text: (cid:88) The Laplace–Beltrami operator is diagonal in the spherical harmonics basis, so we can expand Φ (xµ) Φ(xµ, θ, φ) = lm Y (θ, φ) , (3.8) lm R (cid:82) lm which allows to write the 4D(cid:88)-reduced Lagrangia(cid:0)n (cid:1) L = R2 dΩL Φ4D Φ = Φ† ∂ ∂µΦ − m2 + l(l + 1) Φ† Φ . (3.9) lm µ lm Φ lm lm lm 3.2 Gauge bosons on a sphere The description of the properties of spin-1 gauge bosons can be obtained in a way, which is very similar to the derivation for the scalars. At the same time some diﬀerences are crucial in terms of applications to phenomenology and model building, as discussed in the following. 3.2.1 Degrees of freedom As usual for gauge bosons in extra-dimensional theories, the gauge ﬁeld A breaks into two M parts: a 4D vector spanning the 4 components pointing into ordinary space-time directions, and 4D scalar ﬁelds pointing along the extra space. In the KK expansion, the scalar ﬁelds contain both Goldstone bosons for the massive vectors in the Kaluza-Klein expansion, and physical scalars. The connection between the components A and the physical degrees of θ,φ freedom, however, requires a non-trivial redeﬁnition of the ﬁelds, unlike in the ﬂat case. We use the following redeﬁnitions [31]: A A A˜ = θ , A˜ = φ , θ φ R R sin θ 1 1 1 A = √ (A˜ ± iA˜ ) , A˜ = √ (A + A ) , A˜ = √ (A − A ) , ± θ φ θ + − φ + − 2 2 2i ð 1 A = A† , A = −i Ξ† , Ξ = √ (Φ + iΘ) . (3.10) + − + 2 As we shall see from the Lagrangian, Φ is the physical scalar that will remain in the spectrum, while Θ is a Goldstone boson, whose KK levels will be eaten by the corresponding levels of the vector to give them mass. 3.2.2 Gauge ﬁxing As in standard gauge theories, we need a gauge ﬁxing part for the Lagrangian. The usual Feynman-t’Hooft part is present to ﬁx the gauge of the 4D massless vector, while the rest is determined to eliminate the mixing between the vector and scalar parts in the quadratic Lagrangian [12, 31]: 1 L = tr(f2) (3.11) gf ξ where ξ ξ f = ηµν∂ Aν + ∂ sin θA + ∂ A . (3.12) µ R2 sin θ θ θ R2 sin2 θ φ φ – 11 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: Laplace–Beltrami

text: operator

text: is

text: diagonal

text: in

text: the

text: spherical

text: harmonics

text: basis,

text: so

text: we

text: can

text: expand

text: Φ

text: (xµ)

text: Φ(xµ,

text: θ,

text: φ)

text: lm

text: Y

text: (θ,

text: φ)

text: lm

text: R

text: lm

text: which

text: allows

text: to

text: write

text: the

text: L

text: dΩL

text: Φ

text: Φ†

text: ∂µΦ

text: l(l

text: Φ†

text: Φ

text: lm

text: µ

text: lm

text: Φ

text: lm

text: lm

text: lm

text: Gauge

text: bosons

text: on

text: a

text: sphere

text: The

text: description

text: of

text: the

text: properties

text: of

text: gauge

text: bosons

text: can

text: be

text: obtained

text: in

text: a

text: way,

text: which

text: is

text: very

text: similar

text: to

text: the

text: derivation

text: for

text: the

text: scalars.

text: At

text: the

text: same

text: time

text: some

text: diﬀerences

text: are

text: crucial

text: in

text: terms

text: of

text: applications

text: to

text: phenomenology

text: and

text: model

text: building,

text: as

text: discussed

text: in

text: the

text: following.

text: Degrees

text: of

text: freedom

text: As

text: usual

text: for

text: gauge

text: bosons

text: in

text: extra-dimensional

text: theories,

text: the

text: gauge

text: ﬁeld

text: A

text: breaks

text: into

text: two

text: M

text: parts:

text: a

text: vector

text: spanning

text: the

text: components

text: pointing

text: into

text: ordinary

text: space-time

text: directions,

text: and

text: scalar

text: ﬁelds

text: pointing

text: along

text: the

text: extra

text: space.

text: In

text: the

text: KK

text: expansion,

text: the

text: scalar

text: ﬁelds

text: contain

text: both

text: Goldstone

text: bosons

text: for

text: the

text: massive

text: vectors

text: in

text: the

text: Kaluza-Klein

text: expansion,

text: and

text: physical

text: scalars.

text: The

text: connection

text: between

text: the

text: components

text: A

text: and

text: the

text: physical

text: degrees

text: of

text: θ,φ

text: freedom,

text: however,

text: requires

text: a

text: non-trivial

text: redeﬁnition

text: of

text: the

text: ﬁelds,

text: unlike

text: in

text: the

text: ﬂat

text: case.

text: We

text: use

text: the

text: following

text: redeﬁnitions

text: A

text: A

text: A˜

text: θ

text: A˜

text: φ

text: θ

text: φ

text: R

text: R

text: sin

text: θ

text: A

text: (A˜

text: iA˜

text: A˜

text: (A

text: A

text: A˜

text: (A

text: A

text: θ

text: φ

text: θ

text: φ

text: ð

text: A

text: A†

text: A

text: −i

text: Ξ†

text: Ξ

text: (Φ

text: iΘ)

text: As

text: we

text: shall

text: see

text: from

text: the

text: Lagrangian,

text: Φ

text: is

text: the

text: physical

text: scalar

text: that

text: will

text: remain

text: in

text: the

text: spectrum,

text: while

text: Θ

text: is

text: a

text: Goldstone

text: boson,

text: whose

text: KK

text: levels

text: will

text: be

text: eaten

text: by

text: the

text: corresponding

text: levels

text: of

text: the

text: vector

text: to

text: give

text: them

text: mass.

text: Gauge

text: ﬁxing

text: As

text: in

text: standard

text: gauge

text: theories,

text: we

text: need

text: a

text: gauge

text: ﬁxing

text: part

text: for

text: the

text: Lagrangian.

text: The

text: usual

text: Feynman-t’Hooft

text: part

text: is

text: present

text: to

text: ﬁx

text: the

text: gauge

text: of

text: the

text: massless

text: vector,

text: while

text: the

text: rest

text: is

text: determined

text: to

text: eliminate

text: the

text: mixing

text: between

text: the

text: vector

text: and

text: scalar

text: parts

text: in

text: the

text: quadratic

text: Lagrangian

text: L

text: gf

text: ξ

text: where

text: ξ

text: ξ

text: f

text: ηµν∂

text: Aν

text: sin

text: θA

text: A

text: µ

text: sin

text: θ

text: θ

text: θ

text: θ

text: φ

text: φ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Figure 1. Representation of S2/S as a fundamental domain on S2 with boundary conditions in 4 R green - using standard nomenclature for 2 orbifolds [10, 34]. There is one singular point: the orbit of the North and South poles of the sphere, which are identiﬁed. This ﬁgure helps illustrate that the residual U(1) symmetry of the orbifold is in fact φ translational invariance, which is suggested in purple arrows Note that a zero mode (l, m) = (0, 0) is consistent. This expansion is valid for the scalar (Higgs), and for spin-1 states. For the scalar components of gauge vectors, the constraint that the ﬁeld transforms as a 6D vector, imposes that the transformation of the scalar components are G A = −A and G A = A , which trickles down to imposing that Φ has the following A θ θ A φ φ expansion: • l even: m is an odd integer so m = n, 3n, ... n • l odd: m is an even integer so m = 0, 2n, 4n, ... n In the case n = 2, which we will study in detail, this means that the ﬁrst level l = 1 only contains gauge scalars with m = 0 and the next level is l = 2, with gauge vectors and scalar ﬁelds at m = 0, and gauge scalars at m = ±2. 4 The lightest rotating particle As we have stated in Section , we are interested in the orbifold S2/S for our model 4 because of its geometric features. Indeed, we found that besides its single ﬁxed point, there is a residual global U(1) symmetry of this space. Due to Noether’s Theorem, this symmetry is associated with a quantum number that is conserved, which in turn means that the lightest – 13 – ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine##

text: Figure

label: paragraph

text: Representation

text: of

text: as

text: a

text: fundamental

text: domain

text: on

text: with

text: boundary

text: conditions

text: in

text: R

text: green

text: using

text: standard

text: nomenclature

text: for

text: orbifolds

text: There

text: is

text: one

text: singular

text: point:

text: the

text: orbit

text: of

text: the

text: North

text: and

text: South

text: poles

text: of

text: the

text: sphere,

text: which

text: are

text: identiﬁed.

text: This

text: ﬁgure

text: helps

text: illustrate

text: that

text: the

text: residual

text: symmetry

text: of

text: the

text: orbifold

text: is

text: in

text: fact

text: φ

text: translational

text: invariance,

text: which

text: is

text: suggested

text: in

text: purple

text: arrows

text: Note

text: that

text: a

text: zero

text: mode

text: (l,

text: m)

text: is

text: consistent.

text: This

text: expansion

text: is

text: valid

text: for

text: the

text: scalar

text: (Higgs),

text: and

text: for

text: states.

text: For

text: the

text: scalar

text: components

text: of

text: gauge

text: vectors,

text: the

text: constraint

text: that

text: the

text: ﬁeld

text: transforms

text: as

text: a

text: vector,

text: imposes

text: that

text: the

text: transformation

text: of

text: the

text: scalar

text: components

text: are

text: G

text: A

text: −A

text: and

text: G

text: A

text: A

text: which

text: trickles

text: down

text: to

text: imposing

text: that

text: Φ

text: has

text: the

text: following

text: A

text: θ

text: θ

text: A

text: φ

text: φ

text: expansion:

text: l

text: even:

text: m

text: is

text: an

text: odd

text: integer

text: so

text: m

text: n,

text: n

text: l

text: odd:

text: m

text: is

text: an

text: even

text: integer

text: so

text: m

text: n

text: In

text: the

text: case

text: n

text: which

text: we

text: will

text: study

text: in

text: detail,

text: this

text: means

text: that

text: the

text: ﬁrst

text: level

text: l

text: only

text: contains

text: gauge

text: scalars

text: with

text: m

text: and

text: the

text: next

text: level

text: is

text: l

text: with

text: gauge

text: vectors

text: and

text: scalar

text: ﬁelds

text: at

text: m

text: and

text: gauge

text: scalars

text: at

text: m

text: The

text: lightest

text: rotating

text: particle

text: As

text: we

text: have

text: stated

text: in

text: Section

text: we

text: are

text: interested

text: in

text: the

text: orbifold

text: for

text: our

text: model

text: because

text: of

text: its

text: geometric

text: features.

text: Indeed,

text: we

text: found

text: that

text: besides

text: its

text: single

text: ﬁxed

text: point,

text: there

text: is

text: a

text: residual

text: global

text: symmetry

text: of

text: this

text: space.

