text: M. Livesu, S. Ellero, J. Martínez, S. Lefebvre & M. Attene / From 3D Models to 3D Prints : an Overview of the Processing Pipeline METHOD OPTIMIZES FOR Cost Fidelity Functionality Slice number Supports volume Cusp height Volume loss Surface ﬁnish Stress resiliency [DTS16] (cid:35) (cid:35) (cid:35) (cid:35) (cid:32) (cid:35) ∗ [MCJ 16] (cid:35) (cid:32) (cid:35) (cid:35) (cid:35) (cid:35) [WZK16] (cid:35) (cid:32) (cid:32) (cid:35) (cid:35) (cid:35) [EME15] (cid:35) (cid:32) (cid:35) (cid:35) (cid:35) (cid:35) ∗ [UKY 15] (cid:35) (cid:35) (cid:35) (cid:35) (cid:35) (cid:32) ∗ [ZLP 15] (cid:35) (cid:35) (cid:35) (cid:35) (cid:32) (cid:35) [US13] (cid:35) (cid:35) (cid:35) (cid:35) (cid:35) (cid:32) [HBA13] (cid:35) (cid:35) (cid:35) (cid:32) (cid:35) (cid:35) [PP12] (cid:32) (cid:32) (cid:32) (cid:35) (cid:32) (cid:35) [AKL07] (cid:35) (cid:35) (cid:32) (cid:35) (cid:32) (cid:35) [BL06a] (cid:32) (cid:32) (cid:32) (cid:35) (cid:35) (cid:35) [BL06b] (cid:32) (cid:32) (cid:32) (cid:35) (cid:35) (cid:35) [TPR04] (cid:32) (cid:32) (cid:32) (cid:35) (cid:35) (cid:35) [MRI03] (cid:35) (cid:35) (cid:35) (cid:32) (cid:35) (cid:35) [MRI00] (cid:35) (cid:35) (cid:35) (cid:32) (cid:35) (cid:35) [PDG99] (cid:32) (cid:32) (cid:32) (cid:35) (cid:35) (cid:35) [HL98] (cid:32) (cid:32) (cid:32) (cid:35) (cid:35) (cid:35) Table 2: Techniques for orientation optimization and their properties, see Section 3.2. faces in need for support, and the generation of the support struc- ture itself. For detecting surfaces in overhang a ﬁrst family of ap- proaches consider the down-facing facets of the input mesh having ∗ an angle too steep to print correctly (e.g. [KJAB 91, AD95]). A second family of approaches consist in performing a boolean dif- ∗ ference between two successive slices (e.g. [ACC 88, CJR95]). The width of the difference determines which regions are self- supporting [CJR95]. This can also be detected conveniently and ef- Figure 6: Left: The M letter is decomposed into layers. The down- ﬁciently with two-dimensional morphological operations in image space [HYML09,CLQ13,HWC14]. As discussed in [HWC14] care ward facing tip becomes an island with respect to the build direc- tion. Right: Each layer is solidiﬁed from the outside towards the must be taken however with some speciﬁc conﬁguration, where some protruding regions might be mis-classiﬁed as supported (see inside. The hatched rectangles are the cross sections of material Figure 1.8 in this publication). [DHL14] performs the detection di- deposition paths. After a critical angle, the deposited material no rectly at the toolpath level, verifying whether each deposited seg- longer bonds to the layer below and falls. ment is supported by at least half its width from below. This ﬁrst analysis generally leads to a compact set of points to Overhangs also can produce problematic cases with material de- be supported. Several approaches then select a subset by down- position. The material is typically added progressively along depo- sampling [ER07,CLQ13,DHL14,HWC14]. sition paths. The right illustration of Figure 6 shows a cross section of the deposited paths for an overhang region. Due to the layering, Generating a support structure. Once the surfaces requiring sup- at some excess overhangs the deposited material will simply fall port are determined, the support geometry is computed. The main (red rectangles in the Figure). It is interesting to note, however, that trade-off is between print time, material use, and reliability. until some threshold angle the deposited material will have a suf- ﬁcient bonding surface with the layer below. This self-supporting For instance, for ﬁlament printers the traditional approach con- property stems from the bonding between successive layers and al- sists in extruding the mesh facets requiring support downward, thus lows for overhangs to exist up to some maximum angle without deﬁning a large support volume. The support volume is usually requiring support. Overhangs are less of an issue for technologies printed with a weak inﬁll pattern (see also Section 3.3.2.3). This employing layer solidiﬁcation – even though excessive overhangs still uses a signiﬁcant amount of material and time, but it is very may distort due to change in material properties during solidiﬁca- reliable: the support typically has a large area of contact with both tion, or to auxiliary motion during preparation of the next layers. the part and the print bed, ensuring the print stability in most cases. In this context several approaches modify the support volume to re- ∗ Detecting surfaces requiring support. Generating supports for duce its size. Huang et al. [HYW 09] use sloped walls instead of areas in overhang requires two main steps: the detection of the sur- straight walls for the sides, shrinking the support volumes in their (cid:13) c 2017TheAuthor(s) (cid:13) ComputerGraphicsForum c 2017TheEurographicsAssociationandJohnWiley&SonsLtd. ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: M.

text: Livesu,

text: S.

text: Ellero,

text: J.

text: Martínez,

text: S.

text: Lefebvre

text: M.

text: Attene

text: From

text: Models

text: to

text: Prints

text: an

text: Overview

text: of

text: the

text: Processing

text: Pipeline

text: METHOD

text: OPTIMIZES

text: FOR

text: Cost

text: Fidelity

text: Functionality

text: Slice

text: number

text: Supports

text: volume

text: Cusp

text: height

text: Volume

text: loss

text: Surface

text: ﬁnish

text: Stress

text: resiliency

text: [MCJ

text: [UKY

text: [ZLP

text: Table

text: Techniques

text: for

text: orientation

text: optimization

text: and

text: their

text: properties,

text: see

text: Section

text: faces

text: in

text: need

text: for

text: support,

text: and

text: the

text: generation

text: of

text: the

text: support

text: struc-

text: ture

text: itself.

text: For

text: detecting

text: surfaces

text: in

text: overhang

text: a

text: ﬁrst

text: family

text: of

text: ap-

text: proaches

text: consider

text: the

text: down-facing

text: facets

text: of

text: the

text: input

text: mesh

text: having

text: an

text: angle

text: too

text: steep

text: to

text: print

text: correctly

text: (e.g.

text: [KJAB

text: A

text: second

text: family

text: of

text: approaches

text: consist

text: in

text: performing

text: a

text: boolean

text: dif-

text: ference

text: between

text: two

text: successive

text: slices

text: (e.g.

text: [ACC

text: The

text: width

text: of

text: the

text: difference

text: determines

text: which

text: regions

text: are

text: self-

text: supporting

text: This

text: can

text: also

text: be

text: detected

text: conveniently

text: and

text: ef-

text: Figure

text: Left:

text: The

text: M

text: letter

text: is

text: decomposed

text: into

text: layers.

text: The

text: down-

text: ﬁciently

text: with

text: two-dimensional

text: morphological

text: operations

text: in

text: image

text: space

text: As

text: discussed

text: in

text: care

text: ward

text: facing

text: tip

text: becomes

text: an

text: island

text: with

text: respect

text: to

text: the

text: build

text: direc-

text: tion.

text: Right:

text: Each

text: layer

text: is

text: solidiﬁed

text: from

text: the

text: outside

text: towards

text: the

text: must

text: be

text: taken

text: however

text: with

text: some

text: speciﬁc

text: conﬁguration,

text: where

text: some

text: protruding

text: regions

text: might

text: be

text: mis-classiﬁed

text: as

text: supported

text: (see

text: inside.

text: The

text: hatched

text: rectangles

text: are

text: the

text: cross

text: sections

text: of

text: material

text: Figure

text: in

text: this

text: publication).

text: performs

text: the

text: detection

text: di-

text: deposition

text: paths.

text: After

text: a

text: critical

text: angle,

text: the

text: deposited

text: material

text: no

text: rectly

text: at

text: the

text: toolpath

text: level,

text: verifying

text: whether

text: each

text: deposited

text: seg-

text: longer

text: bonds

text: to

text: the

text: layer

text: below

text: and

text: falls.

text: ment

text: is

text: supported

text: by

text: at

text: least

text: half

text: its

text: width

text: from

text: below.

text: This

text: ﬁrst

text: analysis

text: generally

text: leads

text: to

text: a

text: compact

text: set

text: of

text: points

text: to

text: Overhangs

text: also

text: can

text: produce

text: problematic

text: cases

text: with

text: material

text: de-

text: be

text: supported.

text: Several

text: approaches

text: then

text: select

text: a

text: subset

text: by

text: down-

text: position.

text: The

text: material

text: is

text: typically

text: added

text: progressively

text: along

text: depo-

text: sampling

text: sition

text: paths.

text: The

text: right

text: illustration

text: of

text: Figure

text: shows

text: a

text: cross

text: section

text: of

text: the

text: deposited

text: paths

text: for

text: an

text: overhang

text: region.

text: Due

text: to

text: the

text: layering,

text: Generating

text: a

text: support

text: structure.

text: Once

text: the

text: surfaces

text: requiring

text: sup-

text: at

text: some

text: excess

text: overhangs

text: the

text: deposited

text: material

text: will

text: simply

text: fall

text: port

text: are

text: determined,

text: the

text: support

text: geometry

text: is

text: computed.

text: The

text: main

text: (red

text: rectangles

text: in

text: the

text: Figure).

text: It

text: is

text: interesting

text: to

text: note,

text: however,

text: that

text: trade-off

text: is

text: between

text: print

text: time,

text: material

text: use,

text: and

text: reliability.

text: until

text: some

text: threshold

text: angle

text: the

text: deposited

text: material

text: will

text: have

text: a

text: suf-

text: ﬁcient

text: bonding

text: surface

text: with

text: the

text: layer

text: below.

