to construct a polygon ∆ which has those vectors
as inwards pointing normal vectors, and by the definition of semitoric fans (∆, (`λj ,+1, 0)cj=1)
(where λj is the x-value of the location of the jth hidden or fake corner) is a primitive semi-
toric polygon with the prescribed associated semitoric fan. Suppose that (∆, (`λj ,+1, kj)
mf
j=1) is
a compact primitive semitoric polygon with associated semitoric fan F = (v0, . . . , vd−1) ∈
(
Z2
)d
and fix some i ∈ {0, . . . , d − 1}. Let v−1 := vd−1 and vd := v0 so that the formulas used in
this proof will be valid if i = 0 or i = d − 1. Throughout, let p ∈ ∆ be the corner adjacent to
the edges with inwards pointing normal vectors vi and vi+1. Let ui ∈ Z2, i = 1, 2, denote the
primitive vectors along which the edges adjacent to p are aligned (oriented to point outwards
from p), ordered so that det(u1, u2) > 0.
For w1, w2 ∈ Z2 let Hεp(w1, w2) denote the half-space given by
Hεp(w1, w2) = {p+ t1w1 + t2w2 : t1 + t2 > ε}.
First we consider the corner chop operation. Suppose p is a Delzant corner. Fix some ε0 > 0
smaller than the length of the edges incident at p.
For t ∈ [0, 1] let
∆t = ∆ ∩Htε0p (u1, u2).
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 vi + vi+1 we see that the
semitoric fan of ∆t is the corner chop of the semitoric fan for ∆ for t ∈ (0, 1]. Since a reverse
corner chop is the inverse of this operation, we can use the same path backwards.
Now suppose that p is a hidden corner. Let β ∈ Z be the second component of vi+1. For
ε0 > 0 smaller than the length of the adjacent edges and t ∈ [0, 1] let
∆t = ∆ ∩Htε0p
(
u1, β
2u2
)
.
The integral normal vector to the new edge of the polygon is Tvi+1. Indeed let vi = ( ab ) and
vi+1 = ( αβ ), so the new vector should be parallel to
β2vi + vi+1 = β2vi + (aβ − αb− bβ)vi+1 =
(
β2a+ α(aβ − αb− bβ)
β2b+ β(aβ − αb− bβ)
)
= (aβ − αb)Tvi+1
using the fact that det(vi, T vi+1) = aβ−αb− bβ = 1 since p is a hidden corner. We see that ∆t
is a continuous family and by construction it has the desired semitoric fan. Thus ∆t is the
required family for the operation of removing hidden corners.
Finally, suppose that p is a fake corner and the next corner, which has adjacent edges which
have inwards pointing normal vectors vi+1 and vi+2, is a Delzant corner on the top boundary.
Since vi, vi+1, and vi+2 are all in the lower half plane we see d > 3 since a semitoric fan must
38 D.M. Kane, J. Palmer and Á. Pelayo
Figure 5. For each fan transformation there is a continuous path of semitoric polygons which transitions
between the semitoric fans. In (a) we show the corner chop and in (b) we show the removal of a hidden
corner (which is replaced by a fake and a Delzant corner).
always include at least one vector in the upper half plane. Viewing the convex polygon as
an intersection of half-planes, remove the half-plane corresponding to the edge with inwards
pointing normal vector vi+1 from the intersection to produce a new polygon ∆′ which has one
fewer edge. That is, extend the edges with inwards pointing normal vectors vi and vi+2 until
they intersect; call the new corner where they intersect p′. The polygon ∆′ has semitoric fan
F ′ = (v0, . . . , vi, vi+2, . . . , vd−1) ∈
(
Z2
)d−1
.
Since (vi, vi+1) is fake and (vi+1, vi+2) is Delzant by assumption we have
det(vi, T vi+2) = det(Tvi+1, T vi+2) = det(vi+1, vi+2) = 1,
so F ′ is a semitoric fan and the corner (vi, vi+2) in F ′ is a hidden corner. Removing the hidden
corner (vi, vi+2) from F ′ produces the required semitoric fan (v0, . . . , vi, T vi+2, vi+2, . . . , vd−1),
and since removing a hidden corner can be achieved by a continuous family (as shown above)
all that remains is to produce a continuous family from a polygon with semitoric fan F to ∆′.
