text: Manifestation of chirality in the vortex lattice in a two-dimensional topological ∗ superconductor † Evan D. B. Smith, K. Tanaka, and Yuki Nagai ‡ § (Dated: May 29, 2020) We study the vortex lattice in a two-dimensional s-wave topological superconductor with Rashba spin-orbit coupling and Zeeman ﬁeld by solving the Bogoliubov-de Gennes equations self-consistently for the superconducting order parameter. We ﬁnd that when spin-orbit coupling is relatively weak, one of the two underlying chiralities in the topological superconducting state can be strongly manifest in the vortex core structure and govern the response of the system to vorticity and a nonmagnetic impurity where the vortex is pinned. The Majorana zero mode in the vortex core is found to be robust against the nonmagnetic impurity in that it remains eﬀectively a zero-energy bound state regardless of the impurity potential strength and the major chirality. The spin polarization of the Majorana bound state depends on the major chirality for weak spin-orbit coupling, while it is determined simply by the vorticity when spin-orbit coupling is relatively strong. PACS numbers: 74.20.-z, 74.20.Rp, 74.25.Uv, 75.70.Tj I. INTRODUCTION becomes eﬀectively spinless – condition required for TSC that can support Majorana zero modes.5 Although the Fermi surface has a certain chirality (spin winding), in One of the most promising models proposed so far for principle the two intrinsic chiralities are always present platforms to realise topological quantum computation is in a 2D s-wave TSC state.15 the two-dimensional (2D) s-wave topological supercon- It is known that in a spin-triplet chiral p-wave super- ductivity (TSC) model with Rashba spin-orbit (SO) cou- − pling and Zeeman ﬁeld.1–5 Sato, Takahashi, and Fujimoto conductor, vortices in the px + ipy and px ipy states, have proposed the tight-binding model1,2 that can de- which are degenerate at zero ﬁeld, are not equivalent.17,18 The free energy is lower and the upper critical ﬁeld is scribe an s-wave superﬂuid of ultracold fermionic atoms higher when the vorticity, i.e., the angular momentum in an optical lattice, where an eﬀective SO interaction carried by the supercurrent, is antiparallel to the chiral- can be generated by spatially varying laser ﬁelds, or 2D s- ity, i.e., the angular momentum of Cooper pairs in the wave TSC in a solid device. Such TSC can be realised, as proposed by Sau et al. in terms of the continuum model,3 condensate.18 Moreover, bound states in such an antipar- allel vortex are more robust against nonmagnetic impu- in a semiconductor heterostructure, where a semiconduc- rities due to cancellation of angular momenta between tor with Rashba SO coupling is sandwiched between a the supercurrent and the condensate; which makes the conventional s-wave superconductor and a ferromagnetic vortex core region s-wave-like and allows the Anderson insulator. While s-wave superconductivity is induced by theorem19 to take eﬀect.20,21 Most notably, in each of the the proximity eﬀect, the ferromagnetic insulator can gen- chirality domains, the order parameter of the other chi- erate Zeeman coupling via exchange interactions, thus rality is induced around the vortex centre, more promi- aﬀecting only the spin degree of freedom in the semicon- nently for lower applied ﬁeld.17,18 ductor. A vortex in the model hosts a zero-energy Ma- jorana bound state3,6 and hence vortices in a 2D s-wave The eﬀects of nonmagnetic impurities on vortex bound topological superconductor obey non-Abelian exchange states in a 2D s-wave topological superconductor, de- statistics, like those in chiral p-wave superconductors.7–11 pending on the major chirality with respect to the vortic- ity, have been studied recently for a single vortex in the Inherent in the 2D s-wave TSC model are the two chi- ralities, ∼ sin k ±i sin k in the tight-binding model12,13 continuum model3 by means of a Green-function tech- or k ±ik in thxe continuyum model4,14,15 (also for |k| (cid:28) 1 nique for calculating the impurity self-energy.22,23 It has x y been found that when the major chirality is opposite to in the tight-binding model), where k = (k , k ) is the x y the vorticity, vortex bound states are less inﬂuenced by wavevector. The noninteracting Fermi surface is split nonmagnetic impurities for relatively weak SO coupling, sideways by the Rashba SO interaction, which causes winding of spin as one goes around each Fermi surface,16 compared to the case where the major chirality is i±n the same direction as the vorticity. As to spinless p ip and the Zeeman ﬁeld perpendicular to the 2D system x y superconducting states per se, Ivanov has stated that favours one spin component along its direction over though with slightly diﬀerent structure of the quasipar- the other, resulting in two separate bands. With large ticle eigenfunctions, the two types of vortices have the enough Zeeman splitting and the chemical potential in same low-energy spectra and braiding statistics.9 the gap between the two bands, the system has a single Fermi surface on which spin is ﬁxed for each k and thus Volovik has studied the eﬀects of a single nonmagnetic

text: Manifestation

label: title

text: of

text: chirality

text: in

text: the

text: vortex

text: lattice

text: in

text: a

text: two-dimensional

text: topological

text: superconductor

text: Evan

label: author

text: D.

text: B.

text: Smith,

text: K.

text: Tanaka,

text: and

text: Yuki

text: Nagai

text: (Dated:

text: May

label: date

text: We

label: abstract

text: study

text: the

text: vortex

text: lattice

text: in

text: a

text: two-dimensional

text: s-wave

text: topological

text: superconductor

text: with

text: Rashba

text: spin-orbit

text: coupling

text: and

text: Zeeman

text: ﬁeld

text: by

text: solving

text: the

text: Bogoliubov-de

text: Gennes

text: equations

text: self-consistently

text: for

text: the

text: superconducting

text: order

text: parameter.

text: We

text: ﬁnd

text: that

text: when

text: spin-orbit

text: coupling

text: is

text: relatively

text: weak,

text: one

text: of

text: the

text: two

text: underlying

text: chiralities

text: in

text: the

text: topological

text: superconducting

text: state

text: can

text: be

text: strongly

text: manifest

text: in

text: the

text: vortex

text: core

text: structure

text: and

text: govern

text: the

text: response

text: of

text: the

text: system

text: to

text: vorticity

text: and

text: a

text: nonmagnetic

text: impurity

text: where

text: the

text: vortex

text: is

text: pinned.

text: The

text: Majorana

text: zero

text: mode

text: in

text: the

text: vortex

text: core

text: is

text: found

text: to

text: be

text: robust

text: against

text: the

text: nonmagnetic

text: impurity

text: in

text: that

text: it

text: remains

text: eﬀectively

text: a

text: zero-energy

text: bound

text: state

text: regardless

text: of

text: the

text: impurity

text: potential

text: strength

text: and

text: the

text: major

text: chirality.

text: The

text: spin

text: polarization

text: of

text: the

text: Majorana

text: bound

text: state

text: depends

text: on

text: the

text: major

text: chirality

text: for

text: weak

text: spin-orbit

text: coupling,

text: while

text: it

text: is

text: determined

text: simply

text: by

text: the

text: vorticity

text: when

text: spin-orbit

text: coupling

text: is

text: relatively

text: strong.

text: PACS

text: numbers:

text: I.

text: INTRODUCTION

text: becomes

text: eﬀectively

text: spinless

text: condition

text: required

text: for

text: TSC

text: that

text: can

text: support

text: Majorana

text: zero

text: Although

text: the

text: Fermi

text: surface

text: has

text: a

text: certain

text: chirality

text: (spin

text: winding),

text: in

text: One

text: of

text: the

text: most

text: promising

text: models

text: proposed

text: so

text: far

text: for

text: principle

text: the

text: two

text: intrinsic

text: chiralities

text: are

text: always

text: present

text: platforms

text: to

text: realise

text: topological

text: quantum

text: computation

text: is

text: in

text: a

text: s-wave

text: TSC

text: the

text: two-dimensional

text: s-wave

text: topological

text: supercon-

text: It

text: is

text: known

text: that

text: in

text: a

text: spin-triplet

text: chiral

text: p-wave

text: super-

text: ductivity

text: (TSC)

text: model

text: with

text: Rashba

text: spin-orbit

text: (SO)

text: cou-

text: pling

text: and

text: Zeeman

text: Sato,

text: Takahashi,

text: and

text: Fujimoto

text: conductor,

text: vortices

text: in

text: the

text: px

text: ipy

text: and

text: px

text: ipy

text: states,

text: have

text: proposed

text: the

text: tight-binding

text: that

text: can

text: de-

text: which

text: are

text: degenerate

text: at

text: zero

text: ﬁeld,

text: are

text: not

text: The

text: free

text: energy

text: is

text: lower

text: and

text: the

text: upper

text: critical

text: ﬁeld

text: is

text: scribe

text: an

text: s-wave

text: superﬂuid

text: of

text: ultracold

text: fermionic

text: atoms

text: higher

text: when

text: the

text: vorticity,

text: i.e.,

text: the

text: angular

text: momentum

text: in

text: an

text: optical

text: lattice,

text: where

text: an

text: eﬀective

text: SO

text: interaction

text: carried

text: by

text: the

text: supercurrent,

text: is

text: antiparallel

text: to

text: the

text: chiral-

text: can

text: be

text: generated

text: by

text: spatially

text: varying

text: laser

text: ﬁelds,

text: or

text: s-

text: ity,

text: i.e.,

text: the

text: angular

text: momentum

text: of

text: Cooper

text: pairs

text: in

text: the

text: wave

text: TSC

text: in

text: a

text: solid

text: device.

text: Such

text: TSC

text: can

text: be

text: realised,

text: as

text: proposed

text: by

text: Sau

text: et

text: al.

text: in

text: terms

text: of

text: the

text: continuum

text: Moreover,

text: bound

text: states

text: in

text: such

text: an

text: antipar-

text: allel

text: vortex

text: are

text: more

text: robust

text: against

text: nonmagnetic

text: impu-

text: in

text: a

text: semiconductor

text: heterostructure,

text: where

text: a

text: semiconduc-

text: rities

text: due

text: to

text: cancellation

text: of

text: angular

text: momenta

text: between

text: tor

text: with

text: Rashba

text: SO

text: coupling

text: is

text: sandwiched

text: between

text: a

text: the

text: supercurrent

text: and

text: the

text: condensate;

text: which

text: makes

text: the

text: conventional

text: s-wave

text: superconductor

text: and

text: a

text: ferromagnetic

text: vortex

text: core

text: region

text: s-wave-like

text: and

text: allows

text: the

text: Anderson

text: insulator.

text: While

text: s-wave

text: superconductivity

text: is

text: induced

text: by

text: to

text: take

text: Most

text: notably,

text: in

text: each

text: of

text: the

text: the

text: proximity

text: eﬀect,

text: the

text: ferromagnetic

text: insulator

text: can

text: gen-

text: chirality

text: domains,

text: the

text: order

text: parameter

text: of

text: the

text: other

text: chi-

text: erate

text: Zeeman

text: coupling

text: via

text: exchange

text: interactions,

text: thus

text: rality

text: is

text: induced

text: around

text: the

text: vortex

text: centre,

text: more

text: promi-

text: aﬀecting

text: only

text: the

text: spin

text: degree

text: of

text: freedom

text: in

text: the

text: semicon-

text: nently

text: for

text: lower

text: applied

text: ductor.

text: A

text: vortex

text: in

text: the

text: model

text: hosts

text: a

text: zero-energy

text: Ma-

text: jorana

text: bound

text: and

text: hence

text: vortices

text: in

text: a

text: s-wave

text: The

text: eﬀects

text: of

text: nonmagnetic

text: impurities

text: on

text: vortex

text: bound

text: topological

text: superconductor

text: obey

text: non-Abelian

text: exchange

text: states

text: in

text: a

text: s-wave

text: topological

text: superconductor,

text: de-

text: statistics,

text: like

text: those

text: in

text: chiral

text: p-wave

text: pending

text: on

text: the

text: major

text: chirality

text: with

text: respect

text: to

text: the

text: vortic-

text: ity,

text: have

text: been

text: studied

text: recently

text: for

text: a

text: single

text: vortex

text: in

text: the

text: Inherent

text: in

text: the

text: s-wave

text: TSC

text: model

text: are

text: the

text: two

text: chi-

text: ralities,

text: sin

text: k

text: ±i

text: sin

text: k

text: in

text: the

text: tight-binding

text: continuum

text: by

text: means

text: of

text: a

text: Green-function

text: tech-

text: or

text: k

text: ±ik

text: in

text: thxe

text: continuyum

text: (also

text: for

text: |k|

text: nique

text: for

text: calculating

text: the

text: impurity

text: It

text: has

text: x

text: y

text: been

text: found

text: that

text: when

text: the

text: major

text: chirality

text: is

text: opposite

text: to

text: in

text: the

text: tight-binding

text: model),

text: where

text: k

text: (k

text: k

text: is

text: the

text: x

text: y

text: the

text: vorticity,

text: vortex

text: bound

text: states

text: are

text: less

text: inﬂuenced

text: by

text: wavevector.