text: This

text: self-supporting

text: For

text: instance,

text: for

text: ﬁlament

text: printers

text: the

text: traditional

text: approach

text: con-

text: property

text: stems

text: from

text: the

text: bonding

text: between

text: successive

text: layers

text: and

text: al-

text: sists

text: in

text: extruding

text: the

text: mesh

text: facets

text: requiring

text: support

text: downward,

text: thus

text: lows

text: for

text: overhangs

text: to

text: exist

text: up

text: to

text: some

text: maximum

text: angle

text: without

text: deﬁning

text: a

text: large

text: support

text: volume.

text: The

text: support

text: volume

text: is

text: usually

text: requiring

text: support.

text: Overhangs

text: are

text: less

text: of

text: an

text: issue

text: for

text: technologies

text: printed

text: with

text: a

text: weak

text: inﬁll

text: pattern

text: (see

text: also

text: Section

text: This

text: employing

text: layer

text: solidiﬁcation

text: even

text: though

text: excessive

text: overhangs

text: still

text: uses

text: a

text: signiﬁcant

text: amount

text: of

text: material

text: and

text: time,

text: but

text: it

text: is

text: very

text: may

text: distort

text: due

text: to

text: change

text: in

text: material

text: properties

text: during

text: solidiﬁca-

text: reliable:

text: the

text: support

text: typically

text: has

text: a

text: large

text: area

text: of

text: contact

text: with

text: both

text: tion,

text: or

text: to

text: auxiliary

text: motion

text: during

text: preparation

text: of

text: the

text: next

text: layers.

text: the

text: part

text: and

text: the

text: print

text: bed,

text: ensuring

text: the

text: print

text: stability

text: in

text: most

text: cases.

text: In

text: this

text: context

text: several

text: approaches

text: modify

text: the

text: support

text: volume

text: to

text: re-

text: Detecting

text: surfaces

text: requiring

text: support.

text: Generating

text: supports

text: for

text: duce

text: its

text: size.

text: Huang

text: et

text: al.

text: [HYW

text: use

text: sloped

text: walls

text: instead

text: of

text: areas

text: in

text: overhang

text: requires

text: two

text: main

text: steps:

text: the

text: detection

text: of

text: the

text: sur-

text: straight

text: walls

text: for

text: the

text: sides,

text: shrinking

text: the

text: support

text: volumes

text: in

text: their

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text: 14 FRANCESCA ARICI, RUBEN STIENSTRA, AND WALTER D. VAN SUIJLEKOM since the Haar measure is left-invariant and normalized. Here, µ and ν (cid:4) denote the Haar measures on G and GΛ1i\{e0}, respectively. Proposition 16. There exist two covariant functors from Reﬁne to the category K G \K of Hilbert spaces that send a graph Λ to the spaces L2( ) and L2( ), and a i i i i reﬁnement (Λ , Λ , ι ) to the linear isometries u and ured, respectively. i j i,j i,j i,j Proof. Let Λ , Λ and Λ be three graphs, with corresponding spaces of connections K K Ki j k G G G , and and gauge groups , and . Suppose in addition that we are i j k i j k given reﬁnements (Λ , Λ , ι ) and (Λ , Λ , ι ). i j i,j j k j,k We need to prove that the corresponding maps between Hilbert spaces and ob- servable algebras satisfy ◦ u = u u ; i,k j,k i,j ◦ ured = ured ured; i,k j,k i,j ◦ The fact that u = u u follows from Remark 8 and the deﬁnition of the i,k j,k i,j ◦ ≤ map R. To prove ured = ured ured, note that for Λ Λ , the maps p∗ and i,k j,k i,j G \K i Gj \K i p∗ are isometries by deﬁnition of the measure on and . Thus p p∗ = j i i j j i i Id and p p∗ = Id . L2(Gi\Ki) j j L2(Gj\Kj) Commutativity of the ﬁrst square in Proposition 15 and the fact that p∗ and ◦ ◦ i p∗ are sections of p and p , respectively, imply that p u p∗ = ured, and that j G i j G i i u maps -invariant functions to -invariant functions. Observing that p∗p and i j G G i i p∗p are the orthogonal projections onto the spaces of - and -invariant functions, j j i j respectively, we infer that ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ured = p u p∗ = p u u p∗ = p u p∗p u p∗ = ured ured, i,k k i,k i k j,k i,j i k j,k j j i,j i j,k i,j (cid:4) which proves the claim. The isometries between the Hilbert spaces constructed in the previous subsection naturally induce maps between the observables. In fact, we have the following: Proposition 17. The maps K → K (cid:55)→ v: B (L2( )) B (L2( )), b ubu∗; 0 iG \K 0 → j G \K (cid:55)→ vred : B (L2( )) B (L2( )), b uredb(ured)∗, 0 i i 0 j j are injective ∗-homomorphisms. Proof. It is clear that v and vred respect its linear structures as well as the involu- tions. Since u and ured are isometries, we have u∗u = Id , and (ured)∗ured = Id , L2(Ki) L2(Gi\Ki) from which it readily follows that the maps v and vred are injective and respect the (cid:4) algebra structures. Thus the maps u and v are embeddings of the ‘coarse’ Hilbert space and observable algebra into the corresponding ‘ﬁner’ structures, respectively, and the maps ured and vred are their ‘reduced’ counterparts. We can now formulate the analogue of Proposition 15 for the observable algebras:

label: paragraph

text: FRANCESCA

text: ARICI,

text: RUBEN

text: STIENSTRA,

text: AND

text: WALTER

text: D.

text: VAN

text: SUIJLEKOM

text: since

label: list

text: the

text: Haar

text: measure

text: is

text: left-invariant

text: and

text: normalized.

text: Here,

text: µ

text: and

text: ν

text: denote

text: the

text: Haar

text: measures

text: on

text: G

text: and

text: respectively.

text: Proposition

text: There

text: exist

text: two

text: covariant

text: functors

text: from

text: Reﬁne

text: to

text: the

text: category

text: K

text: G

text: \K

text: of

text: Hilbert

text: spaces

text: that

text: send

text: a

text: graph

text: Λ

text: to

text: the

text: spaces

text: and

text: and

text: a

text: i

text: i

text: i

text: i

text: reﬁnement

text: (Λ

text: Λ

text: ι

text: to

text: the

text: linear

text: isometries

text: u

text: and

text: ured,

text: respectively.

text: i

text: j

text: i,j

text: i,j

text: i,j

text: Proof.

text: Let

text: Λ

text: Λ

text: and

text: Λ

text: be

text: three

text: graphs,

text: with

text: corresponding

text: spaces

text: of

text: connections

text: K

text: K

text: Ki

text: j

text: k

text: G

text: G

text: G

text: and

text: and

text: gauge

text: groups

text: and

text: Suppose

text: in

text: addition

text: that

text: we

text: are

text: i

text: j

text: k

text: i

text: j

text: k

text: given

text: reﬁnements

text: (Λ

text: Λ

text: ι

text: and

text: (Λ

text: Λ

text: ι

text: i

text: j

text: i,j

text: j

text: k

text: j,k

text: We

text: need

text: to

text: prove

text: that

text: the

text: corresponding

text: maps

text: between

text: Hilbert

text: spaces

text: and

text: ob-

text: servable

text: algebras

text: satisfy

text: u

text: u

text: u

text: i,k

text: j,k

text: i,j

text: ured

text: ured

text: ured;

text: i,k

text: j,k

text: i,j

text: The

text: fact

text: that

text: u

text: u

text: u

text: follows

text: from

text: Remark

text: and

text: the

text: deﬁnition

text: of

text: the

text: i,k

text: j,k

text: i,j

text: map

text: R.

text: To

text: prove

text: ured

text: ured

text: ured,

text: note

text: that

text: for

text: Λ

text: Λ

text: the

text: maps

text: p∗

text: and

text: i,k

text: j,k

text: i,j

text: G

text: \K

text: i

text: Gj

text: \K

text: i

text: p∗

text: are

text: isometries

text: by

text: deﬁnition

text: of

text: the

text: measure

text: on

text: and

text: Thus

text: p

text: p∗

text: j

text: i

text: i

text: j

text: j

text: i

text: i

text: Id

text: and

text: p

text: p∗

text: Id

text: j

text: j

text: Commutativity

text: of

text: the

text: ﬁrst

text: square

text: in

text: Proposition

text: and

text: the

text: fact

text: that

text: p∗

text: and

text: i

text: p∗

text: are

text: sections

text: of

text: p

text: and

text: p

text: respectively,

text: imply

text: that

text: p

text: u

text: p∗

text: ured,

text: and

text: that

text: j

text: G

text: i

text: j

text: G

text: i

text: i

text: u

text: maps

text: -invariant

text: functions

text: to

text: -invariant

text: functions.

text: Observing

text: that

text: p∗p

text: and

text: i

text: j

text: G

text: G

text: i

text: i

text: p∗p

text: are

text: the

text: orthogonal

text: projections

text: onto

text: the

text: spaces

text: of

text: and

text: -invariant

text: functions,

text: j

text: j

text: i

text: j

text: respectively,

text: we

text: infer

text: that

text: ured

text: p

text: u

text: p∗

text: p

text: u

text: u

text: p∗

text: p

text: u

text: p∗p

text: u

text: p∗

text: ured

text: ured,

text: i,k

text: k

text: i,k

text: i

text: k

text: j,k

text: i,j

text: i

text: k

text: j,k

text: j

text: j

text: i,j

text: i

text: j,k

text: i,j

text: which

text: proves

text: the

text: claim.

text: The

text: isometries

text: between

text: the

text: Hilbert

text: spaces

text: constructed

text: in

text: the

text: previous

text: subsection

text: naturally

text: induce

text: maps

text: between

text: the

text: observables.

text: In

text: fact,

text: we

text: have

text: the

text: following:

text: Proposition

text: The

text: maps

text: K

text: K

text: v:

text: B

text: B

text: b

text: ubu∗;

text: iG

text: \K

text: j

text: G

text: \K

text: vred

text: B

text: B

text: b

text: uredb(ured)∗,

text: i

text: i

text: j

text: j

text: are

text: injective

text: ∗-homomorphisms.

