text: 45 IX. INDUCTIVE LIMIT HILBERT SPACE AND CYLINDRICAL-CONSISTENCY OF THE INNER PRODUCT With the reﬁnement maps at hand, we can ﬁnally deﬁne the inductive limit Hilbert space H∞. Such an inductive limit construction is based on a partially-ordered directed set, which is here provided by the set of triangulations {∆, ≺}. The reﬁnement maps (8.13)) satisfy (i) R = id ∀ ∆, (9.1a) ∆,∆ ∆ (cid:48) (cid:48)(cid:48) (ii) R (cid:48) (cid:48)(cid:48) ◦ R (cid:48) = R (cid:48)(cid:48) ∀ ∆ ≺ ∆ ≺ ∆ . (9.1b) ∆ ,∆ ∆,∆ ∆,∆ The inductive limit is deﬁned as the following disjoint union on which an equivalence relation is imposed: H∞ = ∪ H / ∼, (9.2) ∆ ∆ where ψ ∈ H and ψ (cid:48) ∈ H (cid:48) are equivalent, i.e. ψ ∼ ψ (cid:48), if there exists a triangulation (cid:48)(cid:48) ∆ ∆ ∆ ∆ ∆ ∆ ∆ such that R (cid:48)(cid:48)(ψ ) = R (cid:48) (cid:48)(cid:48)(ψ (cid:48)). In words, the two states are equivalent if they become ∆,∆ ∆ ∆ ,∆ ∆ eventually equal under reﬁnement. The inductive limit carries a linear vector space structure which is inherited from the vector space structure on H together with the linearity of the reﬁnement maps. The inductive limit can ∆ also inherit the inner product from the Hilbert spaces H . For this, we need the reﬁnement maps ∆ to be isometric, i.e. to satisfy the condition (cid:48) (cid:48) (cid:104)R (cid:48)(ψ )|R (cid:48)(ψ )(cid:105) (cid:48) = (cid:104)ψ |ψ (cid:105) (cid:48) = (cid:104)ψ |ψ (cid:105) , (9.3) ∆,∆ 1 ∆,∆ 2 ∆ 1 2 ∆ 1 2 ∆ which is also known as cylindrical-consistency for the inner product. This condition makes the inner product well-deﬁned on the equivalence classes [ψ] of states deﬁned above. In order to compute the inner product between two states ψ ∈ H and ψ (cid:48) ∈ H (cid:48), one chooses a common reﬁning (cid:48)(cid:48) ∆ ∆ ∆ ∆ triangulation ∆ and takes the inner product between the states R (cid:48)(cid:48)(ψ ) and R (cid:48) (cid:48)(cid:48)(ψ (cid:48)) in ∆,∆ ∆ ∆ ,∆ ∆ H (cid:48)(cid:48). The condition (9.3) ensures that the result does not depend on the choice of the common ∆ (cid:48)(cid:48) reﬁning triangulation ∆ . Let us therefore show that the reﬁnement maps are indeed isometric. It is suﬃcient to show this for a set of basis states. In (8.23), we have considered the following reﬁnement of the basis states: R (cid:48)ψ{ } = ψ{ (cid:48) }, (9.4) ∆,∆ α(cid:96) α(cid:96)(cid:48) (cid:48) where {α (cid:48)} is determined by (cid:96) (cid:48) α (cid:48) = 1 if P (cid:48) (γ (cid:48)) = r, (9.5a) (cid:96) ∆ ,∆ (cid:96) (cid:48) α (cid:48) = α if P (cid:48) (γ (cid:48)) = γ , (9.5b) (cid:96) (cid:96) ∆ ,∆ (cid:96) (cid:96) (cid:48) − − α (cid:48) = α 1 if P (cid:48) (γ (cid:48)) = γ 1. (9.5c) (cid:96) (cid:96) ∆ ,∆ (cid:96) (cid:96) This reﬁnement operation does in(cid:89)deed leave the inner product (5.8) between basis states invari- ant in the sense that (cid:104)ψ{ }|ψ{ }(cid:105) = δ(α , β ) = (cid:104)R (cid:48)ψ{ }|R (cid:48)ψ{ }(cid:105) (cid:48), (9.6) α β ∆ (cid:96) (cid:96) ∆,∆ α ∆,∆ β ∆ (cid:96) (cid:96) (cid:96) (cid:96) (cid:96) where we have made explicit the triangulation on which the inner product is computed. This gives an inner product which is well-deﬁned (with respect to the equivalence relation) on the inductive limit Hilbert space H∞. ##LTLine##

text: IX.

label: section

text: INDUCTIVE

text: LIMIT

text: HILBERT

text: SPACE

text: AND

text: CYLINDRICAL-CONSISTENCY

text: OF

text: THE

text: INNER

text: PRODUCT

text: With

text: the

text: reﬁnement

text: maps

text: at

text: hand,

text: we

text: can

text: ﬁnally

text: deﬁne

text: the

text: inductive

text: limit

text: Hilbert

text: space

text: H∞.

text: Such

text: an

text: inductive

text: limit

text: construction

text: is

text: based

text: on

text: a

text: partially-ordered

text: directed

text: set,

text: which

text: is

text: here

text: provided

text: by

text: the

text: set

text: of

text: triangulations

text: The

text: reﬁnement

text: maps

text: satisfy

text: (i)

text: R

text: id

text: (ii)

text: R

text: R

text: R

text: The

text: inductive

text: limit

text: is

text: deﬁned

text: as

text: the

text: following

text: disjoint

text: union

text: on

text: which

text: an

text: equivalence

text: relation

text: is

text: imposed:

text: H∞

text: H

text: where

text: ψ

text: H

text: and

text: ψ

text: H

text: are

text: equivalent,

text: i.e.

text: ψ

text: ψ

text: if

text: there

text: exists

text: a

text: triangulation

text: such

text: that

text: R

text: R

text: In

text: words,

text: the

text: two

text: states

text: are

text: equivalent

text: if

text: they

text: become

text: eventually

text: equal

text: under

text: reﬁnement.

text: The

text: inductive

text: limit

text: carries

text: a

text: linear

text: vector

text: space

text: structure

text: which

text: is

text: inherited

text: from

text: the

text: vector

text: space

text: structure

text: on

text: H

text: together

text: with

text: the

text: linearity

text: of

text: the

text: reﬁnement

text: maps.

text: The

text: inductive

text: limit

text: can

text: also

text: inherit

text: the

text: inner

text: product

text: from

text: the

text: Hilbert

text: spaces

text: H

text: For

text: this,

text: we

text: need

text: the

text: reﬁnement

text: maps

text: to

text: be

text: isometric,

text: i.e.

text: to

text: satisfy

text: the

text: condition

text: )|R

text: |ψ

text: |ψ

text: which

text: is

text: also

text: known

text: as

text: cylindrical-consistency

text: for

text: the

text: inner

text: product.

text: This

text: condition

text: makes

text: the

text: inner

text: product

text: well-deﬁned

text: on

text: the

text: equivalence

text: classes

text: [ψ]

text: of

text: states

text: deﬁned

text: above.

text: In

text: order

text: to

text: compute

text: the

text: inner

text: product

text: between

text: two

text: states

text: ψ

text: H

text: and

text: ψ

text: H

text: one

text: chooses

text: a

text: common

text: reﬁning

text: triangulation

text: and

text: takes

text: the

text: inner

text: product

text: between

text: the

text: states

text: R

text: and

text: R

text: in

text: H

text: The

text: condition

text: ensures

text: that

text: the

text: result

text: does

text: not

text: depend

text: on

text: the

text: choice

text: of

text: the

text: common

text: reﬁning

text: triangulation

text: Let

text: us

text: therefore

text: show

text: that

text: the

text: reﬁnement

text: maps

text: are

text: indeed

text: isometric.

text: It

text: is

text: suﬃcient

text: to

text: show

text: this

text: for

text: a

text: set

text: of

text: basis

text: states.

text: In

text: we

text: have

text: considered

text: the

text: following

text: reﬁnement

text: of

text: the

text: basis

text: states:

text: R

text: ψ{

text: where

text: {α

text: is

text: determined

text: by

text: α

text: if

text: P

text: (γ

text: r,

text: α

text: α

text: if

text: P

text: (γ

text: γ

text: α

text: α

text: if

text: P

text: (γ

text: γ

text: This

text: reﬁnement

text: operation

text: does

text: leave

text: the

text: inner

text: product

text: between

text: basis

text: states

text: invari-

text: ant

text: in

text: the

text: sense

text: that

text: }|ψ{

text: δ(α

text: β

text: }|R

text: α

text: β

text: α

text: β

text: where

text: we

text: have

text: made

text: explicit

text: the

text: triangulation

text: on

text: which

text: the

text: inner

text: product

text: is

text: computed.

text: This

text: gives

text: an

text: inner

text: product

text: which

text: is

text: well-deﬁned

text: (with

text: respect

text: to

text: the

text: equivalence

text: relation)

text: on

text: the

text: inductive

text: limit

text: Hilbert

text: space

text: H∞.