text: Due

text: to

text: Noether’s

text: Theorem,

text: this

text: symmetry

text: is

text: associated

text: with

text: a

text: quantum

text: number

text: that

text: is

text: conserved,

text: which

text: in

text: turn

text: means

text: that

text: the

text: lightest

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: The symmetries of the model also allow to write localised interactions on the ﬁxed point: as we have argued above, such terms do not break the U(1) protecting the Dark Matter candidate. Furthermore, localised operators are always required as counter–terms of the divergent 6D loops, thus they are necessary for a correct renormalisation of the theory. 4.2 Mass splittings: electroweak and loop eﬀects Now that we have singled out which Kaluza-Klein tier will contain the dark matter candidate of our model, we need to ﬁnd which ﬁeld in this tier is actually the lightest, and will hence be stable. There are two sources of mass splitting for same-tier Kaluza-Klein particles: loop eﬀects, which are usually dominant, and electroweak eﬀects from the Higgs sector, which are most often negligible. In our case, however, the situation may be reversed as the dominant log- divergent contribution to the loop vanishes and one is left with subleading ﬁnite contributions. 4.2.1 Electroweak symmetry breaking: tree-level spectrum At tree level, the only source of mass splitting among the U(1) and SU(2) ﬁelds is the Y L Higgs mechanism. Let us ﬁrst work out the electroweak symmetry breaking mechanism in this geometry. Electroweak symmetry breaking. The 6D Higgs Lagrangian is deﬁned as L6D = (D φ)†(DMφ) + µ2 φ†φ − λ (φ†φ)2 (4.2) H M 6D 6D (cid:32) (cid:33) where φ is a SU(2) doublet: 1 φ + iφ φ = √ 1 2 . (4.3) 2 v6D + H + iφ3 (cid:88) The Kaluza-Klein expansion is formulated in terms of spherical harmonics: Y (Ω) φ(xµ, Ω) = lm φ (xµ) . (4.4) lm R The vacuum expectation value of the ﬁeld in the minimum energy state has to have a (cid:115) constant wavefunction to minimise kinetic energy, so its value is determined by the minimum of the potential as: µ2 v v = 6D = √ 4D . (4.5) 6D 2λ6D 4πR Because the situation is the same as in the Standard Model in the lowest Kaluza-Klein mode (0, 0), we know what to expect in the Unitary gauge: the Higgs doublet will only contain one physical state, H , which is the Standard Model Higgs boson. In the higher modes, however, 00 the situation is more complex as there are already Goldstone bosons for the excitations of the W ± and Z bosons, coming from the gauge scalar levels. Counting degrees of freedom, it appears that there should be 4 physical scalar excitations per KK level which are a mixture – 15 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: symmetries

text: of

text: the

text: model

text: also

text: allow

text: to

text: write

text: localised

text: interactions

text: on

text: the

text: ﬁxed

text: point:

text: as

text: we

text: have

text: argued

text: above,

text: such

text: terms

text: do

text: not

text: break

text: the

text: protecting

text: the

text: Dark

text: Matter

text: candidate.

text: Furthermore,

text: localised

text: operators

text: are

text: always

text: required

text: as

text: counter–terms

text: of

text: the

text: divergent

text: loops,

text: thus

text: they

text: are

text: necessary

text: for

text: a

text: correct

text: renormalisation

text: of

text: the

text: theory.

text: Mass

text: splittings:

text: electroweak

text: and

text: loop

text: eﬀects

text: Now

text: that

text: we

text: have

text: singled

text: out

text: which

text: Kaluza-Klein

text: tier

text: will

text: contain

text: the

text: dark

text: matter

text: candidate

text: of

text: our

text: model,

text: we

text: need

text: to

text: ﬁnd

text: which

text: ﬁeld

text: in

text: this

text: tier

text: is

text: actually

text: the

text: lightest,

text: and

text: will

text: hence

text: be

text: stable.

text: There

text: are

text: two

text: sources

text: of

text: mass

text: splitting

text: for

text: same-tier

text: Kaluza-Klein

text: particles:

text: loop

text: eﬀects,

text: which

text: are

text: usually

text: dominant,

text: and

text: electroweak

text: eﬀects

text: from

text: the

text: Higgs

text: sector,

text: which

text: are

text: most

text: often

text: negligible.

text: In

text: our

text: case,

text: however,

text: the

text: situation

text: may

text: be

text: reversed

text: as

text: the

text: dominant

text: log-

text: divergent

text: contribution

text: to

text: the

text: loop

text: vanishes

text: and

text: one

text: is

text: left

text: with

text: subleading

text: ﬁnite

text: contributions.

text: Electroweak

text: symmetry

text: breaking:

text: tree-level

text: spectrum

text: At

text: tree

text: level,

text: the

text: only

text: source

text: of

text: mass

text: splitting

text: among

text: the

text: and

text: ﬁelds

text: is

text: the

text: Y

text: L

text: Higgs

text: mechanism.

text: Let

text: us

text: ﬁrst

text: work

text: out

text: the

text: electroweak

text: symmetry

text: breaking

text: mechanism

text: in

text: this

text: geometry.

text: Electroweak

text: symmetry

text: breaking.

text: The

text: Higgs

text: Lagrangian

text: is

text: deﬁned

text: as

text: (D

text: φ)†(DMφ)

text: µ2

text: φ†φ

text: λ

text: (φ†φ)2

text: H

text: M

text: where

text: φ

text: is

text: a

text: doublet:

text: φ

text: iφ

text: φ

text: H

text: The

text: Kaluza-Klein

text: expansion

text: is

text: formulated

text: in

text: terms

text: of

text: spherical

text: harmonics:

text: Y

text: (Ω)

text: φ(xµ,

text: Ω)

text: lm

text: φ

text: (xµ)

text: lm

text: R

text: The

text: vacuum

text: expectation

text: value

text: of

text: the

text: ﬁeld

text: in

text: the

text: minimum

text: energy

text: state

text: has

text: to

text: have

text: a

text: constant

text: wavefunction

text: to

text: minimise

text: kinetic

text: energy,

text: so

text: its

text: value

text: is

text: determined

text: by

text: the

text: minimum

text: of

text: the

text: potential

text: as:

text: µ2

text: v

text: v

text: Because

text: the

text: situation

text: is

text: the

text: same

text: as

text: in

text: the

text: Standard

text: Model

text: in

text: the

text: lowest

text: Kaluza-Klein

text: mode

text: we

text: know

text: what

text: to

text: expect

text: in

text: the

text: Unitary

text: gauge:

text: the

text: Higgs

text: doublet

text: will

text: only

text: contain

text: one

text: physical

text: state,

text: H

text: which

text: is

text: the

text: Standard

text: Model

text: Higgs

text: boson.

text: In

text: the

text: higher

text: modes,

text: however,

text: the

text: situation

text: is

text: more

text: complex

text: as

text: there

text: are

text: already

text: Goldstone

text: bosons

text: for

text: the

text: excitations

text: of

text: the

text: W

text: and

text: Z

text: bosons,

text: coming

text: from

text: the

text: gauge

text: scalar

text: levels.

text: Counting

text: degrees

text: of

text: freedom,

text: it

text: appears

text: that

text: there

text: should

text: be

text: physical

text: scalar

text: excitations

text: per

text: KK

text: level

text: which

text: are

text: a

text: mixture

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: of the Higgs and scalar gauge ﬁelds, and we can ﬁnd their exact content by looking at the gauge-ﬁxing Lagrangian to ﬁnd mass eigenstates, as was done in [10]. The gauge-ﬁxing Lagrangian for the broken SU(2) × U(1) is obtained by expanding on the pure Yang-Mills gauge-ﬁxing term with the requirement that the quadratic terms mixing the Higgs ﬁeld and gauge bosons be cancelled, yielding the expression: 1 L = − tr(faf ) (4.6) gf a ξ ξ ξ fa =ηµν∂ Aa + ∂ (sin θAa) + ∂ Aa (4.7) µ ν R2 sin θ θ θ R2 sin θ φ φ − ξgF aφ (4.8) i i where a ∈ {1, 2, 3, Y } and gF a = g F a (cid:104)φ(cid:105)j is the the same matrix as in the typical R i (a) ij ξ gauge-ﬁxing in broken 4D gauge theories. The resulting Lagrangian terms can be sorted by their order in ξ • O(ξ−1): correction to the vector propagators that vanish in the unitary limit • O(ξ0): terms that kill mixed quadratic terms between the Higgs, gauge vectors and gauge scalars • O(ξ1): ξ-dependent mass matrix that ﬁxes which combination of the Higgs components and gauge scalars are Goldstone bosons and which are physical ﬁelds. This matrix must be taken KK-level by KK-level and yields l-dependent admixtures of the two components in the physical state, which is its null vector. To give a simpliﬁed picture (cid:32) (cid:112) (cid:33) of this mixing, we show the mixing matrix in the case of an Abelian Higgs mechanism: (cid:112) ξ g2ζ2 gζ l (l + 1) M2 = (4.9) gf R2 gζ l (l + 1) l (l + 1) where ζ = vR. This matrix acts on the vector (φ , Θ)T where the ﬁrst entry is the Higgs 0 component whose zero mode is a Goldstone boson and the second entry is the imaginary part of the canonically normalised gauge scalar Ξ. One can note that M2 is of rank 1, showing indeed that one of its eigenvector is integrated gf out in the Unitary limit ξ → ∞ while the other does not get a ξ-dependent mass and therefore (cid:32) (cid:33) is physical. A null vector of this matrix, corresponding to the physical state is 1 −√ gζ . (4.10) l(l+1) The Higgs component in the Goldstone boson goes to zero at large l, which is expected due to the fact that the mass of the associated vector becomes very large compared to the Higgs vacuum expectation value. In particular, already at l = 2, which is the level of interest for this – 16 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: of

text: the

text: Higgs

text: and

text: scalar

text: gauge

text: ﬁelds,

text: and

text: we

text: can

text: ﬁnd

text: their

text: exact

text: content

text: by

text: looking

text: at

text: the

text: gauge-ﬁxing

text: Lagrangian

text: to

text: ﬁnd

text: mass

text: eigenstates,

text: as

text: was

text: done

text: in

text: The

text: gauge-ﬁxing

text: Lagrangian

text: for

text: the

text: broken

text: is

text: obtained

text: by

text: expanding

text: on

text: the

text: pure

text: Yang-Mills

text: gauge-ﬁxing

text: term

text: with

text: the

text: requirement

text: that

text: the

text: quadratic

text: terms

text: mixing

text: the

text: Higgs

text: ﬁeld

text: and

text: gauge

text: bosons

text: be

text: cancelled,

text: yielding

text: the

text: expression:

text: L

text: tr(faf

text: gf

text: a

text: ξ

text: ξ

text: ξ

text: fa

text: =ηµν∂

text: Aa

text: (sin

text: θAa)

text: Aa

text: µ

text: ν

text: sin

text: θ

text: θ

text: θ

text: sin

text: θ

text: φ

text: φ

text: ξgF

text: aφ

text: i

text: i

text: where

text: a

text: Y

text: and

text: gF

text: a

text: g

text: F

text: a

text: is

text: the

text: the

text: same

text: matrix

text: as

text: in

text: the

text: typical

text: R

text: i

text: (a)

text: ij

text: ξ

text: gauge-ﬁxing

text: in

text: broken

text: gauge

text: theories.

text: The

text: resulting

text: Lagrangian

text: terms

text: can

text: be

text: sorted

text: by

text: their

text: order

text: in

text: ξ

text: correction

text: to

text: the

text: vector

text: propagators

text: that

text: vanish

text: in

text: the

text: unitary

text: limit

text: terms

text: that

text: kill

text: mixed

text: quadratic

text: terms

text: between

text: the

text: Higgs,

text: gauge

text: vectors

text: and

text: gauge

text: scalars

text: ξ-dependent

text: mass

text: matrix

text: that

text: ﬁxes

text: which

text: combination

text: of

text: the

text: Higgs

text: components

text: and

text: gauge

text: scalars

text: are

text: Goldstone

text: bosons

text: and

text: which

text: are

text: physical

text: ﬁelds.