text: Proof.

text: It

text: is

text: clear

text: that

text: v

text: and

text: vred

text: respect

text: its

text: linear

text: structures

text: as

text: well

text: as

text: the

text: involu-

text: tions.

text: Since

text: u

text: and

text: ured

text: are

text: isometries,

text: we

text: have

text: u∗u

text: Id

text: and

text: (ured)∗ured

text: Id

text: from

text: which

text: it

text: readily

text: follows

text: that

text: the

text: maps

text: v

text: and

text: vred

text: are

text: injective

text: and

text: respect

text: the

text: algebra

text: structures.

text: Thus

text: the

text: maps

text: u

text: and

text: v

text: are

text: embeddings

text: of

text: the

text: ‘coarse’

text: Hilbert

text: space

text: and

text: observable

text: algebra

text: into

text: the

text: corresponding

text: ‘ﬁner’

text: structures,

text: respectively,

text: and

text: the

text: maps

text: ured

text: and

text: vred

text: are

text: their

text: ‘reduced’

text: counterparts.

text: We

text: can

text: now

text: formulate

text: the

text: analogue

text: of

text: Proposition

text: for

text: the

text: observable

text: algebras:


text: ∗ QUANTUM LATTICE GAUGE FIELDS AND GROUPOID C -ALGEBRAS 15 Proposition 18. Deﬁne the maps P , P , Π and Π by i j i j K → G \K (cid:55)→ P : B (L2( )) B (L2( )), b p bp∗; i 0 Ki → 0 Gi \Ki (cid:55)→ i i P : B (L2( )) B (L2( )), b p bp∗; j 0 Gj\K →0 j K j (cid:55)→ j j Π : B (L2( )) B (L2( )), b p∗bp ; i 0 Gi \Ki → 0 Ki (cid:55)→ i i Π : B (L2( )) B (L2( )), b p∗bp . j 0 j j 0 j j j Then the following squares B0(L2(cid:79)(cid:79) (Ki)) v (cid:47)(cid:47) B0(L2(cid:79)(cid:79) (Kj)) Πi Πj B (L2(G \K )) vred (cid:47)(cid:47) B (L2(G \K )) 0 i i 0 j j and B (L2(K )) v (cid:47)(cid:47) B (L2(K )) 0 i 0 j Pi (cid:15)(cid:15) (cid:15)(cid:15) Pj B (L2(G \K )) vred (cid:47)(cid:47) B (L2(G \K )) 0 i i 0 j j commute. Proof. We shall only present a proof of commutativity of the ﬁrst square; com- ∈ mutativity of the second square can be proved in a similar fashion. Let b G \K B (L2( )). Then using the commutativity of the ﬁrst square in Proposition 0 i i 15, we obtain ◦ v Π (b) = up∗bp u∗ = (up∗)b(up∗)∗ i i i i i ◦ = (p∗ured)b(p∗ured)∗ = p∗uredb(ured)∗p = Π vred(b), j j j j j (cid:4) as desired. Proposition 19. There exist two covariant functors from Reﬁne to the category K G \K of C∗-algebras that send a graph Λ to the spaces B (L2( )) and B (L2( )), i 0 i 0 i i and a reﬁnement (Λ , Λ , ι ) to the injective ∗-homomorphisms v and vred, re- i j i,j K G \K i,j i,j spectively. The collections (B (L2( )), v ) and (B (L2( )), vred) form direct 0 i i,j 0 i i i,j systems of C∗-algebras. Proof. This follows from the fact that ◦ ◦ v = v v ; vred = vred vred, i,k j,k i,j i,k j,k i,j (cid:4) which is a direct consequence of Proposition 16. We are also interested in describing the reﬁnements of the observable algebras in (cid:16) (cid:17) K × K purely geometric terms, that is to say, in terms of the pair groupoids G = K → K i i i that we associated to a graph Λ . A map R : canonically gives rise i i,j j i → to a groupoid morphism R = R(0), R(1) : G G , where R(0) = R and i,j i,j i,j j i i,j i,j × → R(1) = R R . Similarly, we obtain a groupoid morphism Rred : Gred Gred i,j i,j i,j i,j j i between the pair groupoids associated to the reduced conﬁguration spaces. The following proposition is then an immediate consequence of Proposition 10: ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: QUANTUM

text: LATTICE

text: GAUGE

text: FIELDS

text: AND

text: GROUPOID

text: C

text: -ALGEBRAS

text: Proposition

text: Deﬁne

text: the

text: maps

text: P

text: P

text: Π

text: and

text: Π

text: by

text: i

text: j

text: i

text: j

text: K

text: G

text: \K

text: P

text: B

text: B

text: b

text: p

text: bp∗;

text: i

text: Ki

text: Gi

text: \Ki

text: i

text: i

text: P

text: B

text: B

text: b

text: p

text: bp∗;

text: j

text: Gj\K

text: →0

text: j

text: K

text: j

text: j

text: j

text: Π

text: B

text: B

text: b

text: p∗bp

text: i

text: Gi

text: \Ki

text: Ki

text: i

text: i

text: Π

text: B

text: B

text: b

text: p∗bp

text: j

text: j

text: j

text: j

text: j

text: j

text: Then

text: the

text: following

text: squares

text: (Ki))

text: v

text: (Kj))

text: Πi

text: Πj

text: B

text: \K

text: vred

text: B

text: \K

text: i

text: i

text: j

text: j

text: and

text: B

text: v

text: B

text: i

text: j

text: Pi

text: Pj

text: B

text: \K

text: vred

text: B

text: \K

text: i

text: i

text: j

text: j

text: commute.

text: Proof.

text: We

text: shall

text: only

text: present

text: a

text: proof

text: of

text: commutativity

text: of

text: the

text: ﬁrst

text: square;

text: com-

text: mutativity

text: of

text: the

text: second

text: square

text: can

text: be

text: proved

text: in

text: a

text: similar

text: fashion.

text: Let

text: b

text: G

text: \K

text: B

text: Then

text: using

text: the

text: commutativity

text: of

text: the

text: ﬁrst

text: square

text: in

text: Proposition

text: i

text: i

text: we

text: obtain

text: v

text: Π

text: (b)

text: up∗bp

text: u∗

text: (up∗)b(up∗)∗

text: i

text: i

text: i

text: i

text: i

text: (p∗ured)b(p∗ured)∗

text: p∗uredb(ured)∗p

text: Π

text: vred(b),

text: j

text: j

text: j

text: j

text: j

text: as

text: desired.

text: Proposition

text: There

text: exist

text: two

text: covariant

text: functors

text: from

text: Reﬁne

text: to

text: the

text: category

text: K

text: G

text: \K

text: of

text: C∗-algebras

text: that

text: send

text: a

text: graph

text: Λ

text: to

text: the

text: spaces

text: B

text: and

text: B

text: i

text: i

text: i

text: i

text: and

text: a

text: reﬁnement

text: (Λ

text: Λ

text: ι

text: to

text: the

text: injective

text: ∗-homomorphisms

text: v

text: and

text: vred,

text: re-

text: i

text: j

text: i,j

text: K

text: G

text: \K

text: i,j

text: i,j

text: spectively.

text: The

text: collections

text: (B

text: v

text: and

text: (B

text: vred)

text: form

text: direct

text: i

text: i,j

text: i

text: i

text: i,j

text: systems

text: of

text: C∗-algebras.

text: Proof.

text: This

text: follows

text: from

text: the

text: fact

text: that

text: v

text: v

text: v

text: vred

text: vred

text: vred,

text: i,k

text: j,k

text: i,j

text: i,k

text: j,k

text: i,j

text: which

text: is

text: a

text: direct

text: consequence

text: of

text: Proposition

text: We

text: are

text: also

text: interested

text: in

text: describing

text: the

text: reﬁnements

text: of

text: the

text: observable

text: algebras

text: in

text: K

text: K

text: purely

text: geometric

text: terms,

text: that

text: is

text: to

text: say,

text: in

text: terms

text: of

text: the

text: pair

text: groupoids

text: G

text: K

text: K

text: i

text: i

text: i

text: that

text: we

text: associated

text: to

text: a

text: graph

text: Λ

text: A

text: map

text: R

text: canonically

text: gives

text: rise

text: i

text: i,j

text: j

text: i

text: to

text: a

text: groupoid

text: morphism

text: R

text: G

text: G

text: where

text: R

text: and

text: i,j

text: i,j

text: i,j

text: j

text: i

text: i,j

text: i,j

text: R

text: R

text: Similarly,

text: we

text: obtain

text: a

text: groupoid

text: morphism

text: Rred

text: Gred

text: Gred

text: i,j

text: i,j

text: i,j

text: i,j

text: j

text: i

text: between

text: the

text: pair

text: groupoids

text: associated

text: to

text: the

text: reduced

text: conﬁguration

text: spaces.