text: ##LTLine##


text: 52 the task of specifying the tree for the parallel transport inside a given triangle t. We construct the parallel transport on t\V (∆2) as follows. Since the t-root r(t) lies in the inside of one of the induced triangles σ2, the parallel transport for a point p of this triangle σ2 to r(t) is 0 therefore uniquely speciﬁed along the geodesic from p to r(t). Let us consider some other induced triangle σ2 of t. Then we can have the following two cases: i 1. The parallel transports along straight lines of all the points in σ2 are homotopy-equivalent in i S\V (∆2). This means that for any two points p and p lying inside this induced triangle, the 1 2 piecewise-geodesic path from r(t) to p , then to p , and back to r(t), is homotopy-equivalent 1 2 to the trivial cycle. We can then choose any of the inside points p, and adopt as a parallel transport for all points of this triangle a representative path which crosses through all the triangles which are crossed by the path from p to r(t). 2. There are points p and p of the inside of the induced triangle for which the piecewise 1 2 (cid:0) (cid:1) geodesic path from r(t) to p , then to p , and back to r(t), is not homotopy-equivalent (in 1 2 t\V (∆2)) to the trivial cycle. In this case, we have to subdivide the triangulation ∆ further, (cid:48) until this situation eventually does not appear (now with respect to t\V (∆ )2 ). To see that this procedure does indeed stabilize after a ﬁnite number of steps (and does not produce further and further subdivisions), consider the example of ﬁgure 4. There, we construct the subdivision of t into regions which are homotopy-equivalently transported to r(t). To this end, we need to draw straight lines (appearing as dashed lines in the middle panel of ﬁgure 4) from the t-root r(t) to all the vertices V (∆2), and continue these lines until they hit the boundary of t. We can ignore the parts of the lines which lie in σ2 (which are in dashed green in ﬁgure 4). The 0 remaining parts of the lines (in dashed red) will subdivide the other triangles σ2 into homotopy- i equivalent regions. We then need to subdivide the triangulation further, until these regions are (cid:48) unions of triangles (σ )2 of the reﬁned triangulation. This introduces the ﬁnely dashed red line in the middle panel of ﬁgure 4. However, note that the lines emanating from r(t) do not cross themselves, and therefore only form new vertices with the edges which are already subdividing the triangle t (this is the vertex w in the middle panel). These new vertices can in principle reﬁne the notion of homotopy-equivalent regions, which would then require an iteration of the procedure. All the new vertices are however exactly on the boundary of the homotopy-equivalent regions which have just been deﬁned. Therefore, an iteration of this procedure is not necessary. We now have a triangulation of t which is such that any two points in the inside of any given triangle are homotopy-equivalent as far as the transport to r(t) is concerned. This allows to deﬁne a sub-tree, which speciﬁes the parallel transport for the triangle t itself (see the right panel of ﬁgure 4). As a (cid:48) last step, one needs to ﬁnd a reﬁned triangulation ∆ of Σ which induces a triangulation of t that coincides with the one we have just constructed. In summary, if the above assumptions are satisﬁed, we will eventually be able to deﬁne the (cid:48) parallel transport on a possibly reﬁned triangulation ∆ , which is required in order to specify the translation operator (on H (cid:48)). We can also apply this translation operator to states in H since ∆ (cid:48) ∆ we just need to reﬁne these states to ∆ . (cid:48) The discussion on alternative extensions to ∆ for a given translation operator deﬁned on a triangulation ∆ in section included in particular the change of parallel transport. Therefore, we (cid:48) can adjust the parallel transport of a given extension to ∆ so that it matches the one given by the projection of the continuum translation operator. Consider the (integrated) ﬂux associated to a ﬁxed choice of a triangulated surface S, a choice of t-roots, and a choice of S-surface tree. To deﬁne the corresponding translation operator on the

label: paragraph

text: the

text: task

text: of

text: specifying

text: the

text: tree

text: for

text: the

text: parallel

text: transport

text: inside

text: a

text: given

text: triangle

text: t.

text: We

text: construct

text: the

text: parallel

text: transport

text: on

text: t\V

text: as

text: follows.

text: Since

text: the

text: t-root

text: r(t)

text: lies

text: in

text: the

text: inside

text: of

text: one

text: of

text: the

text: induced

text: triangles

text: the

text: parallel

text: transport

text: for

text: a

text: point

text: p

text: of

text: this

text: triangle

text: σ2

text: to

text: r(t)

text: is

text: therefore

text: uniquely

text: speciﬁed

text: along

text: the

text: geodesic

text: from

text: p

text: to

text: r(t).

text: Let

text: us

text: consider

text: some

text: other

text: induced

text: triangle

text: σ2

text: of

text: t.

text: Then

text: we

text: can

text: have

text: the

text: following

text: two

text: cases:

text: i

label: list

text: The

text: parallel

text: transports

text: along

text: straight

text: lines

text: of

text: all

text: the

text: points

text: in

text: σ2

text: are

text: homotopy-equivalent

text: in

text: i

text: S\V

text: This

text: means

text: that

text: for

text: any

text: two

text: points

text: p

text: and

text: p

text: lying

text: inside

text: this

text: induced

text: triangle,

text: the

text: piecewise-geodesic

text: path

text: from

text: r(t)

text: to

text: p

text: then

text: to

text: p

text: and

text: back

text: to

text: r(t),

text: is

text: homotopy-equivalent

text: to

text: the

text: trivial

text: cycle.

text: We

text: can

text: then

text: choose

text: any

text: of

text: the

text: inside

text: points

text: p,

text: and

text: adopt

text: as

text: a

text: parallel

text: transport

text: for

text: all

text: points

text: of

text: this

text: triangle

text: a

text: representative

text: path

text: which

text: crosses

text: through

text: all

text: the

text: triangles

text: which

text: are

text: crossed

text: by

text: the

text: path

text: from

text: p

text: to

text: r(t).

text: There

text: are

text: points

text: p

text: and

text: p

text: of

text: the

text: inside

text: of

text: the

text: induced

text: triangle

text: for

text: which

text: the

text: piecewise

text: geodesic

text: path

text: from

text: r(t)

text: to

text: p

text: then

text: to

text: p

text: and

text: back

text: to

text: r(t),

text: is

text: not

text: homotopy-equivalent

text: (in

text: t\V

text: to

text: the

text: trivial

text: cycle.

text: In

text: this

text: case,

text: we

text: have

text: to

text: subdivide

text: the

text: triangulation

text: further,

text: until

text: this

text: situation

text: eventually

text: does

text: not

text: appear

text: (now

text: with

text: respect

text: to

text: t\V

text: To

text: see

text: that

text: this

text: procedure

text: does

text: indeed

text: stabilize

text: after

text: a

text: ﬁnite

text: number

text: of

text: steps

text: (and

text: does

text: not

text: produce

text: further

text: and

text: further

text: subdivisions),

text: consider

text: the

text: example

text: of

text: ﬁgure

text: There,

text: we

text: construct

text: the

text: subdivision

text: of

text: t

text: into

text: regions

text: which

text: are

text: homotopy-equivalently

text: transported

text: to

text: r(t).

text: To

text: this

text: end,

text: we

text: need

text: to

text: draw

text: straight

text: lines

text: (appearing

text: as

text: dashed

text: lines

text: in

text: the

text: middle

text: panel

text: of

text: ﬁgure

text: from

text: the

text: t-root

text: r(t)

text: to

text: all

text: the

text: vertices

text: V

text: and

text: continue

text: these

text: lines

text: until

text: they

text: hit

text: the

text: boundary

text: of

text: t.

text: We

text: can

text: ignore

text: the

text: parts

text: of

text: the

text: lines

text: which

text: lie

text: in

text: σ2

text: (which

text: are

text: in

text: dashed

text: green

text: in

text: ﬁgure

text: The

text: remaining

text: parts

text: of

text: the

text: lines

text: (in

text: dashed

text: red)

text: will

text: subdivide

text: the

text: other

text: triangles

text: σ2

text: into

text: homotopy-

text: i

text: equivalent

text: regions.

text: We

text: then

text: need

text: to

text: subdivide

text: the

text: triangulation

text: further,

text: until

text: these

text: regions

text: are

text: unions

text: of

text: triangles

text: (σ

text: of

text: the

text: reﬁned

text: triangulation.

text: This

text: introduces

text: the

text: ﬁnely

text: dashed

text: red

text: line

text: in

text: the

text: middle

text: panel

text: of

text: ﬁgure

text: However,

text: note

text: that

text: the

text: lines

text: emanating

text: from

text: r(t)

text: do

text: not

text: cross

text: themselves,

text: and

text: therefore

text: only

text: form

text: new

text: vertices

text: with

text: the

text: edges

text: which

text: are

text: already

text: subdividing

text: the

text: triangle

text: t

text: (this

text: is

text: the

text: vertex

text: w

text: in

text: the

text: middle

text: panel).

text: These

text: new

text: vertices

text: can

text: in

text: principle

text: reﬁne

text: the

text: notion

text: of

text: homotopy-equivalent

text: regions,

text: which

text: would

text: then

text: require

text: an

text: iteration

text: of

text: the

text: procedure.

text: All

text: the

text: new

text: vertices

text: are

text: however

text: exactly

text: on

text: the

text: boundary

text: of

text: the

text: homotopy-equivalent

text: regions

text: which

text: have

text: just

text: been

text: deﬁned.

text: Therefore,

text: an

text: iteration

text: of

text: this

text: procedure

text: is

text: not

text: necessary.

text: We

text: now

text: have

text: a

text: triangulation

text: of

text: t

text: which

text: is

text: such

text: that

text: any

text: two

text: points

text: in

text: the

text: inside

text: of

text: any

text: given

text: triangle

text: are

text: homotopy-equivalent

text: as

text: far

text: as

text: the

text: transport

text: to

text: r(t)

text: is

text: concerned.

text: This

text: allows

text: to

text: deﬁne

text: a

text: sub-tree,

text: which

text: speciﬁes

text: the

text: parallel

text: transport

text: for

text: the

text: triangle

text: t

text: itself

text: (see

text: the

text: right

text: panel

text: of

text: ﬁgure

text: As

text: a

text: last

text: step,

text: one

text: needs

text: to

text: ﬁnd

text: a

text: reﬁned

text: triangulation

text: of

text: Σ

text: which

text: induces

text: a

text: triangulation

text: of

text: t

text: that

text: coincides

text: with

text: the

text: one

text: we

text: have

text: just

text: constructed.

text: In

text: summary,

text: if

text: the

text: above

text: assumptions

text: are

text: satisﬁed,

text: we

text: will

text: eventually

text: be

text: able

text: to

text: deﬁne

text: the

text: parallel

text: transport

text: on

text: a

text: possibly

text: reﬁned

text: triangulation

text: which

text: is

text: required

text: in

text: order

text: to

text: specify

text: the

text: translation

text: operator

text: (on

text: H

text: We

text: can

text: also

text: apply

text: this

text: translation

text: operator

text: to

text: states

text: in

text: H

text: since

text: we

text: just

text: need

text: to

text: reﬁne

text: these

text: states

text: to

text: The

text: discussion

text: on

text: alternative

text: extensions

text: to

text: for

text: a

text: given

text: translation

text: operator

text: deﬁned

text: on

text: a

text: triangulation

text: in

text: section

text: included

text: in

text: particular

text: the

text: change

text: of

text: parallel

text: transport.

text: Therefore,

text: we

text: can

text: adjust

text: the

text: parallel

text: transport

text: of

text: a

text: given

text: extension

text: to

text: so

text: that

text: it

text: matches

text: the

text: one

text: given

text: by

text: the

text: projection

text: of

text: the

text: continuum

text: translation

text: operator.

