EXAMPLES OF FOUR-DIMENSIONAL GEOMETRIC TRANSITON

Joint with S. Riolo

Wroclaw, 18th September 2018

WHAT IS GEOMETRIC TRANSITION?

Roughly speaking, geometric transition is a deformation of geometric structures on a manifold, which at some point “transitions” from one geometry to another.

That is, one has a path Pt of (G,X)-structures on a manifold M, for t in (-ε,ε), where G and X suddenly change for some t0.

A classical example of geometric transition is obtained as a deformation from hyperbolic structures to spherical structures, “going through” Euclidean structures:

X = ℍn, G = Isom(ℍn) for t < 0X = 𝔼n, G = Isom(𝔼n) for t = 0X = 𝕊n, G = Isom(𝕊n) for t > 0

A simple example of this transition is obtained by considering a hyperbolic sphere with three cone points, i.e. the double of a (regular, say) hyperbolic triangle.

A FIRST EXAMPLE

HYPERBOLIC-EUCLIDEAN-SPHERICAL TRANSITION

One can then “zoom" around this limit point, and obtain a transition of the geometric structure to a Euclidean sphere with cone points.

2 CHAPTER 1. INTRODUCTION

of a geometric transition. The focus of this thesis is the study of geometric transitions in the

context of a di↵erent degeneration behavior, that of three-dimensional structures collapsing

down onto a hyperbolic plane.

Figure 1.2: Fundamental domains for hyperbolic cone manifolds collapse onto a hyperbolicplane.

To begin the discussion, we focus our attention on hyperbolic structures on a compact

three-manifold M with boundary. Let Dt : fM ! H3 be a family of developing maps de-

fined for t > 0 and suppose Dt converges to D0 a local submersion onto a two dimensional

hyperbolic plane P. Then D0 defines a co-dimension one transversely hyperbolic foliation.

The problem of regenerating hyperbolic structures from this data was examined by Hodg-

son [Hod86], and later in a specific case by Porti [Por10]. However, it had not yet been

established how to construct a geometric transition in this context. Our point of view,

based on projective geometry, is that such a degeneration naturally suggests a transition

to AdS geometry. As motivation, note that the isometry groups Isom(H3) = SO(3, 1) and

Isom(AdS3) = SO(2, 2) are naturally subgroups of the projective group PGL(4,R). The

intersection SO(3, 1) \ SO(2, 2) is, with suitable choice of coordinates, exactly the copy of

O(2, 1) that preserves the hyperbolic plane P.

Roughly, the transition from hyperbolic to AdS geometry is constructed similarly to

the transition from hyperbolic to spherical, alluded to above. As the hyperbolic structures

degenerate, the collapsing direction is rescaled so that the structures converge to a robust

three-dimensional structure called a half-pipe (HP) structure. Interpreting the hyperbolic

structures as projective structures, this rescaling is really a projective change of coordinates.

This construction is given in Chapter 3.

The key for constructing transitions is that HP structures contain precisely the informa-

tion needed to regenerate to both hyperbolic and AdS structures. In fact, if a regeneration

can be constructed on the level of representations, then an HP structure can be “expo-

nentiated” to produce a regeneration to robust geometric structures. We simply note here

that HP geometry plays the same central role in the transition between hyperbolic and AdS

In this talk, we will be interested in hyperbolic structures which “collapse" to a co-dimension one totally geodesic hyperplane.

An example (from Cooper-Hodgson-Kerckhoff's book):

By glueing opposite faces, and then doubling, one constructs a hyperbolic structure on T3, singular along a link, which collapses to a hyperbolic torus with one cone point.

We want to study "rescaled limits” for this type of degenerations.

2 CHAPTER 1. INTRODUCTION

of a geometric transition. The focus of this thesis is the study of geometric transitions in the

context of a di↵erent degeneration behavior, that of three-dimensional structures collapsing

down onto a hyperbolic plane.

Figure 1.2: Fundamental domains for hyperbolic cone manifolds collapse onto a hyperbolicplane.

To begin the discussion, we focus our attention on hyperbolic structures on a compact

three-manifold M with boundary. Let Dt : fM ! H3 be a family of developing maps de-

fined for t > 0 and suppose Dt converges to D0 a local submersion onto a two dimensional

hyperbolic plane P. Then D0 defines a co-dimension one transversely hyperbolic foliation.