text: The

text: following

text: proposition

text: is

text: then

text: an

text: immediate

text: consequence

text: of

text: Proposition

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:83) ∗ QUANTUM LATTICE GAUGE FIELDS AND GROUPOID C -ALGEBRAS 21 H H By deﬁnition of the direct limit we have that = . So for every 0 < (cid:15) < 1 ∈ ∈ H ∞(cid:107) − i(cid:107)∈I≤ i (cid:107) − (cid:107) ≤ and i I we can ﬁnd ξ(cid:48), η(cid:48) u ( ) with ξ ξ(cid:48) (cid:15)/3 and ξ ξ(cid:48) (cid:15)/3, (cid:107) − (cid:107) i≤,∞ i H which implies that θ θ (cid:48) (cid:48) (cid:15). This means that the closed set −li→m B ( ) ξ,η ξ ,η H i∈HI 0 (cid:4)i contains all ﬁnite rank operators on , hence it contains the whole of B ( ). 0 ∞ In our case of interest, we have: Corollary 27. The direct limit of the observable algebras is given by (cid:39) K l−i→m A B (L2( )). i 0 ∞ i∈I Next, we determine the inverse limit of the groupoids G and show that the direct K i limit C∗-algebra A = B (L2( , µ )) agrees with the C∗-algebra C∗(G ) of the ∞ 0 ∞ ∞ ∞ inverse limit groupoid G . ∞ → Given the simple structure of the groupoid morphisms R : G G one easily i,j j i checks that the limit groupoid G is also a pair groupoid and is given by ∞ K × K G = . ∞ ∞ ∞ It is by deﬁnition a free and transitive groupoid. Remark 28. More generally, the limit of an inverse family of compact transitive groupoids such that all groupoid homomorphisms are surjective is also transitive. Moreover, for inverse families of compact free groupoids, the limit is also a free groupoid. The proofs rely on the fact that source and target in the limit groupoid are deﬁned component-wise. K × K On the groupoid G = we have a natural Haar system given by ∞ ∞ ∞ { × | ∈ K } µ δ x , x ∞ K where δ is the unit point mass at x and µ is a positive Radon measure on of x ∞ full support. K × K Theorem 29. The groupoid C∗-algebra C∗( ) is isomorphic to the limit ∞ ∞ K observable algebra A , which in turn is isomorphic to B (L2( , µ)), where µ is ∞ K 0 ∞ the injective limit of the measures on . i K Proof. Since the limit measure µ on the space is a positive Radon measure ∞ ∞ of full support, the result follows from the second isomorphism in Equation (4) and (cid:4) Corollary 27 above. Remark 30. The question whether the C∗-algebras associated with two Haar system on a given groupoid are isomorphic was answered positively by Muhly, Renault and Williams for the case of transitive groupoids (cf. [29, Theorem 3.1]), of which pair groupoids are a special case. Hence the choice of Haar system does not aﬀect, in our setting, the structure of the groupoid C∗-algebra. For a more in depth discussion on the dependence of the groupoid C∗-algebra on the choice of Haar system we refer the reader to [6, Section 5] and [31, Section 3.1]. Thus we have (cid:39) K (cid:39) K (cid:39) C∗(G ) = C∗(l←im− G ) B (L2( , µ)) l−i→m B (L2( , µ )) l−i→m C∗(G ), ∞ i 0 ∞ 0 i i i i∈I i∈I i∈I ##LTLine##

text: QUANTUM

text: LATTICE

text: GAUGE

text: FIELDS

text: AND

text: GROUPOID

text: C

text: -ALGEBRAS

text: H

text: H

text: By

text: deﬁnition

text: of

text: the

text: direct

text: limit

text: we

text: have

text: that

text: So

text: for

text: every

text: H

text: i

text: and

text: i

text: I

text: we

text: can

text: ﬁnd

text: u

text: with

text: ξ

text: and

text: ξ

text: i≤,∞

text: i

text: H

text: which

text: implies

text: that

text: θ

text: θ

text: This

text: means

text: that

text: the

text: closed

text: set

text: −li→m

text: B

text: ξ,η

text: ξ

text: ,η

text: H

text: i∈HI

text: contains

text: all

text: ﬁnite

text: rank

text: operators

text: on

text: hence

text: it

text: contains

text: the

text: whole

text: of

text: B

text: In

text: our

text: case

text: of

text: interest,

text: we

text: have:

text: Corollary

text: The

text: direct

text: limit

text: of

text: the

text: observable

text: algebras

text: is

text: given

text: by

text: K

text: l−i→m

text: A

text: B

text: i

text: i∈I

text: Next,

text: we

text: determine

text: the

text: inverse

text: limit

text: of

text: the

text: groupoids

text: G

text: and

text: show

text: that

text: the

text: direct

text: K

text: i

text: limit

text: C∗-algebra

text: A

text: B

text: µ

text: agrees

text: with

text: the

text: C∗-algebra

text: C∗(G

text: of

text: the

text: inverse

text: limit

text: groupoid

text: G

text: Given

text: the

text: simple

text: structure

text: of

text: the

text: groupoid

text: morphisms

text: R

text: G

text: G

text: one

text: easily

text: i,j

text: j

text: i

text: checks

text: that

text: the

text: limit

text: groupoid

text: G

text: is

text: also

text: a

text: pair

text: groupoid

text: and

text: is

text: given

text: by

text: K

text: K

text: G

text: It

text: is

text: by

text: deﬁnition

text: a

text: free

text: and

text: transitive

text: groupoid.

text: Remark

text: More

text: generally,

text: the

text: limit

text: of

text: an

text: inverse

text: family

text: of

text: compact

text: transitive

text: groupoids

text: such

text: that

text: all

text: groupoid

text: homomorphisms

text: are

text: surjective

text: is

text: also

text: transitive.

text: Moreover,

text: for

text: inverse

text: families

text: of

text: compact

text: free

text: groupoids,

text: the

text: limit

text: is

text: also

text: a

text: free

text: groupoid.

text: The

text: proofs

text: rely

text: on

text: the

text: fact

text: that

text: source

text: and

text: target

text: in

text: the

text: limit

text: groupoid

text: are

text: deﬁned

text: component-wise.

text: K

text: K

text: On

text: the

text: groupoid

text: G

text: we

text: have

text: a

text: natural

text: Haar

text: system

text: given

text: by

text: K

text: µ

text: δ

text: x

text: x

text: K

text: where

text: δ

text: is

text: the

text: unit

text: point

text: mass

text: at

text: x

text: and

text: µ

text: is

text: a

text: positive

text: Radon

text: measure

text: on

text: of

text: x

text: full

text: support.

text: K

text: K

text: Theorem

text: The

text: groupoid

text: C∗-algebra

text: C∗(

text: is

text: isomorphic

text: to

text: the

text: limit

text: K

text: observable

text: algebra

text: A

text: which

text: in

text: turn

text: is

text: isomorphic

text: to

text: B

text: µ)),

text: where

text: µ

text: is

text: K

text: the

text: injective

text: limit

text: of

text: the

text: measures

text: on

text: i

text: K

text: Proof.

text: Since

text: the

text: limit

text: measure

text: µ

text: on

text: the

text: space

text: is

text: a

text: positive

text: Radon

text: measure

text: of

text: full

text: support,

text: the

text: result

text: follows

text: from

text: the

text: second

text: isomorphism

text: in

text: Equation

text: and

text: Corollary

text: above.

text: Remark

text: The

text: question

text: whether

text: the

text: C∗-algebras

text: associated

text: with

text: two

text: Haar

text: system

text: on

text: a

text: given

text: groupoid

text: are

text: isomorphic

text: was

text: answered

text: positively

text: by

text: Muhly,

text: Renault

text: and

text: Williams

text: for

text: the

text: case

text: of

text: transitive

text: groupoids

text: (cf.

text: Theorem

text: of

text: which

text: pair

text: groupoids

text: are

text: a

text: special

text: case.

text: Hence

text: the

text: choice

text: of

text: Haar

text: system

text: does

text: not

text: aﬀect,

text: in

text: our

text: setting,

text: the

text: structure

text: of

text: the

text: groupoid

text: C∗-algebra.

text: For

text: a

text: more

text: in

text: depth

text: discussion

text: on

text: the

text: dependence

text: of

text: the

text: groupoid

text: C∗-algebra

text: on

text: the

text: choice

text: of

text: Haar

text: system

text: we

text: refer

text: the

text: reader

text: to

text: Section

text: and

text: Section

text: Thus

text: we

text: have

text: K

text: K

text: C∗(G

text: C∗(l←im−

text: G

text: B

text: µ))

text: l−i→m

text: B

text: µ

text: l−i→m

text: C∗(G

text: i

text: i

text: i

text: i

text: i∈I

text: i∈I

text: i∈I

text: ##LTLine##


text: ∗ QUANTUM LATTICE GAUGE FIELDS AND GROUPOID C -ALGEBRAS 5 × By deﬁnition the reduced groupoid C∗-algebra C∗(X X) is the closure in r B(L2(X)) of the image of the above representation. This is actually isomorphic to the full groupoid C∗-algebra and one has × (cid:39) × (cid:39) (4) C∗(X X) C∗(X X) B (L2(X)). r 0 We refer to [32] for full details on the construction of groupoid C∗-algebras, see also [20, III.3.4 and III.3.6]. The relation of this construction to strict quantization can be found in [20, III.3.12]. K If we specialise to our case for which = GΛ1 is our conﬁguration space, this K × K leads us to consider the pair groupoid G := , whose space of morphism is K × K K G(1) = and whose space of objects is G(0) = . Thus the observable algebra A is isomorphic to C∗(G). We will now discuss the reduction of the quantum system with respect to the gauge group. For this we use a procedure known as Rieﬀel induction [34], which in this context can be considered as the quantum analogue of Marsden-Weinstein reduction [19]. G Clearly, there is a unitary representation U of the gauge group = GΛ0 on H K (cid:16) (cid:17) = L2( ), which is given by dualizing the action in Equation (1): (5) U((g ) )ψ((a ) ) = ψ (g a g−1 ) , x x∈Λ0 e e∈Λ1 s(e) e t(e) e∈Λ1 ∈ H G for all ψ . Because is compact, we can use this representation to endow H G E G with the structure of a right Hilbert C∗( )-module , where C∗( ) denotes the G G group C∗-algebra of . Indeed, a C∗( )-valued inner product is determined by (cid:104) (cid:105) (cid:104) (cid:105) φ, ψ (g) = ψ, U((g ) )ψ , x x∈Λ0 H G ⊂ G thus deﬁning an element in C( ) C∗( ). The reduced Hilbert space is then given by the balanced tensor product E ⊗ ∗ V, C (G) G with a representation module V of . In our case of interest V is the trivial H G representation, and one arrives at the Hilbert space G of -invariant vectors in H . G G\K Remark 4. Since the group is compact, the quotient space is also a com- pact Hausdorﬀ space. Using the Riesz–Markov representation theorem, which es- tablishes a one-to-one correspondence between positive functionals on C (X) and 0 Radon measures, we can push forward the measure µ to a measure µ , to obtain G\K G H the Hilbert space L2( , µ ). This Hilbert space is naturally isomorphic to G. G At the level of the observable algebra, one ﬁrst considers the algebra AG of elements G H of A that commute with the unitary representation of . Since the space G is invariant under these observables, we obtain a representation π of the C∗-algebra H AG on G. It can b∼e showHn that this representation is not necessarily faithful but that AG/ ker(π) = B ( G) and that the generators of ker(π) implement the 0 local Gauss law [15, 38]. In view of equation (4), it is natural to associate the pair G\K × G\K groupoid ( ) ( ) to the reduced system.