text: Consider

text: the

text: (integrated)

text: ﬂux

text: associated

text: to

text: a

text: ﬁxed

text: choice

text: of

text: a

text: triangulated

text: surface

text: S,

text: a

text: choice

text: of

text: t-roots,

text: and

text: a

text: choice

text: of

text: S-surface

text: tree.

text: To

text: deﬁne

text: the

text: corresponding

text: translation

text: operator

text: on

text: the


text: 60 Appendix B: Reconstruction of branch ﬂuxes We explain in this appendix how to reconstruct the ﬂuxes (and operators) associated to the branches from the ones associated to the leaves.(cid:88)To this en(cid:88)d we have to employ the Gauß constraints29. Recall that the Gauß constraint for a node n takes the form ! G = X + X − = 0, (B1) n l l 1 | | l l(0)=n l l(1)=n where the rooted ﬂuxes are given by − − − − Xl = grl(10)Xlgrl(0), Xl−1 = grl−11(0)Xl−1grl−1(0) = −grl−11(0)hlXlhl 1grl−1(0). (B2) The last deﬁnition implies that for branches l = b we simply have X − = −X , while for leaves − b 1 b l = (cid:96) we have X − = −g X g 1, where g = g h g . (cid:96) 1 (cid:96) (cid:96) (cid:96) (cid:96) r(cid:96)(1) (cid:96) r(cid:96)(0) Given a choice of tree, one can successively solve for all the ﬂuxes by starting from the end nodes of the tree, where the Gauß constraints each involve only one ﬂux associated to a branch. By solving for these ﬂuxes associated to the end nodes of the tree, we can go to the nodes which are only one step away from the end nodes along the tree. Again, for every node there is only one unknown branch ﬂux. One can then iterate this procedure and solve all the Gauß constraints except for the one at the root. The ﬂuxes associated to branches are then given by linear combinations of the ﬂuxes associated to the leaves and to the inverse leaves. To give the result of this procedure in a compact form, let us denote by N T the set of ∂ >b(1) end nodes of the tree which lie after the node b(1) when following the orientation of the tree. For each end node n in N T there is a unique path γ going from the target node b(1) of the e ∂ >b(1) b(1)ne branch to the end node n . Considering the set of all such paths for a given b, we can then denote e by L (b) the set of leaves which have their source node (cid:96)(0) along the paths γ , and by L (b) s b(1)ne t the set of leaves which have their target node (cid:96)(1) along the paths γ . As suggested by the b(1)ne notation, a choice of branch b automatica(cid:88)lly determin(cid:88)es the sets Ls(b) and Lt(b) (which can have a non-empty intersection). The ﬂux is then given by X = X + X − . (B3) b (cid:96) (cid:96) 1 ∈L ∈L (cid:96) s(b) (cid:96) t(b) In this formula, each leaf (cid:96) can appear at m(cid:89)ost once w(cid:89)ith the orientation (cid:96) and at most once with − the orientation (cid:96) 1. The right translations corresponding to a branch can therefore be deﬁned as Rα = Rα Lα. (B4) b (cid:96) (cid:96) ∈L ∈L (cid:96) s(b) (cid:96) t(b) Appendix C: The spin representation and measure functional In most of this article we have worked in the holonomy representation for the wave functions. In this appendix we discuss a group Fourier transform with respect to a discrete measure on the group, which deﬁnes a spin representation. 1. The group U(1) | | | | − | | | | − | | 29 Recall that a maximal spanning tree has b = n 1 branches and (cid:96) = l b leaves. As far as the counting is concerned, it is therefore possible to reconstruct all the ﬂuxes associated with the links l from the knowledge of | | − the ﬂuxes associated to the leaves (cid:96) and n 1 Gauß constraints. ##LTLine##

text: Appendix

text: B:

text: Reconstruction

text: of

text: branch

text: ﬂuxes

text: We

text: explain

text: in

text: this

text: appendix

text: how

text: to

text: reconstruct

text: the

text: ﬂuxes

text: (and

text: operators)

text: associated

text: to

text: the

text: branches

text: from

text: the

text: ones

text: associated

text: to

text: the

label: equation

text: this

text: we

text: have

text: to

text: employ

text: the

text: Gauß

text: Recall

text: that

text: the

text: Gauß

text: constraint

text: for

text: a

text: node

text: n

text: takes

text: the

text: form

text: G

text: X

text: X

text: n

text: l

text: l

text: l

text: l

text: where

text: the

text: rooted

text: ﬂuxes

text: are

text: given

text: by

text: Xl

text: The

text: last

text: deﬁnition

text: implies

text: that

text: for

text: branches

text: l

text: b

text: we

text: simply

text: have

text: X

text: −X

text: while

text: for

text: leaves

text: b

text: b

text: l

text: we

text: have

text: X

text: −g

text: X

text: g

text: where

text: g

text: g

text: h

text: g

text: Given

text: a

text: choice

text: of

text: tree,

text: one

text: can

text: successively

text: solve

text: for

text: all

text: the

text: ﬂuxes

text: by

text: starting

text: from

text: the

text: end

text: nodes

text: of

text: the

text: tree,

text: where

text: the

text: Gauß

text: constraints

text: each

text: involve

text: only

text: one

text: ﬂux

text: associated

text: to

text: a

text: branch.

text: By

text: solving

text: for

text: these

text: ﬂuxes

text: associated

text: to

text: the

text: end

text: nodes

text: of

text: the

text: tree,

text: we

text: can

text: go

text: to

text: the

text: nodes

text: which

text: are

text: only

text: one

text: step

text: away

text: from

text: the

text: end

text: nodes

text: along

text: the

text: tree.

text: Again,

text: for

text: every

text: node

text: there

text: is

text: only

text: one

text: unknown

text: branch

text: ﬂux.

text: One

text: can

text: then

text: iterate

text: this

text: procedure

text: and

text: solve

text: all

text: the

text: Gauß

text: constraints

text: except

text: for

text: the

text: one

text: at

text: the

text: root.

text: The

text: ﬂuxes

text: associated

text: to

text: branches

text: are

text: then

text: given

text: by

text: linear

text: combinations

text: of

text: the

text: ﬂuxes

text: associated

text: to

text: the

text: leaves

text: and

text: to

text: the

text: inverse

text: leaves.

text: To

text: give

text: the

text: result

text: of

text: this

text: procedure

text: in

text: a

text: compact

text: form,

text: let

text: us

text: denote

text: by

text: N

text: T

text: the

text: set

text: of

text: end

text: nodes

text: of

text: the

text: tree

text: which

text: lie

text: after

text: the

text: node

text: when

text: following

text: the

text: orientation

text: of

text: the

text: tree.

text: For

text: each

text: end

text: node

text: n

text: in

text: N

text: T

text: there

text: is

text: a

text: unique

text: path

text: γ

text: going

text: from

text: the

text: target

text: node

text: of

text: the

text: e

text: branch

text: to

text: the

text: end

text: node

text: n

text: Considering

text: the

text: set

text: of

text: all

text: such

text: paths

text: for

text: a

text: given

text: b,

text: we

text: can

text: then

text: denote

text: e

text: by

text: L

text: (b)

text: the

text: set

text: of

text: leaves

text: which

text: have

text: their

text: source

text: node

text: along

text: the

text: paths

text: γ

text: and

text: by

text: L

text: (b)

text: s

text: t

text: the

text: set

text: of

text: leaves

text: which

text: have

text: their

text: target

text: node

text: along

text: the

text: paths

text: γ

text: As

text: suggested

text: by

text: the

text: notation,

text: a

text: choice

text: of

text: branch

text: b

text: the

text: sets

text: Ls(b)

text: and

text: Lt(b)

text: (which

text: can

text: have

text: a

text: non-empty

text: intersection).

text: The

text: ﬂux

text: is

text: then

text: given

text: by

text: X

text: X

text: X

text: b

text: ∈L

text: ∈L

text: s(b)

text: t(b)

text: In

text: this

text: formula,

text: each

text: leaf

text: can

text: appear

text: at

text: once

text: the

text: orientation

text: and

text: at

text: most

text: once

text: with

text: the

text: orientation

text: The

text: right

text: translations

text: corresponding

text: to

text: a

text: branch

text: can

text: therefore

text: be

text: deﬁned

text: as

text: Rα

text: Rα

text: Lα.

text: b

text: ∈L

text: ∈L

text: s(b)

text: t(b)

text: Appendix

text: C:

text: The

text: spin

text: representation

text: and

text: measure

text: functional

text: In

text: most

text: of

text: this

text: article

text: we

text: have

text: worked

text: in

text: the

text: holonomy

text: representation

text: for

text: the

text: wave

text: functions.

text: In

text: this

text: appendix

text: we

text: discuss

text: a

text: group

text: Fourier

text: transform

text: with

text: respect

text: to

text: a

text: discrete

text: measure

text: on

text: the

text: group,

text: which

text: deﬁnes

text: a

text: spin

text: representation.

text: The

text: group

text: Recall

text: that

text: a

text: maximal

text: spanning

text: tree

text: has

text: b

text: n

text: branches

text: and

text: l

text: b

text: leaves.

text: As

text: far

text: as

text: the

text: counting

text: is

text: concerned,

text: it

text: is

text: therefore

text: possible

text: to

text: reconstruct

text: all

text: the

text: ﬂuxes

text: associated

text: with

text: the

text: links

text: l

text: from

text: the

text: knowledge

text: of

text: the

text: ﬂuxes

text: associated

text: to

text: the

text: leaves

text: and

text: n

text: Gauß

text: constraints.