text: The

text: noninteracting

text: Fermi

text: surface

text: is

text: split

text: nonmagnetic

text: impurities

text: for

text: relatively

text: weak

text: SO

text: coupling,

text: sideways

text: by

text: the

text: Rashba

text: SO

text: interaction,

text: which

text: causes

text: winding

text: of

text: spin

text: as

text: one

text: goes

text: around

text: each

text: Fermi

text: compared

text: to

text: the

text: case

text: where

text: the

text: major

text: chirality

text: is

text: i±n

text: the

text: same

text: direction

text: as

text: the

text: vorticity.

text: As

text: to

text: spinless

text: p

text: ip

text: and

text: the

text: Zeeman

text: ﬁeld

text: perpendicular

text: to

text: the

text: system

text: x

text: y

text: superconducting

text: states

text: per

text: se,

text: Ivanov

text: has

text: stated

text: that

text: favours

text: one

text: spin

text: component

text: along

text: its

text: direction

text: over

text: though

text: with

text: slightly

text: diﬀerent

text: structure

text: of

text: the

text: quasipar-

text: the

text: other,

text: resulting

text: in

text: two

text: separate

text: bands.

text: With

text: large

text: ticle

text: eigenfunctions,

text: the

text: two

text: types

text: of

text: vortices

text: have

text: the

text: enough

text: Zeeman

text: splitting

text: and

text: the

text: chemical

text: potential

text: in

text: same

text: low-energy

text: spectra

text: and

text: braiding

text: the

text: gap

text: between

text: the

text: two

text: bands,

text: the

text: system

text: has

text: a

text: single

text: Fermi

text: surface

text: on

text: which

text: spin

text: is

text: ﬁxed

text: for

text: each

text: k

text: and

text: thus

text: Volovik

text: has

text: studied

text: the

text: eﬀects

text: of

text: a

text: single

text: nonmagnetic


text: Algorithms to Compute the Lyndon Array(cid:63) Frantisek Franek1, A. S. M. Sohidull Islam2, M. Sohel Rahman3, and W. F. Smyth1,3,4 1 Algorithms Research Group Department of Computing & Software McMaster University, Hamilton, Canada { } franek/smyth @mcmaster.ca 2 School of Computational Science & Engineering McMaster University, Hamilton, Canada sohansayed@gmail.com 3 Department of Computer Science & Engineering Bangladesh University of Engineering & Technology msrahman@cse.buet.ac.bd 4 School of Engineering & Information Technology Murdoch University, Perth, Australia Abstract. In the Lyndon array λ = λx[1..n] of a string x = x[1..n], λ[i] is the length of the longest Lyndon word starting at position i of x. The computation of λ has recently become of great interest, since it was shown (Bannai et al., The “Runs” Theorem [2]) that the runs in x are computable in linear time from λ . Here we ﬁrst describe three x algorithms for computing λ that have been suggested in the literature, x but for which no structured exposition has been given. Two of these O algorithms execute in (n2) time in the worst case; the third achieves Θ(n) time, but at the expense of prior computation of both the suﬃx array and the inverse suﬃx array of x. We then go on to describe two variants of a new algorithm that avoids prior computation of global data O structures and executes in worst-case (n log n) time. Experimental ev- idence suggests that all but one of these ﬁve algorithms require only linear execution time in practice, with the two new algorithms faster by a small factor. We conjecture that there exists a fast and worst-case linear-time algorithm to compute the Lyndon array that is also “elemen- tary” (making no use of global data structures such as the suﬃx array). (cid:63) This work was supported in part by the Natural Sciences & Engineering Research Council of Canada. The authors wish to thank Maxime Crochemore and Hideo Bannai for helpful discussions. ##LTLine##

text: Algorithms

text: to

text: Compute

text: the

text: Lyndon

text: Frantisek

text: A.

text: S.

text: M.

text: Sohidull

text: M.

text: Sohel

text: and

text: W.

text: F.

text: Algorithms

text: Research

text: Group

text: Department

text: of

text: Computing

text: Software

text: McMaster

text: University,

text: Hamilton,

text: Canada

label: footer

text: franek/smyth

text: @mcmaster.ca

text: School

text: of

text: Computational

text: Science

text: Engineering

text: McMaster

text: University,

text: Hamilton,

text: Canada

text: sohansayed@gmail.com

text: Department

text: of

text: Computer

text: Science

text: Engineering

text: Bangladesh

text: University

text: of

text: Engineering

text: Technology

text: msrahman@cse.buet.ac.bd

text: School

text: of

text: Engineering

text: Information

text: Technology

text: Murdoch

text: University,

text: Perth,

text: Australia

text: Abstract.

label: paragraph

text: In

text: the

text: Lyndon

text: array

text: λ

text: of

text: a

text: string

text: x

text: λ[i]

text: is

text: the

text: length

text: of

text: the

text: longest

text: Lyndon

text: word

text: starting

text: at

text: position

text: i

text: of

text: x.

text: The

text: computation

text: of

text: λ

text: has

text: recently

text: become

text: of

text: great

text: interest,

text: since

text: it

text: was

text: shown

text: (Bannai

text: et

text: al.,

text: The

text: “Runs”

text: Theorem

text: that

text: the

text: runs

text: in

text: x

text: are

text: computable

text: in

text: linear

text: time

text: from

text: λ

text: Here

text: we

text: ﬁrst

text: describe

text: three

text: x

text: algorithms

text: for

text: computing

text: λ

text: that

text: have

text: been

text: suggested

text: in

text: the

text: literature,

text: x

text: but

text: for

text: which

text: no

text: structured

text: exposition

text: has

text: been

text: given.

text: Two

text: of

text: these

text: O

text: algorithms

text: execute

text: in

text: time

text: in

text: the

text: worst

text: case;

text: the

text: third

text: achieves

text: Θ(n)

text: time,

text: but

text: at

text: the

text: expense

text: of

text: prior

text: computation

text: of

text: both

text: the

text: suﬃx

text: array

text: and

text: the

text: inverse

text: suﬃx

text: array

text: of

text: x.

text: We

text: then

text: go

text: on

text: to

text: describe

text: two

text: variants

text: of

text: a

text: new

text: algorithm

text: that

text: avoids

text: prior

text: computation

text: of

text: global

text: data

text: O

text: structures

text: and

text: executes

text: in

text: worst-case

text: (n

text: log

text: n)

text: time.

text: Experimental

text: ev-

text: idence

text: suggests

text: that

text: all

text: but

text: one

text: of

text: these

text: ﬁve

text: algorithms

text: require

text: only

text: linear

text: execution

text: time

text: in

text: practice,

text: with

text: the

text: two

text: new

text: algorithms

text: faster

text: by

text: a

text: small

text: factor.

text: We

text: conjecture

text: that

text: there

text: exists

text: a

text: fast

text: and

text: worst-case

text: linear-time

text: algorithm

text: to

text: compute

text: the

text: Lyndon

text: array

text: that

text: is

text: also

text: “elemen-

text: tary”

text: (making

text: no

text: use

text: of

text: global

text: data

text: structures

text: such

text: as

text: the

text: suﬃx

text: array).

text: This

text: work

text: was

text: supported

text: in

text: part

text: by

text: the

text: Natural

text: Sciences

text: Engineering

text: Research

text: Council

text: of

text: Canada.

text: The

text: authors

text: wish

text: to

text: thank

text: Maxime

text: Crochemore

text: and

text: Hideo

text: Bannai

text: for

text: helpful

text: discussions.

text: ##LTLine##


text: The so-called Fourier-Laplace transf(cid:16)ormed Brieskorn lattice o(cid:17)f the FL-transformed Gauß-Manin G system + is given by the following RC × ◦-module: z Λ • − ∧ H0 ΩY+×dΛ◦/Λ◦[z], zd dyφ . If the semigroup NA is saturated we have the following identiﬁcation: Proposition 3.9. [RS12, Prop(cid:16)osition 3.20] Let NA be a (cid:17)saturated semigroup. There exists the following RC × ◦-linear isomorphism (cid:99) z Λ • − ∧ (cid:39) ◦ H0 ΩY+×dΛ◦/Λ◦[z], zd dyφA 0MA . Proposition 3.10. [RS10, Corollary 2.19] (cid:16) (cid:17) (cid:16) (cid:17) (1) There is a non-degene(cid:99)rate ﬂat (−1)d-sym(cid:99)metric pairing ◦M ⊗ ∗ ◦M → O P : (cid:99) (cid:99) ι C∗× ◦. (cid:99) A |C∗×Λ◦ A |C∗×Λ◦ z Λ z z ◦M ◦M ⊂ O ◦M (2) We have that P( , (cid:34) ) zd (cid:35)C × (cid:34)◦, and P (cid:35)is non-degenerate on . i.e., it induces a non-deg0eneArat0e syAmme(cid:99)tric pazirΛing (cid:99) 0 A ◦M(cid:99) ◦M(cid:99) − ⊗ → O [z dP] : 0· ◦MA 0· ◦MA Λ◦. z z 0 A 0 A 4. Construction of the Landau-Ginzburg model X Let be a projective toric Deligne-Mumford stack with fan Σ = (N, Σ, a). In this section we explain the construction of a (Zariski local) Landau-Ginzburg model which will serve as a mirror X partner for . Recall the sequence 2.4 0 −→ L −→t Zn −a→ N −→ 0 . We apply the functor Hom(−, C∗) to the sequence above which gives the following sequence of algebraic tori: 1 −→ Hom(N, C∗) −→ Y(cid:48) := Hom(Zn, C∗) −ψ→t T := Hom(L, C∗) −→ 1 . Y The basis e , . . . , e equips with coordinates w , . . . , w . Consider the map 1 n 1 n (cid:48) Y(cid:48) −→ C(cid:88)× T W : , t (cid:55)→ − n (w , . . . , w ) ( w , ψ (w)) . 1 n i t i=1 (cid:48) X Usually this map W is called the Landau-Ginzburg model of the toric orbild . However to ensure a correct limit behavior we need to consider a covering of this model. Consider the 14 ##LTLine## ##LTLine## ##LTLine##

text: The

text: so-called

text: Fourier-Laplace

text: Brieskorn

text: lattice

text: the

text: FL-transformed

text: Gauß-Manin

text: G

text: system

text: is

text: given

text: by

text: the

text: following

text: RC

text: ◦-module:

text: z

text: Λ

text: ΩY+×dΛ◦/Λ◦[z],

text: zd

text: dyφ

text: If

text: the

text: semigroup

text: is

text: saturated

text: we

text: have

text: the

text: following

text: identiﬁcation:

text: Proposition

text: Let

text: be

text: a

text: semigroup.

text: There

text: exists

text: the

text: following

text: RC

text: ◦-linear

text: isomorphism

text: z

text: Λ

text: ΩY+×dΛ◦/Λ◦[z],

text: zd

text: dyφA

text: Proposition

text: Corollary

label: list

text: There

text: is

text: a

text: ﬂat

text: pairing

text: ◦M

text: ◦M

text: O

text: P

text: ι

text: C∗×

text: A

text: |C∗×Λ◦

text: A

text: |C∗×Λ◦

text: z

text: Λ

text: z

text: z

text: ◦M

text: ◦M

text: O

text: ◦M

text: We

text: have

text: that

text: P(

text: zd

text: and

text: P

text: non-degenerate

text: on

text: i.e.,

text: it

text: induces

text: a

text: pazirΛing

text: A

text: O

text: [z

text: dP]

text: 0·

text: ◦MA

text: 0·

text: ◦MA

text: Λ◦.

text: z

text: z

text: A

text: A

text: Construction

text: of

text: the

text: Landau-Ginzburg

text: model

text: X

text: Let

text: be

text: a

text: projective

text: toric

text: Deligne-Mumford

text: stack

text: with

text: fan

text: Σ

text: (N,

text: Σ,

text: a).