text: QUANTUM

text: LATTICE

text: GAUGE

text: FIELDS

text: AND

text: GROUPOID

text: C

text: -ALGEBRAS

text: By

text: deﬁnition

text: the

text: reduced

text: groupoid

text: C∗-algebra

text: C∗(X

text: X)

text: is

text: the

text: closure

text: in

text: r

text: of

text: the

text: image

text: of

text: the

text: above

text: representation.

text: This

text: is

text: actually

text: isomorphic

text: to

text: the

text: full

text: groupoid

text: C∗-algebra

text: and

text: one

text: has

label: equation

text: C∗(X

text: X)

text: C∗(X

text: X)

text: B

text: r

text: We

text: refer

text: to

text: for

text: full

text: details

text: on

text: the

text: construction

text: of

text: groupoid

text: C∗-algebras,

text: see

text: also

text: and

text: The

text: relation

text: of

text: this

text: construction

text: to

text: strict

text: quantization

text: can

text: be

text: found

text: in

text: K

text: If

text: we

text: specialise

text: to

text: our

text: case

text: for

text: which

text: is

text: our

text: conﬁguration

text: space,

text: this

text: K

text: K

text: leads

text: us

text: to

text: consider

text: the

text: pair

text: groupoid

text: G

text: whose

text: space

text: of

text: morphism

text: is

text: K

text: K

text: K

text: and

text: whose

text: space

text: of

text: objects

text: is

text: Thus

text: the

text: observable

text: algebra

text: A

text: is

text: isomorphic

text: to

text: C∗(G).

text: We

text: will

text: now

text: discuss

text: the

text: reduction

text: of

text: the

text: quantum

text: system

text: with

text: respect

text: to

text: the

text: gauge

text: group.

text: For

text: this

text: we

text: use

text: a

text: procedure

text: known

text: as

text: Rieﬀel

text: induction

text: which

text: in

text: this

text: context

text: can

text: be

text: considered

text: as

text: the

text: quantum

text: analogue

text: of

text: Marsden-Weinstein

text: reduction

text: G

text: Clearly,

text: there

text: is

text: a

text: unitary

text: representation

text: U

text: of

text: the

text: gauge

text: group

text: on

text: H

text: K

text: which

text: is

text: given

text: by

text: dualizing

text: the

text: action

text: in

text: Equation

text: U((g

text: )ψ((a

text: ψ

text: (g

text: a

text: x

text: x∈Λ0

text: e

text: e∈Λ1

text: s(e)

text: e

text: t(e)

text: e∈Λ1

text: H

text: G

text: for

text: all

text: ψ

text: Because

text: is

text: compact,

text: we

text: can

text: use

text: this

text: representation

text: to

text: endow

text: H

text: G

text: E

text: G

text: with

text: the

text: structure

text: of

text: a

text: right

text: Hilbert

text: C∗(

text: )-module

text: where

text: C∗(

text: denotes

text: the

text: G

text: G

text: group

text: C∗-algebra

text: of

text: Indeed,

text: a

text: C∗(

text: )-valued

text: inner

text: product

text: is

text: determined

text: by

text: φ,

text: ψ

text: (g)

text: ψ,

text: U((g

text: )ψ

text: x

text: x∈Λ0

text: H

text: G

text: G

text: thus

text: deﬁning

text: an

text: element

text: in

text: C(

text: C∗(

text: The

text: reduced

text: Hilbert

text: space

text: is

text: then

text: given

text: by

text: the

text: balanced

text: tensor

text: product

text: E

text: V,

text: C

text: (G)

text: G

text: with

text: a

text: representation

text: module

text: V

text: of

text: In

text: our

text: case

text: of

text: interest

text: V

text: is

text: the

text: trivial

text: H

text: G

text: representation,

text: and

text: one

text: arrives

text: at

text: the

text: Hilbert

text: space

text: G

text: of

text: -invariant

text: vectors

text: in

text: H

text: G

text: G\K

text: Remark

text: Since

text: the

text: group

text: is

text: compact,

text: the

text: quotient

text: space

text: is

text: also

text: a

text: com-

text: pact

text: Hausdorﬀ

text: space.

text: Using

text: the

text: Riesz–Markov

text: representation

text: theorem,

text: which

text: es-

text: tablishes

text: a

text: one-to-one

text: correspondence

text: between

text: positive

text: functionals

text: on

text: C

text: (X)

text: and

text: Radon

text: measures,

text: we

text: can

text: push

text: forward

text: the

text: measure

text: µ

text: to

text: a

text: measure

text: µ

text: to

text: obtain

text: G\K

text: G

text: H

text: the

text: Hilbert

text: space

text: µ

text: This

text: Hilbert

text: space

text: is

text: naturally

text: isomorphic

text: to

text: G.

text: G

text: At

text: the

text: level

text: of

text: the

text: observable

text: algebra,

text: one

text: ﬁrst

text: considers

text: the

text: algebra

text: AG

text: of

text: elements

text: G

text: H

text: of

text: A

text: that

text: commute

text: with

text: the

text: unitary

text: representation

text: of

text: Since

text: the

text: space

text: G

text: is

text: invariant

text: under

text: these

text: observables,

text: we

text: obtain

text: a

text: representation

text: π

text: of

text: the

text: C∗-algebra

text: H

text: AG

text: on

text: G.

text: It

text: can

text: b∼e

text: showHn

text: that

text: this

text: representation

text: is

text: not

text: necessarily

text: faithful

text: but

text: that

text: AG/

text: ker(π)

text: B

text: G)

text: and

text: that

text: the

text: generators

text: of

text: ker(π)

text: implement

text: the

text: local

text: Gauss

text: law

text: In

text: view

text: of

text: equation

text: it

text: is

text: natural

text: to

text: associate

text: the

text: pair

text: G\K

text: G\K

text: groupoid

text: to

text: the

text: reduced

text: system.


text: 8 FRANCESCA ARICI, RUBEN STIENSTRA, AND WALTER D. VAN SUIJLEKOM ... Figure 1. The notion of reﬁniments of graphs as introduced above allows to simultaneously take the thermodynamic and the contin- uum limit of the system. → K → Given a reﬁnement ι : Λ Λ of two graphs Λ , Λ , we introduce a map R : K i j i j i,j ∈j between the corresponding conﬁguration spaces as follows. Given an edge e Λ , i i we let · · · (7) R (a) = a a , i,j e e1 en where ι(1)(e) = (e , . . . , e ). The compatibility of these maps under composition i,j 1 n is readily checked and we arrive at the following result. Proposition 10. There exists a contravariant functor from Reﬁne to the category K of compact Hausdorﬀ spaces that sends a graph Λ to the space , and a reﬁnement i i (Λ , Λ , ι ) to the map R .. i j i,j i,j Remark 11. A particular consequence of the above proposition is that a direct system ((Λ ) , (ι ) ) in Reﬁne induces an inverse system K i i∈I i,j i,j∈I, i≤j (( ) , (R ) ) in the category of compact Hausdorﬀ spaces. In what i i∈I i,j i,j∈I, i≤j follows, we will construct various other co- and contravariant functors from Reﬁne to certain categories, which induce direct and inverse systems in these categories, respectively. For the sake of brevity, we will write the above direct system as (Λ , ι ), and do the same with other direct and inverse systems. i i,j In what follows we need to carry out a number of computations, some of which are rather tedious to write out for arbitrary reﬁnements. We simplify our compu- tations by making use of the fact that any reﬁnement can be decomposed into the composition of elementary reﬁnements. This is in line with [4, Lemma 4], although we do not admit the reversal of the orientation of an edge. More precisely, given an arbitrary reﬁnement (Λ , Λ , ι), there exists a sequence (Λ , Λ , ι )n−1 of i j ◦ · · ·k◦ k+1 k,k+1 k=0 reﬁnements such that Λ = Λ , Λ = Λ , ι = ι ι , and for each ∈ { − } 0 i n j n−1,n 0,1 i 0, . . . , n 1 , the reﬁnement (Λ , Λ , ι ) falls into one of the following k k+1 k,k+1 two classes of examples: • The graph Λ is obtained from Λ by adding an extra edge k+1 k ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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text: FRANCESCA

text: ARICI,

text: RUBEN

text: STIENSTRA,

text: AND

text: WALTER

text: D.

text: VAN

text: SUIJLEKOM

font: default

text: Figure

text: The

label: caption

text: notion

text: of

text: reﬁniments

text: of

text: graphs

text: as

text: introduced

text: above

text: allows

text: to

text: simultaneously

text: take

text: the

text: thermodynamic

text: and

text: the

text: contin-

text: uum

text: limit

text: of

text: the

text: system.

text: K

text: Given

text: a

text: reﬁnement

text: ι

text: Λ

text: Λ

text: of

text: two

text: graphs

text: Λ

text: Λ

text: we

text: introduce

text: a

text: map

text: R

text: K

text: i

text: j

text: i

text: j

text: i,j

text: ∈j

text: between

text: the

text: corresponding

text: conﬁguration

text: spaces

text: as

text: follows.

text: Given

text: an

text: edge

text: e

text: Λ

text: i

text: i

text: we

text: let

text: R

text: (a)

text: a

text: a

text: i,j

text: e

text: en

text: where

text: (e

text: e

text: The

text: compatibility

text: of

text: these

text: maps

text: under

text: composition

text: i,j

text: n

text: is

text: readily

text: checked

text: and

text: we

text: arrive

text: at

text: the

text: following

text: result.