text: This

text: matrix

text: must

text: be

text: taken

text: KK-level

text: by

text: KK-level

text: and

text: yields

text: l-dependent

text: admixtures

text: of

text: the

text: two

text: components

text: in

text: the

text: physical

text: state,

text: which

text: is

text: its

text: vector.

text: To

text: give

text: a

text: simpliﬁed

text: picture

text: of

text: this

text: mixing,

text: we

text: show

text: the

text: mixing

text: matrix

text: in

text: the

text: case

text: of

text: an

text: Abelian

text: Higgs

text: mechanism:

text: ξ

text: gζ

text: l

text: (l

text: gf

text: gζ

text: l

text: (l

text: l

text: (l

text: where

text: ζ

text: vR.

text: This

text: matrix

text: acts

text: on

text: the

text: vector

text: (φ

text: Θ)T

text: where

text: the

text: ﬁrst

text: entry

text: is

text: the

text: Higgs

text: component

text: whose

text: zero

text: mode

text: is

text: a

text: Goldstone

text: boson

text: and

text: the

text: second

text: entry

text: is

text: the

text: imaginary

text: part

text: of

text: the

text: canonically

text: normalised

text: gauge

text: scalar

text: Ξ.

text: One

text: can

text: note

text: that

text: is

text: of

text: rank

text: showing

text: indeed

text: that

text: one

text: of

text: its

text: eigenvector

text: is

text: integrated

text: gf

text: out

text: in

text: the

text: Unitary

text: limit

text: ξ

text: while

text: the

text: other

text: does

text: not

text: get

text: a

text: ξ-dependent

text: mass

text: and

text: therefore

text: is

text: physical.

text: A

text: vector

text: of

text: this

text: matrix,

text: corresponding

text: to

text: the

text: physical

text: state

text: is

text: gζ

text: The

text: Higgs

text: component

text: in

text: the

text: Goldstone

text: boson

text: goes

text: to

text: zero

text: at

text: large

text: l,

text: which

text: is

text: expected

text: due

text: to

text: the

text: fact

text: that

text: the

text: mass

text: of

text: the

text: associated

text: vector

text: becomes

text: very

text: large

text: compared

text: to

text: the

text: Higgs

text: vacuum

text: expectation

text: value.

text: In

text: particular,

text: already

text: at

text: l

text: which

text: is

text: the

text: level

text: of

text: interest

text: for

text: this

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6.2 Thermal relic density results In the following, we will consider two extreme cases: ﬁrst, the case where only one state contributes to the DM relic abundance, and later we consider all the co-annihilation processes. The latter is a more realistic case, as the mass splittings we expect in this model are very small. We numerically checked that eﬀects due to the splitting induced by the electroweak symmetry breaking are negligible, so that the results are shown in the most favourable case where such splittings are set to zero. The case without co-annihilation can be seen as a least favourable scenario. 6.2.1 Annihilation process There is only one relevant channel for the annihilation of dark matter: ϕ ϕ¯ → W +W −. A A Due to phase space eﬀects, the production of W excitations is largely suppressed and can therefore be neglected. Using the CalcHEP analytical expression for the matrix element, we could compute the non-relativistic expansion of the associated (cid:104)σv(cid:105), which in turn allowed us to predict the dark matter density in our model, taking into account only annihilation. This prediction is formulated as a function of the radius of the extra-dimension, and Ωh2 increases with the mass scale of the extra-dimensions as expected. The result is presented in Figure 6 where the Planck preferred value is represented to show that the upper bound on the inverse radius inferred from this calculation would be 1 ≤ 100 GeV which in turns means R m ≤ 250 GeV. This limit, however is too stringent as co–annihilation eﬀects with other l=2 gauge scalars change the prediction for the relic density signiﬁcantly. 6.2.2 Co–annihilation processes As we show in the Appendices 6.2.2 and B.3, there are several channels available for co– annihilation of our dark matter candidate, once the other electroweak gauge bosons are in- cluded in the calculation. A list of all the processes we considered can be found in B.3 and we show in Figure 4 a sample of thermal cross-sections computed using the dedicated CalcHEP module. These processes can be sorted into several classes, as shown in Figure 5. As we will show below, the dominant contribution comes from co–annihilation to pairs of tier-0 vector bosons (Figure 5(a)). Annihilation to fermions (Figure 5(b)) is a lot less important due to the absence of s-wave processes and the associated production of a vector boson and excited vector bosons (Figure 5(c)) is phase-space suppressed. It is worth mentioning that we considered the possible contribution of near-resonant s- channel diagrams involving (4, 0) excitations like in Figure 5(d), but that the resonance is still too far-oﬀ to have a signiﬁcant impact on the thermal cross-sections. We should however note that a signiﬁcant change can be introduced by ﬁne-tuning the localised Higgs mass operator so that Higgs excitations hit precisely the resonance (Figure 5(e)). A quick estimate using a subset of all processes showed that this changes signiﬁcantly the thermal cross-section – possibly providing a small funnel in parameter space where Dark matter is allowed to be very heavy. We however leave aside the study of this ﬁne-tuned possibility in the present work. – 24 – ##LTLine##

text: Thermal

text: relic

text: density

text: results

text: In

text: the

text: following,

text: we

text: will

text: consider

text: two

text: extreme

text: cases:

text: ﬁrst,

text: the

text: case

text: where

text: only

text: one

text: state

text: contributes

text: to

text: the

text: DM

text: relic

text: abundance,

text: and

text: later

text: we

text: consider

text: all

text: the

text: co-annihilation

text: processes.

text: The

text: latter

text: is

text: a

text: more

text: realistic

text: case,

text: as

text: the

text: mass

text: splittings

text: we

text: expect

text: in

text: this

text: model

text: are

text: very

text: small.

text: We

text: numerically

text: checked

text: that

text: eﬀects

text: due

text: to

text: the

text: splitting

text: induced

text: by

text: the

text: electroweak

text: symmetry

text: breaking

text: are

text: negligible,

text: so

text: that

text: the

text: results

text: are

text: shown

text: in

text: the

text: most

text: favourable

text: case

text: where

text: such

text: splittings

text: are

text: set

text: to

text: zero.

text: The

text: case

text: without

text: co-annihilation

text: can

text: be

text: seen

text: as

text: a

text: least

text: favourable

text: scenario.

text: Annihilation

text: process

text: There

text: is

text: only

text: one

text: relevant

text: channel

text: for

text: the

text: annihilation

text: of

text: dark

text: matter:

text: ϕ

text: ϕ¯

text: W

text: +W

text: A

text: A

text: Due

text: to

text: phase

text: space

text: eﬀects,

text: the

text: production

text: of

text: W

text: excitations

text: is

text: largely

text: suppressed

text: and

text: can

text: therefore

text: be

text: neglected.

text: Using

text: the

text: CalcHEP

text: analytical

text: expression

text: for

text: the

text: matrix

text: element,

text: we

text: could

text: compute

text: the

text: non-relativistic

text: expansion

text: of

text: the

text: associated

text: which

text: in

text: turn

text: allowed

text: us

text: to

text: predict

text: the

text: dark

text: matter

text: density

text: in

text: our

text: model,

text: taking

text: into

text: account

text: only

text: annihilation.

text: This

text: prediction

text: is

text: formulated

text: as

text: a

text: function

text: of

text: the

text: radius

text: of

text: the

text: extra-dimension,

text: and

text: increases

text: with

text: the

text: mass

text: scale

text: of

text: the

text: extra-dimensions

text: as

text: expected.

text: The

text: result

text: is

text: presented

text: in

text: Figure

text: where

text: the

text: Planck

text: preferred

text: value

text: is

text: represented

text: to

text: show

text: that

text: the

text: upper

text: bound

text: on

text: the

text: inverse

text: radius

text: inferred

text: from

text: this

text: calculation

text: would

text: be

text: GeV

text: which

text: in

text: turns

text: means

text: R

text: m

text: GeV.

text: This

text: limit,

text: however

text: is

text: too

text: stringent

text: as

text: co–annihilation

text: eﬀects

text: with

text: other

text: gauge

text: scalars

text: change

text: the

text: prediction

text: for

text: the

text: relic

text: density

text: signiﬁcantly.

text: Co–annihilation

text: processes

text: As

text: we

text: show

text: in

text: the

text: Appendices

text: and

text: there

text: are

text: several

text: channels

text: available

text: for

text: co–

text: annihilation

text: of

text: our

text: dark

text: matter

text: candidate,

text: once

text: the

text: other

text: electroweak

text: gauge

text: bosons

text: are

text: in-

text: cluded

text: in

text: the

text: calculation.

text: A

text: list

text: of

text: all

text: the

text: processes

text: we

text: considered

text: can

text: be

text: found

text: in

text: and

text: we

text: show

text: in

text: Figure

text: a

text: sample

text: of

text: thermal

text: cross-sections

text: computed

text: using

text: the

text: dedicated

text: CalcHEP

text: module.

text: These

text: processes

text: can

text: be

text: sorted

text: into

text: several

text: classes,

text: as

text: shown

text: in

text: Figure

text: As

text: we

text: will

text: show

text: below,

text: the

text: dominant

text: contribution

text: comes

text: from

text: co–annihilation

text: to

text: pairs

text: of

text: vector

text: bosons

text: (Figure

text: Annihilation

text: to

text: fermions

text: (Figure

text: is

text: a

text: lot

text: less

text: important

text: due

text: to

text: the

text: absence

text: of

text: s-wave

text: processes

text: and

text: the

text: associated

text: production

text: of

text: a

text: vector

text: boson

text: and

text: excited

text: vector

text: bosons

text: (Figure

text: is

text: phase-space

text: suppressed.

text: It

text: is

text: worth

text: mentioning

text: that

text: we

text: considered

text: the

text: possible

text: contribution

text: of

text: near-resonant

text: s-

text: channel

text: diagrams

text: involving

text: excitations

text: like

text: in

text: Figure

text: but

text: that

text: the

text: resonance

text: is

text: still

text: too

text: far-oﬀ

text: to

text: have

text: a

text: signiﬁcant

text: impact

text: on

text: the

text: thermal

text: cross-sections.

text: We

text: should

text: however

text: note

text: that

text: a

text: signiﬁcant

text: change

text: can

text: be

text: introduced

text: by

text: ﬁne-tuning

text: the

text: localised

text: Higgs

text: mass

text: operator

text: so

text: that

text: Higgs

text: excitations

text: hit

text: precisely

text: the

text: resonance

text: (Figure

text: A

text: quick

text: estimate

text: using

text: a

text: subset

text: of

text: all

text: processes

text: showed

text: that

text: this

text: changes

text: signiﬁcantly

text: the

text: thermal

text: cross-section

text: possibly

text: providing

text: a

text: small

text: funnel

text: in

text: parameter

text: space

text: where

text: Dark

text: matter

text: is

text: allowed

text: to

text: be

text: very

text: heavy.

text: We

text: however

text: leave

text: aside

text: the

text: study

text: of

text: this

text: ﬁne-tuned

text: possibility

text: in

text: the

text: present

text: work.