text: In

text: this

text: section

text: we

text: explain

text: the

text: construction

text: of

text: a

text: (Zariski

text: local)

text: Landau-Ginzburg

text: model

text: which

text: will

text: serve

text: as

text: a

text: mirror

text: X

text: partner

text: for

text: Recall

text: the

text: sequence

text: L

text: −→t

text: Zn

text: −a→

text: N

text: We

text: apply

text: the

text: functor

text: Hom(−,

text: C∗)

text: to

text: the

text: sequence

text: above

text: which

text: gives

text: the

text: following

text: sequence

text: of

text: algebraic

text: tori:

text: Hom(N,

text: C∗)

text: Hom(Zn,

text: C∗)

text: −ψ→t

text: T

text: Hom(L,

text: C∗)

text: Y

text: The

text: basis

text: e

text: e

text: equips

text: with

text: coordinates

text: w

text: w

text: Consider

text: the

text: map

text: n

text: n

text: T

text: W

text: t

text: n

text: (w

text: w

text: w

text: ψ

text: (w))

text: n

text: i

text: t

text: X

text: Usually

text: this

text: map

text: W

text: is

text: called

text: the

text: Landau-Ginzburg

text: model

text: of

text: the

text: toric

text: orbild

text: However

text: to

text: ensure

text: a

text: correct

text: limit

text: behavior

text: we

text: need

text: to

text: consider

text: a

text: covering

text: of

text: this

text: model.

text: Consider

text: the

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ◦QM R Proposition 4.19. Let log,X be the -module deﬁned above. We have the following iso- 0 A morphism of ﬁnite-dimensional commutative algebras: ◦QMlog,X| (cid:39) H∗ (X, C) . 0 A z=χ=0 orb Proof. Let l ∈ L be a cone relation. The corresponding box operator in the limit z = χ = 0 is equal to (cid:89) (cid:89) m+e − − m+e (4.20) ((cid:50)X)| := D li Dli l z=χ=0 i i i=1 i=1 li<0 li>0 where we have used the fact that for a cone relation l we have p (l) = 0 for a = 1, . . . , r. a ⊂ { } Now suppose that I 1, . . . , m+e (cid:88)is a gener(cid:88)alized primitive collection and consider a primitive relation l : I − m+e a l a = 0 . i j j ∈ i I j=∈1 ≥ aj σI Notice that we have p (l ) 0 for a = 1, . . . , r which can be shown similarly to (4.16). a I ∈ We now claim that there exists an a 1, . . . , r such that p (l ) > 0. Notice that the kernel of a I the map L −→ Z r , (cid:55)→ l (p (l), . . . , p (l)) . 1 r is e-dimensional, since p , . . . , p is part of a basis of Pice(X). Because the p vanish on all cone 1 r a relations for a = 1, . . . , r (cf. (4.13) and the space of cone relations has rank e, the claim follows by dimensional reasons. We therefore get (cid:89) − (4.21) ((cid:50)X)| = .D l z=χ=0 i I ∈ i I Using Lemma 2.6 and the formulas (4.18), (4.20) and (4.21) we get the following surjective map (4.22) H∗ (X, C) (cid:16) ◦QMlog,X| , orb 0 A z=χ=0 by sending D to D . i i ◦QM Notice that log,X is coherent by the theorem above and its generic rank is equal to vol(Q) by Propositio0n 3.8A. Since H∗ (X, C) is also vol(Q)-dimensional and the dimension of the ﬁbers orb of a coherent sheaf is upper-semi-continuous, we conclude that the map above is an isomorphism. (cid:3) O ◦QM Corollary 4.23. The Cz×UX -module 0 lAog,X is locally free of rank µ. 26 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ◦QM

text: R

text: Proposition

text: Let

text: log,X

text: be

text: the

text: -module

text: deﬁned

text: above.

text: We

text: have

text: the

text: following

text: iso-

text: A

text: morphism

text: of

text: ﬁnite-dimensional

text: commutative

text: algebras:

text: ◦QMlog,X|

text: H∗

text: (X,

text: C)

text: A

text: orb

text: Proof.

text: Let

text: l

text: L

text: be

text: a

text: cone

text: relation.

text: The

text: corresponding

text: box

text: operator

text: in

text: the

text: limit

text: z

text: χ

text: is

text: equal

text: to

label: equation

text: m+e

text: m+e

text: D

text: li

text: Dli

text: l

text: i

text: i

text: where

text: we

text: have

text: used

text: the

text: fact

text: that

text: for

text: a

text: cone

text: relation

text: l

text: we

text: have

text: p

text: (l)

text: for

text: a

text: r.

text: a

text: Now

text: suppose

text: that

text: I

text: m+e

text: a

text: primitive

text: collection

text: and

text: consider

text: a

text: primitive

text: relation

text: l

text: I

text: m+e

text: a

text: l

text: a

text: i

text: j

text: j

text: i

text: I

text: aj

text: σI

text: Notice

text: that

text: we

text: have

text: p

text: (l

text: for

text: a

text: r

text: which

text: can

text: be

text: shown

text: similarly

text: to

text: a

text: I

text: We

text: now

text: claim

text: that

text: there

text: exists

text: an

text: a

text: r

text: such

text: that

text: p

text: (l

text: Notice

text: that

text: the

text: kernel

text: of

text: a

text: I

text: the

text: map

text: L

text: Z

text: r

text: l

text: (p

text: (l),

text: p

text: (l))

text: r

text: is

text: e-dimensional,

text: since

text: p

text: p

text: is

text: part

text: of

text: a

text: basis

text: of

text: Pice(X).

text: Because

text: the

text: p

text: vanish

text: on

text: all

text: cone

text: r

text: a

text: relations

text: for

text: a

text: r

text: (cf.

text: and

text: the

text: space

text: of

text: cone

text: relations

text: has

text: rank

text: e,

text: the

text: claim

text: follows

text: by

text: dimensional

text: reasons.

text: We

text: therefore

text: get

text: .D

text: l

text: i

text: I

text: i

text: I

text: Using

text: Lemma

text: and

text: the

text: formulas

text: and

text: we

text: get

text: the

text: following

text: surjective

text: map

text: H∗

text: (X,

text: C)

text: ◦QMlog,X|

text: orb

text: A

text: by

text: sending

text: D

text: to

text: D

text: i

text: i

text: ◦QM

text: Notice

text: that

text: log,X

text: is

text: coherent

text: by

text: the

text: theorem

text: above

text: and

text: its

text: generic

text: rank

text: is

text: equal

text: to

text: vol(Q)

text: by

text: Since

text: H∗

text: (X,

text: C)

text: is

text: also

text: vol(Q)-dimensional

text: and

text: the

text: dimension

text: of

text: the

text: ﬁbers

text: orb

text: of

text: a

text: coherent

text: sheaf

text: is

text: upper-semi-continuous,

text: we

text: conclude

text: that

text: the

text: map

text: above

text: is

text: an

text: isomorphism.

text: O

text: ◦QM

text: Corollary

text: The

text: Cz×UX

text: -module

text: lAog,X

text: is

text: locally

text: free

text: of

text: rank

text: µ.

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: ( ◦QMlog,X)|C∗× on C∗×B resp. (Eˆ )|P \{ } on P1\{0}×{χ = 0}. We therefore can glue Q(cid:92)M a0nd ( ◦QAMlog,zX)B|C × zto a holomorpXhic1zbu0ndle onz P1 × B, which is trivial on on P1 × {χ = 0A} since i0ts resAtrictionzisBisomorphic to E . Since trivialitzy is an open condition there exizsts a subset V ⊂ B such that the restriction of thXe glued bundle to P1 ×V is trivial. This shows the points X z X ∞ 1. to 3. . For the fourth point notice that the ﬂat pairing P gives rise to a pairing on L which ∞ E ∞ in turn gives rise to a pairing on E . The pole order property of this pairing on ˆ at z = X can be encoded by an orthogonality property of the ﬁltration F• with respect to that pairing (see .e.g. [Her03, Theorem 7.17, Deﬁnition 7.18]). Hence the same property must hold for P and F• seen as deﬁned on L∞, so we conclude P : Q(cid:91)Mlog,X ⊗O ι∗ Q(cid:91)Mlog,X → OP × (−n, n) as required. 0 A P1z×VX 0 A 1z VX ∇ ∞ The last statement follows from the fact that the residue connection res,z= deﬁned on Q(cid:91)Mlog,X/z−1 Q(cid:91)Mlog,X has trivial monodromy around D ∩{z = ∞}×B if for any a = 1, . . . , r 0 A 0 A L (cid:48) X ⊂ the nilpotent part N of the monodromy of kills gr•F , i.e. N F• F•− . Using the identiﬁca- ∞ (cid:48) a ∞ • a 1 tion (L , F•) with (E , F ) this has been shown in Lemma 4.31. From this follows that the element 1 is a global sections over P1 × V and ﬂat with respect to z X the residue connection. (cid:3) We begin with a deﬁnition from [Rei09] which formalizes the structure which we obtained in Proposition 4.32. Deﬁnition 4.33. Let M be a complex manifold of dimension bigger or equal than one and ⊂ D M be a simple normal crossing divisor. (1) A log-TEP(n)-structure on M is a holomorphic vector bundle H → P1 × M which is ∇ z { } × equipped with an integrable connection with a pole of type one along 0 M and a logarithmic pole along (C × D) and a ﬂat, (−1)n-symmetric, non(cid:98)-degenerate pairing H⊗ ∗H → O z P : ι (cid:98) (cid:98)zn C × . If D is empty we will simply denote it as a TEP(n)-structure. (2) A log-trTLEP(n)-strzucMture on M is a holomorphic vector bundle H → P1 × M such that p∗p∗H = H (where p : P1 × M (cid:16) M is the projection) which is equipzpe(cid:98)d with(cid:98) an ∇ z { }× integrable connection with a pole of type 1 along 0 M and a logarithmic pole along (P1×D)∪({∞}×M) and a ﬂat, (−1)n-symmetric, non-degenerate pairing P : H⊗ι∗H → O z − P × ( n, n). If D is empty we will simply denote it as a trTLEP(n)-structure. 1 M z X PstarcokpyofsaitnioΣne 4w.i3t4h.SL=et Ge(nΣ()Σb)eaand(cid:100)prloetjeWctiv:eYto×ricMDXeli−g→ne-CM×umMforXd tshteacckorwrietshpoanninSg-eLxatnenddaeud- Ginzburg model. Then the tuple ( QMlog,X, ∇, P) from Proposition 4.32 is a log-trTLEP(n)- ⊂ M 0 A structure on V aXn. X (cid:3) Proof. This follows from Proposition 4.32. The following theorem which is a combination of Proposition 1.20 and Theorem 1.22 in [Rei09] gives suﬃcient conditions when a given log-trTLEP(n)-structure can be unfolded to a logarithmic Frobenius manifold. 32 ##LTLine## ##LTLine##

text: ◦QMlog,X)|C∗×

text: on

text: C∗×B

text: resp.

text: (Eˆ

text: )|P

text: on

text: We

text: therefore

text: can

text: glue

text: ◦QAMlog,zX)B|C

text: zto

text: a

text: onz

text: B,

text: which

text: is

text: trivial

text: on

text: on

text: {χ

text: since

text: resAtrictionzisBisomorphic

text: to

text: E

text: Since

text: trivialitzy

text: is

text: an

text: open

text: condition

text: there

text: exizsts

text: a

text: subset

text: V

text: B

text: such

text: that

text: the

text: restriction

text: of

text: thXe

text: glued

text: bundle

text: to

text: ×V

text: is

text: trivial.

text: This

text: shows

text: the

text: points

text: X

text: z

text: X

text: to

text: For

text: the

text: fourth

text: point

text: notice

text: that

text: the

text: ﬂat

text: pairing

text: P

text: gives

text: rise

text: to

text: a

text: pairing

text: on

text: L

text: which

text: E

text: in

text: turn

text: gives

text: rise

text: to

text: a

text: pairing

text: on

text: E

text: The

text: pole

text: order

text: property

text: of

text: this

text: pairing

text: on

text: ˆ

text: at

text: z

text: X

text: can

text: be

text: encoded

text: by

text: an

text: orthogonality

text: property

text: of

text: the

text: ﬁltration

text: F•

text: with

text: respect

text: to

text: that

text: pairing

text: (see

text: .e.g.

text: Theorem

text: Deﬁnition

text: Hence

text: the

text: same

text: property

text: must

text: hold

text: for

text: P

text: and

text: F•

text: seen

text: as

text: deﬁned

text: on

text: L∞,

text: so

text: we

text: conclude

text: P

text: ⊗O

text: ι∗

text: OP

text: (−n,

text: n)

text: as

text: required.