text: Proposition

text: There

text: exists

text: a

text: contravariant

text: functor

text: from

text: Reﬁne

text: to

text: the

text: category

text: K

text: of

text: compact

text: Hausdorﬀ

text: spaces

text: that

text: sends

text: a

text: graph

text: Λ

text: to

text: the

text: space

text: and

text: a

text: reﬁnement

text: i

text: i

text: (Λ

text: Λ

text: ι

text: to

text: the

text: map

text: R

text: i

text: j

text: i,j

text: i,j

text: Remark

text: A

text: particular

text: consequence

text: of

text: the

text: above

text: proposition

text: is

text: that

text: a

text: direct

text: system

text: ((Λ

text: (ι

text: in

text: Reﬁne

text: induces

text: an

text: inverse

text: system

text: K

text: i

text: i∈I

text: i,j

text: i,j∈I,

text: i≤j

text: (R

text: in

text: the

text: category

text: of

text: compact

text: Hausdorﬀ

text: spaces.

text: In

text: what

text: i

text: i∈I

text: i,j

text: i,j∈I,

text: i≤j

text: follows,

text: we

text: will

text: construct

text: various

text: other

text: co-

text: and

text: contravariant

text: functors

text: from

text: Reﬁne

text: to

text: certain

text: categories,

text: which

text: induce

text: direct

text: and

text: inverse

text: systems

text: in

text: these

text: categories,

text: respectively.

text: For

text: the

text: sake

text: of

text: brevity,

text: we

text: will

text: write

text: the

text: above

text: direct

text: system

text: as

text: (Λ

text: ι

text: and

text: do

text: the

text: same

text: with

text: other

text: direct

text: and

text: inverse

text: systems.

text: i

text: i,j

text: In

text: what

text: follows

text: we

text: need

text: to

text: carry

text: out

text: a

text: number

text: of

text: computations,

text: some

text: of

text: which

text: are

text: rather

text: tedious

text: to

text: write

text: out

text: for

text: arbitrary

text: reﬁnements.

text: We

text: simplify

text: our

text: compu-

text: tations

text: by

text: making

text: use

text: of

text: the

text: fact

text: that

text: any

text: reﬁnement

text: can

text: be

text: decomposed

text: into

text: the

text: composition

text: of

text: elementary

text: reﬁnements.

text: This

text: is

text: in

text: line

text: with

text: Lemma

text: although

text: we

text: do

text: not

text: admit

text: the

text: reversal

text: of

text: the

text: orientation

text: of

text: an

text: edge.

text: More

text: precisely,

text: given

text: an

text: arbitrary

text: reﬁnement

text: (Λ

text: Λ

text: ι),

text: there

text: exists

text: a

text: sequence

text: (Λ

text: Λ

text: ι

text: of

text: i

text: j

text: ·k◦

text: reﬁnements

text: such

text: that

text: Λ

text: Λ

text: Λ

text: Λ

text: ι

text: ι

text: ι

text: and

text: for

text: each

text: i

text: n

text: j

text: i

text: n

text: the

text: reﬁnement

text: (Λ

text: Λ

text: ι

text: falls

text: into

text: one

text: of

text: the

text: following

text: k

text: two

text: classes

text: of

text: examples:

text: The

text: graph

text: Λ

text: is

text: obtained

text: from

text: Λ

text: by

text: adding

text: an

text: extra

text: edge

text: k

text: ##LTFigure##

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##

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text: 10 FRANCESCA ARICI, RUBEN STIENSTRA, AND WALTER D. VAN SUIJLEKOM Λ k () 1 R R0 k,k+1 k,k+1 Λ Λ0 k+1 k+1 (a a ) (a ) 1 2 1 1 3 1 2 R R0 Λk+2 k+1,k+2 k+1,k+2 (a , a ) 1 2 1 2 3 The maps in the diagram on the right commute, which shows that Equation (10) holds; • Suppose that both pairs of reﬁnements correspond to the subdivision of an edge into two edges but in diﬀerent orders. Then we have Λ k (a a a ) 1 2 3 1 4 R R0 k,k+1 k,k+1 Λ Λ0 k+1 k+1 (a , a a ) (a a , a ) 1 2 3 1 2 3 1 2 4 1 3 4 R R0 Λk+2 k+1,k+2 k+1,k+2 (a , a , a ) 1 2 3 1 2 3 4 from which we infer that also in this case, Equation (10) holds. Throughout the rest of the text, whenever we discuss a reﬁnement (Λ , Λ , ι) of i j graphs, we shall only discuss the cases in which the reﬁnements are elementary reﬁnements, and use the above observation to extend statements to the general case. Let us ﬁx two graphs Λ and Λ together with a reﬁnement (Λ , Λ , ι). i j i j The map ι(0) induces a surjective group homomorphism between the gauge groups given by pull-back: G → G (cid:55)→ (11) (ι(0))∗ : , g = (g ) (g ) , j i x x∈Λ0 ι(0)(x) x∈Λ0 j i Clearly, this map can be directly factorized into products of maps corresponding to elementary reﬁnements. K → K Moreover, one readily veriﬁes that R : satisﬁes the equivariance con- i,j j i dition · · (ι(0))∗(g) R (a) = R (g a), i,j i,j ∈ G ∈ K G \K → G \K for all g and a , hence it descends to a map Rred : . j K → G j\K i,j j j i i If we let π : denote the canonical projection, we obtain a commu- i i i i tative diagram: ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FRANCESCA

text: ARICI,

text: RUBEN

text: STIENSTRA,

text: AND

text: WALTER

text: D.

text: VAN

text: SUIJLEKOM

text: Λ

text: k

text: R

text: Λ

text: Λ0

text: (a

text: a

text: (a

text: R

text: (a

text: a

text: The

text: maps

text: in

text: the

text: diagram

text: on

text: the

text: right

text: commute,

text: which

text: shows

text: that

text: Equation

text: holds;

text: Suppose

text: that

text: both

text: pairs

text: of

text: reﬁnements

text: correspond

text: to

text: the

text: subdivision

text: of

text: an

text: edge

text: into

text: two

text: edges

text: but

text: in

text: diﬀerent

text: orders.

text: Then

text: we

text: have

text: Λ

text: k

text: (a

text: a

text: a

text: R

text: Λ

text: Λ0

text: (a

text: a

text: a

text: (a

text: a

text: a

text: R

text: (a

text: a

text: a

text: from

text: which

text: we

text: infer

text: that

text: also

text: in

text: this

text: case,

text: Equation

text: holds.

text: Throughout

text: the

text: rest

text: of

text: the

text: text,

text: whenever

text: we

text: discuss

text: a

text: reﬁnement

text: (Λ

text: Λ

text: ι)

text: of

text: i

text: j

text: graphs,

text: we

text: shall

text: only

text: discuss

text: the

text: cases

text: in

text: which

text: the

text: reﬁnements

text: are

text: elementary

text: reﬁnements,

text: and

text: use

text: the

text: above

text: observation

text: to

text: extend

text: statements

text: to

text: the

text: general

text: case.

text: Let

text: us

text: ﬁx

text: two

text: graphs

text: Λ

text: and

text: Λ

text: together

text: with

text: a

text: reﬁnement

text: (Λ

text: Λ

text: ι).

text: i

text: j

text: i

text: j

text: The

text: map

text: induces

text: a

text: surjective

text: group

text: homomorphism

text: between

text: the

text: gauge

text: groups

text: given

text: by

text: pull-back:

text: G

text: G

text: g

text: (g

text: (g

text: j

text: i

text: x

text: x∈Λ0

text: x∈Λ0

text: j

text: i

text: Clearly,

text: this

text: map

text: can

text: be

text: directly

text: factorized

text: into

text: products

text: of

text: maps

text: corresponding

text: to

text: elementary

text: reﬁnements.

text: K

text: K

text: Moreover,

text: one

text: readily

text: veriﬁes

text: that

text: R

text: satisﬁes

text: the

text: equivariance

text: con-

text: i,j

text: j

text: i

text: dition

text: R

text: (a)

text: R

text: (g

text: a),

text: i,j

text: i,j

text: G

text: K

text: G

text: \K

text: G

text: \K

text: for

text: all

text: g

text: and

text: a

text: hence

text: it

text: descends

text: to

text: a

text: map

text: Rred

text: j

text: K

text: G

text: j\K

text: i,j

text: j

text: j

text: i

text: i

text: If

text: we

text: let

text: π

text: denote

text: the

text: canonical

text: projection,

text: we

text: obtain

text: a

text: commu-

text: i

text: i

text: i

text: i

text: tative

text: diagram:

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A&A proofs: manuscript no. Damiani_VelOB2 Fig. 3. As in Figure 2, but using data from the external regions in Vela Fig. 5. Smoothed spatial distributions of all datapoints except for con- OB2 (772 stars). centrated sub-populations C and D. Fig. 4. Spatial distributions of kinematical populations C and D (red et al. (2015). Population C is instead not particularly clustered and green respectively). The small black dots indicate the datapoints in or around the ﬁeld studied by Jeﬀries et al. (2014); however, outside of rectangles in Figure 2. The big circles indicate positions of its proper motion is very similar to that of both populations A γ2 Vel (north) and NGC 2547 (south). The dashed arrows indicate their and B, from UCAC4 data (Jeﬀries et al. 2014), which have them- inferred positions 1.1 Myr ago. selves indistinguishable proper motions within the UCAC4 er- ± ± r±ors (mean (µ±α,µδ) equal to (-5.9 0.8, +8.5 1.0) and (-4.6 − 1.0, +8.7 0.9) mas/yr for populations A and B, respec- (122.4305, 48.27079). This sample comprises of 5785 TGAS tively). We therefore tentatively associate population C with the objects. A diagram showing proper motion along Declination γ2 Vel populations A and B cumulatively, and population D with (µδ) vs. parallax (π) is shown in Figure 1. Close to the litera- NGC 2547. Figure 4 also shows that NGC 2547 is not the geo- ture distances of γ2 Vel and NGC 2547 a characteristic pattern metrical centroid of either population C or D, which are spread is found, with stars tending to regroup in two horizontal, nar- more towards the northern half of the surveyed region than south row bands in the diagram. Therefore, we selected the subset of of NGC 2547. Such non-spherical geometry might suggest that data within parallax limits 2.2 < π < 3.5 mas (1084 stars), as radial expansion from a small region was not the origin of the suggested by the horizontal spread of the bands in the Figure. large spatial extent of these populations. These rather generous limits are justiﬁed since the TGAS preci- There is only a very small number of common objects be- sion on parallax is much lower than that anticipated for the ﬁnal tween populations C and D and stars in Jeﬀries et al. (2014: Gaia data releases. The mean uncertainty on π for the parallax- two matches), or Sacco et al. (2015: no matches), because of selected sample is 0.34 mas. However, the TGAS parallaxes are the small overlap between the TGAS and Gaia-ESO magnitude likely to have an additional systematic uncertainty of 0.3 mas ranges. There are no additional matches even considering the (Gaia collaboration 2016b; Lindegren et al. 2016), and if we add Gaia-ESO targets in the NGC 2547 ﬁeld not published in Sacco this in quadrature our chosen parallax interval corresponds to a ± et al. (2015). The two matches with Jeﬀries et al. (2014) are stars 1.4σ range, that would contain about 85 per cent of stars that TYC 8140-2731-1 and TYC 8140-6234-1 (2MASS J08094701- are actually at the distance of γ2 Vel and NGC 2547, if the un- 4744297 and J08092627-4731001, respectively), both belonging certainties are normally distributed. to population C. Jeﬀries et al. assign the second star to their pop- A proper-motion diagram (µ , µ ) for this parallax-selected ulation B (5% probability of belonging to population A), while α δ subset is shown in Figure 2. This shows clearly the existence of the population of the ﬁrst one is undetermined. two localized overdensities of datapoints against a much broader The contrast between the centrally-condensed distributions distribution of points (ﬁeld stars). We therefore deﬁne from this of members of γ2 Vel and NGC 2547 clusters, and the sparseness diagram (and fro−m the previou−s parallax limits) two populations, of both populations C and D renders questionable a one-to-one termed here C ( −9 < µα < 3− mas/yr and 7.5 < µδ < 10.5 correspondence between the former and the latter. One possibil- mas/yr), and D ( 13 < µ < 7 mas/yr and 2.5 < µ < 5.5 ity is then that each of populations C and D is actually composed α δ mas/yr). We use C and D to avoid confusion with the two pop- of a concentrated and a more sparse component. To this aim we ulations (A and B) deﬁned by Jeﬀries et al. (2014); we discuss have considered separately, for each of populations C and D, below the possible connection between these pairs. Population C stars within and outside a 1◦-radius circle around their respec- (D) comprises 148 (101) stars. We have tried to estimate the tive centers (assumed to be the centers of γ2 Vel and NGC 2547 ﬁeld-star contamination in these samples by considering TGAS clusters). In doing so, the number statistics is reduced, and the stars, within the same parallax limits, falling in a rectangular re- relative level of ﬁeld-star contamination becomes more impor- gion of the (µα, µδ) diagram interm−ediate betwee−n those enclos- tant. The just-deﬁned concentrated (sparse) population C con- ing populations C and D, that is 13 < µ < 3 mas/yr and tains 24 (124) stars, which likely contain 4 (49) contaminants. α 5.6 < µ < 7.4 mas/yr: 53 stars are found. This region has the For population D we obtain 27 (74) concentrated (sparse) stars, δ same area in the (µ , µ ) plane as those enclosing populations C of which 2 (51) are likely to be contaminants. It can therefore α δ and D, so no area corrections are∼required. Therefore, the ex- be seen that, although the sparse sub-populations contain in both p∼ected level of contamination is 36% for population C, and cases the majority of members, their contamination level is far 52% for population D. above those of the concentrated sub-populations. The number of To help understand the spatial region where Populations C ﬁeld contaminants for each of the sparse sub-populations can be ∼ and D exist, we have repeated the same selection of TGAS stars◦, mo∼deled as a random distribution with mean µ 50 and std.dev. bu◦t now from an annulus surrounding the previous region, of 4 σ 7 stars. The contamination-corrected numbers of stars in (6 ) inner (outer) radius, which yielded the proper-motion dia- sparse sub-populations C and D (74 and 24 stars, respectively) gram (µα, µδ) shown in Figure 3. Here, the two density peaks would be signiﬁcant at the 10.6σ and 3.4σ level, respectively. are no longer clearly recognizable: the two populations C and D Although the existence of the sparse population D is only slightly ◦ are therefore mostly conﬁned within the 4 -radius circle (nearly above a safe conﬁdence threshold, the distribution of datapoints 50 pc in diameter at the distance of γ2 Vel). in the (µ , µ ) plane after subtraction of the concentrated C and α δ Inside this radius, however, the selected stars from popula- D sub-populations (Figure 5), still provides evidence for both C tions C and D are distributed very inhomogeneously, as shown and D sparse sub-populations. Therefore, even if the properties in Figure 4: the strongest clustering is found for a subset of of the sparse population D are ill-deﬁned because of strong con- population D stars (green dots), spatially coincident with the lo- tamination, its existence is not greatly in doubt. cation of the NGC 2547 cluster (southern black circle in Fig. 4). It may also be of some importance to consider the subsets of This fact suggests that our population D is physically the same as TGAS stars in our populations with Hipparcos data, and char- the main kinematical population of this cluster, found by Sacco acterized by proper-motion measurements that are more precise Article number, page 2 of 3

text: A&A

text: proofs:

text: manuscript

text: no.

text: Fig.

text: As

text: in

text: Figure

text: but

text: using

text: data

text: from

text: the

text: external

text: regions

text: in

text: Vela

text: Fig.

text: Smoothed

text: spatial

text: distributions

text: of

text: all

text: datapoints

text: except

text: for

text: con-

text: stars).

text: centrated

text: sub-populations

text: C

text: and

text: D.

text: Fig.

text: Spatial

text: distributions

text: of

text: kinematical

text: populations

text: C

text: and

text: D

text: (red

text: et

text: al.

text: Population

text: C

text: is

text: instead

text: not

text: particularly

text: clustered

text: and

text: green

text: respectively).

text: The

text: small

text: black

text: dots

text: indicate

text: the

text: datapoints

text: in

text: or

text: around

text: the

text: ﬁeld

text: studied

text: by

text: Jeﬀries

text: et

text: al.

text: however,

text: outside

text: of

text: rectangles

text: in

text: Figure

text: The

text: big

text: circles

text: indicate

text: positions

text: of

text: its

text: proper

text: motion

text: is

text: very

text: similar

text: to

text: that

text: of

text: both

text: populations

text: A

text: γ2

text: Vel

text: (north)

text: and

text: NGC

text: (south).

text: The

text: dashed

text: arrows

text: indicate

text: their

text: and

text: B,

text: from

text: data

text: (Jeﬀries

text: et

text: al.

text: which

text: have

text: them-

text: inferred

text: positions

text: Myr

text: ago.

text: selves

text: indistinguishable

text: proper

text: motions

text: within

text: the

text: er-

text: r±ors

text: (mean

text: (µ±α,µδ)

text: equal

text: to

text: and

text: mas/yr

text: for

text: populations

text: A

text: and

text: B,

text: respec-

text: This

text: sample

text: comprises

text: of

text: TGAS

text: tively).

text: We

text: therefore

text: tentatively

text: associate

text: population

text: C

text: with

text: the

text: objects.

text: A

text: diagram

text: showing

text: proper

text: motion

text: along

text: Declination

text: γ2

text: Vel

text: populations

text: A

text: and

text: B

text: cumulatively,

text: and

text: population

text: D

text: with

text: (µδ)

text: vs.

text: parallax

text: (π)

text: is

text: shown

text: in

text: Figure

text: Close

text: to

text: the

text: litera-

text: NGC

text: Figure

text: also

text: shows

text: that

text: NGC

text: is

text: not

text: the

text: geo-

text: ture

text: distances

text: of

text: γ2

text: Vel

text: and

text: NGC

text: a

text: characteristic

text: pattern

text: metrical

text: centroid

text: of

text: either

text: population

text: C

text: or

text: D,

text: which

text: are

text: spread

text: is

text: found,

text: with

text: stars

text: tending

text: to

text: regroup

text: in

text: two

text: horizontal,

text: nar-

text: more

text: towards

text: the

text: northern

text: half

text: of

text: the

text: surveyed

text: region

text: than

text: south

text: row

text: bands

text: in

text: the

text: diagram.

text: Therefore,

text: we

text: selected

text: the

text: subset

text: of

text: of

text: NGC

text: Such

text: non-spherical

text: geometry

text: might

text: suggest

text: that

text: data

text: within

text: parallax

text: limits

text: π

text: mas

text: stars),

text: as

text: radial

text: expansion

text: from

text: a

text: small

text: region

text: was

text: not

text: the

text: origin

text: of

text: the

text: suggested

text: by

text: the

text: horizontal

text: spread

text: of

text: the

text: bands

text: in

text: the

text: Figure.

text: large

text: spatial

text: extent

text: of

text: these

text: populations.

text: These

text: rather

text: generous

text: limits

text: are

text: justiﬁed

text: since

text: the

text: TGAS

text: preci-

text: There

text: is

text: only

text: a

text: very

text: small

text: number

text: of

text: common

text: objects

text: be-

text: sion

text: on

text: parallax

text: is

text: much

text: lower

text: than

text: that

text: anticipated

text: for

text: the

text: ﬁnal

text: tween

text: populations

text: C

text: and

text: D

text: and

text: stars

text: in

text: Jeﬀries

text: et

text: al.

text: Gaia

text: data

text: releases.

text: The

text: mean

text: uncertainty

text: on

text: π

text: for

text: the

text: parallax-

text: two

text: matches),

text: or

text: Sacco

text: et

text: al.

text: no

text: matches),

text: because

text: of

text: selected

text: sample

text: is

text: mas.

