text: 14 the particle couples√to the quantum metric itself and thus the eﬀective action will also include an extra contribution= ∫Γnl prop−ort[io−nal to m. This (t−er◻m) is given− by (−◻)( + ) m ( ) ( ) ( ) + (−1◻) + (−◻)1 ] 1 Γnl d4x g m log Mρσµν m log Pρσµν Rgρ µ g ν σ m 32π2 µνρσ 32π2 µνρσ 6 A βi R log Tµν B βi Rg log → Tµν → (4.10) 2 µν m 2 µν m with mµνρσ equal to Mµνρσ with the replacem⃗ ent M m and yµ zν − ∇⃗ = In the non-relativistic limit the quantum corrected geodesic equation is given by ⃗ d2z 1 1 δΓnl h m (4.11) dt2 2 00 m δz ⃗ where both Γnl and h depend on the parameterisation. However taking into account both of these m 00 contributions one ﬁnds that d2z is independent of the parameterisation. Explicitly one ﬁnds that the dVt2( ) = − [ + − + ] potential (4.2) is given by G M 43G 5G 7G r N 1 N N N (4.12) r 30πr2 12πr2 12πr2 where the eﬀect of coupling the test particle is to cancel the parameterisation dependence and contribute the last term. Thus we observe that, unlike the beta functions, the Newtonian potential is independent of the parameterisation of the quantum ﬁelds. V. SEMI-CLASSICAL FLOW EQUATION We now return to the main topic of this paper namely obtaining a gauge independent beta function for > Newton’s constant. To this end we will use the results of sections and utilising the parameterisation (3.5). = The starting point is the functional integral (2.27) on an arbitrary Einstein background with the constant mode φ removed (in d 2 this mode will not renormalise the Newton’s constant but should modify 0 curvature squared corrections in d 4). Here we shall use the modern formalism of the non-perturbative renormalisation group [32–34], even though it is not technically needed since we are working at one-loop. This approach can then be compared to the old fashioned renormalisation scheme of simply evaluating (3.8) [ ] = ⋅ R ⋅ and regulating the one loop integrals. To this end we add a regulator term, R 1 < ∆S ϕ ϕ ϕ, (5.1) ≫ k 2 k = { } to the bare action. Here k is an infra red regulator which suppresses l⊥ow momentum modes p2 k2 while vanishing for high momentum modes p2 k2. The scale k plays the role of µ as the scale which deﬁnes = { } the beta functions. In the case where we use (3.8) the ﬁelds ϕ h ,ζ,c¯,c are all ﬁelds appearing in = { } R the bare action after exponentiating the determinants in functional measure. Alternatively we can use the R eﬀective actions (3.16) or (3.20) for the unconstrained ﬁelds in which case we have ϕ hˆ ,σ,C¯ ,C or µν µ µ ϕ h ,C¯ ,C . In any case ϕ can be thought of as a vector in ﬁeld space with components ϕ and =µ{ν µ }µ a k as a matrix with components . ab For the covariant momentum p2 we take the eigenvalues of the diﬀerential operators of the operators ∆ ( ) + R ( ) = ( → ) = + ( / ) n for n 1,2,s (given (in) equations (2.13), (2(.2)3) and (3.13)). The regulator then takes a form such that ( ) ≫ S 2 ∆ ∆ S 2 ∆ P , P ∆ k2C ∆ k2 , (5.2) → n k n n n n n n −∇ where C z is a dimensionless scalar cutoﬀ function which vanishes for z 1 but stays ﬁnite and non-zero for z 0. This constitutes a type-II cutoﬀ, in the nomenclature of [35], where also the potential terms of ∆ n are included in the regulator. Another choice, termed type I, is to replace ∆ by 2 in (5.2), however we n expect a non-trivial cancellation to occur between the traces of the transverse traceless ﬂuctuations and the ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: the

text: particle

text: couples√to

text: the

text: quantum

text: metric

text: itself

text: and

text: thus

text: the

text: eﬀective

text: action

text: will

text: also

text: include

text: an

text: extra

text: contribution=

text: ∫Γnl

text: prop−ort[io−nal

text: to

text: m.

text: This

text: (t−er◻m)

text: is

text: given−

text: by

text: m

text: Γnl

text: g

text: m

text: log

text: Mρσµν

text: m

text: log

text: Pρσµν

text: Rgρ

text: µ

text: g

text: ν

text: σ

text: m

text: µνρσ

text: µνρσ

text: A

text: βi

text: R

text: log

text: Tµν

text: B

text: βi

text: Rg

text: log

text: Tµν

text: µν

text: m

text: µν

text: m

text: with

text: mµνρσ

text: equal

text: to

text: Mµνρσ

text: with

text: the

text: replacem⃗

text: ent

text: M

text: m

text: and

text: yµ

text: zν

text: In

text: the

text: non-relativistic

text: limit

text: the

text: quantum

text: corrected

text: geodesic

text: equation

text: is

text: given

text: by

text: δΓnl

text: h

text: m

text: m

text: δz

text: where

text: both

text: Γnl

text: and

text: h

text: depend

text: on

text: the

text: parameterisation.