Let u′1 be the primitive integral vector directing the edge away from p′ which has inwards
pointing normal vector vi+2 and let u′2 be the primitive integral vector directing the other edge
away from p′. Then for some sufficiently small ε0 > 0 let
∆t = ∆′ ∩Htε0p′ (u′1, u
′
2),
and note that ∆0 = ∆′ and ∆1 has the same semitoric fan as ∆ (for the correct choice of ε0 we
can actually get ∆0 = ∆, but this is not necessary), so we have formed a continuous path from
a polygon with associated semitoric fan F to the polygon ∆′, as desired.
So in any case we have produced a family of polygons ∆t, 0 6 t 6 1. Let
∆t
w =
(
∆t, (`λtj ,+1, kj)
mf
j=1
)
,
where λtj is the x-coordinate of the jth non-Delzant corner of ∆t ordered left to right, to complete
the proof. �
Classifying Toric and Semitoric Fans by Lifting Equations from SL2(Z) 39
Recall that toric polygons are precisely the compact primitive semitoric polygons with defect
zero. Thus, by Proposition 5.6 and Lemma 6.29 we have recovered Theorem 2.6.
In light of Propositions 5.6 and 6.29 the only difficulty remaining to prove the following lemma
is incorporating the case of semitoric systems which have noncompact polygons as invariants.
Recall τD is the topology on DPolyg
mf ,~k
(
R2
)
induced by the metric defined in 6.20.
Lemma 6.30. Let mf ∈ Z>0 and ~k ∈ Zmf . Then DPolyg
mf ,~k
(
R2
)
is path-connected with
respect to the topology τD.
Proof. Every semitoric polygon is an orbit of a primitive semitoric polygon, and to conclude
that there is a continuous path between two semitoric polygons it is sufficient to construct
a continuous (with respect to τ0, Definition 6.28) path between the primitive semitoric polygons
that generate those orbits. This is because the map which takes a primitive semitoric polygon
to its orbit is continuous. Thus, it suffices to consider only primitive semitoric polygons. Any
two compact primitive semitoric polygons in DPolyg
mf ,~k
(
R2
)
with the same semitoric 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
semitoric fan while also changing λj to still correspond to the locations of the fake and hidden
corners. So, by Proposition 5.6 given two elements of DPolyg
mf ,~k
(
R2
)
which are compact we
know that the corresponding semitoric fans are related by the moves listed in that proposition
and then by Lemma 6.29 we know these moves correspond to continuous paths of polygons.
So we have established that any two compact elements of DPolyg
mf ,~k
(
R2
)
are connected by
a continuous path.
Next assume that
[∆w] =
[(
∆, (`λj , εj , kj)
mf
j=1
)]
∈ DPolyg
mf ,~k
(
R2
)
is such that ∆ is noncompact but has only finitely many vertices. Let
π1 : R2 → R
be projection onto the first component. Choose N ∈ R such that all of the vertices of ∆ are
in the region π−1
1 ([−N,N ]). The set ∆ ∩ π−1
1 ([−N − 1, N + 1]) is a polygon but the corners
which intersect `N+1∪`−N−1 may not satisfy the Delzant condition. By [36, Remark 23] we may
change the set on arbitrarily small neighborhoods of these corners to produce a new compact
polygon, ∆′, which is equal to ∆ ∩ π−1
1 ([−N − 1, N + 1]) outside of those small neighborhoods
and has only Delzant corners inside of those neighborhoods. Thus,
∆′w =
(
∆′, (`λj , εj , kj)
mf
j=1
)
is a primitive semitoric polygon and by choosing the neighborhoods small enough we can assure
that ∆ ∩ π−1
1 ([−N,N ]) = ∆′ ∩ π−1
1 ([−N,N ]). For t ∈ [0, 1) let ∆(t) be the polygon with the
same semitoric fan as ∆′, the property that
∆(t) ∩ [−N,N ] = ∆′ ∩ [−N,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 π−1
1 (N) had as one of its endpoints (x0, y0) with
x0 > N then the corresponding edge of ∆(t) would have as its endpoint
(
x0 + 1
t−1 , y0 +m 1
t−1
)
,
where m is the slope of the edge in question. Then define ∆(1) = ∆ and we can see that ∆(t) for
40 D.M. Kane, J. Palmer and Á. Pelayo
Figure 6. The continuous path from a compact primitive semitoric polygon to a noncompact primitive
semitoric polygon with finitely many vertices.
t ∈ [0, 1] is a path from ∆′, which is compact, to ∆ so
[(
∆(t), (`λj ,+1, kj)
mf
j=1
)]
is a continuous
path which connects a compact semitoric polygon to [∆w]. This process is shown in Fig. 6.