text: A

text: A

text: VX

text: The

text: last

text: statement

text: follows

text: from

text: the

text: fact

text: that

text: the

text: residue

text: connection

text: res,z=

text: deﬁned

text: on

text: has

text: trivial

text: monodromy

text: around

text: D

text: ∩{z

text: ∞}×B

text: if

text: for

text: any

text: a

text: r

text: A

text: A

text: L

text: X

text: the

text: nilpotent

text: part

text: N

text: of

text: the

text: monodromy

text: of

text: kills

text: gr•F

text: i.e.

text: N

text: F•

text: F•−

text: Using

text: the

text: identiﬁca-

text: a

text: a

text: tion

text: (L

text: F•)

text: with

text: (E

text: F

text: this

text: has

text: been

text: shown

text: in

text: Lemma

text: From

text: this

text: follows

text: that

text: the

text: element

text: is

text: a

text: global

text: sections

text: over

text: V

text: and

text: ﬂat

text: with

text: respect

text: to

text: z

text: X

text: the

text: residue

text: connection.

text: We

text: begin

text: with

text: a

text: deﬁnition

text: from

text: which

text: formalizes

text: the

text: structure

text: which

text: we

text: obtained

text: in

text: Proposition

text: Deﬁnition

text: Let

text: M

text: be

text: a

text: complex

text: manifold

text: of

text: dimension

text: bigger

text: or

text: equal

text: than

text: one

text: and

text: D

text: M

text: be

text: a

text: simple

text: normal

text: crossing

text: divisor.

text: A

text: log-TEP(n)-structure

text: on

text: M

text: is

text: a

text: holomorphic

text: vector

text: bundle

text: H

text: M

text: which

text: is

text: z

text: equipped

text: with

text: an

text: integrable

text: connection

text: with

text: a

text: pole

text: of

text: type

text: one

text: along

text: M

text: and

text: a

text: logarithmic

text: pole

text: along

text: (C

text: D)

text: and

text: a

text: ﬂat,

text: pairing

text: H⊗

text: ∗H

text: O

text: z

text: P

text: ι

text: C

text: If

text: D

text: is

text: empty

text: we

text: will

text: simply

text: denote

text: it

text: as

text: a

text: TEP(n)-structure.

text: A

text: log-trTLEP(n)-strzucMture

text: on

text: M

text: is

text: a

text: holomorphic

text: vector

text: bundle

text: H

text: M

text: such

text: that

text: p∗p∗H

text: H

text: (where

text: p

text: M

text: M

text: is

text: the

text: projection)

text: which

text: is

text: an

text: z

text: integrable

text: connection

text: with

text: a

text: pole

text: of

text: type

text: along

text: M

text: and

text: a

text: logarithmic

text: pole

text: along

text: and

text: a

text: ﬂat,

text: non-degenerate

text: pairing

text: P

text: H⊗ι∗H

text: O

text: z

text: P

text: n,

text: n).

text: If

text: D

text: is

text: empty

text: we

text: will

text: simply

text: denote

text: it

text: as

text: a

text: trTLEP(n)-structure.

text: M

text: z

text: X

text: PstarcokpyofsaitnioΣne

text: Ginzburg

text: model.

text: Then

text: the

text: tuple

text: QMlog,X,

text: P)

text: from

text: Proposition

text: is

text: a

text: log-trTLEP(n)-

text: M

text: A

text: structure

text: on

text: V

text: aXn.

text: X

text: Proof.

text: This

text: follows

text: from

text: Proposition

text: The

text: following

text: theorem

text: which

text: is

text: a

text: combination

text: of

text: Proposition

text: and

text: Theorem

text: in

text: gives

text: suﬃcient

text: conditions

text: when

text: a

text: given

text: log-trTLEP(n)-structure

text: can

text: be

text: unfolded

text: to

text: a

text: logarithmic

text: Frobenius

text: manifold.

text: ##LTLine##

text: ##LTLine##


text: (cid:88) We now recall the fundamental solution of the Givental connection. Deﬁne − − − 1 (cid:104)e δ/zα (cid:48) (cid:48) (cid:105) L(τ, z)α := e δ/zα , τ , . . . , τ , T eδ(d)Tk l! z + ψ k 0,l+2,d d∈Eff≤X\≤{0} l>0,0 k s (cid:48) where τ = δ + τ . The following proposition summarizes the properties of the fundamental solution. Proposition 5.8 ([Iri09, Proposition 2.4]). (1) L(τ, z) satisﬁes the following diﬀerential equations: ∇ ∇ − ρ L(τ, z)α = 0, L(τ, z)α = L(τ, z)(µα α) ∂tk z∂z z ∈ ∗ X ∈ X where α H ( ), ρ := c (TX) H2( ) and µ is the grading operator from Deﬁnition orb − 1 − 5.2. If we put z µzρ := exp( µ log z) exp(ρ log z), then ∇ − ∇ − L(τ, z)z µzρα = 0, L(τ, z)z µzρα = 0 . (2) L(τ, z) is converg∂etknt and invertible on U ×zC∂z∗. − (3) (L(τ, z)α, L(τ, z)β) = (α, β) − orb orb ∈ ∗ X (4) dG(ξ)L(G(ξ) 1τ, z)α = L(τ, z)e2πiξ0e2πifv(ξ)α for α H ( v). In particular − dG(ξ)L(G(ξ) 1τ, z)α = L(τ, z)α ∈ for ξ Pic(X). − (5) Deﬁne L˜(τ, z) := L(τ, z)z µzρ, then ∇ ∇ L˜(τ, z)α = 0, L˜(τ, z)α = 0 . ∂tk z∂z Since L and therefore also L˜ is invertible, then the sections s := L˜(T ) are a basis of ﬂat sections. X (cid:88) −(cid:88) − i i The J-function of is given by L(τ, z) 11 = L(τ, z) 1T . We set 0 s s − J˜ := J˜T := (s , T ) T = L˜(τ, z) 1(T ) i i i 0 orb i 0 i=0 i=0 and get the formula (cid:88) (5.9) 1 = T = s J˜s on C∗ × U . 0 i i z i=0 6. Mirror correspondance X Let be projective, toric orbifold. In order to state the mirror theorem for toric orbifolds we have to introduce the I-function. (cid:81) (cid:88) X Deﬁnition 6.1. The I(cid:80)-function of a toric orbif(cid:81)old is deﬁned by ∞ (cid:104) (cid:105) − (cid:100)(cid:104) (cid:101)(D + ( D , d ν)z) I(χ, z) = e ra+=e1 epa logχa/z χd ν=∞Di,d i (cid:104) i(cid:105) − 1v(d) . d∈K ν=0(Di + ( Di, d ν)z) We collect a few facts about the I-function. 37 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: We

text: now

text: recall

text: the

text: fundamental

text: solution

text: of

text: the

text: Givental

text: connection.

text: Deﬁne

text: δ/zα

text: L(τ,

text: z)α

text: e

text: δ/zα

text: τ

text: τ

text: T

text: eδ(d)Tk

text: l!

text: z

text: ψ

text: k

text: k

text: s

text: where

text: τ

text: δ

text: τ

text: The

text: following

text: proposition

text: summarizes

text: the

text: properties

text: of

text: the

text: fundamental

text: solution.

text: Proposition

text: Proposition

text: L(τ,

text: z)

text: satisﬁes

text: the

text: following

text: diﬀerential

text: equations:

text: ρ

text: L(τ,

text: z)α

text: L(τ,

text: z)α

text: L(τ,

text: z)(µα

text: α)

text: ∂tk

text: z∂z

text: z

text: X

text: X

text: where

text: α

text: H

text: ρ

text: c

text: (TX)

text: and

text: µ

text: is

text: the

text: grading

text: operator

text: from

text: Deﬁnition

text: orb

text: If

text: we

text: put

text: z

text: µzρ

text: exp(

text: µ

text: log

text: z)

text: exp(ρ

text: log

text: z),

text: then

text: L(τ,

text: z)z

text: µzρα

text: L(τ,

text: z)z

text: µzρα

text: L(τ,

text: z)

text: is

text: converg∂etknt

text: and

text: invertible

text: on

text: U

text: ×zC∂z∗.

text: (L(τ,

text: z)α,

text: L(τ,

text: z)β)

text: (α,

text: β)

text: orb

text: orb

text: X

text: dG(ξ)L(G(ξ)

text: z)α

text: L(τ,

text: for

text: α

text: H

text: v).

text: In

text: particular

text: dG(ξ)L(G(ξ)

text: z)α

text: L(τ,

text: z)α

text: for

text: ξ

text: Pic(X).

text: Deﬁne

text: L˜(τ,

text: z)

text: L(τ,

text: z)z

text: µzρ,

text: then

text: L˜(τ,

text: z)α

text: L˜(τ,

text: z)α

text: ∂tk

text: z∂z

text: Since

text: L

text: and

text: therefore

text: also

text: L˜

text: is

text: invertible,

text: then

text: the

text: sections

text: s

text: L˜(T

text: are

text: a

text: basis

text: of

text: ﬂat

text: sections.

text: X

text: i

text: i

text: The

text: J-function

text: of

text: is

text: given

text: by

text: L(τ,

text: z)

text: L(τ,

text: z)

text: We

text: set

text: s

text: s

text: J˜

text: J˜T

text: (s

text: T

text: T

text: L˜(τ,

text: z)

text: i

text: i

text: i

text: orb

text: i

text: and

text: get

text: the

text: formula

text: T

text: s

text: J˜s

text: on

text: C∗

text: U

text: i

text: i

text: z

text: Mirror

text: correspondance

text: X

text: Let

text: be

text: projective,

text: toric

text: orbifold.

text: In

text: order

text: to

text: state

text: the

text: mirror

text: theorem

text: for

text: toric

text: orbifolds

text: we

text: have

text: to

text: introduce

text: the

text: I-function.

text: X

text: Deﬁnition

text: The

text: of

text: a

text: toric

text: is

text: deﬁned

text: by

text: D

text: d

text: ν)z)

text: I(χ,

text: z)

text: e

text: epa

text: logχa/z

text: χd

text: ν=∞Di,d

text: i

text: d∈K

text: Di,

text: d

text: ν)z)

text: We

text: collect

text: a

text: few

text: facts

text: about

text: the

text: I-function.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: variation pure and polarized TERP-structures. Our main result is that if X admits a crepant ∗ resolution Z than the pure and polarized TERP-structures glue which gives global tt -geometry. M Deﬁnition 7.1 ([Her03, Deﬁnition 2.12], [HS10, Deﬁnition 2.1]). Let be a complex manifold and n ∈ Z. A variation of TERP-structures on M of weight n consists of the following set of data (1) A holomorphic vector bundle H on C × M (2) A R-local system L on C∗ × M, togetzher with an isomorphism z L ⊗ O −→ H R Can∗×M |aCn∗×M z z ∇ H such that the induced connection extends to a meromorphic connection on such that ∇ { } × M has a pole of Poincaré rank 1 along 0 . (3) A polarization P : L ⊗ ι∗L −→ inRC∗×M, which is (−1)n symmetric and which induces a non-degenerate pairing z H ⊗ ∗H −→ O P : C ×M ι zn C ×M z z − H H ⊗ where non-degenerate means that the induced symmetric pairing [z wP] : /z H H → O /z M is non-degenerate. We now state the deﬁnition of a pure and polarized TERP-structure. H L M M Deﬁnition 7.2. Let ( , , P, n) be a variation of TERP-structures on . Let be the complex manifold with the conjugate complex structure and γ : P1 × M → P1 × M be the involution (z, x) (cid:55)→ (z−1, x). Consider γ∗H which is a holomorphic vector bundle on (P1 \ {0}) × M . Let O C C P Man be the subsheaf of Pan×M consisting of functions which are annihilated by ∂ . Deﬁne a loca1lly free OP CMan-module Hˆ1by glueing H and γ∗H via the following identiﬁcation ozn C∗ × M: Let x ∈ M an1d z ∈ C∗ and deﬁne z z H −→ ∗H c : | (γ )| , (z,x) (z,x) (cid:55)→ ∇ − · a -parallel transport of z n a . H ∗H Then c is an anti-linear involution and identiﬁes |C∗×M with γ |C∗×M. The involution c restricts to complex conjugation (with respect to L) in tzhe ﬁbres over S1z× M. (1) (H, L, P, n) is called pure iﬀ Hˆ = p∗p∗Hˆ, where p : P1 × M → M. H L (2) Let ( , , P, n) be pure, then H ⊗ H −→ C h : p∗ ˆ C p∗ ˆ Man Man (cid:55)→ − (s, t) z nP(s, c(t)) H H L is a hermetian form on p∗ ˆ. We call ( , , P, n) a pure and polarized TERP- structure if this form is positive deﬁnite. Theorem 7.3. The restriction of the quantum D-module G of a toric orbifold to C × W \ D z X X underlies a variation of pure and polarized TERP-structures of weight n. 40 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: variation

text: pure

text: and

text: polarized

text: TERP-structures.