The problem of regenerating hyperbolic structures from this data was examined by Hodg-

son [Hod86], and later in a specific case by Porti [Por10]. However, it had not yet been

established how to construct a geometric transition in this context. Our point of view,

based on projective geometry, is that such a degeneration naturally suggests a transition

to AdS geometry. As motivation, note that the isometry groups Isom(H3) = SO(3, 1) and

Isom(AdS3) = SO(2, 2) are naturally subgroups of the projective group PGL(4,R). The

intersection SO(3, 1) \ SO(2, 2) is, with suitable choice of coordinates, exactly the copy of

O(2, 1) that preserves the hyperbolic plane P.

Roughly, the transition from hyperbolic to AdS geometry is constructed similarly to

the transition from hyperbolic to spherical, alluded to above. As the hyperbolic structures

degenerate, the collapsing direction is rescaled so that the structures converge to a robust

three-dimensional structure called a half-pipe (HP) structure. Interpreting the hyperbolic

structures as projective structures, this rescaling is really a projective change of coordinates.

This construction is given in Chapter 3.

The key for constructing transitions is that HP structures contain precisely the informa-

tion needed to regenerate to both hyperbolic and AdS structures. In fact, if a regeneration

can be constructed on the level of representations, then an HP structure can be “expo-

nentiated” to produce a regeneration to robust geometric structures. We simply note here

that HP geometry plays the same central role in the transition between hyperbolic and AdS

In dimension 3, J. Danciger introduced the “right” limit geometry (half-pipe) and understood a fairly general condition which ensures the existence of transition on a compact manifold.

What is bad of this talk:

• In dimension 4, we “only" produce a class of finite-volume examples of this geometric transition.

What is good of this talk:

• Many pictures.

• Some movies.

A REMINDER

REAL PROJECTIVE STRUCTURES

A real projective structure on a manifold M is given by an atlas with values in n-dimensional projective space, where transition functions are the restrictions of projective transformations.

Ui Uj

'i(Ui)⊆X 'j(Uj)⊆X

'i :Ui!X 'j :Uj!X'j◦'−1

i 2G

This is the definition of (G,X)-structure, for:

X = ℝPn

G = PGL(n + 1,ℝ)

SPECIAL CASES

REAL PROJECTIVE STRUCTURES

Here we are interested in three special types of real projective structures:

1) Hyperbolic structures

-1) Anti-de Sitter structures

0) Half-pipe structures

X1 = ℍn = P {−x20 + x2

1 + … + x2n−1 + x2

n < 0}, G1 = O(n,1)

X−1 = 𝔸d𝕊n = P {−x20 + x2

1 + … + x2n−1 − x2

n < 0}, G−1 = O(n − 1,2)

X0 = ℍℙn = P {−x20 + x2

1 + … + x2n−1+0 ⋅ x2

n < 0},

G0 = {[ A 0vT ±1] : A ∈ O(n − 1,1), v ∈ ℝn}

HALF-PIPE AS A LIMIT GEOMETRY

Consider the projective transformations

Then define

Namely, Xt is defined by the inequality

Then Xt converges to:

rt = [diag (1,…,1,1t )] .

{Xt = rt(ℍn) t > 0Xt = r|t|(𝔸d𝕊n) t < 0

Xt = P {−x20 + x2

1 + … + x2n−1 + t | t |x2

n < 0} .

X0 = ℍℙn = P {−x20 + x2

1 + … + x2n−1 < 0} .

HALF-PIPE AS A LIMIT GEOMETRY

Moreover, Gt converges to:

which is what we define as the isometry group of half-pipe space.

Half-pipe space X0 is topologically the product

However, its geometry is substantially different. For instance, the second f a c t o r re p re s e n t s a d e g e n e r a t e direction.

G0 = {[ A 0vT ±1] : A ∈ O(n − 1,1), v ∈ ℝn}

X0 ≅ ℍn−1 × ℝ .

HOW TO PRODUCE EXAMPLES OF TRANSITION

Here’s my favourite recipe:

• Take a continuous family of finite-volume hyperbolic polytopes Pt, for t in [0,ε), such that P0 collapses to a polytope in a co-dimension one totally geodesic subspace.

• Glue a finite number of copies of Pt, so as to obtain a family of hyperbolic structures on a fixed manifold (with some singularities!)

• Apply the “rescaling” transformations rt=[diag(1,…,1,1/t)].

• Show that the rescaled polytopes glue to a half-pipe structure when t=0, and even continue towards Anti-de Sitter structures for t in (-ε,0].

EXAMPLES OF TRANSITIONA FIRST EXAMPLE IN 3D

2 CHAPTER 1. INTRODUCTION

of a geometric transition. The focus of this thesis is the study of geometric transitions in the

context of a di↵erent degeneration behavior, that of three-dimensional structures collapsing

down onto a hyperbolic plane.