Now we have connected all of the elements except for those with an infinite amount of vertices.
Suppose that
[∆w] =
[(
∆, (`λj , εj , kj)
mf
j=1
)]
∈ DPolyg
mf ,~k
(
R2
)
is such that ∆ is noncompact and has infinitely many vertices. We will connect [∆w] to a semi-
toric 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 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 xn ∈ R such that
ν
(
A ∩ π−1
1 ([xn,∞))
)
< 1/n,
∆ does not have a corner on the line `xn , and xn > |λj | for all j = 1, . . . ,mf . Let ∆n denote
the polygon which satisfies
∆n ∩ π−1
1 ([−∞, xn]) = ∆ ∩ π−1
1 ([−∞, xn])
and has no vertices with x-coordinate greater than xn.
For each n ∈ Z>0 and t ∈ (0, 1] define ∆n(t) to have the same semitoric fan as ∆n+1 and to
have all the same edge lengths as ∆n+1 except for the two edges which intersect `xn . Extend
those two edges horizontally by 1/t − 1. Define ∆n(0) = ∆n. Now ∆n(t) for t ∈ [0, 1] is a C0
path, with respect to the topology on polygons generated by the ν-measure of the symmetric
difference, which takes ∆n to ∆n+1. Moreover,
∆ ∗∆n(t) ⊂ A ∩ π−1
1 ([xn,∞]) so ν(∆ ∗∆n(t)) < 1/n
for each t ∈ [0, 1]. Each of these paths for n ∈ Z>0 can be concatenated to form a continuous
path ∆t, t ∈ [0, 1], from ∆0 to ∆ and we know that ∆0 has only finitely many vertices. It
is important not only that each ∆n be getting closer to ∆ but also that the path from ∆n
to ∆n+1 stays close to ∆. Then we define [(∆(t), (λj ,+1, kj)
mf
j=1)] which is a continuous path of
semitoric polygons from a semitoric polygon with finitely many vertices to [∆w]. This is shown
in Fig. 7. �
Now we can classify the connected components of M and T . Recall that M
mf ,~k
=M
mf ,~k′
if kj = k′j + c for some c ∈ Z so when stating the following lemma we require that the first
Classifying Toric and Semitoric Fans by Lifting Equations from SL2(Z) 41
Figure 7. The continuous path which adds new vertices to a noncompact primitive semitoric polygon.
This process is repeated to form a path to systems with infinitely many vertices.
component of the twisting index be 0. This is done only to make sure that there are no repeats in
the list of components. RecallM0 is the collection of semitoric lists of ingredients with mf = 0
and T0 = Φ−1(M0) is the collection of semitoric systems with no focus-focus singularities.
Finally, also recall that the fact that the different M
mf ,~k
are in different components really
comes from the definition of the constructed topology in [35] and not from any result of the
present paper.
Lemma 6.31. The connected components of M are each element of the set
{M
mf ,~k
|mf ∈ Z>0,~k ∈ Zmf with k1 = 0} ∪ {M0}
and they are each path-connected.
Proof. It is sufficient to prove that M
mf ,~k
is path-connected for each choice of mf ∈ Z>0 and
~k ∈ Zmf . Let m,m′ ∈M
mf ,~k
with
m =
(
[∆w], (hj)
mf
j=1, ((Sj)
∞)
mf
j=1
)
and m′ =
(
[∆′w], (h′j)
mf
j=1, ((S
′
j)
∞)
mf
j=1
)
.
By Lemma 6.30 we know there exists a continuous path
[∆w(t)] =
[(
∆(t), (`λj(t),+1, kj)
mf
j=1
)]
,
t ∈ [0, 1], from [∆w] to [∆′w] and by [35, Proposition 3.2] we know R[[X,Y ]]0 is path-connected
so there exists a continuous path (Sj(t))
∞ from (Sj)
∞ to (S′j)
∞ for each j = 1, . . . ,mf . For
j = 1, . . . ,mf let
lenj = length(π2(∆ ∩ `λj )), len′j = length(π2(∆′ ∩ `λ′j )),
lenj(t) = length(π2(∆(t) ∩ `λj(t)))
for t ∈ [0, 1] and define
hj(t) =
(
(1− t)hj
lenj
+
th′j
len′j
)
lenj(t).