text: ##LTLine##


text: 7 only ﬁx the number of defects, but also their positions. This is achieved by embedding a discrete structure, e.g. a graph or a triangulation, into the spatial manifold. Defects are then conﬁned to this discrete structure, for instance line defects to the links of the graph or to the edges of the triangulation. The degrees of freedom are given by the values of the charges for the defects. The values can also describe a vanishing charge, which is then equivalent to having no defect. This is an important point since it allows to embed states realized in a Hilbert space based on e.g. a coarse triangulation into a Hilbert space based on a reﬁnement of this triangulation. For the continuum theory, one however wishes to allow for an arbitrary number of defects (at arbitrary positions). One can construct such a continuum Hilbert space as an inductive limit of Hilbert spaces. A key point in this construction is the notion of reﬁnement for the discrete structures carrying the defects on the one hand, and for the states on the other hand. Let us consider the example of line defects conﬁned to a graph embedded into the manifold. One possible reﬁnement of a graph is to add links to it. The corresponding state for the reﬁned graph is then deﬁned by transposing over all the charge labels from the coarse state, and by assigning the charge labels describing a vanishing defect for the new links. In other words, one puts the new additional degrees of freedom into a vacuum state [43]. In order to deﬁne the inductive Hilbert space structure, the reﬁnement operator acting on the states needs to be isometric (with respect to the inner product deﬁned on the Hilbert spaces with a ﬁxed discrete structure). In particular, the vacuum state where all the charges describe vanishing defects should have a ﬁnite norm. Moreover, what happens (in both the BF and the AL representation) is that the charges label an orthonormal basis for the Hilbert spaces associated to a ﬁxed discrete structure. Therefore, even if the charge labels are continuous (as it is the case with the BF representation, where the labels are elements of the (Lie) group), we will have to equip the space of charge values with a discrete topology. If we see the charge labels as momentum space, then a discrete topology on momentum space corresponds to a compactiﬁcation of the conﬁguration space. This compactiﬁcation can be under- stood in the following way for the two representations: • In the AL representation, the conﬁguration space associated to a ﬁxed discrete structure is given by group holonomies along the links of embedded graphs, and therefore corresponds to GL, where G is a compact Lie group (usually SU(2)), and L denotes the number of allowed defects, which in this case coincides with the number of links of the graph. The inductive limit of Hilbert spaces amounts to a projective limit for the conﬁguration space, in which compactness is preserved (see for instance [7]). The space of charge labels is constructed through the group Fourier transform and given by representation labels (and magnetic indices). This space does indeed naturally carry a discrete topology (for a compact group G). • In the BF representation, the conﬁguration variables are actually the ﬂuxes6, which corre- spond to the dual of the Lie group. We therefore need to consider a compactiﬁcation of this space. In fact, for the group G = U(1) the (Pontryagin) dual is given by the Abelian group Z , which can be Bohr-compactiﬁed. This in turn leads to a discrete topology on G = U(1). For G = SU(2), it is however easier to start with a discrete topology, and to then derive the compactiﬁcation of the dual space via a (generalized) group Fourier transform. This is described in appendix . 6 We will however mostly work in the holonomy representation. A spin representation is described in the appendices and . ##LTLine##

text: only

text: ﬁx

text: the

text: number

text: of

text: defects,

text: but

text: also

text: their

text: positions.

text: This

text: is

text: achieved

text: by

text: embedding

text: a

text: discrete

text: structure,

text: e.g.

text: a

text: graph

text: or

text: a

text: triangulation,

text: into

text: the

text: spatial

text: manifold.

text: Defects

text: are

text: then

text: conﬁned

text: to

text: this

text: discrete

text: structure,

text: for

text: instance

text: line

text: defects

text: to

text: the

text: links

text: of

text: the

text: graph

text: or

text: to

text: the

text: edges

text: of

text: the

text: triangulation.

text: The

text: degrees

text: of

text: freedom

text: are

text: given

text: by

text: the

text: values

text: of

text: the

text: charges

text: for

text: the

text: defects.

text: The

text: values

text: can

text: also

text: describe

text: a

text: vanishing

text: charge,

text: which

text: is

text: then

text: equivalent

text: to

text: having

text: no

text: defect.

text: This

text: is

text: an

text: important

text: point

text: since

text: it

text: allows

text: to

text: embed

text: states

text: realized

text: in

text: a

text: Hilbert

text: space

text: based

text: on

text: e.g.

text: a

text: coarse

text: triangulation

text: into

text: a

text: Hilbert

text: space

text: based

text: on

text: a

text: reﬁnement

text: of

text: this

text: triangulation.

text: For

text: the

text: continuum

text: theory,

text: one

text: however

text: wishes

text: to

text: allow

text: for

text: an

text: arbitrary

text: number

text: of

text: defects

text: (at

text: arbitrary

text: positions).

text: One

text: can

text: construct

text: such

text: a

text: continuum

text: Hilbert

text: space

text: as

text: an

text: inductive

text: limit

text: of

text: Hilbert

text: spaces.

text: A

text: key

text: point

text: in

text: this

text: construction

text: is

text: the

text: notion

text: of

text: reﬁnement

text: for

text: the

text: discrete

text: structures

text: carrying

text: the

text: defects

text: on

text: the

text: one

text: hand,

text: and

text: for

text: the

text: states

text: on

text: the

text: other

text: hand.

text: Let

text: us

text: consider

text: the

text: example

text: of

text: line

text: defects

text: conﬁned

text: to

text: a

text: graph

text: embedded

text: into

text: the

text: manifold.

text: One

text: possible

text: reﬁnement

text: of

text: a

text: graph

text: is

text: to

text: add

text: links

text: to

text: it.

text: The

text: corresponding

text: state

text: for

text: the

text: reﬁned

text: graph

text: is

text: then

text: deﬁned

text: by

text: transposing

text: over

text: all

text: the

text: charge

text: labels

text: from

text: the

text: coarse

text: state,

text: and

text: by

text: assigning

text: the

text: charge

text: labels

text: describing

text: a

text: vanishing

text: defect

text: for

text: the

text: new

text: links.

text: In

text: other

text: words,

text: one

text: puts

text: the

text: new

text: additional

text: degrees

text: of

text: freedom

text: into

text: a

text: vacuum

text: state

text: In

text: order

text: to

text: deﬁne

text: the

text: inductive

text: Hilbert

text: space

text: structure,

text: the

text: reﬁnement

text: operator

text: acting

text: on

text: the

text: states

text: needs

text: to

text: be

text: isometric

text: (with

text: respect

text: to

text: the

text: inner

text: product

text: deﬁned

text: on

text: the

text: Hilbert

text: spaces

text: with

text: a

text: ﬁxed

text: discrete

text: structure).

text: In

text: particular,

text: the

text: vacuum

text: state

text: where

text: all

text: the

text: charges

text: describe

text: vanishing

text: defects

text: should

text: have

text: a

text: ﬁnite

text: norm.

text: Moreover,

text: what

text: happens

text: (in

text: both

text: the

text: BF

text: and

text: the

text: AL

text: representation)

text: is

text: that

text: the

text: charges

text: label

text: an

text: orthonormal

text: basis

text: for

text: the

text: Hilbert

text: spaces

text: associated

text: to

text: a

text: ﬁxed

text: discrete

text: structure.

text: Therefore,

text: even

text: if

text: the

text: charge

text: labels

text: are

text: continuous

text: (as

text: it

text: is

text: the

text: case

text: with

text: the

text: BF

text: representation,

text: where

text: the

text: labels

text: are

text: elements

text: of

text: the

text: (Lie)

text: group),

text: we

text: will

text: have

text: to

text: equip

text: the

text: space

text: of

text: charge

text: values

text: with

text: a

text: discrete

text: topology.

text: If

text: we

text: see

text: the

text: charge

text: labels

text: as

text: momentum

text: space,

text: then

text: a

text: discrete

text: topology

text: on

text: momentum

text: space

text: corresponds

text: to

text: a

text: compactiﬁcation

text: of

text: the

text: conﬁguration

text: space.

text: This

text: compactiﬁcation

text: can

text: be

text: under-

text: stood

text: in

text: the

text: following

text: way

text: for

text: the

text: two

text: representations:

text: In

text: the

text: AL

text: representation,

text: the

text: conﬁguration

text: space

text: associated

text: to

text: a

text: ﬁxed

text: discrete

text: structure

text: is

text: given

text: by

text: group

text: holonomies

text: along

text: the

text: links

text: of

text: embedded

text: graphs,

text: and

text: therefore

text: corresponds

text: to

text: GL,

text: where

text: G

text: is

text: a

text: compact

text: Lie

text: group

text: (usually

text: and

text: L

text: denotes

text: the

text: number

text: of

text: allowed

text: defects,

text: which

text: in

text: this

text: case

text: coincides

text: with

text: the

text: number

text: of

text: links

text: of

text: the

text: graph.

text: The

text: inductive

text: limit

text: of

text: Hilbert

text: spaces

text: amounts

text: to

text: a

text: projective

text: limit

text: for

text: the

text: conﬁguration

text: space,

text: in

text: which

text: compactness

text: is

text: preserved

text: (see

text: for

text: instance

text: The

text: space

text: of

text: charge

text: labels

text: is

text: constructed

text: through

text: the

text: group

text: Fourier

text: transform

text: and

text: given

text: by

text: representation

text: labels

text: (and

text: magnetic

text: indices).

text: This

text: space

text: does

text: indeed

text: naturally

text: carry

text: a

text: discrete

text: topology

text: (for

text: a

text: compact

text: group

text: G).

text: In

text: the

text: BF

text: representation,

text: the

text: conﬁguration

text: variables

text: are

text: actually

text: the

text: which

text: corre-

text: spond

text: to

text: the

text: dual

text: of

text: the

text: Lie

text: group.

text: We

text: therefore

text: need

text: to

text: consider

text: a

text: compactiﬁcation

text: of

text: this

text: space.

text: In

text: fact,

text: for

text: the

text: group

text: G

text: the

text: (Pontryagin)

text: dual

text: is

text: given

text: by

text: the

text: Abelian

text: group

text: Z

text: which

text: can

text: be

text: Bohr-compactiﬁed.

text: This

text: in

text: turn

text: leads

text: to

text: a

text: discrete

text: topology

text: on

text: G

text: For

text: G

text: it

text: is

text: however

text: easier

text: to

text: start

text: with

text: a

text: discrete

text: topology,

text: and

text: to

text: then

text: derive

text: the

text: compactiﬁcation

text: of

text: the

text: dual

text: space

text: via

text: a

text: (generalized)

text: group

text: Fourier

text: transform.

text: This

text: is

text: described

text: in

text: appendix

text: We

text: will

text: however

text: mostly

text: work

text: in

text: the

text: holonomy

text: representation.