Figure 1.2: Fundamental domains for hyperbolic cone manifolds collapse onto a hyperbolicplane.

To begin the discussion, we focus our attention on hyperbolic structures on a compact

three-manifold M with boundary. Let Dt : fM ! H3 be a family of developing maps de-

fined for t > 0 and suppose Dt converges to D0 a local submersion onto a two dimensional

hyperbolic plane P. Then D0 defines a co-dimension one transversely hyperbolic foliation.

The problem of regenerating hyperbolic structures from this data was examined by Hodg-

son [Hod86], and later in a specific case by Porti [Por10]. However, it had not yet been

established how to construct a geometric transition in this context. Our point of view,

based on projective geometry, is that such a degeneration naturally suggests a transition

to AdS geometry. As motivation, note that the isometry groups Isom(H3) = SO(3, 1) and

Isom(AdS3) = SO(2, 2) are naturally subgroups of the projective group PGL(4,R). The

intersection SO(3, 1) \ SO(2, 2) is, with suitable choice of coordinates, exactly the copy of

O(2, 1) that preserves the hyperbolic plane P.

Roughly, the transition from hyperbolic to AdS geometry is constructed similarly to

the transition from hyperbolic to spherical, alluded to above. As the hyperbolic structures

degenerate, the collapsing direction is rescaled so that the structures converge to a robust

three-dimensional structure called a half-pipe (HP) structure. Interpreting the hyperbolic

structures as projective structures, this rescaling is really a projective change of coordinates.

This construction is given in Chapter 3.

The key for constructing transitions is that HP structures contain precisely the informa-

tion needed to regenerate to both hyperbolic and AdS structures. In fact, if a regeneration

can be constructed on the level of representations, then an HP structure can be “expo-

nentiated” to produce a regeneration to robust geometric structures. We simply note here

that HP geometry plays the same central role in the transition between hyperbolic and AdS

Let us now go back to our first 3d example of geometric transition.

For simplicity, we send some of the vertices at infinity.

Here is an affine picture of the hyperbo l i c po lyhedron we obtain (in the projective model).

EXAMPLES OF TRANSITIONA FIRST EXAMPLE IN 3D

We can now deform this hyperbolic (ideal) polyhedron, by “flattening” towards a hyperbolic quadrilateral.

By glueing faces/doubling, one obtains a deformation of geometric structures (with cone singularities!).

EXAMPLES OF TRANSITIONA FIRST EXAMPLE IN 3D

Let us now apply the projective rescaling rt=[diag(1,…,1,1/t)]. One shows that the polytope limits to a polytope in half-pipe space, and continues to Anti-de Sitter space!

THE STRUCTURE OF THE SINGULAR LOCUS

When we glue opposite faces, and double, we get finite-volume “cusped” structures, with singularities along a two-component link.

• When t>0, this is a cone singularity, for which the cone angle approaches 2π.

• When t<0, this is a tachyon singularity.

• When t=0, the singularity is the infinitesimal version of cone angles and tachyons.

EXAMPLES OF TRANSITIONA SECOND EXAMPLE IN 3D

We shall now see another simple example which will be useful later on.

All the angles of this polytopes are π/2, except at the three top edges.

Let us now send these three dihedral angles to π.

EXAMPLES OF TRANSITIONA SECOND EXAMPLE IN 3D

Again, by applying the projective rescaling rt=[diag(1,…,1,1/t)], one obtains a limit polytope in half-pipe space. As before, the three lateral faces are degenerate.

We c a n a g a i n p r o d u c e a t rans i t ion o f f i n i te -vo lume structures on a manifold, by doubling along lateral faces, and then along top and bottom faces.

The s ingular locus is more complicated here (a θ-graph).

FROM DIMENSION THREE…

As already mentioned, J. Danciger studied geometric transition on a general closed 3-manifold M, with singular locus a knot K.

He proves that a collapsing of hyperbolic structures on M, with cone singularities along K, admits a geometric transition towards Anti-de Sitter structures under the following condition on the character variety:

He also produces examples of regeneration from a half-pipe structure.

In general, it is a difficult problem to determine which manifolds admit a transition of geometric structures.

H1(π1(M∖K ), 𝔰𝔬(2,1)hol) ≅ ℝ

…TO DIMENSION FOUR.

DEFINITION: A cuboctahedral manifold is a hyperbolic finite-volume 3-manifold obtained by gluieing a finite number of copies of an ideal right-angles cuboctahedron. It is good if the glueings preserve a checkerboard coloration of the triangular faces.