text: ##LTLine##


text: 100 ϕZϕ¯Z → W+W− 10-1 ϕZϕ¯W+ → W+Z 10-2 ϕZϕ¯Z → HH 10-3 ϕZϕ¯Z → W+W∗− 10-4 10-5 ϕZϕ¯+W → ff¯ 10-6 ϕZϕ¯+W → W+H 10-7 0 0.1 0.2 0.3 0.4 0.5 0.6 v Figure 4. Characteristic sample of annihilation channels → → → ∗ (a) ϕϕ¯ V µV µ (b) ϕϕ¯ ff¯ (c) ϕϕ¯ V µV µ (d) Vector Resonance (e) Higgs Resonance Figure 5. Examples of classes of processes. Thick lines represent Kaluza-Klein excitations and the fermion gauge coupling is marked to highlight its localised feature. eOur complete calculation also includes 4-point interactions for the relevant processes. With the caveat of the possible Higgs resonance funnel, we could compute (cid:104)σv(cid:105) analyt- eff ically and predict for Ω h2 as a function of 1 . The result of this calculation is presented in DM R Figure 6. As expected, the upper value on 1 is increased signiﬁcantly once co–annihilations R are included. We now have the constraint 1 ≤ 220 GeV, which translates to m ≤ 540 GeV. R l=2 – 25 – ]bp[ vσ ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

font: default

text: ϕZϕ¯Z

text: W+W−

text: ϕZϕ¯W+

text: W+Z

text: ϕZϕ¯Z

text: HH

text: ϕZϕ¯Z

text: W+W∗−

text: ϕZϕ¯+W

text: ff¯

text: ϕZϕ¯+W

text: W+H

text: v

text: Figure

text: Characteristic

text: sample

text: of

text: annihilation

text: channels

text: (a)

text: ϕϕ¯

text: V

text: µV

text: µ

text: (b)

text: ϕϕ¯

text: ff¯

text: (c)

text: ϕϕ¯

text: V

text: µV

text: µ

text: (d)

text: Vector

text: Resonance

text: (e)

text: Higgs

text: Resonance

text: Figure

text: Examples

text: of

text: classes

text: of

text: processes.

text: Thick

text: lines

text: represent

text: Kaluza-Klein

text: excitations

text: and

text: the

text: fermion

text: gauge

text: coupling

text: is

text: marked

text: to

text: highlight

text: its

text: localised

text: feature.

text: eOur

text: complete

text: calculation

text: also

text: includes

text: interactions

text: for

text: the

text: relevant

text: processes.

text: With

text: the

text: caveat

text: of

text: the

text: possible

text: Higgs

text: resonance

text: funnel,

text: we

text: could

text: compute

text: analyt-

text: eff

text: ically

text: and

text: predict

text: for

text: Ω

text: as

text: a

text: function

text: of

text: The

text: result

text: of

text: this

text: calculation

text: is

text: presented

text: in

text: DM

text: R

text: Figure

text: As

text: expected,

text: the

text: upper

text: value

text: on

text: is

text: increased

text: signiﬁcantly

text: once

text: co–annihilations

text: R

text: are

text: included.

text: We

text: now

text: have

text: the

text: constraint

text: GeV,

text: which

text: translates

text: to

text: m

text: GeV.

text: R

text: ]bp[

text: vσ

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 12 THOMAS ECKL AND MICHAEL LOENNE Figure 3.1. Proof. We will construct a topological equivalence of the intersection foliations F ∩ F ∩ S3 and S3. Then the statement follows by the Reconstruction Theorem 2.2. 1 2 F Lemma 3.2 and 3.3 show that every leaf of in the tubular torus { ∈ | | ≤ } i (x, y) S3 : 0 < x 1 is parametrized on the one hand by the absolute value 2 (cid:15) of the x-coordinate, on the other hand by the argument t of the y-coordinate. x The parametrisation by (cid:15) yields the homeomorphisms x × → { ∈ | | ≤ } Φ(i) : T (0, 1/2] (x, y) S3 : 0 < x 1/2 x 1/2 × F ∩ mapping a pair (x, y) (cid:15) to the unique intersection point of the leaf of S3 x i through (x, y) with Tx . ◦ (cid:15)x− Then Φ(2) (Φ(1)) 1 is a homeomorphism of the tubular torus { ∈x x | | ≤ } (x, y) S3 : 0 < x 1 into itself but might not be extendable to a 2 { ∈ ≤ | | ≤ } homeomorphism of the solid torus (x, y) S3 : 0 x 1 . To achieve that 2 we reparametrize the (cid:15) -interval (0, 1] using the second parametrization by the x 2 ∈ argument of the y-coordinate: Every leaf L through a point (x, y) Tx deﬁnes (x,y) 1 → ∞ − 2 an invertible function φ(i) : (0, 1] [0, ), mapping (cid:15) to t t where t is the x 2 x 0 argument of the y-coordinate of the intersection point of L with Tx and t is (x,y) (cid:15)x 0 the argument of y. These(cid:104)functions are the same fo(cid:105)r all such leaves because of the × S1 S1-invariance, and we always have φ(i)(1) = 0. Then x 2 ◦ × − ◦ ◦ − Φ(2) id ((φ(2)) 1 φ(1)) (Φ(1)) 1 x Tx 1 x x x 2 ≤ maps Dx and Tx , 0 < (cid:15) 1 onto themselves. This implies that the identity t{,12 } ∩(cid:15)x x 2 map on x = 0 S3 extends this composition of maps to a homeomorphism Φ { ∈ | | ≤ } F ∩ x of the solid torus (x, y) S3 : 0 < x 1 mapping leaves of S3 to leaves of F ∩ 2 1 S3. Furthermore, the restriction of Φ to Tx is the identity map. 2 x 1 2 In the same way we can construct a homeomorphism Φ of the solid torus { ∈ | | ≤ } F ∩ y F ∩ (x, y) S3 : 0 < y 1 mapping leaves of S3 to leaves of S3. 2 1 2 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: THOMAS

text: ECKL

text: AND

text: MICHAEL

text: LOENNE

text: Figure

text: Proof.

text: We

text: will

text: construct

text: a

text: topological

text: equivalence

text: of

text: the

text: intersection

text: foliations

text: F

text: F

text: and

text: Then

text: the

text: statement

text: follows

text: by

text: the

text: Reconstruction

text: Theorem

text: F

text: Lemma

text: and

text: show

text: that

text: every

text: leaf

text: of

text: in

text: the

text: tubular

text: torus

text: i

text: (x,

text: y)

text: x

text: is

text: parametrized

text: on

text: the

text: one

text: hand

text: by

text: the

text: absolute

text: value

text: of

text: the

text: x-coordinate,

text: on

text: the

text: other

text: hand

text: by

text: the

text: argument

text: t

text: of

text: the

text: y-coordinate.

text: x

text: The

text: parametrisation

text: by

text: yields

text: the

text: homeomorphisms

text: x

text: Φ(i)

text: T

text: (x,

text: y)

text: x

text: x

text: F

text: mapping

text: a

text: pair

text: (x,

text: y)

text: to

text: the

text: unique

text: intersection

text: point

text: of

text: the

text: leaf

text: of

text: x

text: i

text: through

text: (x,

text: y)

text: with

text: Tx

text: Then

text: is

text: a

text: homeomorphism

text: of

text: the

text: tubular

text: torus

text: ∈x

text: x

text: (x,

text: y)

text: x

text: into

text: itself

text: but

text: might

text: not

text: be

text: extendable

text: to

text: a

text: homeomorphism

text: of

text: the

text: solid

text: torus

text: (x,

text: y)

text: x

text: To

text: achieve

text: that

text: we

text: reparametrize

text: the

text: -interval

text: using

text: the

text: second

text: parametrization

text: by

text: the

text: x

text: argument

text: of

text: the

text: y-coordinate:

text: Every

text: leaf

text: L

text: through

text: a

text: point

text: (x,

text: y)

text: Tx

text: deﬁnes

text: (x,y)

text: an

text: invertible

text: function

text: φ(i)

text: mapping

text: to

text: t

text: t

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text: t

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text: x

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text: of

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text: Tx

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text: the

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text: of

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text: are

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text: same

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text: x

text: maps

text: Dx

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text: Tx

text: onto

text: themselves.

text: This

text: implies

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text: identity

text: x

text: map

text: on

text: x

text: extends

text: this

text: composition

text: of

text: maps

text: to

text: a

text: homeomorphism

text: Φ

text: F

text: x

text: of

text: the

text: solid

text: torus

text: (x,

text: y)

text: x

text: mapping

text: leaves

text: of

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text: Furthermore,

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text: restriction

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text: Tx

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text: identity

text: map.

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text: the

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text: construct

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text: F

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text: 16 THOMAS ECKL AND MICHAEL LOENNE Figure 4.1. R 5. The non-resonant case of -linearly dependent eigenvalues in dimension 2 F ∈ C In this section, we only consider holomorphic foliation germs around 0 2 λ represented by vector ﬁelds of the form ∂ ∂ ∈ R λx + y , λ . ∂x ∂y >0 As in Section these foliation germs are invariant under rescaling with positive real constants. Hence it is enough to consider the real intersection foliations F ∩ F ∩ S3 = S3. λ 1 λ F ∩ Lemma 5.1. Every leaf of the intersection foliation S3 lies on a torus Tx , ≤ ≤ λ (cid:15)x 0 (cid:15) 1. x (cid:55)→ Proof. The ﬂow of the vector ﬁeld λx ∂ + y ∂ is given by (a, b, t) (aeλt, bet). ∈ R ∂x ∂y Since λ the intersection of the associated integral manifold through a point ∈ >0 (cid:55)→ F ∩ (a, b) S3 with S3 is parametrised by t (aeλit, beit). Thus the leaf of S3 | | λ (cid:3) through (a, b) lies on the torus T ( a ). x ∈ N Assume that λ = p, where p, q are relatively prime. q F ∩ Proposition 5.2. Every leaf of the intersection foliation S3 is closed. A leaf λ on the torus T ((cid:15) ), 0 < (cid:15) < 1, is a curve of type (p, q), where p describes the x x x { } ∩ winding number of the leaf around x = 0 S3 and q the winding number around { } ∩ ∈ { } ∩ y = 0 S3. The holonomy in a point (0, eit) x = 0 S3 following the leaf → in counter-clockwise direction is given by the germ of the map Dt ((cid:15) ) Dt ((cid:15) ), (cid:28) · x x x x 0 < (cid:15)x 1, multiplying the x-coordinate by e2πi pq . Similarly, the holonomy of the ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: THOMAS

text: ECKL

text: AND

text: MICHAEL

text: LOENNE

text: Figure

text: R

label: section

text: The

text: non-resonant

text: case

text: of

text: -linearly

text: dependent

text: eigenvalues

text: in

text: dimension

text: F

text: C

text: In

text: this

text: section,

text: we

text: only

text: consider

text: holomorphic

text: foliation

text: germs

text: around

text: λ

text: represented

text: by

text: vector

text: ﬁelds

text: of

text: the

text: form

text: R

text: λx

text: y

text: λ

text: ∂x

text: ∂y

text: As

text: in

text: Section

text: these

text: foliation

text: germs

text: are

text: invariant

text: under

text: rescaling

text: with

text: positive

text: real

text: constants.

text: Hence

text: it

text: is

text: enough

text: to

text: consider

text: the

text: real

text: intersection

text: foliations

text: F

text: F

text: λ

text: λ

text: F

text: Lemma

text: Every

text: leaf

text: of

text: the

text: intersection

text: foliation

text: lies

text: on

text: a

text: torus

text: Tx

text: λ

text: x

text: Proof.

text: The

text: ﬂow

text: of

text: the

text: vector

text: ﬁeld

text: λx

text: y

text: is

text: given

text: by

text: (a,

text: b,

text: t)

text: (aeλt,

text: bet).

text: R

text: ∂x

text: ∂y

text: Since

text: λ

text: the

text: intersection

text: of

text: the

text: associated

text: integral

text: manifold

text: through

text: a

text: point

text: F

text: (a,

text: b)

text: with

text: is

text: parametrised

text: by

text: t

text: (aeλit,

text: beit).