text: Our

text: main

text: result

text: is

text: that

text: if

text: X

text: admits

text: a

text: crepant

text: resolution

text: Z

text: than

text: the

text: pure

text: and

text: polarized

text: TERP-structures

text: glue

text: which

text: gives

text: global

text: tt

text: -geometry.

text: M

text: Deﬁnition

text: Deﬁnition

text: Deﬁnition

text: Let

text: be

text: a

text: complex

text: manifold

text: and

text: n

text: Z.

text: A

text: variation

text: of

text: TERP-structures

text: on

text: M

text: of

text: weight

text: n

text: consists

text: of

text: the

text: following

text: set

text: of

text: data

text: A

text: holomorphic

text: vector

text: bundle

text: H

text: on

text: C

text: M

text: A

text: R-local

text: system

text: L

text: on

text: C∗

text: M,

text: togetzher

text: with

text: an

text: isomorphism

text: z

text: L

text: O

text: H

text: R

text: Can∗×M

text: |aCn∗×M

text: z

text: z

text: H

text: such

text: that

text: the

text: induced

text: connection

text: extends

text: to

text: a

text: meromorphic

text: connection

text: on

text: such

text: that

text: M

text: has

text: a

text: pole

text: of

text: Poincaré

text: rank

text: along

text: A

text: polarization

text: P

text: L

text: ι∗L

text: inRC∗×M,

text: which

text: is

text: symmetric

text: and

text: which

text: induces

text: a

text: non-degenerate

text: pairing

text: z

text: H

text: ∗H

text: O

text: P

text: C

text: ×M

text: ι

text: zn

text: C

text: ×M

text: z

text: z

text: H

text: H

text: where

text: non-degenerate

text: means

text: that

text: the

text: induced

text: symmetric

text: pairing

text: [z

text: wP]

text: /z

text: H

text: H

text: O

text: /z

text: M

text: is

text: non-degenerate.

text: We

text: now

text: state

text: the

text: deﬁnition

text: of

text: a

text: pure

text: and

text: polarized

text: TERP-structure.

text: H

text: L

text: M

text: M

text: Deﬁnition

text: Let

text: P,

text: n)

text: be

text: a

text: variation

text: of

text: TERP-structures

text: on

text: Let

text: be

text: the

text: complex

text: manifold

text: with

text: the

text: conjugate

text: complex

text: structure

text: and

text: γ

text: M

text: M

text: be

text: the

text: involution

text: (z,

text: x)

text: x).

text: Consider

text: γ∗H

text: which

text: is

text: a

text: holomorphic

text: vector

text: bundle

text: on

text: M

text: Let

text: O

text: C

text: C

text: P

text: Man

text: be

text: the

text: subsheaf

text: of

text: Pan×M

text: consisting

text: of

text: functions

text: which

text: are

text: annihilated

text: by

text: Deﬁne

text: a

text: free

text: OP

text: CMan-module

text: glueing

text: H

text: and

text: γ∗H

text: via

text: the

text: following

text: identiﬁcation

text: ozn

text: C∗

text: M:

text: Let

text: x

text: M

text: z

text: C∗

text: and

text: deﬁne

text: z

text: z

text: H

text: ∗H

text: c

text: (γ

text: (z,x)

text: (z,x)

text: a

text: -parallel

text: transport

text: of

text: z

text: n

text: a

text: H

text: ∗H

text: Then

text: c

text: is

text: an

text: anti-linear

text: involution

text: and

text: identiﬁes

text: |C∗×M

text: with

text: γ

text: |C∗×M.

text: The

text: involution

text: c

text: restricts

text: to

text: complex

text: conjugation

text: (with

text: respect

text: to

text: L)

text: in

text: tzhe

text: ﬁbres

text: over

text: M.

text: (H,

text: L,

text: P,

text: n)

text: is

text: called

text: pure

text: iﬀ

text: Hˆ

text: p∗p∗Hˆ,

text: where

text: p

text: M

text: M.

text: H

text: L

text: Let

text: P,

text: n)

text: be

text: pure,

text: then

text: H

text: H

text: C

text: h

text: p∗

text: ˆ

text: C

text: p∗

text: ˆ

text: Man

text: Man

text: (s,

text: t)

text: z

text: nP(s,

text: c(t))

text: H

text: H

text: L

text: is

text: a

text: hermetian

text: form

text: on

text: p∗

text: ˆ.

text: We

text: call

text: P,

text: n)

text: a

text: pure

text: and

text: polarized

text: TERP-

text: structure

text: if

text: this

text: form

text: is

text: positive

text: deﬁnite.

text: Theorem

text: The

text: restriction

text: of

text: the

text: quantum

text: D-module

text: G

text: of

text: a

text: toric

text: orbifold

text: to

text: C

text: W

text: D

text: z

text: X

text: X

text: underlies

text: a

text: variation

text: of

text: pure

text: and

text: polarized

text: TERP-structures

text: of

text: weight

text: n.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 Ahmad Sabihi (cid:81) (cid:81) (cid:81) − − − 4e γ (1 −1 ) | p−1(1 + O( 1 )) p>2 (p 1)2 p>2 ,p N p 2 log(N) log(N) = √ − (11-3) (1 −1 ) p> N,gcd(p,N)=1 p 1 Having the conjecture the(cid:88)relation (10-3) gives: − × (cid:89) η(cid:89)(n)Λ(n) = 4e γ ≤ n x − − 1 p 1 1 x (1 − ) − (1 + O( )) (12-3) (p 1)2 | p 2 log(N) log(N) p>2 p>2 ,p N (cid:80) → ∞ Assume we hold N to a constant value and x then could be written: | | log ≤ η(n)Λ(n) σ = lim n x = 1 (13-3) c x→∞ log(x) Therefore, σ = σ = 1 and the Lemma is proven. c a Lemma 3.4 Let Dirichlet’s series in Lemma 3.3 have a pole at s = 1,then it has a residue of the following form at same pole: (cid:89) (cid:89) − − − 1 p 1 1 1 4e γ (1 − ) − (1 + O( )) (14-3) (p 1)2 | p 2 log(N) log(N) p>2 p>2 ,p N Proof: (cid:88) It is well-known ∞ − Λ(n) lim(1 s) = 1 (15-3) s→1 ns n=1 (cid:88) (cid:88) On the other hand ∞ Λ(n) 1 Λ(n) lim = (16-3) →∞ x ≤ ≡ ns φ(k) ns n x, n Nmod(k) n=1 Consequently (cid:88) (cid:88) (cid:89) (cid:88) ∞ ∞ η(n)Λ(n) − 1 Λ(n) lim = (1 − ) x→∞ ≤ ≡ ns √ p 1 ns n x, n Nmod(k) n=1 p> N, gcd(p,N)=1 n=1 (17-3) and this is equal to (cid:89) (cid:89) (cid:88) − ∞ − − 1 p 1 1 1 Λ(n) 4e γ (1 − ) − (1 + O( )) (18-3) (p 1)2 | p 2 log(N) log(N) ns p>2 p>2 ,p N n=1 Just, one could obtain the residue of the Dirichlet’s series as below: ##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: Ahmad

text: Sabihi

text: γ

text: −1

text: O(

text: (p

text: ,p

text: N

text: p

text: log(N)

text: log(N)

text: −1

text: p>

text: p

text: Having

text: the

text: conjecture

text: gives:

text: γ

text: n

text: x

text: p

text: x

text: O(

text: (p

text: p

text: log(N)

text: log(N)

text: ,p

text: N

text: Assume

text: we

text: hold

text: N

text: to

text: a

text: constant

text: value

text: and

text: x

text: then

text: could

text: be

text: written:

text: log

text: η(n)Λ(n)

text: σ

text: lim

text: n

text: x

text: c

text: x→∞

text: log(x)

text: Therefore,

text: σ

text: σ

text: and

text: the

text: Lemma

text: is

text: proven.

text: c

text: a

text: Lemma

text: Let

text: Dirichlet’s

text: series

text: in

text: Lemma

text: have

text: a

text: pole

text: at

text: s

text: it

text: has

text: a

text: residue

text: of

text: the

text: following

text: form

text: at

text: same

text: pole:

text: p

text: γ

text: O(

text: (p

text: p

text: log(N)

text: log(N)

text: ,p

text: N

text: Proof:

text: It

text: is

text: well-known

text: Λ(n)

text: s)

text: ns

text: On

text: the

text: other

text: hand

text: Λ(n)

text: Λ(n)

text: lim

text: x

text: ns

text: φ(k)

text: ns

text: n

text: x,

text: n

text: Nmod(k)

text: Consequently

text: η(n)Λ(n)

text: Λ(n)

text: lim

text: x→∞

text: ns

text: p

text: ns

text: n

text: x,

text: n

text: Nmod(k)

text: p>

text: N,

text: and

text: this

text: is

text: equal

text: to

text: p

text: Λ(n)

text: γ

text: O(

text: (p

text: p

text: log(N)

text: log(N)

text: ns

text: ,p

text: N

text: Just,

text: one

text: could

text: obtain

text: the

text: residue

text: of

text: the

text: Dirichlet’s

text: series

text: as

text: below:

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4 InAs GaAs AlAs GaAs AlAs InAs Al Ga As In Ga As 0.3 0.7 0.2 0.8 αa 0.5603 0.0000 0.9720 E 1.519 3.130 0.417 2.000 1.207 s g αa 1.9539 1.6257 1.8880 m 0.0665 0.158 0.0235 0.0904 0.0519 p − e αa 1.7005 2.4531 2.0600 γ 24.21 9.12 45.39 16.04 28.61 d c − − − − − αc 0.5603 0.0000 0.9720 a 7.17 5.64 5.08 6.74 6.62 s c αc 1.9539 1.6257 1.8880 a 1.60 2.47 1.00 1.63 1.22 p v − − − − − αc 1.7005 2.4531 2.0600 b 2.00 2.30 1.80 0.74 1.35 d − − − − − n 5.4002 4.5619 2.0880 d 4.80 3.40 3.60 4.39 4.53 ssσ − − n 4.4014 3.0363 5.7560 C 8.12 3.34 104.5 2.65 15.10 spσ 3 n 6.8053 3.1594 4.4720 sdσ nss∗σ 5.8401 3.2676 2.8600 TABLE III. The band gaps Eg (in eV), the eﬀective elec- tron masses at the Γ point m (in the units of free electron n ∗ 6.8116 6.9229 3.2240 e · s pσ mass), the bulk Dresselhaus constants γ (in eV ˚A3) [40], the c n 6.9787 6.2602 5.1560 ppσ deformation-potential constants a , a , b and d (in eV), and · c v nppπ 6.0189 7.0824 2.7960 C3 (in eV ˚A) computed using the tight-binding parameters n 2.7559 3.5344 5.5920 from Table I and Table II. pdσ n 6.0212 7.3976 4.8080 pdπ π 0.0952 0.1476 0.1000 001 tight-binding parameters to to reproduce the recom- π 0.1456 0.1588 0.1160 111 mended values of the deformation-potential constants in the Brillouin zone center given in Ref. 37. The obtain TABLE II. Deformation tight-binding parameters used in cal- parameters are presented in Table II. Table III summa- culations. Other parameters are α ∗ = 2.0 and n ∗ ∗ = s s s σ n ∗ = n = n = n = 2.0 rize the band gaps Eg, the eﬀective electron masses at s dσ ddσ ddπ ddδ the Γ point m , the Dresselhaus constants γ , and the e c deformation-potential constants a , a , b, d, and C for c v 3 parameters λ for anions and cations are connected to some bulk binary and ternary compounds obtained from j each other by the tight-binding calculations with the parameters listed in Table I and Table II. 1 − 1 − For alloys, the tight-binding needs a special care to be λa1 = 2(Epa Eraef)π100 , λc1 = 2(Epc Ercef)π100 , (10) taken to reproduce the band gap bowing properly [38]. −8 − −8 − Here, we use an original interpolation scheme [39] to con- λa2 = 3(Epa Eraef)π111 , λc2 = 3(Epc Ercef)π111 . struct the alloy AxB1−xC tight-binding parameters in the virtual crystal approximation from the tight-binding pa- rameters of the binary compounds AC and BC. First, The deformation parameters π and π for sev- 100 111 the lattice constant of the alloy is found as the linear in- eral binary compounds are listed in Table II. We note terpolation between the binaries. Then, we calculate the that similar splitting occurs for the d orbitals as well. strain contributions as described above and construct the However, to ﬁt the conduction-band and valence-band parameters of the AC and BC materials strained to the deformation-potential constants at the Γ point of the lattice constant of the alloy. Finally, the tight-binding Brillouin zone it is suﬃcient to consider the splitting of parameters of the alloy are determined as the linear in- the p orbitals. Therefore, to simplify calculations we ne- terpolations of the parameters of the strained binary ma- glect the splitting of the d orbitals. We keep in mind terials. We note that, unlike the approach from Ref. 38, that for ﬁtting simultaneously the deformation-potential this procedure provides the correct bowing without the constants at the Γ, X and L points a complete set of introduction of additional parameters. parameters should be used. ∗ We also note that the standard sp3d5s tight-binding The determination of the strain-related parameters model [28] does not take into account the spin-orbit cou- of the tight-binding Hamiltonian is a challenging task pling of the p and d orbitals which yields the major contri- because of the small number of available well docu- (cid:48) (cid:48) bution to the C constant [23]. The missing of C has the mented deformation-potential constants. In fact, the 3 3 same origin as the missing of the k-linear spin splitting deformation-potential constants at the Brillouin zone of the Γ valence band in the bulk crystal [23, 41]. Pos- center do not provide enough information to uniquely 8 sible solution of this problem proposed by Boykin [41] is determine all the tight-binding parameters. We expect based on the consideration of second-nearest neighboring that any parametrization providing the correct values atoms and has no straightforward extension to strained of the conduction-band deformation-potential constant heterostructures. a and the valence-band deformation-potential constants c IV. RESULTS a , b and d yields satisfactory results the strain-induced v spin-orbit coupling. Therefore, we adopt the approach described in Ref. 36 and numerically ﬁt the strain-related We use the procedure described in Ref. 25 to extract ##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: InAs