text: A

text: spin

text: representation

text: is

text: described

text: in

text: the

text: appendices

text: and

text: ##LTLine##


text: 64 (cid:88) and { } { } UIj { }UIj { (cid:48) (cid:48)} = δ{ } { (cid:48)}δ{ } { (cid:48)}. (C30) ,J,L; M,N ,J,L; M ,N M , M N , N I ,J,L (cid:88) We can then change our spin representation to { } ψ({j}, I, J) := UIj { }ψ{j}{ }, (C31) L ,J,L; M,N MN { } MN and also rewrite the inner product (C25) as a sum over the {j}, I, J, L indices. To obtain the (completely) gauge-invariant part of a wave function ψ({j}, I, J) , one would L set J = 0 (and hence L = 0). Note however that a wave function with (Kronecker) δ behavior J,0 would have vanishing norm in the (rewritten) inner product (C25). This is due to the scaling factor | | N (Λ) (cid:96) in (C25)). However, it is not possible to change this scaling factor such that a suﬃciently large class of wave functions have ﬁnite norm. The reason for this is that the conjugacy classes {α} require diﬀerent scaling behaviors depending on which type of adjoint orbit they describe. We can nevertheless apply the rigging map t(cid:32)o a basis state in the recouple(cid:33)d spin representation: (cid:90) (cid:0) (cid:1) (cid:88) (cid:89) (cid:112) − − { } η ψ{α} ({j}, I, J)L = G/stab(u) dµd(u) { } dj(cid:96) Dj(cid:96)(uα(cid:96) 1u 1)M(cid:96)N(cid:96) UI,jJ,L;{M,N}. (C32) M,N (cid:96) (cid:0) (cid:1) Note that this will not give a regular function in the indices (J, L) since the J = 0 is given by a (discrete measure) integral over a constant. This integral only makes sense if one applies η ψ{ } α to a state in the kinematical Hilbert space. Therefore, the physical inner product between two basis states can also be expressed as a sum over I, J, L and {j}. The sum over I, J, L leads, due to (C30), to Kronecker delta symbols in the indices {M, N}, and the sum over these indices leads to the representation of the (Kronecker) delta (cid:90) in the group elem(cid:0)ents,(cid:1)i.e. (cid:89) (cid:88) Λ (cid:90) 1 − − η ψ{ } ψ = dµ (u) (cid:89) l(cid:0)im (cid:1) d χ (uα u 1β 1) α β G/stab(u) d Λ→∞ N (Λ) j j (cid:96) (cid:96) (cid:96) j=0 (cid:0) (cid:1) − = dµ (u) δ uα u 1, β d (cid:96) (cid:96) G/stab(u) (cid:96) = δ [{α}], [{β}] . (C33) (cid:0) (cid:1) Appendix D: Spectrum of the translation operator for irrational angle In this appendix, we prove that the translation operator Rφ acting on L2 U(1), dµ has a spectrum (cid:0) (cid:1) (cid:0) (cid:1) d given by U(1) when the angle φ is irrational. This actually holds when equipping U(1) with either a continuous or a discrete topology. The diﬀerence between these two cases is just that the spectral measure for L2 U(1), dµ is continuous, while for L2 U(1), dµ it is discrete. d Haar Z Let us introduce the ket |n(cid:105) := ψ − for n ∈ . Note that this is not the usual eigenvector α nφ of the momentum operator for eigenvalue n. The translation operator Rφ acts as a right shift, i.e. Rφ|n(cid:105) = |n + 1(cid:105). Let eiρ ∈ U(1) for ρ ∈ [0, 2π). We are now going to show that eiρ is in the spectrum of Rφ. This is equivalent to saying that the operator Rφ − eiρ1 does not have a bounded inverse. First, if this operator has no inverse, we have the desired result and there is nothing to ##LTLine## ##LTLine## ##LTLine##

text: and

text: UIj

text: }UIj

text: δ{

text: ,J,L;

text: M,N

text: ,J,L;

text: M

text: ,N

text: M

text: M

text: N

text: N

text: I

text: ,J,L

text: We

text: can

text: then

text: change

text: our

text: spin

text: representation

text: to

text: ψ({j},

text: I,

text: J)

text: UIj

text: }ψ{j}{

text: L

text: ,J,L;

text: M,N

text: MN

text: MN

text: and

text: also

text: rewrite

text: the

text: inner

text: product

text: as

text: a

text: sum

text: over

text: the

text: {j},

text: I,

text: J,

text: L

text: indices.

text: To

text: obtain

text: the

text: (completely)

text: gauge-invariant

text: part

text: of

text: a

text: wave

text: function

text: ψ({j},

text: I,

text: J)

text: one

text: would

text: L

text: set

text: J

text: (and

text: hence

text: L

text: Note

text: however

text: that

text: a

text: wave

text: function

text: with

text: (Kronecker)

text: δ

text: behavior

text: would

text: have

text: vanishing

text: norm

text: in

text: the

text: (rewritten)

text: inner

text: product

text: This

text: is

text: due

text: to

text: the

text: scaling

text: factor

text: N

text: (Λ)

text: in

text: However,

text: it

text: is

text: not

text: possible

text: to

text: change

text: this

text: scaling

text: factor

text: such

text: that

text: a

text: suﬃciently

text: large

text: class

text: of

text: wave

text: functions

text: have

text: ﬁnite

text: norm.

text: The

text: reason

text: for

text: this

text: is

text: that

text: the

text: conjugacy

text: classes

text: {α}

text: require

text: diﬀerent

text: scaling

text: behaviors

text: depending

text: on

text: which

text: type

text: of

text: adjoint

text: orbit

text: they

text: describe.

text: We

text: can

text: nevertheless

text: apply

text: the

text: rigging

text: map

text: a

text: basis

text: state

text: in

text: the

text: spin

text: representation:

text: η

text: ψ{α}

text: ({j},

text: I,

text: J)L

text: G/stab(u)

text: dµd(u)

text: UI,jJ,L;{M,N}.

text: M,N

text: Note

text: that

text: this

text: will

text: not

text: give

text: a

text: regular

text: function

text: in

text: the

text: indices

text: (J,

text: L)

text: since

text: the

text: J

text: is

text: given

text: by

text: a

text: (discrete

text: measure)

text: integral

text: over

text: a

text: constant.

text: This

text: integral

text: only

text: makes

text: sense

text: if

text: one

text: applies

text: η

text: ψ{

text: α

text: to

text: a

text: state

text: in

text: the

text: kinematical

text: Hilbert

text: space.

text: Therefore,

text: the

text: physical

text: inner

text: product

text: between

text: two

text: basis

text: states

text: can

text: also

text: be

text: expressed

text: as

text: a

text: sum

text: over

text: I,

text: J,

text: L

text: and

text: {j}.

text: The

text: sum

text: over

text: I,

text: J,

text: L

text: leads,

text: due

text: to

text: to

text: Kronecker

text: delta

text: symbols

text: in

text: the

text: indices

text: {M,

text: N},

text: and

text: the

text: sum

text: over

text: these

text: indices

text: leads

text: to

text: the

text: representation

text: of

text: the

text: (Kronecker)

text: delta

text: in

text: the

text: group

text: Λ

text: η

text: ψ{

text: ψ

text: dµ

text: (u)

text: d

text: χ

text: (uα

text: u

text: 1β

text: α

text: β

text: G/stab(u)

text: d

text: Λ→∞

text: N

text: (Λ)

text: j

text: j

text: dµ

text: (u)

text: δ

text: uα

text: u

text: β

text: d

text: G/stab(u)

text: δ

text: [{α}],

text: [{β}]

text: Appendix

text: D:

text: Spectrum

text: of

text: the

text: translation

text: operator

text: for

text: irrational

text: angle

text: In

text: this

text: appendix,

text: we

text: prove

text: that

text: the

text: translation

text: operator

text: Rφ

text: acting

text: on

text: dµ

text: has

text: a

text: spectrum

text: d

text: given

text: by

text: when

text: the

text: angle

text: φ

text: is

text: irrational.

text: This

text: actually

text: holds

text: when

text: equipping

text: with

text: either

text: a

text: continuous

text: or

text: a

text: discrete

text: topology.

text: The

text: diﬀerence

text: between

text: these

text: two

text: cases

text: is

text: just

text: that

text: the

text: spectral

text: measure

text: for

text: dµ

text: is

text: continuous,

text: while

text: for

text: dµ

text: it

text: is

text: discrete.

text: d

text: Haar

text: Z

text: Let

text: us

text: introduce

text: the

text: ket

text: ψ

text: for

text: n

text: Note

text: that

text: this

text: is

text: not

text: the

text: usual

text: eigenvector

text: α

text: nφ

text: of

text: the

text: momentum

text: operator

text: for

text: eigenvalue

text: n.

text: The

text: translation

text: operator

text: Rφ

text: acts

text: as

text: a

text: right

text: shift,

text: i.e.

text: |n

text: Let

text: eiρ

text: for

text: ρ

text: We

text: are

text: now

text: going

text: to

text: show

text: that

text: eiρ

text: is

text: in

text: the

text: spectrum

text: of

text: Rφ.

text: This

text: is

text: equivalent

text: to

text: saying

text: that

text: the

text: operator

text: Rφ

text: does

text: not

text: have

text: a

text: bounded

text: inverse.