text: Thus

text: the

text: leaf

text: of

text: λ

text: through

text: (a,

text: b)

text: lies

text: on

text: the

text: torus

text: T

text: a

text: x

text: N

text: Assume

text: that

text: λ

text: p,

text: where

text: p,

text: q

text: are

text: relatively

text: prime.

text: q

text: F

text: Proposition

text: Every

text: leaf

text: of

text: the

text: intersection

text: foliation

text: is

text: closed.

text: A

text: leaf

text: λ

text: on

text: the

text: torus

text: T

text: is

text: a

text: curve

text: of

text: type

text: (p,

text: q),

text: where

text: p

text: describes

text: the

text: x

text: x

text: x

text: winding

text: number

text: of

text: the

text: leaf

text: around

text: x

text: and

text: q

text: the

text: winding

text: number

text: around

text: y

text: The

text: holonomy

text: in

text: a

text: point

text: eit)

text: x

text: following

text: the

text: leaf

text: in

text: counter-clockwise

text: direction

text: is

text: given

text: by

text: the

text: germ

text: of

text: the

text: map

text: Dt

text: Dt

text: x

text: x

text: x

text: x

text: multiplying

text: the

text: x-coordinate

text: by

text: pq

text: Similarly,

text: the

text: holonomy

text: of

text: the

text: ##LTFigure##

text: ##LTLine##

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text: 18 THOMAS ECKL AND MICHAEL LOENNE (cid:18) →(cid:19) (cid:18) (cid:19) F ∩ Proposition 5.5. If Φ : S3 S3 is a topological equivalence of S3 with F ∩ ∈ R − Q → λ S3, λ, µ then the restriction Φ| : T ((cid:15) ) T (φ((cid:15) )) is µ >0 ± >0 ± Tx((cid:15)x) x x x x 1 0 0 1 of one of the types ± or ± in the mapping class group of a 0 1 1 0 2-dimensional torus, for all 0 < (cid:15) < 1. x (cid:18) (cid:19) Proof. Interchanging the coordinates yields a homeomorphism → (cid:55)→ Ψ : S3 S3, (x, y) (y, x) whose restriction to tori T ((cid:15) ) is of type x x 0 1 in the mapping class group of a 2-dimensional torus. Composing Ψ with 1 0 F ∩ F ∩ a topological equivalence Φ of S3 with S3 such that φ(0) = 1, φ(1) = 0 λ (cid:48) F ∩ µ F ∩ (cid:48) yields a topological equivalence Φ of S3 with S3 such that φ (0) = 0, λ 1 (cid:83)(cid:48) µ (cid:83) φ (1) = 1. Hence, from now on we will only consider that case. (cid:83) (cid:83) For all 0 < (cid:15) < 1 the topological equivalence Φ maps the solid torus 0 ≤ ≤ T ((cid:15) ) homeomorphically onto the solid torus ≤ ≤ T (φ((cid:15) )) and 0 (cid:15)x (cid:15)0 x x 0 (cid:15)x (cid:15)0 x x ≤ ≤ T ((cid:15) ) onto ≤ ≤ T (φ((cid:15) )), always mapping the tori T ((cid:15) ) onto (cid:15)0 (cid:15)x 1 x x (cid:15)0 (cid:15)x 1 x x x x T (φ((cid:15) )). The fundamental groups of these solid tori are generated by x x{ } ∩ { } ∩ L := x = 0 S3 resp. L := y = 0 S3, and a curve of type (p, q) on x y · the torus T ((cid:15) ) (for the notation, see Prop. 5.2) is mapped to the class of q L · x x x resp. p L by the inclusion into the solid tori. Consequently, the homeomorphism Φ y ± ± must map a curve of type (p, q) on T ((cid:15) ) to a curve of type ( p, q) on T (φ((cid:15) )). x x x x (cid:3) This implies the claim on the isotopy classes of Φ| . Tx((cid:15)x) F ∈ R − Q To ﬁnally classify the holomorphic foliation germs , λ , we consider ∈ R λ >0 >0 × Kronecker foliations F , λ , on the 2-dimensional torus T2 = S1 S1. These λ >0 foliations are given by the orbits of the ﬂow · ∈ R t (eia, eib) = (ei(a+λt), ei(b+t)), t, a, b . λ (cid:18) (cid:19) ∈ R Proposition 5.6. Two Kronecker foliations F and F , λ, µ , are topologically λ µ a b ∈ Z equivalent if µ = aλ+b, where GL(2, ). (cid:18)cλ+d (cid:19) c d a b ∈ Z Proof. Let Q := GL(2, ). Then c d → (cid:55)→ φ : T2 T2, (eix, eiy) (ei(ax+by), ei(cx+dy)) Q is a homeomorphism with inverse map φ − . Since for s = (cλ + d)t, Q 1 · φ (t (eix, eiy)) = (ei(ax+by+(aλ+b)t), ei(cx+dy+(cλ+d)t)) = Q λ = (ei(ax+by+µs), ei(cx+dy+s)) = · = s φ (eix, eiy), µ Q (cid:3) φ is a topological equivalence of F and F . Q λ µ ##LTLine## ##LTLine##

text: THOMAS

text: ECKL

text: AND

text: MICHAEL

text: LOENNE

text: F

text: Proposition

text: If

text: Φ

text: is

text: a

text: topological

text: equivalence

text: of

text: with

text: F

text: R

text: Q

text: λ

text: λ,

text: µ

text: then

text: the

text: restriction

text: Φ|

text: T

text: T

text: is

text: µ

text: x

text: x

text: x

text: x

text: of

text: one

text: of

text: the

text: types

text: or

text: in

text: the

text: mapping

text: class

text: group

text: of

text: a

text: torus,

text: for

text: all

text: x

text: Proof.

text: Interchanging

text: the

text: coordinates

text: yields

text: a

text: homeomorphism

text: Ψ

text: (x,

text: y)

text: (y,

text: x)

text: whose

text: restriction

text: to

text: tori

text: T

text: is

text: of

text: type

text: x

text: x

text: in

text: the

text: mapping

text: class

text: group

text: of

text: a

text: torus.

text: Composing

text: Ψ

text: with

text: F

text: F

text: a

text: topological

text: equivalence

text: Φ

text: of

text: with

text: such

text: that

text: λ

text: F

text: µ

text: F

text: yields

text: a

text: topological

text: equivalence

text: Φ

text: of

text: with

text: such

text: that

text: φ

text: λ

text: µ

text: φ

text: Hence,

text: from

text: now

text: on

text: we

text: will

text: only

text: consider

text: that

text: case.

text: For

text: all

text: the

text: topological

text: equivalence

text: Φ

text: maps

text: the

text: solid

text: torus

text: T

text: homeomorphically

text: onto

text: the

text: solid

text: torus

text: T

text: and

text: x

text: x

text: x

text: x

text: T

text: onto

text: T

text: always

text: mapping

text: the

text: tori

text: T

text: onto

text: x

text: x

text: x

text: x

text: x

text: x

text: T

text: The

text: fundamental

text: groups

text: of

text: these

text: solid

text: tori

text: are

text: generated

text: by

text: x

text: x{

text: L

text: x

text: resp.

text: L

text: y

text: and

text: a

text: curve

text: of

text: type

text: (p,

text: q)

text: on

text: x

text: y

text: the

text: torus

text: T

text: (for

text: the

text: notation,

text: see

text: Prop.

text: is

text: mapped

text: to

text: the

text: class

text: of

text: q

text: L

text: x

text: x

text: x

text: resp.

text: p

text: L

text: by

text: the

text: inclusion

text: into

text: the

text: solid

text: tori.

text: Consequently,

text: the

text: homeomorphism

text: Φ

text: y

text: must

text: map

text: a

text: curve

text: of

text: type

text: (p,

text: q)

text: on

text: T

text: to

text: a

text: curve

text: of

text: type

text: p,

text: q)

text: on

text: T

text: x

text: x

text: x

text: x

text: This

text: implies

text: the

text: claim

text: on

text: the

text: isotopy

text: classes

text: of

text: Φ|

text: F

text: R

text: Q

text: To

text: ﬁnally

text: classify

text: the

text: holomorphic

text: foliation

text: germs

text: λ

text: we

text: consider

text: R

text: λ

text: Kronecker

text: foliations

text: F

text: λ

text: on

text: the

text: torus

text: These

text: λ

text: foliations

text: are

text: given

text: by

text: the

text: orbits

text: of

text: the

text: ﬂow

text: R

text: t

text: (eia,

text: eib)

text: (ei(a+λt),

text: ei(b+t)),

text: t,

text: a,

text: b

text: λ

text: R

text: Proposition

text: Two

text: Kronecker

text: foliations

text: F

text: and

text: F

text: λ,

text: µ

text: are

text: topologically

text: λ

text: µ

text: a

text: b

text: Z

text: equivalent

text: if

text: µ

text: aλ+b,

text: where

text: c

text: d

text: a

text: b

text: Z

text: Proof.

text: Let

text: Q

text: Then

text: c

text: d

text: φ

text: (eix,

text: eiy)

text: (ei(ax+by),

text: ei(cx+dy))

text: Q

text: is

text: a

text: homeomorphism

text: with

text: inverse

text: map

text: φ

text: Since

text: for

text: s

text: (cλ

text: d)t,

text: Q

text: φ

text: (t

text: (eix,

text: eiy))

text: (ei(ax+by+(aλ+b)t),

text: ei(cx+dy+(cλ+d)t))

text: Q

text: λ

text: (ei(ax+by+µs),

text: ei(cx+dy+s))

text: s

text: φ

text: (eix,

text: eiy),

text: µ

text: Q

text: φ

text: is

text: a

text: topological

text: equivalence

text: of

text: F

text: and

text: F

text: Q

text: λ

text: µ

text: ##LTLine##

text: ##LTLine##


text: 20 THOMAS ECKL AND MICHAEL LOENNE (cid:72) (cid:8) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) γ(λ) P,(cid:15) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1)(cid:65)(cid:113) (cid:1) (cid:1) (cid:1)(cid:65)(cid:113) (cid:113) (cid:1) (cid:1) (cid:1) (cid:1) (cid:48) (cid:1) (cid:1) (cid:48) P (cid:1) (cid:72) (cid:1) (cid:1) P (cid:8) P B ((cid:15)) P Figure 5.1. in the point (1, 1) ∈ T2. Let [γ(λ)] ∈ H (T2, Z) denote the homological 1-class P,(cid:15) 1 · represented by γ(λ), and consider the topological intersection numbers [γ(λ)] [C ] P,(cid:15) P,(cid:15) i (see [SZ94, 14.6]). · Claim: λ = lim(cid:15)→0 [[γγPP((λλ,,(cid:15)(cid:15)))]]·[[CC21]]. Proof. We calculate the intersection numbers using their diﬀerential-topological interpretation, for smoothly embedded paths γ(λ) (see [Hir94, 5.2]). Since L(λ) P,(cid:15) P intersects C and C everywhere with the same orientation, we just need to count 1 2 ∩ (cid:54)∈ ∪ the intersection points in γ(λ) C . Assuming for the moment that P C C , P,(cid:15) i 1 2 for small enough (cid:15) we only need to count the intersection points of the part of γ(λ) P,(cid:15) lyi(cid:20)ng on(cid:21) L(λ) with C . This(cid:20)part(cid:21)is the ima(cid:20)ge p(cid:21)([0, b ]) of an interva(cid:20)l [0,(cid:21)b ] ⊂ R under the pParametrisaition p : R → L(λ) introduced ab(cid:15)ove. Then (cid:15) P λb ≤ | ∩ | ≤ λb b ≤ | ∩ | ≤ b (cid:15) p([0, b ]) C (cid:15) + 1 and (cid:15) p([0, b ]) C (cid:15) + 1, 2π (cid:15) 2 2π 2π (cid:15) 1 2π ≤ ∈ R | ∩ | where [x] denotes the maximal integer x and p([0, b ]) C the number of (cid:15) → i∞ → intersection points of p([0, b ]) and C . As discussed above, b if (cid:15) 0, and (cid:15) i (cid:15) the claim follows. ∈ ∪ (cid:48) If P C C the path in γ(λ) connecting P with P can be chosen to intersect 1 2 P,(cid:15) C only in a number of points bounded from above independently of (cid:15). Hence the i (cid:3) claim also holds in that case. Now, we calculate: · · · [γ(λ)] [C ] [φ(γ(λ))] [φ(C )] [φ(γ(λ))] [C ] λ = l→im P,(cid:15) · 2 = l→im P,(cid:15) · 2 = l→im P,(cid:15) · 2 , (cid:15) 0 [γ(λ)] [C ] (cid:15) 0 [φ(γ(λ))] [φ(C )] (cid:15) 0 [φ(γ(λ))] [C ] P,(cid:15) 1 P,(cid:15) 1 P,(cid:15) 1 → by the Claim and since φ : T2 T2 is a homeomorphism assumed to be homotopic to the identity. But φ(L(λ)) = L(µ) , hence φ(γ(λ)) is a path constructed as above P φ(P) P,(cid:15) for the leaf L(µ) of F and the neighborhood basis U := φ(B ((cid:15))) of φ(P), so the φ(P) µ (cid:15) P above limit is equal to · [γ(µ) ] [C ] l→im φ(P),U(cid:15) · 2 = µ, (cid:15) 0 [γ(µ) ] [C ] φ(P),U(cid:15) 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: THOMAS