text: GaAs

text: AlAs

text: GaAs

label: table

text: AlAs

text: InAs

text: Al

text: Ga

text: As

text: In

text: Ga

text: As

text: αa

text: E

text: s

text: g

text: αa

text: m

text: p

text: e

text: αa

text: γ

text: d

text: c

text: αc

text: a

text: s

text: c

text: αc

text: a

text: p

text: v

text: αc

text: b

text: d

text: n

text: d

text: ssσ

text: n

text: C

text: spσ

text: n

text: sdσ

text: nss∗σ

text: TABLE

text: III.

text: The

text: band

text: gaps

text: Eg

text: (in

text: eV),

text: the

text: eﬀective

text: elec-

text: tron

text: masses

text: at

text: the

text: Γ

text: point

text: m

text: (in

text: the

text: units

text: of

text: free

text: electron

text: n

text: e

text: s

text: pσ

text: mass),

text: the

text: bulk

text: Dresselhaus

text: constants

text: γ

text: (in

text: eV

text: the

text: c

text: n

text: ppσ

text: deformation-potential

text: constants

text: a

text: a

text: b

text: and

text: d

text: (in

text: eV),

text: and

text: c

text: v

text: nppπ

text: (in

text: eV

text: ˚A)

text: computed

text: using

text: the

text: tight-binding

text: parameters

text: n

text: from

text: Table

text: I

text: and

text: Table

text: II.

text: pdσ

text: n

text: pdπ

text: π

text: tight-binding

text: parameters

text: to

text: to

text: reproduce

text: the

text: recom-

text: π

text: mended

text: values

text: of

text: the

text: deformation-potential

text: constants

text: in

text: the

text: Brillouin

text: zone

text: center

text: given

text: in

text: Ref.

text: The

text: obtain

text: TABLE

text: II.

text: Deformation

text: tight-binding

text: parameters

text: used

text: in

text: cal-

text: parameters

text: are

text: presented

text: in

text: Table

text: II.

text: Table

text: III

text: summa-

text: culations.

text: Other

text: parameters

text: are

text: α

text: and

text: n

text: s

text: s

text: s

text: σ

text: n

text: n

text: n

text: n

text: rize

text: the

text: band

text: gaps

text: Eg,

text: the

text: eﬀective

text: electron

text: masses

text: at

text: s

text: dσ

text: ddσ

text: ddπ

text: ddδ

text: the

text: Γ

text: point

text: m

text: the

text: Dresselhaus

text: constants

text: γ

text: and

text: the

text: e

text: c

text: deformation-potential

text: constants

text: a

text: a

text: b,

text: d,

text: and

text: C

text: for

text: c

text: v

text: parameters

text: λ

text: for

text: anions

text: and

text: cations

text: are

text: connected

text: to

text: some

text: bulk

text: binary

text: and

text: ternary

text: compounds

text: obtained

text: from

text: j

text: each

text: other

text: by

text: the

text: tight-binding

text: calculations

text: with

text: the

text: parameters

text: listed

text: in

text: Table

text: I

text: and

text: Table

text: II.

text: For

text: alloys,

text: the

text: tight-binding

text: needs

text: a

text: special

text: care

text: to

text: be

text: taken

text: to

text: reproduce

text: the

text: band

text: gap

text: bowing

text: properly

text: −8

text: −8

text: Here,

text: we

text: use

text: an

text: original

text: interpolation

text: scheme

text: to

text: con-

text: struct

text: the

text: alloy

text: tight-binding

text: parameters

text: in

text: the

text: virtual

text: crystal

text: approximation

text: from

text: the

text: tight-binding

text: pa-

text: rameters

text: of

text: the

text: binary

text: compounds

text: AC

text: and

text: BC.

text: First,

text: The

text: deformation

text: parameters

text: π

text: and

text: π

text: for

text: sev-

text: the

text: lattice

text: constant

text: of

text: the

text: alloy

text: is

text: found

text: as

text: the

text: linear

text: in-

text: eral

text: binary

text: compounds

text: are

text: listed

text: in

text: Table

text: II.

text: We

text: note

text: terpolation

text: between

text: the

text: binaries.

text: Then,

text: we

text: calculate

text: the

text: that

text: similar

text: splitting

text: occurs

text: for

text: the

text: d

text: orbitals

text: as

text: well.

text: strain

text: contributions

text: as

text: described

text: above

text: and

text: construct

text: the

text: However,

text: to

text: ﬁt

text: the

text: conduction-band

text: and

text: valence-band

text: parameters

text: of

text: the

text: AC

text: and

text: BC

text: materials

text: strained

text: to

text: the

text: deformation-potential

text: constants

text: at

text: the

text: Γ

text: point

text: of

text: the

text: lattice

text: constant

text: of

text: the

text: alloy.

text: Finally,

text: the

text: tight-binding

text: Brillouin

text: zone

text: it

text: is

text: suﬃcient

text: to

text: consider

text: the

text: splitting

text: of

text: parameters

text: of

text: the

text: alloy

text: are

text: determined

text: as

text: the

text: linear

text: in-

text: the

text: p

text: orbitals.

text: Therefore,

text: to

text: simplify

text: calculations

text: we

text: ne-

text: terpolations

text: of

text: the

text: parameters

text: of

text: the

text: strained

text: binary

text: ma-

text: glect

text: the

text: splitting

text: of

text: the

text: d

text: orbitals.

text: We

text: keep

text: in

text: mind

text: terials.

text: We

text: note

text: that,

text: unlike

text: the

text: approach

text: from

text: Ref.

text: that

text: for

text: ﬁtting

text: simultaneously

text: the

text: deformation-potential

text: this

text: procedure

text: provides

text: the

text: correct

text: bowing

text: without

text: the

text: constants

text: at

text: the

text: Γ,

text: X

text: and

text: L

text: points

text: a

text: complete

text: set

text: of

text: introduction

text: of

text: additional

text: parameters.

text: parameters

text: should

text: be

text: used.

text: We

text: also

text: note

text: that

text: the

text: standard

text: tight-binding

text: The

text: determination

text: of

text: the

text: strain-related

text: parameters

text: model

text: does

text: not

text: take

text: into

text: account

text: the

text: spin-orbit

text: cou-

text: of

text: the

text: tight-binding

text: Hamiltonian

text: is

text: a

text: challenging

text: task

text: pling

text: of

text: the

text: p

text: and

text: d

text: orbitals

text: which

text: yields

text: the

text: major

text: contri-

text: because

text: of

text: the

text: small

text: number

text: of

text: available

text: well

text: docu-

text: bution

text: to

text: the

text: C

text: constant

text: The

text: missing

text: of

text: C

text: has

text: the

text: mented

text: deformation-potential

text: constants.

text: In

text: fact,

text: the

text: same

text: origin

text: as

text: the

text: missing

text: of

text: the

text: k-linear

text: spin

text: splitting

text: deformation-potential

text: constants

text: at

text: the

text: Brillouin

text: zone

text: of

text: the

text: Γ

text: valence

text: band

text: in

text: the

text: bulk

text: crystal

text: Pos-

text: center

text: do

text: not

text: provide

text: enough

text: information

text: to

text: uniquely

text: sible

text: solution

text: of

text: this

text: problem

text: proposed

text: by

text: Boykin

text: is

text: determine

text: all

text: the

text: tight-binding

text: parameters.

text: We

text: expect

text: based

text: on

text: the

text: consideration

text: of

text: second-nearest

text: neighboring

text: that

text: any

text: parametrization

text: providing

text: the

text: correct

text: values

text: atoms

text: and

text: has

text: no

text: straightforward

text: extension

text: to

text: strained

text: of

text: the

text: conduction-band

text: deformation-potential

text: constant

text: heterostructures.

text: a

text: and

text: the

text: valence-band

text: deformation-potential

text: constants

text: c

text: IV.

text: RESULTS

text: a

text: b

text: and

text: d

text: yields

text: satisfactory

text: results

text: the

text: strain-induced

text: v

text: spin-orbit

text: coupling.

text: Therefore,

text: we

text: adopt

text: the

text: approach

text: described

text: in

text: Ref.

text: and

text: numerically

text: ﬁt

text: the

text: strain-related

text: We

text: use

text: the

text: procedure

text: described

text: in

text: Ref.

text: to

text: extract

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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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##

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text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: 5 FIG. 1. Dresselhaus constant β as a function of FIG. 2. Dresselhaus constant β as a function of the QW thick- the QW thickness (in monolayers, ML) calculated for ness calculated for GaAs/In0.2Ga0.8As/GaAs QWs grown on Al0.3Ga0.7As/GaAs/Al0.3Ga0.7As QWs with diﬀerent strain GaAs buﬀer layer (solid lines) and In0.2Ga0.8As buﬀer layer conﬁgurations. Solid line shows the results for QW structures (dashed lines). with the lattice constant corresponding to GaAs, a = 5.65 ˚A. 0 Dashed line shows the results for QWs lattice-matched to AlAs with the lattice constant a0 = 5.66. redistribution of electron presence probability between the well and the barriers. B. InGaAs/GaAs quantum wells the constants of the spin-orbit Hamiltonian (1) from the tight-binding calculations for GaAs-based and InGaAs- While the strain contribution to the spin splitting in based QW structures. As distinct from previous calcu- GaAs/AlGaAs QWs is still a correction, it is natural to lations, we now include the atomistic strain as described expect that in heterostructures grown from compounds in Section . The results show that the constants α± re- with a signiﬁcant lattice mismatch, like InGaAs/GaAs lated to structure inversion asymmetry are almost inde- QWs, the strain contribution dominates the spin split- pendent of the strain present in the QW. Therefore, we ting. focus below on the Dresselhaus parameter β and consider In addition to larger lattice mismatch, the deformation symmetric QWs. constant C in InAs is about an order of magnitude larger A. GaAs/AlGaAs quantum wells 3 than that in GaAs, see Table III. The large value of C · 3 can be explained from the k p perturbation theory for The Dresselhaus constant β as a function of the QW bulk crystals where the major contribution to C is given thickness determined for AlGaAs/GaAs/AlGaAs QWs 3 by [23] with diﬀerent strain conﬁgurations is shown in Fig. 1. Here, we present the results for the same structures but lattice-matched either to GaAs (solid line) or to AlAs 4 C P∆ (dashed line) 1. C3 = 3 E (E2 + ∆) , (11) From Fig. 1 one may conclude that the widely adopted g g consideration of GaAs/AlGaAs heterostructures as un- strained systems is not completely satisﬁed for the anal- where C2 is the interband deformation-potential con- ysis of spin splitting. The ratio of the Dresselhaus con- stant, P is the Kane matrix element, and ∆ is the spin- stants β for the structures lattice-matched to GaAs and orbit splitting of the valence band. The growth of C3 AlAs would exceed a factor 2 for a 50-˚A-wide QW while for InAs as compared to that for GaAs is caused by the · − the strain itself is only 1 10 3. Note that the observed de- decrease in Eg and the increase in C2. pendence of β on the QW thickness actually reﬂects the To elaborate this expectation we calculate the Dressel- haus constant β for GaAs/In Ga As/GaAs QW struc- 0.2 0.8 tures lattice-matched to GaAs and In Ga As. The de- 0.2 0.8 pendence of the Dresselhaus constants on the QW thick- 1 The coordinate system is changed with respect to the one used ness for both strain conﬁgurations are shown in Fig. 2. in Ref. [25] which results in the opposite sign of the Dresselhaus The range of spin splittings is signiﬁcantly larger than constant. that in GaAs/AlGaAs structures and the Dresselhaus ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: FIG.