text: First,

text: if

text: this

text: operator

text: has

text: no

text: inverse,

text: we

text: have

text: the

text: desired

text: result

text: and

text: there

text: is

text: nothing

text: to

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 67 (cid:48)(cid:48) T is also a reﬁned tree for T . (cid:48)(cid:48) In order to prove this lemma, we need to show that the tree T satisﬁes the properties (i) and (ii) of deﬁnition VIII.1 with respect to the tree T . (cid:48) Property (i) follows from result (8.1) of lemma VIII.1, and from the fact that the trees T and (cid:48)(cid:48) (cid:48) (cid:48) (cid:48)(cid:48) (cid:48) T are reﬁnements of T and T respectively. Indeed, since P (cid:48) (T ) = T and P (cid:48)(cid:48) (cid:48)(T ) = T , (cid:48)(cid:48) ∆ ,∆ ∆ ,∆ by virtue of (8.1) we also have that P (cid:48)(cid:48) (T ) = T . (cid:0) (cid:1) ∆ ,∆ (cid:48)(cid:48) (cid:48)(cid:48) For property (ii), we have to show that any two nodes n and n , such that the dual simplices (cid:48)(cid:48) (cid:48)(cid:48) − 1 2 (cid:48)(cid:48) σd (cid:48)(cid:48)(n ) and σd (cid:48)(cid:48)(n ) are in P 1(cid:48)(cid:48) (σd ) for a ﬁxed σd , can be co(cid:0)nnected(cid:1) by a path in T without ∆ 1 ∆ 2 − ∆ ,∆ ∆ − ∆ (cid:48) crossing the boundary ∂ P 1(cid:48)(cid:48) (σd ) = P 1(cid:48)(cid:48) (∂σd ). Let (σd (cid:48)) be the simplex in ∆ containing (cid:48)(cid:48) ∆ ,∆(cid:48)(cid:48) ∆ ∆ ,∆ ∆ (cid:48)(cid:48) −∆ 1 the simplex σd (cid:48)(cid:48)(n ) dual to n , i.e. such that σd (cid:48)(cid:48)(n ) ∈ P 1(cid:48) (σd (cid:48)) (cid:0), and l(cid:1)ikewise for(cid:0) (σd (cid:48)) (cid:1). ∆ 1− 1 ∆ 1 ∆ ,∆ ∆ 1 ∆ 2 We have (σd (cid:48)) ∈ P 1(cid:48) (σd ) for i = 1, 2. Then the following two situations can occur: ∆ i ∆ ,∆ ∆ (cid:48)(cid:48) (cid:48)(cid:48) − − 1. If (σd (cid:48)) = (σd (cid:48)) , the tree path from n to n is “inside” P 1(cid:48) (σd (cid:48)) = P 1(cid:48) (σd (cid:48)) ∆ 1 ∆ 2 1 2 (cid:48) (cid:48)(cid:48) ∆ ,∆ ∆ 1 ∆ ,∆ ∆ 2 by virtue of property (ii) with respect to the pair (T , T ) of trees. Therefore, this path − does also not cross the boundary of P 1(cid:48)(cid:48) (σd ), which shows that (ii) holds in this case also (cid:48)(cid:48) ∆ ,∆ ∆ for the pair (T , T ). (cid:48)(cid:48) (cid:48)(cid:48) (cid:48)(cid:48) 2. If (σd (cid:48)) (cid:54)= (σd (cid:48)) , the path from n to n along T projects under P (cid:48)(cid:48) (cid:48) to a path along ∆ 1 (cid:48) ∆ 2 1 2 ∆ ,∆ the tree T , which is from the node dual to (σd (cid:48)) to the node dual to (σd (cid:48)) . By virtue of (cid:48) ∆ 1 ∆ 2 − property (ii) for the pair (T , T ) of trees, the path does not cross the boundary of P 1(cid:48) (σd ). ∆ (cid:48),(cid:48)∆ ∆ (cid:48)(cid:48) Together with the ﬁrst point above, this shows that also the original path from n to n (cid:48)(cid:48) − 1 2 along T does not cross the boundary of P 1(cid:48)(cid:48) (∂σd ). ∆ ,∆ ∆ (cid:48)(cid:48) (cid:4) Therefore, (ii) holds for the pair (T , T ) in general, which ﬁnishes the proof. Appendix F: Splitting of the observable algebra Lemma VIII.4. The coarse observables (8.10) and the ﬁne observables (8.11) form two mutually- commuting subalgebras of the rooted holonomy-ﬂux algebra Ar, which we denote respectively by Ar c and Ar. f The proof of lemma VIII.4 can be obtained by a direct calculation. First of all, it is clear that ∗ Ar and Ar each form a subalgebra which reproduces the Poisson structure on T G. The nontrivial c f statement is to show that these subalgebras are mutually commuting. Since holonomies are always Poisson-commuting, the holonomies in Ar have vanishing Poisson c (cid:48) (cid:48) brackets with the holonomies in Ar. In addition to this, since the leaves (cid:96) and (cid:96) are always f i I distinct, the holonomies in Ar commute with all the ﬂuxes in Ar. c f Let us now turn to the commutation relations between the ﬂuxes in Ar and the holonomies in (cid:48) − c (cid:48) Ar. In (8.11a), in addition to the leaf (cid:96) which always exists in P 1((cid:96) ), we have a leaf (cid:96) in the f i L i (cid:48) I same pre-image. Therefore, for a given (cid:96) the Poisson bracket between X and C contains two (cid:8) (cid:9) (cid:8) i (cid:9) (cid:96)i I contributions, which are in turn vanishing since (cid:48) − − − (X )k, C = Xk(cid:48) + Xk(cid:48) , g (cid:48)g (cid:48)1 = g (cid:48)τkg (cid:48)1 − g (cid:48)τkg (cid:48)1 = 0. (F1) (cid:96)i I (cid:96)i (cid:96)I (cid:96)i (cid:96)I (cid:96)i (cid:96)I (cid:96)i (cid:96)I (cid:8) (cid:9) (cid:8) (cid:9) Similarly, for (8.11b) we have that (cid:48) (X )k, C = Xk(cid:48) + Xk (cid:48) − , g (cid:48)g (cid:48) = g (cid:48)τkg (cid:48) − g (cid:48)τkg (cid:48) = 0. (F2) (cid:96)i I (cid:96)i ((cid:96)I) 1 (cid:96)i (cid:96)I (cid:96)i (cid:96)I (cid:96)i (cid:96)I

text: T

text: is

text: also

text: a

text: reﬁned

text: tree

text: for

text: T

text: In

text: order

text: to

text: prove

text: this

text: lemma,

text: we

text: need

text: to

text: show

text: that

text: the

text: tree

text: T

text: satisﬁes

text: the

text: properties

text: (i)

text: and

text: (ii)

text: of

text: deﬁnition

text: with

text: respect

text: to

text: the

text: tree

text: T

text: Property

text: (i)

text: follows

text: from

text: result

text: of

text: lemma

text: and

text: from

text: the

text: fact

text: that

text: the

text: trees

text: T

text: and

text: T

text: are

text: reﬁnements

text: of

text: T

text: and

text: T

text: respectively.

text: Indeed,

text: since

text: P

text: (T

text: T

text: and

text: P

text: T

text: by

text: virtue

text: of

text: we

text: also

text: have

text: that

text: P

text: (T

text: T

text: For

text: property

text: (ii),

text: we

text: have

text: to

text: show

text: that

text: any

text: two

text: nodes

text: n

text: and

text: n

text: such

text: that

text: the

text: dual

text: simplices

text: σd

text: and

text: σd

text: are

text: in

text: P

text: (σd

text: for

text: a

text: ﬁxed

text: σd

text: can

text: be

text: by

text: a

text: path

text: in

text: T

text: without

text: crossing

text: the

text: boundary

text: P

text: (σd

text: P

text: (∂σd

text: Let

text: (σd

text: be

text: the

text: simplex

text: in

text: containing

text: the

text: simplex

text: σd

text: dual

text: to

text: n

text: i.e.

text: such

text: that

text: σd

text: P

text: (σd

text: and

text: (σd

text: 1−

text: We

text: have

text: (σd

text: P

text: (σd

text: for

text: i

text: Then

text: the

text: following

text: two

text: situations

text: can

text: occur:

text: i

text: If

text: (σd

text: (σd

text: the

text: tree

text: path

text: from

text: n

text: to

text: n

text: is

text: “inside”

text: P

text: (σd

text: P

text: (σd

text: by

text: virtue

text: of

text: property

text: (ii)

text: with

text: respect

text: to

text: the

text: pair

text: (T

text: T

text: of

text: trees.

text: Therefore,

text: this

text: path

text: does

text: also

text: not

text: cross

text: the

text: boundary

text: of

text: P

text: (σd

text: which

text: shows

text: that

text: (ii)

text: holds

text: in

text: this

text: case

text: also

text: for

text: the

text: pair

text: (T

text: T

text: If

text: (σd

text: (σd

text: the

text: path

text: from

text: n

text: to

text: n

text: along

text: T

text: projects

text: under

text: P

text: to

text: a

text: path

text: along

text: the

text: tree

text: T

text: which

text: is

text: from

text: the

text: node

text: dual

text: to

text: (σd

text: to

text: the

text: node

text: dual

text: to

text: (σd

text: By

text: virtue

text: of

text: property

text: (ii)

text: for

text: the

text: pair

text: (T

text: T

text: of

text: trees,

text: the

text: path

text: does

text: not

text: cross

text: the

text: boundary

text: of

text: P

text: (σd

text: Together

text: with

text: the

text: ﬁrst

text: point

text: above,

text: this

text: shows

text: that

text: also

text: the

text: original

text: path

text: from

text: n

text: to

text: n

text: along

text: T

text: does

text: not

text: cross

text: the

text: boundary

text: of

text: P

text: (∂σd

text: Therefore,

text: (ii)

text: holds

text: for

text: the

text: pair

text: (T

text: T

text: in

text: general,

text: which

text: ﬁnishes

text: the

text: proof.

text: Appendix

text: F:

text: Splitting

text: of

text: the

text: observable

text: algebra

text: Lemma

text: The

text: coarse

text: observables

text: and

text: the

text: ﬁne

text: observables

text: form

text: two

text: mutually-

text: commuting

text: subalgebras

text: of

text: the

text: rooted

text: holonomy-ﬂux

text: algebra

text: Ar,

text: which

text: we

text: denote

text: respectively

text: by

text: Ar

text: c

text: and

text: Ar.

text: f

text: The

text: proof

text: of

text: lemma

text: can

text: be

text: obtained

text: by

text: a

text: direct

text: calculation.

text: First

text: of

text: all,

text: it

text: is

text: clear

text: that

text: Ar

text: and

text: Ar

text: each

text: form

text: a

text: subalgebra

text: which

text: reproduces

text: the

text: Poisson

text: structure

text: on

text: T

text: G.

text: The

text: nontrivial

text: c

text: f

text: statement

text: is

text: to

text: show

text: that

text: these

text: subalgebras

text: are

text: mutually

text: commuting.

text: Since

text: holonomies

text: are

text: always

text: Poisson-commuting,

text: the

text: holonomies

text: in

text: Ar

text: have

text: vanishing

text: Poisson

text: c

text: brackets

text: with

text: the

text: holonomies

text: in

text: Ar.