text: ECKL

text: AND

text: MICHAEL

text: LOENNE

text: γ(λ)

text: P

text: P

text: P

text: B

text: P

text: Figure

text: in

text: the

text: point

text: Let

text: [γ(λ)]

text: H

text: Z)

text: denote

text: the

text: homological

text: represented

text: by

text: γ(λ),

text: and

text: consider

text: the

text: topological

text: intersection

text: numbers

text: [γ(λ)]

text: [C

text: i

text: (see

text: Claim:

text: λ

text: Proof.

text: We

text: calculate

text: the

text: intersection

text: numbers

text: using

text: their

text: diﬀerential-topological

text: interpretation,

text: for

text: smoothly

text: embedded

text: paths

text: γ(λ)

text: (see

text: Since

text: L(λ)

text: P

text: intersects

text: C

text: and

text: C

text: everywhere

text: with

text: the

text: same

text: orientation,

text: we

text: just

text: need

text: to

text: count

text: the

text: intersection

text: points

text: in

text: γ(λ)

text: C

text: Assuming

text: for

text: the

text: moment

text: that

text: P

text: C

text: C

text: i

text: for

text: small

text: enough

text: we

text: only

text: need

text: to

text: count

text: the

text: intersection

text: points

text: of

text: the

text: part

text: of

text: γ(λ)

text: L(λ)

text: with

text: C

text: the

text: b

text: of

text: an

text: R

text: under

text: the

text: pParametrisaition

text: p

text: R

text: L(λ)

text: introduced

text: Then

text: P

text: λb

text: λb

text: b

text: b

text: b

text: C

text: and

text: b

text: C

text: 2π

text: 2π

text: 2π

text: 2π

text: R

text: where

text: [x]

text: denotes

text: the

text: maximal

text: integer

text: x

text: and

text: b

text: C

text: the

text: number

text: of

text: i∞

text: intersection

text: points

text: of

text: b

text: and

text: C

text: As

text: discussed

text: above,

text: b

text: if

text: and

text: i

text: the

text: claim

text: follows.

text: If

text: P

text: C

text: C

text: the

text: path

text: in

text: γ(λ)

text: connecting

text: P

text: with

text: P

text: can

text: be

text: chosen

text: to

text: intersect

text: C

text: only

text: in

text: a

text: number

text: of

text: points

text: bounded

text: from

text: above

text: independently

text: of

text: Hence

text: the

text: i

text: claim

text: also

text: holds

text: in

text: that

text: case.

text: Now,

text: we

text: calculate:

text: [γ(λ)]

text: [C

text: [φ(γ(λ))]

text: [φ(C

text: [φ(γ(λ))]

text: [C

text: λ

text: l→im

text: l→im

text: l→im

text: [γ(λ)]

text: [C

text: [φ(γ(λ))]

text: [φ(C

text: [φ(γ(λ))]

text: [C

text: by

text: the

text: Claim

text: and

text: since

text: φ

text: is

text: a

text: homeomorphism

text: assumed

text: to

text: be

text: homotopic

text: to

text: the

text: identity.

text: But

text: φ(L(λ))

text: L(µ)

text: hence

text: φ(γ(λ))

text: is

text: a

text: path

text: constructed

text: as

text: above

text: P

text: φ(P)

text: for

text: the

text: leaf

text: L(µ)

text: of

text: F

text: and

text: the

text: neighborhood

text: basis

text: U

text: φ(B

text: of

text: φ(P),

text: so

text: the

text: φ(P)

text: µ

text: P

text: above

text: limit

text: is

text: equal

text: to

text: [γ(µ)

text: [C

text: l→im

text: µ,

text: [γ(µ)

text: [C

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text: Topological equivalence of holomorphic foliation germs of rank 1 5 F ∩ − F S2n 1 and called the real intersection foliation or trace foliation of with − (cid:15) S2n 1. (cid:15) ∈ − Its leaves can be canonically oriented: In each point p S2n 1 choose those vectors F ∩ (cid:15) − in the tangent subspace given by the distribution S2n 1 in p which together F (cid:15) ∈ C with the tangent vectors of the leaf of through p pointing away from 0 n represent the positive orientation of the complex structure on the leaf. Taking in ∈ − each point p S2n 1 a unit vector oriented in that way yields a nowhere vanishing (cid:15) − vector ﬁeld on S2n 1 whose ﬂow, denoted by ΦF, has integral curves coinciding (cid:15) F ∩ − with the leaves of S2n 1. (cid:15) F G − Deﬁnition 2.1. Two real 1-dimensional foliations , on the sphere S2n 1 are − → − called topologically equivalent if there exists a homeomorphism φ : S2n 1 S2n 1 F G mapping the leaves of onto the leaves of . (cid:48) F∩ − In [Arn69], Arnol d continued to show that for (cid:15) small enough the foliation B2n 1 − (cid:15) is topologically equivalent to the foliation of B2n 1 by the cones over the leaves of F ∩ − ∈ C (cid:15) S2n 1 with vertex 0 n, and this equivalence is realised by a homeomorphism (cid:15) − F G of B2n 1 on itself. This obviously implies that topological equivalence of and (cid:15) F ∩ − G ∩ − follows from topological equivalence of S2n 1 and S2n 1. (cid:15) (cid:15) It seems to be a well-accepted fact that the converse is also true, but because of lack of reference (and in particular of a recipe to construct the topological equivalence of the intersection foliations) we decided to present this proof and construction in all details. Theorem 2.2 (Reconstruction Theorem). Two normalized holomorphic foliation F G ∈ C germs , with an isolated singularity in 0 n of Poincar´e type are topologically F ∩ − G ∩ − equivalent if, and only if the real intersection foliations S2n 1 and S2n 1, (cid:28) (cid:15) (cid:15) 0 < (cid:15) 1, are topologically equivalent. (cid:98) (cid:98) (cid:48) F ∩ − G ∩ − Proof. As discussed above Arnol d showed that B2n 1 resp. B2n 1 is F G (cid:15)− (cid:15) topologically equivalent to the foliation resp. of B2n 1 by the cones over the F ∩ − G ∩ − ∈ C(cid:15) leaves of S2n 1 resp. S2n 1 with vertex 0 n. As already noticed that (cid:15) (cid:15) shows one direction of the Theorem. F G To prove that a topological equivalence of and induces a topologi(cid:98)cal equ(cid:98)ivalence F ∩ − G ∩ − of the intersection foliations S2n 1 and S2n 1, it is enough to construct (cid:15) (cid:15) F G the latter one from a topological equivalence of the cone foliations and , in the sense of Def. 1.2. (cid:48) By possibly(cid:98) decreasing (cid:15) to (cid:15) this topological equivalence can be reali(cid:98)sed by an → embeddingFH∩C : B−(cid:15)2(cid:48)n (cid:44) B(cid:15)2n such that HC(0) = 0 and leaves of thGe ∩foliat−ion cone over S2(cid:48)n 1 are mapped into leaves of the foliation cone over S2n 1. (cid:15) − { } → − (cid:15) Composing H with the radial projection r : B2n 0 S2n 1 produces a C (cid:15) (cid:15) continuous map h : S2(cid:48)n−1 H(cid:44)→C B2n − {0} →r S2n−1 − (cid:15) (cid:15) (cid:15) Since HC may map S(cid:15)2(cid:48)n 1 to a topological manifold in B(cid:15)2n intersecting the same radial line more than once, h may not be injective, an(cid:98)d hence not th(cid:98)e wanted homeomorphism. But from h we will be able to c(cid:98)onstruct a hom(cid:98)eomorphism − → − F ∩ − G ∩ − g : S2(cid:48)n 1 S2n 1 deﬁning(cid:98)a topological equivalence of (cid:98)S2(cid:48)n 1 with S2n 1. (cid:15) (cid:15) F (cid:15) F ∩ (cid:15)− This (cid:98)shows the theorem since the cone structure of implies that S2(cid:48)n 1 is F ∩ − F ∩ − F ∩(cid:15) − topologically equivalent to S2n 1, and by construction, S2n 1 = S2n 1 G ∩ − G ∩ − (cid:15) (cid:15) (cid:15) and S2n 1 = S2n 1. (cid:15) (cid:15)

text: Topological

text: equivalence

text: of

text: holomorphic

text: foliation

text: germs

text: of

text: rank

text: F

text: F

text: and

text: called

text: the

text: real

text: intersection

text: foliation

text: or

text: trace

text: foliation

text: of

text: with

text: Its

text: leaves

text: can

text: be

text: canonically

text: oriented:

text: In

text: each

text: point

text: p

text: choose

text: those

text: vectors

text: F

text: in

text: the

text: tangent

text: subspace

text: given

text: by

text: the

text: distribution

text: in

text: p

text: which

text: together

text: F

text: C

text: with

text: the

text: tangent

text: vectors

text: of

text: the

text: leaf

text: of

text: through

text: p

text: pointing

text: away

text: from

text: n

text: represent

text: the

text: positive

text: orientation

text: of

text: the

text: complex

text: structure

text: on

text: the

text: leaf.

text: Taking

text: in

text: each

text: point

text: p

text: a

text: unit

text: vector

text: oriented

text: in

text: that

text: way

text: yields

text: a

text: nowhere

text: vanishing

text: vector

text: ﬁeld

text: on

text: whose

text: ﬂow,

text: denoted

text: by

text: ΦF,

text: has

text: integral

text: curves

text: coinciding

text: F

text: with

text: the

text: leaves

text: of

text: F

text: G

text: Deﬁnition

text: Two

text: real

text: foliations

text: on

text: the

text: sphere

text: are

text: called

text: topologically

text: equivalent

text: if

text: there

text: exists

text: a

text: homeomorphism

text: φ

text: F

text: G

text: mapping

text: the

text: leaves

text: of

text: onto

text: the

text: leaves

text: of

text: F∩

text: In

text: Arnol

text: d

text: continued

text: to

text: show

text: that

text: for

text: small

text: enough

text: the

text: foliation

text: is

text: topologically

text: equivalent

text: to

text: the

text: foliation

text: of

text: by

text: the

text: cones

text: over

text: the

text: leaves

text: of

text: F

text: C

text: with

text: vertex

text: n,

text: and

text: this

text: equivalence

text: is

text: realised

text: by

text: a

text: homeomorphism

text: F

text: G

text: of

text: on

text: itself.