text: Dresselhaus

label: caption

text: constant

text: β

text: as

text: a

text: function

text: of

text: FIG.

text: Dresselhaus

text: constant

text: β

text: as

text: a

text: function

text: of

text: the

text: QW

text: thick-

text: the

text: QW

text: thickness

text: (in

text: monolayers,

text: ML)

text: calculated

text: for

text: ness

text: calculated

text: for

text: QWs

text: grown

text: on

text: QWs

text: with

text: diﬀerent

text: strain

text: GaAs

text: buﬀer

text: layer

text: (solid

text: lines)

text: and

text: buﬀer

text: layer

text: conﬁgurations.

text: Solid

text: line

text: shows

text: the

text: results

text: for

text: QW

text: structures

text: (dashed

text: lines).

text: with

text: the

text: lattice

text: constant

text: corresponding

text: to

text: GaAs,

text: a

text: ˚A.

text: Dashed

text: line

text: shows

text: the

text: results

text: for

text: QWs

text: lattice-matched

text: to

text: AlAs

text: with

text: the

text: lattice

text: constant

text: redistribution

text: of

text: electron

text: presence

text: probability

text: between

text: the

text: well

text: and

text: the

text: barriers.

text: B.

text: InGaAs/GaAs

text: quantum

text: wells

text: the

text: constants

text: of

text: the

text: spin-orbit

text: Hamiltonian

text: from

text: the

text: tight-binding

text: calculations

text: for

text: GaAs-based

text: and

text: InGaAs-

text: While

text: the

text: strain

text: contribution

text: to

text: the

text: spin

text: splitting

text: in

text: based

text: QW

text: structures.

text: As

text: distinct

text: from

text: previous

text: calcu-

text: GaAs/AlGaAs

text: QWs

text: is

text: still

text: a

text: correction,

text: it

text: is

text: natural

text: to

text: lations,

text: we

text: now

text: include

text: the

text: atomistic

text: strain

text: as

text: described

text: expect

text: that

text: in

text: heterostructures

text: grown

text: from

text: compounds

text: in

text: Section

text: The

text: results

text: show

text: that

text: the

text: constants

text: α±

text: re-

text: with

text: a

text: signiﬁcant

text: lattice

text: mismatch,

text: like

text: InGaAs/GaAs

text: lated

text: to

text: structure

text: inversion

text: asymmetry

text: are

text: almost

text: inde-

text: QWs,

text: the

text: strain

text: contribution

text: dominates

text: the

text: spin

text: split-

text: pendent

text: of

text: the

text: strain

text: present

text: in

text: the

text: QW.

text: Therefore,

text: we

text: ting.

text: focus

text: below

text: on

text: the

text: Dresselhaus

text: parameter

text: β

text: and

text: consider

text: In

text: addition

text: to

text: larger

text: lattice

text: mismatch,

text: the

text: deformation

text: symmetric

text: QWs.

text: constant

text: C

text: in

text: InAs

text: is

text: about

text: an

text: order

text: of

text: magnitude

text: larger

text: A.

text: GaAs/AlGaAs

text: quantum

text: wells

text: than

text: that

text: in

text: GaAs,

text: see

text: Table

text: III.

text: The

text: large

text: value

text: of

text: C

text: can

text: be

text: explained

text: from

text: the

text: k

text: p

text: perturbation

text: theory

text: for

text: The

text: Dresselhaus

text: constant

text: β

text: as

text: a

text: function

text: of

text: the

text: QW

text: bulk

text: crystals

text: where

text: the

text: major

text: contribution

text: to

text: C

text: is

text: given

text: thickness

text: determined

text: for

text: AlGaAs/GaAs/AlGaAs

text: QWs

text: by

text: with

text: diﬀerent

text: strain

text: conﬁgurations

text: is

text: shown

text: in

text: Fig.

text: Here,

text: we

text: present

text: the

text: results

text: for

text: the

text: same

text: structures

text: but

text: lattice-matched

text: either

text: to

text: GaAs

text: (solid

text: line)

text: or

text: to

text: AlAs

text: C

text: P∆

text: (dashed

text: line)

text: E

text: From

text: Fig.

text: one

text: may

text: conclude

text: that

text: the

text: widely

text: adopted

text: g

text: g

text: consideration

text: of

text: GaAs/AlGaAs

text: heterostructures

text: as

text: un-

text: strained

text: systems

text: is

text: not

text: completely

text: satisﬁed

text: for

text: the

text: anal-

text: where

text: is

text: the

text: interband

text: deformation-potential

text: con-

text: ysis

text: of

text: spin

text: splitting.

text: The

text: ratio

text: of

text: the

text: Dresselhaus

text: con-

text: stant,

text: P

text: is

text: the

text: Kane

text: matrix

text: element,

text: and

text: is

text: the

text: spin-

text: stants

text: β

text: for

text: the

text: structures

text: lattice-matched

text: to

text: GaAs

text: and

text: orbit

text: splitting

text: of

text: the

text: valence

text: band.

text: The

text: growth

text: of

text: AlAs

text: would

text: exceed

text: a

text: factor

text: for

text: a

text: QW

text: while

text: for

text: InAs

text: as

text: compared

text: to

text: that

text: for

text: GaAs

text: is

text: caused

text: by

text: the

text: the

text: strain

text: itself

text: is

text: only

text: Note

text: that

text: the

text: observed

text: de-

text: decrease

text: in

text: Eg

text: and

text: the

text: increase

text: in

text: pendence

text: of

text: β

text: on

text: the

text: QW

text: thickness

text: actually

text: reﬂects

text: the

text: To

text: elaborate

text: this

text: expectation

text: we

text: calculate

text: the

text: Dressel-

text: haus

text: constant

text: β

text: for

text: GaAs/In

text: Ga

text: As/GaAs

text: QW

text: struc-

text: tures

text: lattice-matched

text: to

text: GaAs

text: and

text: In

text: Ga

text: As.

text: The

text: de-

text: pendence

text: of

text: the

text: Dresselhaus

text: constants

text: on

text: the

text: QW

text: thick-

text: The

text: coordinate

text: system

text: is

text: changed

text: with

text: respect

text: to

text: the

text: one

text: used

text: ness

text: for

text: both

text: strain

text: conﬁgurations

text: are

text: shown

text: in

text: Fig.

text: in

text: Ref.

text: which

text: results

text: in

text: the

text: opposite

text: sign

text: of

text: the

text: Dresselhaus

text: The

text: range

text: of

text: spin

text: splittings

text: is

text: signiﬁcantly

text: larger

text: than

text: constant.

text: that

text: in

text: GaAs/AlGaAs

text: structures

text: and

text: the

text: Dresselhaus

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 YONATAN GUTMAN, FREDDIE MANNERS AND PE´TER P. VARJU´ A cubespaces is a structure consisting of a compact metric space X, together with a closed collection ⊆ ≥ of “cubes” Ck(X) X2k for each integer k 0, satisfying certain axioms that we will recall later. The structure (X, Ck(X)) is further called a nilspace if it also satisﬁes certain extra rigidity conditions. The notion of nilspaces has its origins in the work of Host and Kra [HK08], where these objects appeared under the name of “parallelepiped structures”. The study of these objects was furthered by Antol´ın Camarena and Szegedy [ACS12], who in the same work formulated a strong structure theorem for nilspaces, subject to certain further hypotheses. The papers of Candela [Can16b, Can16] expand on [ACS12], providing more detailed proofs. He also includes several additional results implicit in [ACS12], particularly about continuous systems of measures. The purpose of our project is to provide a new exposition of this theory, and also to derive new applications to topological dynamics. Although we rely heavily on the ideas contained in earlier work [HK05, HM07, ACS12, HK08, HKM10, GT10], our proofs diﬀer from the existing literature in many respects and we also obtain new results. The study of nilspaces might be motivated in three diﬀerent ways. First, it can be a useful tool in the area of higher order Fourier analysis, and in particular, forms the basis of Szegedy’s approach [Sze12] to proving an inverse theorem for the Gowers norms (another approach being due to Green, Tao and Ziegler [GTZ12]). Second, nilspaces can be used in topological dynamics. For example, we use them in this paper to generalize a result of Host, Kra and Maass [HKM10] characterizing the largest pronilfactor of a minimal group action. Third, nilspaces can be used in the context of ergodic theory, in particular to give a new and more general proof of the structure theorem for characteristic factors, introduced by Host and Kra [HK05] and by Ziegler [Zie07] (using a diﬀerent framework). For details, see [Gut15]. Given the close connection of the subject to the study of nilsequences arising in additive number theory and ergodic theory [BHK05,GT12], we might expect further applications to arise. In this paper, we explain the notion of nilspaces, and outline our project, from the point of view of topological dynamics. The reader whose main interest lies in combinatorics or higher order Fourier analysis may wish to consult the paper [GMV16], where an outline is given from that perspective. She may then continue directly to Section of this paper, where the main result of this paper is presented in a fashion that does not require familiarity with the dynamical material introduced in Section . Let (T, X) be a topological dynamical system, which for our purposes means that X is a compact metric space and T is a homeomorphism on X. ∈ We begin by recalling some deﬁnitions. We say that a pair of points (x, y) X2 is proximal, if { } ∈ there is a sequence of integers ni such that lim dist(Tnix, Tniy) = 0. We say that (x, y) X2 is { } { } ⊆ { } regionally proximal, if there are sequences of points x , y X and a sequence of integers n i i i

text: YONATAN

text: GUTMAN,

text: FREDDIE

text: MANNERS

text: AND

text: PE´TER

text: P.

text: VARJU´

text: A

text: cubespaces

text: is

text: a

text: structure

text: consisting

text: of

text: a

text: compact

text: metric

text: space

text: X,

text: together

text: with

text: a

text: closed

text: collection

text: of

text: “cubes”

text: Ck(X)

text: for

text: each

text: integer

text: k

text: satisfying

text: certain

text: axioms

text: that

text: we

text: will

text: recall

text: later.

text: The

text: structure

text: (X,

text: Ck(X))

text: is

text: further

text: called

text: a

text: nilspace

text: if

text: it

text: also

text: satisﬁes

text: certain

text: extra

text: rigidity

text: conditions.

text: The

text: notion

text: of

text: nilspaces

text: has

text: its

text: origins

text: in

text: the

text: work

text: of

text: Host

text: and

text: Kra

text: where

text: these

text: objects

text: appeared

text: under

text: the

text: name

text: of

text: “parallelepiped

text: structures”.

text: The

text: study

text: of

text: these

text: objects

text: was

text: furthered

text: by

text: Antol´ın

text: Camarena

text: and

text: Szegedy

text: who

text: in

text: the

text: same

text: work

text: formulated

text: a

text: strong

text: structure

text: theorem

text: for

text: nilspaces,

text: subject

text: to

text: certain

text: further

text: hypotheses.

text: The

text: papers

text: of

text: Candela

text: expand

text: on

text: providing

text: more

text: detailed

text: proofs.

text: He

text: also

text: includes

text: several

text: additional

text: results

text: implicit

text: in

text: particularly

text: about

text: continuous

text: systems

text: of

text: measures.

text: The

text: purpose

text: of

text: our

text: project

text: is

text: to

text: provide

text: a

text: new

text: exposition

text: of

text: this

text: theory,

text: and

text: also

text: to

text: derive

text: new

text: applications

text: to

text: topological

text: dynamics.

text: Although

text: we

text: rely

text: heavily

text: on

text: the

text: ideas

text: contained

text: in

text: earlier

text: work

text: our

text: proofs

text: diﬀer

text: from

text: the

text: existing

text: literature

text: in

text: many

text: respects

text: and

text: we

text: also

text: obtain

text: new

text: results.