text: In

text: addition

text: to

text: this,

text: since

text: the

text: leaves

text: and

text: are

text: always

text: f

text: i

text: I

text: distinct,

text: the

text: holonomies

text: in

text: Ar

text: commute

text: with

text: all

text: the

text: ﬂuxes

text: in

text: Ar.

text: c

text: f

text: Let

text: us

text: now

text: turn

text: to

text: the

text: commutation

text: relations

text: between

text: the

text: ﬂuxes

text: in

text: Ar

text: and

text: the

text: holonomies

text: in

text: c

text: Ar.

text: In

text: in

text: addition

text: to

text: the

text: leaf

text: which

text: always

text: exists

text: in

text: P

text: we

text: have

text: a

text: leaf

text: in

text: the

text: f

text: i

text: L

text: i

text: I

text: same

text: pre-image.

text: Therefore,

text: for

text: a

text: given

text: the

text: Poisson

text: bracket

text: between

text: X

text: and

text: C

text: contains

text: two

text: i

text: I

text: contributions,

text: which

text: are

text: in

text: turn

text: vanishing

text: since

text: (X

text: )k,

text: C

text: g

text: g

text: g

text: I

text: Similarly,

text: for

text: we

text: have

text: that

text: (X

text: )k,

text: C

text: Xk

text: g

text: g

text: g

text: I


text: 68 (cid:8) (cid:9) (cid:48) For (8.11c), since the leaf (cid:96) does not belong to the pre-image of any leaf (cid:96) , it does not appear in (cid:48) I (cid:48) − i the coarse observable X and therefore the Poisson brackets X , g (cid:48)1 are identically vanishing. (cid:96)i (cid:96)i (cid:96)I This shows that the ﬂuxes in Ar commute with the holonomies in Ar. c f Finally, let us look at the commutation relations between the ﬂuxes in Ar and the ﬂuxes in Ar. (cid:48) (cid:8) − (cid:9) c f For (8.11a), since (cid:96) ∈ P 1((cid:96) ), the unique contribution coming from this ﬁner leaf to (8.9) will be I L i a ﬂux X (cid:48) . However, because of the reversed orientation of the leaf in (8.11a), the Poisson brackets (cid:96) I are vanishing since X (cid:48) , X (cid:48) − = 0. A similar reasoning applies to (8.11b). For (8.11c), since (cid:48) (cid:48) (cid:96)I ((cid:96)I) 1 (cid:96) ∈ L , the ﬂux X (cid:48) − never appears in (8.9) and the Poisson brackets are therefore identically I 0 ((cid:96)I) 1 (cid:4) vanishing. [1] T. Regge, “General Relativity Without Coordinates”, Nuovo Cim. 19, 558 (1961). [2] B. Bahr and B. Dittrich, “(Broken) gauge symmetries and constraints in Regge calculus”, Class. Quant. Grav. 26 225011 (2009), arXiv:0905.1670 [gr-qc]. [3] B. Dittrich and S. Steinhaus, “Path integral measure and triangulation independence in discrete gravity”, Phys. Rev. D 85 044032 (2012), arXiv:1110.6866 [gr-qc]. [4] B. Dittrich, W. Kaminski and S. Steinhaus, “Discretization independence implies non-locality in 4D discrete quantum gravity,” Class. Quant. Grav. 31 245009 (2014), arXiv:1404.5288[gr-qc]. [5] C. Rovelli, Quantum Gravity, (Cambridge University Press, Cambridge, 2004). [6] A. Ashtekar and J. Lewandowski, “Background independent quantum gravity: a status report”, Class. Quant. Grav. 21 R53 (2004), arXiv:gr-qc/0404018. [7] T. Thiemann, Introduction to Modern Canonical Quantum General Relativity, (Cambridge University Press, Cambridge, 2007). [8] A. Ashtekar and C. J. Isham, “Representations of the holonomy algebras of gravity and non-Abelian gauge theories”, Class. Quantum Grav. 9 1433 (1992), arXiv:hep-th/9202053. [9] A. Ashtekar and J. Lewandowski, “Representation theory of analytic holonomy C* algebras”, in J. Baez, ed., Knots and Quantum Gravity, (Oxford University Press, 1994), arXiv:gr-qc/9311010. [10] A. Ashtekar and J. Lewandowski, “Projective techniques and functional integration for gauge theories”, J. Math. Phys. 36 (1995) 2170, arXiv:gr-qc/9411046. [11] A. Ashtekar and J. Lewandowski, “Diﬀerential geometry on the space of connections via graphs and projective limits”, J. Geom. Phys. 17 (1995) 191, arXiv:hep-th/9412073. [12] J. F. Barbero, “Real Ashtekar variables for Lorentzian signature spacetimes”, Phys. Rev. D 51 5507 (1995). [13] G. Immirzi, “Real and complex connections for canonical gravity”, Class. Quant. Grav. 14 L177 (1997), arXiv:gr-qc/9612030. [14] A. Ashtekar, “Some surprising implications of background independence in canonical quantum grav- ity”, Gen. Rel. Grav. 41 1927 (2009), arXiv:0904.0184 [gr-qc]. [15] T. Thiemann, “Anomaly-free formulation of nonperturbative, four-dimensional Lorentzian quantum gravity”, Phys. Lett. B 380 257 (1996), arXiv:gr-qc/9606088. [16] T. Thiemann, “Quantum spin dynamics (QSD)”, Class. Quant. Grav. 15 839 (1998), arXiv:gr-qc/9606089. [17] B. Dittrich and M. Geiller, “A new vacuum for loop quantum gravity”, Class. Quant. Grav. (2014), arXiv:1401.6441 [gr-qc]. [18] B. Dittrich and M. Geiller, “Flux formulation of loop quantum gravity: Classical framework”, Class. Quant. Grav. (2015), arXiv:1412.3752 [gr-qc]. [19] B. Dittrich and J. P. Ryan, “Phase space descriptions for simplicial 4d geometries”, Class. Quant. Grav. 28 065006 (2011), arXiv:0807.2806 [gr-qc]. [20] B. Dittrich and J. P. Ryan, “Simplicity in simplicial phase space”, Phys. Rev. D 82 064026 (2010), arXiv:1006.4295 [gr-qc]. [21] L. Freidel and S. Speziale, “Twisted geometries: A geometric parametrisation of SU(2) phase space”, Phys. Rev. D 82 084040 (2010), arXiv:1001.2748 [gr-qc]. [22] L. Freidel, M. Geiller and J. Ziprick, “Continuous formulation of the loop quantum gravity phase ##LTLine## ##LTLine## ##LTLine##

text: For

text: since

text: the

text: leaf

text: does

text: not

text: belong

text: to

text: the

text: pre-image

text: of

text: any

text: leaf

text: it

text: does

text: not

text: appear

text: in

text: I

text: i

text: the

text: coarse

text: observable

text: X

text: and

text: therefore

text: the

text: Poisson

text: brackets

text: X

text: g

text: are

text: identically

text: vanishing.

text: This

text: shows

text: that

text: the

text: ﬂuxes

text: in

text: Ar

text: commute

text: with

text: the

text: holonomies

text: in

text: Ar.

text: c

text: f

text: Finally,

text: let

text: us

text: look

text: at

text: the

text: commutation

text: relations

text: between

text: the

text: ﬂuxes

text: in

text: Ar

text: and

text: the

text: ﬂuxes

text: in

text: Ar.

text: c

text: f

text: For

text: since

text: P

text: the

text: unique

text: contribution

text: coming

text: from

text: this

text: ﬁner

text: leaf

text: to

text: will

text: be

text: I

text: L

text: i

text: a

text: ﬂux

text: X

text: However,

text: because

text: of

text: the

text: reversed

text: orientation

text: of

text: the

text: leaf

text: in

text: the

text: Poisson

text: brackets

text: I

text: are

text: vanishing

text: since

text: X

text: X

text: A

text: similar

text: reasoning

text: applies

text: to

text: For

text: since

text: L

text: the

text: ﬂux

text: X

text: never

text: appears

text: in

text: and

text: the

text: Poisson

text: brackets

text: are

text: therefore

text: identically

text: I

text: vanishing.

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


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text: of

text: loop

text: quantum

text: gravity

text: a

text: framework

text: for

text: solving

text: the

text: theory”,

text: in

text: A.

text: Ashtekar

text: and

text: J.

text: Pullin,

text: ed.,

text: to

text: be

text: published

text: in

text: the

text: World

text: Scientiﬁc

text: series

text: Years

text: of

text: General

text: Relativity”,

text: [gr-qc].

text: B.

text: Bahr

text: and

text: B.

text: Dittrich,

text: “Improved

text: and

text: Perfect

text: Actions

text: in

text: Discrete

text: Gravity”,

text: Phys.

text: Rev.

text: D

text: [gr-qc].

text: B.

text: Bahr,

text: B.

text: Dittrich

text: and

text: S.

text: Steinhaus,

text: “Perfect

text: discretization

text: of

text: reparametrization

text: invariant

text: path

text: integrals”,

text: Phys.

text: Rev.

text: D

text: [gr-qc].

text: B.

text: Bahr

text: and

text: B.