text: This

text: obviously

text: implies

text: that

text: topological

text: equivalence

text: of

text: and

text: F

text: G

text: follows

text: from

text: topological

text: equivalence

text: of

text: and

text: It

text: seems

text: to

text: be

text: a

text: well-accepted

text: fact

text: that

text: the

text: converse

text: is

text: also

text: true,

text: but

text: because

text: of

text: lack

text: of

text: reference

text: (and

text: in

text: particular

text: of

text: a

text: recipe

text: to

text: construct

text: the

text: topological

text: equivalence

text: of

text: the

text: intersection

text: foliations)

text: we

text: decided

text: to

text: present

text: this

text: proof

text: and

text: construction

text: in

text: all

text: details.

text: Theorem

text: (Reconstruction

text: Theorem).

text: Two

text: normalized

text: holomorphic

text: foliation

text: F

text: G

text: C

text: germs

text: with

text: an

text: isolated

text: singularity

text: in

text: n

text: of

text: Poincar´e

text: type

text: are

text: topologically

text: F

text: G

text: equivalent

text: if,

text: and

text: only

text: if

text: the

text: real

text: intersection

text: foliations

text: and

text: are

text: topologically

text: equivalent.

text: F

text: G

text: Proof.

text: As

text: discussed

text: above

text: Arnol

text: d

text: showed

text: that

text: resp.

text: is

text: F

text: G

text: topologically

text: equivalent

text: to

text: the

text: foliation

text: resp.

text: of

text: by

text: the

text: cones

text: over

text: the

text: F

text: G

text: leaves

text: of

text: resp.

text: with

text: vertex

text: n.

text: As

text: already

text: noticed

text: that

text: shows

text: one

text: direction

text: of

text: the

text: Theorem.

text: F

text: G

text: To

text: prove

text: that

text: a

text: topological

text: equivalence

text: of

text: and

text: induces

text: a

text: F

text: G

text: of

text: the

text: intersection

text: foliations

text: and

text: it

text: is

text: enough

text: to

text: construct

text: F

text: G

text: the

text: latter

text: one

text: from

text: a

text: topological

text: equivalence

text: of

text: the

text: cone

text: foliations

text: and

text: in

text: the

text: sense

text: of

text: Def.

text: By

text: decreasing

text: to

text: this

text: topological

text: equivalence

text: can

text: be

text: by

text: an

text: embeddingFH∩C

text: such

text: that

text: and

text: leaves

text: of

text: thGe

text: ∩foliat−ion

text: cone

text: over

text: are

text: mapped

text: into

text: leaves

text: of

text: the

text: foliation

text: cone

text: over

text: Composing

text: H

text: with

text: the

text: radial

text: projection

text: r

text: produces

text: a

text: C

text: continuous

text: map

text: h

text: →r

text: Since

text: HC

text: may

text: map

text: to

text: a

text: topological

text: manifold

text: in

text: intersecting

text: the

text: same

text: radial

text: line

text: more

text: than

text: once,

text: h

text: may

text: not

text: be

text: injective,

text: hence

text: not

text: wanted

text: homeomorphism.

text: But

text: from

text: h

text: we

text: will

text: be

text: able

text: to

text: a

text: F

text: G

text: g

text: topological

text: equivalence

text: of

text: with

text: F

text: F

text: This

text: the

text: theorem

text: since

text: the

text: cone

text: structure

text: of

text: implies

text: that

text: is

text: F

text: F

text: F

text: topologically

text: equivalent

text: to

text: and

text: by

text: construction,

text: G

text: G

text: and


text: 2 where P(t) is the accumulated CPI at time t and ∆t is the period of the measurements. In the academic ﬁ- and nancial literature, the simplest and most robust way to account for inﬂation is to take logarithm. Hence, the ∗ P (t + ∆t) P(t) continuous rate of change in prices is deﬁned as ∗ = − = 1 + i(t) . (2.8) P (t) P(t ∆t) ∗ C(t) = ∂ ln P(t) . (2.2) Actually P (t)/P(t) indicates that the process induces a non exact proportional response of adaptation due to ∂t ∗ the fact that the expected inﬂation P (t) expands the Usually the derivative of Eq. (2.2) is expressed in a response to the price level P(t) in order to forecast and meet the inﬂation of the next period. Now, one may discrete way as introduce the expected GRI (cid:20) (cid:21) (cid:20) − (cid:21) ∆t [ln P(t + ∆t) ln P(t)] ∗ C(t + ) = ∗ ∆t ≡ ∗ ∆t P (t + ∆t) 2 ∆t r (t + ) C (t + ) ∆t = ln ∗ . (2.9) 2 2 P (t) 1 P(t + ∆t) = ln . (2.3) ∆t P(t) So, expressions (2.7) and (2.8) are equivalent, respec- tively, to The growth rate index (GRI) over one period is deﬁned as (cid:20) (cid:21) ∆t ∗ − ∆t r(t + ) = r (t ) , (2.10) 2 2 ∆t ≡ ∆t P(t + ∆t) r(t + ) C(t + ) ∆t = ln 2 2 P(t) and − = ln[1 + i(t + ∆t)] = p(t + ∆t) p(t) , (2.4) ∗ ∆t − ∆t r (t + ) = r(t ) . (2.11) 2 2 where a widely utilized notation These relations imply p(t) = ln P(t) , (2.5) r(t + ∆t) = r(t − ∆t) , (2.12) was introduced. It is straightforward to show that the giving a constant ﬁnite GRI equal to its initial value accumulated CPI is given by(cid:20) (cid:90) (cid:21) r(t) = r(t ) = r . The accumulated CPI evaluated using 0 0 Eq. (2.6) leads to an expone(cid:20)ntia(cid:18)l law (cid:19)(cid:21) 1 t (cid:48) (cid:48) P(t) = P(t ) exp r(t )dt . (2.6) − 0 ∆t t0 P(t) = P0 exp r0 t ∆tt0 , (2.13) A. Cagan’s model of inﬂation where P = P(t ). 0 0 B. Feedback contribution to the equation for In his pioneering work, Cagan has proposed [9] a model inﬂation of inﬂation based on the mechanism of “adaptive inﬂa- tionary expectation” with positive feedback between re- alized growth of the market price P(t) and the growth Due to the fact that the CPI during spirals of hyperin- of people’s averaged expectation price P∗(t). These two ﬂation grows more rapidly than the exponential law given prices are thought to evolve due to a positive feedback by Eq. (2.13), the Cagan’s model for inﬂation has been mechanism: an upward change of market price P(t) in generalized by Mizuno, Takayasu, and Takayasu (MTT) a unit time ∆t induces a rise in the people’s expecta- [10] including a linear feedback (LF) process. For this tion price P∗(t), and such an anticipation pushes on the purpose, the relation (2.7) was kept, while Eq. (2.8) was market price. Cagan’s assumptions may be cast into the replace(cid:20)d by (cid:21) (cid:20) (cid:21) following equations: ∗ P (t + ∆t) P(t) P(t + ∆t) P∗(t) P∗(t) ln P∗(t) = (1 + 2 ap) ln P(t − ∆t) , 1+i(t+∆t) = = = ∗ − , (2.7) (2.14) P(t) P(t) P (t ∆t) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: P(t)

text: is

text: the

text: accumulated

text: CPI

text: at

text: time

text: t

text: and

text: ∆t

text: is

text: the

text: period

text: of

text: the

text: measurements.

text: In

text: the

text: academic

text: ﬁ-

text: and

text: nancial

text: literature,

text: the

text: simplest

text: and

text: most

text: robust

text: way

text: to

text: account

text: for

text: inﬂation

text: is

text: to

text: take

text: logarithm.

text: Hence,

text: the

label: equation

text: P

text: (t

text: ∆t)

text: P(t)

text: continuous

text: rate

text: of

text: change

text: in

text: prices

text: is

text: deﬁned

text: as

text: i(t)

text: P

text: (t)

text: P(t

text: ∆t)

text: C(t)

text: ln

text: P(t)

text: Actually

text: P

text: (t)/P(t)

text: indicates

text: that

text: the

text: process

text: induces

text: a

text: non

text: exact

text: proportional

text: response

text: of

text: adaptation

text: due

text: to

text: ∂t

text: the

text: fact

text: that

text: the

text: expected

text: inﬂation

text: P

text: (t)

text: expands

text: the

text: Usually

text: the

text: derivative

text: of

text: Eq.

text: is

text: expressed

text: in

text: a

text: response

text: to

text: the

text: price

text: level

text: P(t)

text: in

text: order

text: to

text: forecast

text: and

text: meet

text: the

text: inﬂation

text: of

text: the

text: next

text: period.

text: Now,

text: one

text: may

text: discrete

text: way

text: as

text: introduce

text: the

text: expected

text: GRI

text: ∆t

text: [ln

text: P(t

text: ∆t)

text: ln

text: P(t)]

text: C(t

text: ∆t

text: ∆t

text: P

text: (t

text: ∆t)

text: ∆t

text: r

text: (t

text: C

text: (t

text: ∆t

text: ln

text: P

text: (t)

text: P(t

text: ∆t)

text: ln

text: ∆t

text: P(t)

text: So,

text: expressions

text: and

text: are

text: equivalent,

text: respec-

text: tively,

text: to

text: The

text: growth

text: rate

text: index

text: (GRI)

text: over

text: one

text: period

text: is

text: deﬁned

text: as

text: ∆t

text: ∆t

text: r(t

text: r

text: (t

text: ∆t

text: ∆t

text: P(t

text: ∆t)

text: r(t

text: C(t

text: ∆t

text: ln

text: P(t)

text: and

text: i(t

text: ∆t)]

text: p(t

text: ∆t)

text: p(t)

text: ∆t

text: ∆t

text: r

text: (t

text: r(t

text: where

text: a

text: widely

text: utilized

text: notation

text: These

text: relations

text: imply

text: p(t)

text: ln

text: P(t)

text: r(t

text: ∆t)

text: r(t

text: ∆t)

text: was

text: introduced.

text: It

text: is

text: straightforward

text: to

text: show

text: that

text: the

text: giving

text: a

text: constant

text: ﬁnite

text: GRI

text: equal

text: to

text: its

text: initial

text: value

text: accumulated

text: CPI

text: is

text: given

text: r(t)

text: r(t

text: r

text: The

text: accumulated

text: CPI

text: evaluated

text: using

text: Eq.

text: leads

text: to

text: an

text: law

text: t

text: P(t)

text: P(t

text: exp

text: r(t

text: )dt

text: ∆t

text: P(t)

text: exp

text: t

text: A.

text: Cagan’s

text: model

text: of

text: inﬂation

text: where

text: P

text: P(t

text: B.

text: Feedback

text: contribution

text: to

text: the

text: equation

text: for

text: In

text: his

text: pioneering

text: work,

text: Cagan

text: has

text: proposed

text: a

text: model

text: inﬂation

text: of

text: inﬂation

text: based

text: on

text: the

text: mechanism

text: of

text: “adaptive

text: inﬂa-

text: tionary

text: expectation”

text: with

text: positive

text: feedback

text: between

text: re-

text: alized

text: growth

text: of

text: the

text: market

text: price

text: P(t)

text: and

text: the

text: growth

text: Due

text: to

text: the

text: fact

text: that

text: the

text: CPI

text: during

text: spirals

text: of

text: hyperin-

text: of

text: people’s

text: averaged

text: expectation

text: price

text: P∗(t).