text: The

text: study

text: of

text: nilspaces

text: might

text: be

text: motivated

text: in

text: three

text: diﬀerent

text: ways.

text: First,

text: it

text: can

text: be

text: a

text: useful

text: tool

text: in

text: the

text: area

text: of

text: higher

text: order

text: Fourier

text: analysis,

text: and

text: in

text: particular,

text: forms

text: the

text: basis

text: of

text: Szegedy’s

text: approach

text: to

text: proving

text: an

text: inverse

text: theorem

text: for

text: the

text: Gowers

text: norms

text: (another

text: approach

text: being

text: due

text: to

text: Green,

text: Tao

text: and

text: Ziegler

text: Second,

text: nilspaces

text: can

text: be

text: used

text: in

text: topological

text: dynamics.

text: For

text: example,

text: we

text: use

text: them

text: in

text: this

text: paper

text: to

text: generalize

text: a

text: result

text: of

text: Host,

text: Kra

text: and

text: Maass

text: characterizing

text: the

text: largest

text: pronilfactor

text: of

text: a

text: minimal

text: group

text: action.

text: Third,

text: nilspaces

text: can

text: be

text: used

text: in

text: the

text: context

text: of

text: ergodic

text: theory,

text: in

text: particular

text: to

text: give

text: a

text: new

text: and

text: more

text: general

text: proof

text: of

text: the

text: structure

text: theorem

text: for

text: characteristic

text: factors,

text: introduced

text: by

text: Host

text: and

text: Kra

text: and

text: by

text: Ziegler

text: (using

text: a

text: diﬀerent

text: framework).

text: For

text: details,

text: see

text: Given

text: the

text: close

text: connection

text: of

text: the

text: subject

text: to

text: the

text: study

text: of

text: nilsequences

text: arising

text: in

text: additive

text: number

text: theory

text: and

text: ergodic

text: theory

text: we

text: might

text: expect

text: further

text: applications

text: to

text: arise.

text: In

text: this

text: paper,

text: we

text: explain

text: the

text: notion

text: of

text: nilspaces,

text: and

text: outline

text: our

text: project,

text: from

text: the

text: point

text: of

text: view

text: of

text: topological

text: dynamics.

text: The

text: reader

text: whose

text: main

text: interest

text: lies

text: in

text: combinatorics

text: or

text: higher

text: order

text: Fourier

text: analysis

text: may

text: wish

text: to

text: consult

text: the

text: paper

text: where

text: an

text: outline

text: is

text: given

text: from

text: that

text: perspective.

text: She

text: may

text: then

text: continue

text: directly

text: to

text: Section

text: of

text: this

text: paper,

text: where

text: the

text: main

text: result

text: of

text: this

text: paper

text: is

text: presented

text: in

text: a

text: fashion

text: that

text: does

text: not

text: require

text: familiarity

text: with

text: the

text: dynamical

text: material

text: introduced

text: in

text: Section

text: Let

text: (T,

text: X)

text: be

text: a

text: topological

text: dynamical

text: system,

text: which

text: for

text: our

text: purposes

text: means

text: that

text: X

text: is

text: a

text: compact

text: metric

text: space

text: and

text: T

text: is

text: a

text: homeomorphism

text: on

text: X.

text: We

text: begin

text: by

text: recalling

text: some

text: deﬁnitions.

text: We

text: say

text: that

text: a

text: pair

text: of

text: points

text: (x,

text: y)

text: is

text: proximal,

text: if

text: there

text: is

text: a

text: sequence

text: of

text: integers

text: ni

text: such

text: that

text: lim

text: dist(Tnix,

text: Tniy)

text: We

text: say

text: that

text: (x,

text: y)

text: is

text: regionally

text: proximal,

text: if

text: there

text: are

text: sequences

text: of

text: points

text: x

text: y

text: X

text: and

text: a

text: sequence

text: of

text: integers

text: n

text: i

text: i

text: i


text: 14 YONATAN GUTMAN, FREDDIE MANNERS AND PE´TER P. VARJU´ ∈ { } → (2) Let c , c C(cid:96)(X) be two cubes such that π − (c ) = π − (c ). Denote by a : 0, 1 (cid:96) A the 1 2 s 1 1 s 1 2 ∈ { } ∈ unique conﬁguration in A such that a(ω).c (ω) = c (ω) for all ω 0, 1 (cid:96). Then c C(cid:96)(X) 1 2 2 ∈ D if and only if a C(cid:96)( (A)). s It is worth spelling out the meaning of i(cid:88)tem 2 in a few special cases. If (cid:96) = s + 1, then the condition ∈ D a C(cid:96)( (A)) is equivalent to s − | | ( 1) ω a = 0. ω ∈{ } ω 0,1 s+1 { } → ∈ If l < s + 1, then any conﬁguration c : 0, 1 (cid:96) X is a cube provided π(c) C(cid:96)(π(X)), because D (A) is s-ergodic. Note this is consistent with Lemma 1.20. s If we consider cubes contained in a single ﬁbre of π − , we see that they admit a free and transitive s 1 D action by C(cid:96)( (A)). Equivalently, the subcubspaces of X induced by the ﬁbres of π − are all s s 1 D isomorphic copies of (A). s D Using the characterization of (A) in terms of Host–Kra cubegroups, we obtain the following. If s ∈ ∈ c C(cid:96)(X) is a cube, and F is a face of codimension at most s, then [a] .c is also a cube for all a A. F This holds in particular when (cid:96) = s + 1 and F is an edge, i.e. a face of dimension 1. The weak structure theory constitutes progress towards a structure theorem for nilspaces, for the following reasons. • By part (1), we have that X is an A-principal bundle over π − (X). Iterating this procedure s 1 on π − (X), we realize X as a tower of extensions by compact abelian groups, terminating in s 1 π (X), the 1-point space, which constitutes fairly strong information about the structure of 0 X. • In order to recover the cubes of X given knowledge of π − (X), it suﬃces to exhibit a single s 1 (s + 1)-cube lying above each (s + 1)-cube of π − (X), since part (2) then gives us all such s 1 cubes. Again, we can iterate this on π − (X) to obtain a full description of the cubespace. s 1 We call the group A = A (X) the top structure group of X. Also, we deﬁne A (X) := A (π (X)), s t t t ≤ ≤ the top structure group of the canonical factor π (X) for 0 t s, and call it the t-th structure t group of X. The proof of Theorem 1.21 is given in the paper [GMV16], we only recall here how the group A is (cid:120) constructed. Recall that we denote by (cid:96)(x; y) the (cid:96)-conﬁguration all of whose vertices are x except for the one labelled by (cid:126)1, which is y. Recall also the notation [c , c ] from Section , which denotes the 1 2 concatenation of the conﬁgurations c and c . 1 2 We consider the set { ∈ × ∼ } Y = (x, y) X X : x − y s 1 ≈ ≈ (cid:48) (cid:48) (cid:120) (cid:120) (cid:48) (cid:48) ∈ and introduce a relation on Y , given by setting (x, y) (x , y ) if and only if [ s(x; y), s(x ; y )] Cs+1(X).

text: YONATAN

text: GUTMAN,

text: FREDDIE

text: MANNERS

text: AND

text: PE´TER

text: P.

text: VARJU´

text: Let

text: c

text: c

text: be

text: two

text: cubes

text: such

text: that

text: π

text: (c

text: π

text: (c

text: Denote

text: by

text: a

text: A

text: the

text: s

text: s

text: unique

text: conﬁguration

text: in

text: A

text: such

text: that

text: a(ω).c

text: (ω)

text: c

text: (ω)

text: for

text: all

text: ω

text: Then

text: c

text: D

text: if

text: and

text: only

text: if

text: a

text: (A)).

text: s

text: It

text: is

text: worth

text: spelling

text: out

text: the

text: meaning

text: of

text: in

text: a

text: few

text: special

text: cases.

text: If

text: s

text: then

text: the

text: condition

text: D

text: a

text: (A))

text: is

text: equivalent

text: to

text: s

text: ω

text: a

text: ω

text: ω

text: If

text: l

text: s

text: then

text: any

text: conﬁguration

text: c

text: X

text: is

text: a

text: cube

text: provided

text: π(c)

text: because

text: D

text: (A)

text: is

text: s-ergodic.

text: Note

text: this

text: is

text: consistent

text: with

text: Lemma

text: s

text: If

text: we

text: consider

text: cubes

text: contained

text: in

text: a

text: single

text: ﬁbre

text: of

text: π

text: we

text: see

text: that

text: they

text: admit

text: a

text: free

text: and

text: transitive

text: s

text: D

text: action

text: by

text: (A)).

text: Equivalently,

text: the

text: subcubspaces

text: of

text: X

text: induced

text: by

text: the

text: ﬁbres

text: of

text: π

text: are

text: all

text: s

text: s

text: D

text: isomorphic

text: copies

text: of

text: (A).

text: s

text: D

text: Using

text: the

text: characterization

text: of

text: (A)

text: in

text: terms

text: of

text: Host–Kra

text: cubegroups,

text: we

text: obtain

text: the

text: following.

text: If

text: s

text: c

text: is

text: a

text: cube,

text: and

text: F

text: is

text: a

text: face

text: of

text: codimension

text: at

text: most

text: s,

text: then

text: [a]

text: .c

text: is

text: also

text: a

text: cube

text: for

text: all

text: a

text: A.

text: F

text: This

text: holds

text: in

text: particular

text: when

text: s

text: and

text: F

text: is

text: an

text: edge,

text: i.e.

text: a

text: face

text: of

text: dimension

text: The

text: weak

text: structure

text: theory

text: constitutes

text: progress

text: towards

text: a

text: structure

text: theorem

text: for

text: nilspaces,

text: for

text: the

text: following

text: reasons.

text: By

text: part

text: we

text: have

text: that

text: X

text: is

text: an

text: A-principal

text: bundle

text: over

text: π

text: (X).

text: Iterating

text: this

text: procedure

text: s

text: on

text: π

text: (X),

text: we

text: realize

text: X

text: as

text: a

text: tower

text: of

text: extensions

text: by

text: compact

text: abelian

text: groups,

text: terminating

text: in

text: s

text: π

text: (X),

text: the

text: space,

text: which

text: constitutes

text: fairly

text: strong

text: information

text: about

text: the

text: structure

text: of

text: X.

text: In

text: order

text: to

text: recover

text: the

text: cubes

text: of

text: X

text: given

text: knowledge

text: of

text: π

text: (X),

text: it

text: suﬃces

text: to

text: exhibit

text: a

text: single

text: s

text: (s

text: lying

text: above

text: each

text: (s

text: of

text: π

text: (X),

text: since

text: part

text: then

text: gives

text: us

text: all

text: such

text: s

text: cubes.

text: Again,

text: we

text: can

text: iterate

text: this

text: on

text: π

text: (X)

text: to

text: obtain

text: a

text: full

text: description

text: of

text: the

text: cubespace.

text: s

text: We

text: call

text: the

text: group

text: A

text: A

text: (X)

text: the

text: top

text: structure

text: group

text: of

text: X.

text: Also,

text: we

text: deﬁne

text: A

text: (X)

text: A

text: (π

text: (X)),

text: s

text: t

text: t

text: t

text: the

text: top

text: structure

text: group

text: of

text: the

text: canonical

text: factor

text: π

text: (X)

text: for

text: t

text: s,

text: and

text: call

text: it

text: the

text: t-th

text: structure

text: t

text: group

text: of

text: X.

text: The

text: proof

text: of

text: Theorem

text: is

text: given

text: in

text: the

text: paper

text: we

text: only

text: recall

text: here

text: how

text: the

text: group

text: A

text: is

text: constructed.

text: Recall

text: that

text: we

text: denote

text: by

text: y)

text: the

text: all

text: of

text: whose

text: vertices

text: are

text: x

text: except

text: for

text: the

text: one

text: labelled

text: by

text: which

text: is

text: y.

text: Recall

text: also

text: the

text: notation

text: [c

text: c

text: from

text: Section

text: which

text: denotes

text: the

text: concatenation

text: of

text: the

text: conﬁgurations

text: c

text: and

text: c

text: We

text: consider

text: the

text: set

text: Y

text: (x,

text: y)

text: X

text: X

text: x

text: y

text: s

text: and

text: introduce

text: a

text: relation

text: on

text: Y

text: given

text: by

text: setting

text: (x,

text: y)

text: (x

text: y

text: if

text: and

text: only

text: if

text: s(x;

text: y),

text: s(x

text: y