text: Dittrich,

text: “Breaking

text: and

text: restoring

text: of

text: diﬀeomorphism

text: symmetry

text: in

text: discrete

text: gravity”,

text: Proceedings

text: of

text: the

text: XXV

text: Max

text: Born

text: Symposium

text: ”The

text: Planck

text: Scale”,

text: Wroclaw,

text: June

text: July,


text: 9 One can also introduce a quantum deformation of the group at root of unity, i.e. work with SU(2) . In this case, the cutoﬀ on the spin is implemented explicitly. The charge labels q are then described by the Drinfeld center of the SU(2) module category, which has only q ﬁnitely many objects [54]. In three spacetime dimensions, the corresponding vacuum is given by the Turaev–Viro state sum model [55] and describes (Euclidean) gravity with a positive cosmological constant. Future work [56] will be concerned with constructing a continuum Hilbert space representation based on quantum groups. III. SETUP In this section we brieﬂy summarize the setup of our construction, and present the notations and notions which are used throughout the rest of the article. We refer the reader to the companion paper [18] for more details. Let Σ be a d-dimensional closed (i.e. compact and without boundary) manifold, which we choose to be real, analytic, orientable, and path-connected. We will be interested in particular in the cases d = 2 and d = 3, which we treat simultaneously. The manifold Σ can be thought of as a spatial boundary, and our aim is to construct states corresponding to d-dimensional boundary quantum geometries on it. For this, we consider a principal G-bundle P : Σ → G, where the gauge group G is an arbitrary compact Lie group whose Lie algebra will be denoted by g. If G is simply-connected (which is the case for SU(2), the gauge group of interest in this work), since we consider the case d ≤ 3 it is always possible to pick a global trivialization of P , which we do for simplicity. The space A of smooth G connections is then aﬃne over Ω1(Σ, g), the space of g-valued one-form ﬁelds over Σ, and the group of local gauge transformations is given by G = Map(Σ, G). ∗ On the phase space of LQG, a connection A ∈ A is canonically conjugated to a g -valued vector density E (the electric ﬁeld) of weight +1. If we write A = Ai τ dxa and E = Eaτi∂ , where a i i a {xa} are coordinates on Σ and {τ } forms a basis of g, the non-vanishing Poisson a=1,...,d (cid:8) i i=1,...,(cid:9)dimg brackets are given by Ai (x), Eb(y) = δbδiδd(x, y). (3.1) a j a j Following the logic of LQG, we are going to consider holonomies of the connection and ﬂuxes of the electric ﬁeld as our elementary variables. However, instead of assigning this data to arbitrary oriented embedded (dual) graphs and (d − 1)-dimensional submanifolds, as is done in the usual construction of the holonomy-ﬂux (cid:63)-algebra, here we adopt a simplicial point of view and focus rather on the triangulation itself. We now explain the type of discrete data which is relevant for our construction. We consider a ﬁducial Riemannian metric on the manifold Σ, and the inﬁnite set of embedded (or geometric) triangulations ∆ deﬁned by this metric. In both d = 2 and d = 3 dimensions, these embeddings refer to the vertices of the triangulations. The edges and (in d = 3) the triangles of a given triangulation are deﬁned respectively as geodesics and minimal surfaces with respect to the auxiliary metric. We furthermore assume that the triangulations are ﬁne enough so that these geodesics and minimal surfaces are unique. To each triangulation ∆ we assign a dual graph Γ, which is the one-skeleton of the cellular complex dual to ∆. This dual graph is such that its nodes n ∈ Γ are dual to the d-dimensional − simplices σd of ∆, while its links l ∈ Γ are dual to the (d − 1)-dimensional simplices σd 1 of ∆. For each link we choose a ﬁducial orientation, and denote by l(0) and l(1) the source and target nodes. To each pair given by a triangulation and its dual graph we then assign a root, which is a choice of a preferred d-dimensional root simplex σd, or equivalently (by duality) a choice of a preferred r

text: One

text: can

text: also

text: introduce

text: a

text: quantum

text: deformation

text: of

text: the

text: group

text: at

text: root

text: of

text: unity,

text: i.e.

text: work

text: with

text: In

text: this

text: case,

text: the

text: cutoﬀ

text: on

text: the

text: spin

text: is

text: implemented

text: explicitly.

text: The

text: charge

text: labels

text: q

text: are

text: then

text: described

text: by

text: the

text: Drinfeld

text: center

text: of

text: the

text: module

text: category,

text: which

text: has

text: only

text: q

text: ﬁnitely

text: many

text: objects

text: In

text: three

text: spacetime

text: dimensions,

text: the

text: corresponding

text: vacuum

text: is

text: given

text: by

text: the

text: Turaev–Viro

text: state

text: sum

text: model

text: and

text: describes

text: (Euclidean)

text: gravity

text: with

text: a

text: positive

text: cosmological

text: constant.

text: Future

text: work

text: will

text: be

text: concerned

text: with

text: constructing

text: a

text: continuum

text: Hilbert

text: space

text: representation

text: based

text: on

text: quantum

text: groups.

text: III.

text: SETUP

text: In

text: this

text: section

text: we

text: brieﬂy

text: summarize

text: the

text: setup

text: of

text: our

text: construction,

text: and

text: present

text: the

text: notations

text: and

text: notions

text: which

text: are

text: used

text: throughout

text: the

text: rest

text: of

text: the

text: article.

text: We

text: refer

text: the

text: reader

text: to

text: the

text: companion

text: paper

text: for

text: more

text: details.

text: Let

text: Σ

text: be

text: a

text: d-dimensional

text: closed

text: (i.e.

text: compact

text: and

text: without

text: boundary)

text: manifold,

text: which

text: we

text: choose

text: to

text: be

text: real,

text: analytic,

text: orientable,

text: and

text: path-connected.

text: We

text: will

text: be

text: interested

text: in

text: particular

text: in

text: the

text: cases

text: d

text: and

text: d

text: which

text: we

text: treat

text: simultaneously.

text: The

text: manifold

text: Σ

text: can

text: be

text: thought

text: of

text: as

text: a

text: spatial

text: boundary,

text: and

text: our

text: aim

text: is

text: to

text: construct

text: states

text: corresponding

text: to

text: d-dimensional

text: boundary

text: quantum

text: geometries

text: on

text: it.

text: For

text: this,

text: we

text: consider

text: a

text: principal

text: G-bundle

text: P

text: Σ

text: G,

text: where

text: the

text: gauge

text: group

text: G

text: is

text: an

text: arbitrary

text: compact

text: Lie

text: group

text: whose

text: Lie

text: algebra

text: will

text: be

text: denoted

text: by

text: g.

text: If

text: G

text: is

text: simply-connected

text: (which

text: is

text: the

text: case

text: for

text: the

text: gauge

text: group

text: of

text: interest

text: in

text: this

text: work),

text: since

text: we

text: consider

text: the

text: case

text: d

text: it

text: is

text: always

text: possible

text: to

text: pick

text: a

text: global

text: trivialization

text: of

text: P

text: which

text: we

text: do

text: for

text: simplicity.

text: The

text: space

text: A

text: of

text: smooth

text: G

text: connections

text: is

text: then

text: aﬃne

text: over

text: g),

text: the

text: space

text: of

text: g-valued

text: one-form

text: ﬁelds

text: over

text: Σ,

text: and

text: the

text: group

text: of

text: local

text: gauge

text: transformations

text: is

text: given

text: by

text: G

text: Map(Σ,

text: G).

text: On

text: the

text: phase

text: space

text: of

text: LQG,

text: a

text: connection

text: A

text: A

text: is

text: canonically

text: conjugated

text: to

text: a

text: g

text: -valued

text: vector

text: density

text: E

text: (the

text: electric

text: ﬁeld)

text: of

text: weight

text: If

text: we

text: write

text: A

text: Ai

text: τ

text: dxa

text: and

text: E

text: Eaτi∂

text: where

text: a

text: i

text: i

text: a

text: {xa}

text: are

text: coordinates

text: on

text: Σ

text: and

text: {τ

text: forms

text: a

text: basis

text: of

text: g,

text: the

text: non-vanishing

text: Poisson

text: i

text: brackets

text: are

text: given

text: by

text: Ai

text: (x),

text: Eb(y)

text: δbδiδd(x,

text: y).

text: a

text: j

text: a

text: j

text: Following

text: the

text: logic

text: of

text: LQG,

text: we

text: are

text: going

text: to

text: consider

text: holonomies

text: of

text: the

text: connection

text: and

text: ﬂuxes

text: of

text: the

text: electric

text: ﬁeld

text: as

text: our

text: elementary

text: variables.

text: However,

text: instead

text: of

text: assigning

text: this

text: data

text: to

text: arbitrary

text: oriented

text: embedded

text: (dual)

text: graphs

text: and

text: (d

text: submanifolds,

text: as

text: is

text: done

text: in

text: the

text: usual

text: construction

text: of

text: the

text: holonomy-ﬂux

text: here

text: we

text: adopt

text: a

text: simplicial

text: point

text: of

text: view

text: and

text: focus

text: rather

text: on

text: the

text: triangulation

text: itself.

text: We

text: now

text: explain

text: the

text: type

text: of

text: discrete

text: data

text: which

text: is

text: relevant

text: for

text: our

text: construction.

text: We

text: consider

text: a

text: ﬁducial

text: Riemannian

text: metric

text: on

text: the

text: manifold

text: Σ,

text: and

text: the

text: inﬁnite

text: set

text: of

text: embedded

text: (or

text: geometric)

text: triangulations

text: deﬁned

text: by

text: this

text: metric.

text: In

text: both

text: d

text: and

text: d

text: dimensions,

text: these

text: embeddings

text: refer

text: to

text: the

text: vertices

text: of

text: the

text: triangulations.

text: The

text: edges

text: and

text: (in

text: d

text: the

text: triangles

text: of

text: a

text: given

text: triangulation

text: are

text: deﬁned

text: respectively

text: as

text: geodesics

text: and

text: minimal

text: surfaces

text: with

text: respect

text: to

text: the

text: auxiliary

text: metric.

text: We

text: furthermore

text: assume

text: that

text: the

text: triangulations

text: are

text: ﬁne

text: enough

text: so

text: that

text: these

text: geodesics

text: and

text: minimal

text: surfaces

text: are

text: unique.

text: To

text: each

text: triangulation

text: we

text: assign

text: a

text: dual

text: graph

text: Γ,

text: which

text: is

text: the

text: one-skeleton

text: of

text: the

text: cellular

text: complex

text: dual

text: to

text: This

text: dual

text: graph

text: is

text: such

text: that

text: its

text: nodes

text: n

text: Γ

text: are

text: dual

text: to

text: the

text: d-dimensional

text: simplices

text: σd

text: of

text: while

text: its

text: links

text: l

text: Γ

text: are

text: dual

text: to

text: the

text: (d

text: simplices

text: σd

text: of

text: For

text: each

text: link

text: we

text: choose

text: a

text: ﬁducial

text: orientation,

text: and

text: denote

text: by

text: and

text: the

text: source

text: and

text: target

text: nodes.

text: To

text: each

text: pair

text: given

text: by

text: a

text: triangulation

text: and

text: its

text: dual

text: graph

text: we

text: then

text: assign

text: a

text: root,

text: which

text: is

text: a

text: choice

text: of

text: a

text: preferred

text: d-dimensional

text: root

text: simplex

text: σd,

text: or

text: equivalently

text: (by

text: duality)

text: a

text: choice

text: of

text: a

text: preferred

text: r