text: However,

text: the

text: TGAS

text: parallaxes

text: are

text: the

text: small

text: overlap

text: between

text: the

text: TGAS

text: and

text: Gaia-ESO

text: magnitude

text: likely

text: to

text: have

text: an

text: additional

text: systematic

text: uncertainty

text: of

text: mas

text: ranges.

text: There

text: are

text: no

text: additional

text: matches

text: even

text: considering

text: the

text: (Gaia

text: collaboration

text: Lindegren

text: et

text: al.

text: and

text: if

text: we

text: add

text: Gaia-ESO

text: targets

text: in

text: the

text: NGC

text: ﬁeld

text: not

text: published

text: in

text: Sacco

text: this

text: in

text: quadrature

text: our

text: chosen

text: parallax

text: interval

text: corresponds

text: to

text: a

text: et

text: al.

text: The

text: two

text: matches

text: with

text: Jeﬀries

text: et

text: al.

text: are

text: stars

text: range,

text: that

text: would

text: contain

text: about

text: per

text: cent

text: of

text: stars

text: that

text: TYC

text: and

text: TYC

text: are

text: actually

text: at

text: the

text: distance

text: of

text: γ2

text: Vel

text: and

text: NGC

text: if

text: the

text: un-

text: and

text: respectively),

text: both

text: belonging

text: certainties

text: are

text: normally

text: distributed.

text: to

text: population

text: C.

text: Jeﬀries

text: et

text: al.

text: assign

text: the

text: second

text: star

text: to

text: their

text: pop-

text: A

text: proper-motion

text: diagram

text: (µ

text: µ

text: for

text: this

text: parallax-selected

text: ulation

text: B

text: probability

text: of

text: belonging

text: to

text: population

text: A),

text: while

text: α

text: δ

text: subset

text: is

text: shown

text: in

text: Figure

text: This

text: shows

text: clearly

text: the

text: existence

text: of

text: the

text: population

text: of

text: the

text: ﬁrst

text: one

text: is

text: undetermined.

text: two

text: localized

text: overdensities

text: of

text: datapoints

text: against

text: a

text: much

text: broader

text: The

text: contrast

text: between

text: the

text: centrally-condensed

text: distributions

text: distribution

text: of

text: points

text: (ﬁeld

text: stars).

text: We

text: therefore

text: deﬁne

text: from

text: this

text: of

text: members

text: of

text: γ2

text: Vel

text: and

text: NGC

text: clusters,

text: and

text: the

text: sparseness

text: diagram

text: (and

text: fro−m

text: the

text: previou−s

text: parallax

text: limits)

text: two

text: populations,

text: of

text: both

text: populations

text: C

text: and

text: D

text: renders

text: questionable

text: a

text: one-to-one

text: termed

text: here

text: C

text: −9

text: µα

text: 3−

text: mas/yr

text: and

text: µδ

text: correspondence

text: between

text: the

text: former

text: and

text: the

text: latter.

text: One

text: possibil-

text: mas/yr),

text: and

text: D

text: µ

text: mas/yr

text: and

text: µ

text: ity

text: is

text: then

text: that

text: each

text: of

text: populations

text: C

text: and

text: D

text: is

text: actually

text: composed

text: α

text: δ

text: mas/yr).

text: We

text: use

text: C

text: and

text: D

text: to

text: avoid

text: confusion

text: with

text: the

text: two

text: pop-

text: of

text: a

text: concentrated

text: and

text: a

text: more

text: sparse

text: component.

text: To

text: this

text: aim

text: we

text: ulations

text: (A

text: and

text: B)

text: deﬁned

text: by

text: Jeﬀries

text: et

text: al.

text: we

text: discuss

text: have

text: considered

text: separately,

text: for

text: each

text: of

text: populations

text: C

text: and

text: D,

text: below

text: the

text: possible

text: connection

text: between

text: these

text: pairs.

text: Population

text: C

text: stars

text: within

text: and

text: outside

text: a

text: circle

text: around

text: their

text: respec-

text: (D)

text: comprises

text: stars.

text: We

text: have

text: tried

text: to

text: estimate

text: the

text: tive

text: centers

text: (assumed

text: to

text: be

text: the

text: centers

text: of

text: γ2

text: Vel

text: and

text: NGC

text: ﬁeld-star

text: contamination

text: in

text: these

text: samples

text: by

text: considering

text: TGAS

text: clusters).

text: In

text: doing

text: so,

text: the

text: number

text: statistics

text: is

text: reduced,

text: and

text: the

text: stars,

text: within

text: the

text: same

text: parallax

text: limits,

text: falling

text: in

text: a

text: rectangular

text: re-

text: relative

text: level

text: of

text: ﬁeld-star

text: contamination

text: becomes

text: more

text: impor-

text: gion

text: of

text: the

text: (µα,

text: µδ)

text: diagram

text: interm−ediate

text: betwee−n

text: those

text: enclos-

text: tant.

text: The

text: just-deﬁned

text: concentrated

text: (sparse)

text: population

text: C

text: con-

text: ing

text: populations

text: C

text: and

text: D,

text: that

text: is

text: µ

text: mas/yr

text: and

text: tains

text: stars,

text: which

text: likely

text: contain

text: contaminants.

text: α

text: µ

text: mas/yr:

text: stars

text: are

text: found.

text: This

text: region

text: has

text: the

text: For

text: population

text: D

text: we

text: obtain

text: concentrated

text: (sparse)

text: stars,

text: δ

text: same

text: area

text: in

text: the

text: (µ

text: µ

text: plane

text: as

text: those

text: enclosing

text: populations

text: C

text: of

text: which

text: are

text: likely

text: to

text: be

text: contaminants.

text: It

text: can

text: therefore

text: α

text: δ

text: and

text: D,

text: so

text: no

text: area

text: corrections

text: are∼required.

text: Therefore,

text: the

text: ex-

text: be

text: seen

text: that,

text: although

text: the

text: sparse

text: sub-populations

text: contain

text: in

text: both

text: p∼ected

text: level

text: of

text: contamination

text: is

text: for

text: population

text: C,

text: and

text: cases

text: the

text: majority

text: of

text: members,

text: their

text: contamination

text: level

text: is

text: far

text: for

text: population

text: D.

text: above

text: those

text: of

text: the

text: concentrated

text: sub-populations.

text: The

text: number

text: of

text: To

text: help

text: understand

text: the

text: spatial

text: region

text: where

text: Populations

text: C

text: ﬁeld

text: contaminants

text: for

text: each

text: of

text: the

text: sparse

text: sub-populations

text: can

text: be

text: and

text: D

text: exist,

text: we

text: have

text: repeated

text: the

text: same

text: selection

text: of

text: TGAS

text: stars◦,

text: mo∼deled

text: as

text: a

text: random

text: distribution

text: with

text: mean

text: µ

text: and

text: std.dev.

text: bu◦t

text: now

text: from

text: an

text: annulus

text: surrounding

text: the

text: previous

text: region,

text: of

text: σ

text: stars.

text: The

text: contamination-corrected

text: numbers

text: of

text: stars

text: in

text: inner

text: (outer)

text: radius,

text: which

text: yielded

text: the

text: proper-motion

text: dia-

text: sparse

text: sub-populations

text: C

text: and

text: D

text: and

text: stars,

text: respectively)

text: gram

text: (µα,

text: µδ)

text: shown

text: in

text: Figure

text: Here,

text: the

text: two

text: density

text: peaks

text: would

text: be

text: signiﬁcant

text: at

text: the

text: and

text: level,

text: respectively.

text: are

text: no

text: longer

text: clearly

text: recognizable:

text: the

text: two

text: populations

text: C

text: and

text: D

text: Although

text: the

text: existence

text: of

text: the

text: sparse

text: population

text: D

text: is

text: only

text: slightly

text: are

text: therefore

text: mostly

text: conﬁned

text: within

text: the

text: -radius

text: circle

text: (nearly

text: above

text: a

text: safe

text: conﬁdence

text: threshold,

text: the

text: distribution

text: of

text: datapoints

text: pc

text: in

text: diameter

text: at

text: the

text: distance

text: of

text: γ2

text: Vel).

text: in

text: the

text: (µ

text: µ

text: plane

text: after

text: subtraction

text: of

text: the

text: concentrated

text: C

text: and

text: α

text: δ

text: Inside

text: this

text: radius,

text: however,

text: the

text: selected

text: stars

text: from

text: popula-

text: D

text: sub-populations

text: (Figure

text: still

text: provides

text: evidence

text: for

text: both

text: C

text: tions

text: C

text: and

text: D

text: are

text: distributed

text: very

text: inhomogeneously,

text: as

text: shown

text: and

text: D

text: sparse

text: sub-populations.

text: Therefore,

text: even

text: if

text: the

text: properties

text: in

text: Figure

text: the

text: strongest

text: clustering

text: is

text: found

text: for

text: a

text: subset

text: of

text: of

text: the

text: sparse

text: population

text: D

text: are

text: ill-deﬁned

text: because

text: of

text: strong

text: con-

text: population

text: D

text: stars

text: (green

text: dots),

text: spatially

text: coincident

text: with

text: the

text: lo-

text: tamination,

text: its

text: existence

text: is

text: not

text: greatly

text: in

text: doubt.

text: cation

text: of

text: the

text: NGC

text: cluster

text: (southern

text: black

text: circle

text: in

text: Fig.

text: It

text: may

text: also

text: be

text: of

text: some

text: importance

text: to

text: consider

text: the

text: subsets

text: of

text: This

text: fact

text: suggests

text: that

text: our

text: population

text: D

text: is

text: physically

text: the

text: same

text: as

text: TGAS

text: stars

text: in

text: our

text: populations

text: with

text: Hipparcos

text: data,

text: and

text: char-

text: the

text: main

text: kinematical

text: population

text: of

text: this

text: cluster,

text: found

text: by

text: Sacco

text: acterized

text: by

text: proper-motion

text: measurements

text: that

text: are

text: more

text: precise

text: Article

text: number,

text: page

text: of