THEOREM (Riolo-S.):

Let M be a good cuboctahedral mani fo ld. Then MxS1 admits a g e o m e t r i c t r a n s i t i o n f r o m a hyperbolic structure to an Anti-de Sitter structure, singular along a 2-dimensional simplicial complex.

…TO DIMENSION FOUR.

The starting point to construct this transition is a family of 4-dimensional polytopes Pt introduced by Kerckhoff and Storm, by starting from the hyperbolic 24-cells and removing two walls.

For all t, the polytope Pt intersects a totally geodesic plane in a cuboctahedron. When t→0, Pt collapses to the cuboctahedron itself.

After rescaling, one can continue the polytopes rt(Pt) in projective space to a path of polytopes.

For t=0, one obtains a half-pipe polytope, for which some of the walls are degenerate, and this continues to Anti-de Sitter space.

Moreover, all isometries of the cuboctahedron extend to isometries of Pt. This permits to “glue" copies and obtain geometric structures.

SOME WORDS ABOUT THE COMBINATORICS

To get a flavour of how this looks like, here is the list of hyperplanes defining the polytope Pt.

FOUR-DIMENSIONAL GEOMETRIC TRANSITION 15

0+ =h�p2 : 1 : 1 : 1 : 1

�|t|i, 0� =

h�p2 : 1 : 1 : 1 : �t

i,

1+ =h�p2 : 1 : �1 : 1 : �1

�|t|i, 1� =

h�p2 : 1 : �1 : 1 : t

i,

2+ =h�p2 : 1 : �1 : �1 : 1

�|t|i, 2� =

h�p2 : 1 : �1 : �1 : �t

i,

3+ =h�p2 : 1 : 1 : �1 : �1

�|t|i, 3� =

h�p2 : 1 : 1 : �1 : t

i,

4+ =h�p2 : �1 : 1 : �1 : 1

�|t|i, 4� =

h�p2 : �1 : 1 : �1 : �t

i,

5+ =h�p2 : �1 : 1 : 1 : �1

�|t|i, 5� =

h�p2 : �1 : 1 : 1 : t

i,

6+ =h�p2 : �1 : �1 : 1 : 1

�|t|i, 6� =

h�p2 : �1 : �1 : 1 : �t

i,

7+ =h�p2 : �1 : �1 : �1 : �1

�|t|i, 7� =

h�p2 : �1 : �1 : �1 : t

i,

A =h�1 :

p2 : 0 : 0 : 0

i, B =

h�1 : 0 :

p2 : 0 : 0

i,

C =h�1 : 0 : 0 :

p2 : 0

i, D =

h�1 : 0 : 0 : �

p2 : 0

i,

E =h�1 : 0 : �

p2 : 0 : 0

i, F =

h�1 : �

p2 : 0 : 0 : 0

i.

Table 1. The half-spaces that define Pt are given by these elements of S4,⇤

and denoted by the same symbols.

It can also be checked that the faces of P (t) are pairwise orthogonal, except the pairs(1+

,3+), (1+,5+) and (3+

,5+) which form an angle ✓(t) (for t > 0) such that:

cos(✓(t)) = �1 +3

2t2.

Hence at time t =p

2/3, the planes bounding the half-spaces 1+, 3+ and 5+ meet in theboundary at infinity of H3.

4. A four-dimensional transition

Recall the notation introduced in Section ?? for half-spaces of the projective sphere Sn.We define

Pt ⇢ S4

to be the intersection of the half-spaces listed in Table 1, depending on the time parameter

t 2✓�1,

q1�3

�.

When t � 0, each element [v] in the list of Table 1 satisfies hv, vi1 > 0, and thus definesa half-space of H

4. For t > 0, the set Pt ⇢ H4 is indeed a hyperbolic 4-polytope, first

introduced in [?] and then studied in [?]. As t ! 0 the polytope collapses to a three-dimensional polyhedron in

H3 = H \H

4 = H \ AdS4 ⇢ S4,

where H ⇢ S4 is the hyperplane defined by [0 : 0 : 0 : 0 : 1]. PROMEMORIA: definire

nei preliminari carta a�ne

• When t→0, the “plus” walls converge to the same totally geodesic hyperplane.

• The “minus" & “letter" walls are timelike when t < 0 , a n d b e c o m e degenerate in the half-pipe limit.

SOME WORDS ABOUT THE COMBINATORICS

We already know some of the walls!

The “letter” walls look like today's first 3d example…

…whereas the “minus" walls are copies of our second example.

We can compare them with the cuboctahedron, which is the “horizontal” slice of t h e f o u r - d i m e n s i o n a l polytope.

THANKS FOR YOUR ATTENTION!