Now we have that 0 < hj(t) < lenj(t) and t 7→ hj(t) is a continuous function from [0, 1] to R
because it is impossible for a semitoric polygon to have a vertical boundary at `λj for any
j ∈ {1, . . . ,mf}. Now define
m(t) =
(
[∆w(t)], (hj(t))
mf
j=1, ((Sj(t))
∞)
mf
j=1
)
for t ∈ [0, 1] which is a continuous path from m to m′. �
Thus we have established the following result.
Theorem 6.32. The set of connected components of T is{
T
mf ,~k
|mf ∈ Z>0, ~k ∈ Zmf with k1 = 0
}
∪ {T0}
and they are each path-connected.
Theorem 2.19 is equivalent to Theorem 6.32.
42 D.M. Kane, J. Palmer and Á. Pelayo
Acknowledgements
We thank the anonymous referees for reading the paper carefully and providing very helpful
suggestions which have improved the paper. JP and ÁP were partially supported by NSF
grants DMS-1055897 and DMS-1518420.
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1 Introduction
2 Fans, symplectic geometry, and winding numbers
2.1 Toric fans
2.2 Semitoric fans
2.3 Algebraic tools: the winding number
2.4 Applications to symplectic geometry
3 Algebraic set-up: matrices and SL2(Z) relations
4 Toric fans
5 Semitoric fans
6 Application to symplectic geometry
6.1 Invariants of semitoric systems
6.1.1 The number of focus-focus points invariant
6.1.2 The Taylor series invariant
6.1.3 The polygon invariant and the twisting index invariant
6.1.4 The action of GmfG
6.1.5 The volume invariant
6.1.6 The complete invariant
6.1.7 The classification theorem
6.2 Metric and topology on the moduli space
6.2.1 The metric on Taylor series
6.2.2 The metric on the polygon invariant
6.2.3 The metric on the moduli space of semitoric systems
6.3 The connectivity of the moduli space of semitoric integrable systems
References
|
| id | nasplib_isofts_kiev_ua-123456789-209448 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T15:57:37Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kane, D.M. Palmer, J. Pelayo, Á. 2025-11-21T19:01:28Z 2018 Classifying Toric and Semitoric Fans by Lifting Equations from SL₂(Z) / D.M. Kane, J. Palmer, Á. Pelayo // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 48 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 52B20; 15B36; 53D05 arXiv: 1502.07698 https://nasplib.isofts.kiev.ua/handle/123456789/209448 https://doi.org/10.3842/SIGMA.2018.016 We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group SL₂(Z) to its preimage in the universal cover of SL₂(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. We thank the anonymous referees for reading the paper carefully and providing very helpful suggestions, which have improved the paper. JP and ÁP were partially supported by NSF grants DMS-1055897 and DMS-1518420. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Classifying Toric and Semitoric Fans by Lifting Equations from SL₂(Z) Article published earlier |
| spellingShingle | Classifying Toric and Semitoric Fans by Lifting Equations from SL₂(Z) Kane, D.M. Palmer, J. Pelayo, Á. |
| title | Classifying Toric and Semitoric Fans by Lifting Equations from SL₂(Z) |
| title_full | Classifying Toric and Semitoric Fans by Lifting Equations from SL₂(Z) |
| title_fullStr | Classifying Toric and Semitoric Fans by Lifting Equations from SL₂(Z) |
| title_full_unstemmed | Classifying Toric and Semitoric Fans by Lifting Equations from SL₂(Z) |
| title_short | Classifying Toric and Semitoric Fans by Lifting Equations from SL₂(Z) |
| title_sort | classifying toric and semitoric fans by lifting equations from sl₂(z) |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209448 |
| work_keys_str_mv | AT kanedm classifyingtoricandsemitoricfansbyliftingequationsfromsl2z AT palmerj classifyingtoricandsemitoricfansbyliftingequationsfromsl2z AT pelayoa classifyingtoricandsemitoricfansbyliftingequationsfromsl2z |