text: These

text: two

text: ﬂation

text: grows

text: more

text: rapidly

text: than

text: the

text: exponential

text: law

text: given

text: prices

text: are

text: thought

text: to

text: evolve

text: due

text: to

text: a

text: positive

text: feedback

text: by

text: Eq.

text: the

text: Cagan’s

text: model

text: for

text: inﬂation

text: has

text: been

text: mechanism:

text: an

text: upward

text: change

text: of

text: market

text: price

text: P(t)

text: in

text: generalized

text: by

text: Mizuno,

text: Takayasu,

text: and

text: Takayasu

text: (MTT)

text: a

text: unit

text: time

text: ∆t

text: induces

text: a

text: rise

text: in

text: the

text: people’s

text: expecta-

text: including

text: a

text: linear

text: feedback

text: (LF)

text: process.

text: For

text: this

text: tion

text: price

text: P∗(t),

text: and

text: such

text: an

text: anticipation

text: pushes

text: on

text: the

text: purpose,

text: the

text: relation

text: was

text: kept,

text: while

text: Eq.

text: was

text: market

text: price.

text: Cagan’s

text: assumptions

text: may

text: be

text: cast

text: into

text: the

text: by

text: following

text: equations:

text: P

text: (t

text: ∆t)

text: P(t)

text: P(t

text: ∆t)

text: P∗(t)

text: P∗(t)

text: ln

text: P∗(t)

text: ap)

text: ln

text: P(t

text: ∆t)

text: P(t)

text: P(t)

text: P

text: (t

text: ∆t)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4 erosion cutting. For preparing Al contacts, the electro- forming [8] and displacement techniques [9] were used to the same extent. The Be contacts were formed pre- dominantly by the electroforming technique which en- sured their noticeably higher mechanical stability. We did not detect any correlation between the method of preparing a point contact and the form of the A (eV ) 1 dependence. The ﬁrst and second IVC derivatives were recorded by the standard modulation technique, and the data were registered directly on a computer. The follow- ing characteristics were recorded during measurements: | d2V/dI2(V ); dV/dI(V ); dV/dI(B) . The tem- V =const perature was maintained at 1.3 K throughout the mea- surements. By way of an example, Fig.2 shows the PC spectrum of a Be contact, the oscillating component of the magnetoresistance, and its Fourier spectrum. The PC spectrum d2I/dV 2 (V ) is obtained from the exper- imental formulas d2V/dI2 (V ) and dV/dI (V ) by calcu- lations and has a well-deﬁned structure in the interval 40-80 meV , which is associated with the EPI function α2 (ω)F(ω). The magnetoquantum oscillations of resis- pc tance with a natural frequency 985 T correspond to the third electron band (cigar) and are known from the de Haas-van Alphen eﬀect investigations [10]. The Fourier spectrum also contains the second harmonic. The follow- ing procedure was used for determining the dependence of amplitude of magnetoresistance oscillations of the PC on bias voltage. For a ﬁxed bias voltage across the con- tact, the oscillations R (B) = dV/dI(B) were recorded d − in the same interval of magnetic ﬁelds (8.5 10 T as a rule). Figure 3 shows fragments of the recording for two − diﬀerent bias voltages applied to an Al Al point contact. For Al contacts, the investigated oscillations correspond to the γ-pocket of the third band with a frequency 290 T [11]. Two recordings were made for each bias voltage, FIG. 2: (a) Point-contact spectrum of a Be point contact, viz., in increasing and decreasing magnetic ﬁelds. After R = 16.24 Ω. 0 recording of the curves for the entire range of the values (b) Oscillations of the resistance of a Be point contact in a (cid:107) (cid:107) of V , a recording was made at V = 0 and compared with magnetic ﬁeld at zero bias voltage B c z, R = 16.24 Ω; ≈ · − ∼ 0 the initial recording to ensure that the sample remained ∆R/r 2 10 2 at B 9 T. 0 unchanged in the course of measurements. Contacts with (c) I and F are the intensity and the frequency of oscillations. diﬀerent resistances (from 0.6 to 20 Ω), background levels (30 to 85%) and levels of blurring of phonon singulari- ties in the spectrum were studied. The complete set of istics dV/dI(B) (V =const) were recorded for a constant characteristics was obtained for ﬁve Al contacts and ﬁve value of the modulating current, and the diﬀerential resis- Be contacts. The results of experimental measurements tance of the contact increases (up to 10% in the interval were subjected to Fourier analysis for determining the of Debye energies) upon an increase in the bias voltage amplitudes of oscillations in a ﬁxed interval of magnetic across the contact, the obtained results were reduced to ﬁelds for the entire set of curves. Since the character- a ﬁxed modulating voltage and normalized to the value obtained for zero bias voltage across the contact. Figures ##LTFigure## ##LTFigure##

text: erosion

text: cutting.

text: For

text: preparing

text: Al

text: contacts,

text: the

text: electro-

text: forming

text: and

text: displacement

text: techniques

text: were

text: used

text: to

text: the

text: same

text: extent.

text: The

text: Be

text: contacts

text: were

text: formed

text: pre-

text: dominantly

text: by

text: the

text: electroforming

text: technique

text: which

text: en-

text: sured

text: their

text: noticeably

text: higher

text: mechanical

text: stability.

text: We

text: did

text: not

text: detect

text: any

text: correlation

text: between

text: the

text: method

text: of

text: preparing

text: a

text: point

text: contact

text: and

text: the

text: form

text: of

text: the

text: A

text: (eV

text: dependence.

text: The

text: ﬁrst

text: and

text: second

text: IVC

text: derivatives

text: were

text: recorded

text: by

text: the

text: standard

text: modulation

text: technique,

text: and

text: the

text: data

text: were

text: registered

text: directly

text: on

text: a

text: computer.

text: The

text: follow-

text: ing

text: characteristics

text: were

text: recorded

text: during

text: measurements:

text: dV/dI(V

text: dV/dI(B)

text: The

text: tem-

text: V

text: =const

text: perature

text: was

text: maintained

text: at

text: K

text: throughout

text: the

text: mea-

text: surements.

text: By

text: way

text: of

text: an

text: example,

text: shows

text: the

text: PC

text: spectrum

text: of

text: a

text: Be

text: contact,

text: the

text: oscillating

text: component

text: of

text: the

text: magnetoresistance,

text: and

text: its

text: Fourier

text: spectrum.

text: The

text: PC

text: spectrum

text: (V

text: is

text: obtained

text: from

text: the

text: exper-

text: imental

text: formulas

text: (V

text: and

text: dV/dI

text: (V

text: by

text: calcu-

text: lations

text: and

text: has

text: a

text: well-deﬁned

text: structure

text: in

text: the

text: interval

text: meV

text: which

text: is

text: associated

text: with

text: the

text: EPI

text: function

text: α2

text: (ω)F(ω).

text: The

text: magnetoquantum

text: oscillations

text: of

text: resis-

text: pc

text: tance

text: with

text: a

text: natural

text: frequency

text: T

text: correspond

text: to

text: the

text: third

text: electron

text: band

text: (cigar)

text: and

text: are

text: known

text: from

text: the

text: de

text: Haas-van

text: Alphen

text: eﬀect

text: investigations

text: The

text: Fourier

text: spectrum

text: also

text: contains

text: the

text: second

text: harmonic.

text: The

text: follow-

text: ing

text: procedure

text: was

text: used

text: for

text: determining

text: the

text: dependence

text: of

text: amplitude

text: of

text: magnetoresistance

text: oscillations

text: of

text: the

text: PC

text: on

text: bias

text: voltage.

text: For

text: a

text: ﬁxed

text: bias

text: voltage

text: across

text: the

text: con-

text: tact,

text: the

text: oscillations

text: R

text: (B)

text: dV/dI(B)

text: were

text: recorded

text: d

text: in

text: the

text: same

text: interval

text: of

text: magnetic

text: ﬁelds

text: T

text: as

text: a

text: rule).

text: Figure

text: shows

text: fragments

text: of

text: the

text: recording

text: for

text: two

text: diﬀerent

text: bias

text: voltages

text: applied

text: to

text: an

text: Al

text: Al

text: point

text: contact.

text: For

text: Al

text: contacts,

text: the

text: investigated

text: oscillations

text: correspond

text: to

text: the

text: γ-pocket

text: of

text: the

text: third

text: band

text: with

text: a

text: frequency

text: T

text: Two

text: recordings

text: were

text: made

text: for

text: each

text: bias

text: voltage,

text: FIG.

text: (a)

text: Point-contact

text: spectrum

text: of

text: a

text: Be

text: point

text: contact,

text: viz.,

text: in

text: increasing

text: and

text: decreasing

text: magnetic

text: ﬁelds.

text: After

text: R

text: Ω.

text: recording

text: of

text: the

text: curves

text: for

text: the

text: entire

text: range

text: of

text: the

text: values

text: (b)

text: Oscillations

text: of

text: the

text: resistance

text: of

text: a

text: Be

text: point

text: contact

text: in

text: a

text: of

text: V

text: a

text: recording

text: was

text: made

text: at

text: V

text: and

text: compared

text: with

text: magnetic

text: ﬁeld

text: at

text: zero

text: bias

text: voltage

text: B

text: c

text: z,

text: R

text: Ω;

text: the

text: initial

text: recording

text: to

text: ensure

text: that

text: the

text: sample

text: remained

text: ∆R/r

text: at

text: B

text: T.

text: unchanged

text: in

text: the

text: course

text: of

text: measurements.

text: Contacts

text: with

text: (c)

text: I

text: and

text: F

text: are

text: the

text: intensity

text: and

text: the

text: frequency

text: of

text: oscillations.

text: diﬀerent

text: resistances

text: (from

text: to

text: Ω),

text: background

text: levels

text: to

text: and

text: levels

text: of

text: blurring

text: of

text: phonon

text: singulari-

text: ties

text: in

text: the

text: spectrum

text: were

text: studied.

text: The

text: complete

text: set

text: of

text: istics

text: dV/dI(B)

text: (V

text: =const)

text: were

text: recorded

text: for

text: a

text: constant

text: characteristics

text: was

text: obtained

text: for

text: ﬁve

text: Al

text: contacts

text: and

text: ﬁve

text: value

text: of

text: the

text: modulating

text: current,

text: and

text: the

text: diﬀerential

text: resis-

text: Be

text: contacts.

text: The

text: results

text: of

text: experimental

text: measurements

text: tance

text: of

text: the

text: contact

text: increases

text: (up

text: to

text: in

text: the

text: interval

text: were

text: subjected

text: to

text: Fourier

text: analysis

text: for

text: determining

text: the

text: of

text: Debye

text: energies)

text: upon

text: an

text: increase

text: in

text: the

text: bias

text: voltage

text: amplitudes

text: of

text: oscillations

text: in

text: a

text: ﬁxed

text: interval

text: of

text: magnetic

text: across

text: the

text: contact,

text: the

text: obtained

text: results

text: were

text: reduced

text: to

text: ﬁelds

text: for

text: the

text: entire

text: set

text: of

text: curves.

text: Since

text: the

text: character-

text: a

text: ﬁxed

text: modulating

text: voltage

text: and

text: normalized

text: to

text: the

text: value

text: obtained

text: for

text: zero

text: bias

text: voltage

text: across

text: the

text: contact.

text: Figures

text: ##LTFigure##

text: ##LTFigure##