text: However

text: taking

text: into

text: account

text: both

text: of

text: these

text: m

text: contributions

text: one

text: ﬁnds

text: that

text: is

text: independent

text: of

text: the

text: parameterisation.

text: Explicitly

text: one

text: ﬁnds

text: that

text: the

text: potential

text: is

text: given

text: by

text: G

text: M

text: r

text: N

text: N

text: N

text: N

text: r

text: where

text: the

text: eﬀect

text: of

text: coupling

text: the

text: test

text: particle

text: is

text: to

text: cancel

text: the

text: parameterisation

text: dependence

text: and

text: contribute

text: the

text: last

text: term.

text: Thus

text: we

text: observe

text: that,

text: unlike

text: the

text: beta

text: functions,

text: the

text: Newtonian

text: potential

text: is

text: independent

text: of

text: the

text: parameterisation

text: of

text: the

text: quantum

text: ﬁelds.

text: V.

text: SEMI-CLASSICAL

text: FLOW

text: EQUATION

text: We

text: now

text: return

text: to

text: the

text: main

text: topic

text: of

text: this

text: paper

text: namely

text: obtaining

text: a

text: gauge

text: independent

text: beta

text: function

text: for

text: Newton’s

text: constant.

text: To

text: this

text: end

text: we

text: will

text: use

text: the

text: results

text: of

text: sections

text: and

text: utilising

text: the

text: parameterisation

text: The

text: starting

text: point

text: is

text: the

text: functional

text: integral

text: on

text: an

text: arbitrary

text: Einstein

text: background

text: with

text: the

text: constant

text: mode

text: φ

text: removed

text: (in

text: d

text: this

text: mode

text: will

text: not

text: renormalise

text: the

text: Newton’s

text: constant

text: but

text: should

text: modify

text: curvature

text: squared

text: corrections

text: in

text: d

text: Here

text: we

text: shall

text: use

text: the

text: modern

text: formalism

text: of

text: the

text: non-perturbative

text: renormalisation

text: group

text: even

text: though

text: it

text: is

text: not

text: technically

text: needed

text: since

text: we

text: are

text: working

text: at

text: one-loop.

text: This

text: approach

text: can

text: then

text: be

text: compared

text: to

text: the

text: old

text: fashioned

text: renormalisation

text: scheme

text: of

text: simply

text: evaluating

text: R

text: and

text: regulating

text: the

text: one

text: loop

text: integrals.

text: To

text: this

text: end

text: we

text: add

text: a

text: regulator

text: term,

text: R

text: ∆S

text: ϕ

text: ϕ

text: ϕ,

text: k

text: k

text: to

text: the

text: bare

text: action.

text: Here

text: k

text: is

text: an

text: infra

text: red

text: regulator

text: which

text: suppresses

text: l⊥ow

text: momentum

text: modes

text: while

text: vanishing

text: for

text: high

text: momentum

text: modes

text: The

text: scale

text: k

text: plays

text: the

text: role

text: of

text: µ

text: as

text: the

text: scale

text: which

text: deﬁnes

text: the

text: beta

text: functions.

text: In

text: the

text: case

text: where

text: we

text: use

text: the

text: ﬁelds

text: ϕ

text: h

text: ,ζ,c¯,c

text: are

text: all

text: ﬁelds

text: appearing

text: in

text: R

text: the

text: bare

text: action

text: after

text: exponentiating

text: the

text: determinants

text: in

text: functional

text: measure.

text: Alternatively

text: we

text: can

text: use

text: the

text: R

text: eﬀective

text: actions

text: or

text: for

text: the

text: unconstrained

text: ﬁelds

text: in

text: which

text: case

text: we

text: have

text: ϕ

text: hˆ

text: ,σ,C¯

text: ,C

text: or

text: µν

text: µ

text: µ

text: ϕ

text: h

text: ,C¯

text: ,C

text: In

text: any

text: case

text: ϕ

text: can

text: be

text: thought

text: of

text: as

text: a

text: vector

text: in

text: ﬁeld

text: space

text: with

text: components

text: ϕ

text: and

text: =µ{ν

text: µ

text: }µ

text: a

text: k

text: as

text: a

text: matrix

text: with

text: components

text: ab

text: For

text: the

text: covariant

text: momentum

text: we

text: take

text: the

text: eigenvalues

text: of

text: the

text: diﬀerential

text: operators

text: of

text: the

text: operators

text: R

text: n

text: for

text: n

text: (given

text: (in)

text: equations

text: and

text: The

text: regulator

text: then

text: takes

text: a

text: form

text: such

text: that

text: S

text: S

text: P

text: P

text: n

text: k

text: n

text: n

text: n

text: n

text: n

text: n

text: where

text: C

text: z

text: is

text: a

text: dimensionless

text: scalar

text: cutoﬀ

text: function

text: which

text: vanishes

text: for

text: z

text: but

text: stays

text: ﬁnite

text: and

text: non-zero

text: for

text: z

text: This

text: constitutes

text: a

text: type-II

text: cutoﬀ,

text: in

text: the

text: nomenclature

text: of

text: where

text: also

text: the

text: potential

text: terms

text: of

text: n

text: are

text: included

text: in

text: the

text: regulator.

text: Another

text: choice,

text: termed

text: type

text: I,

text: is

text: to

text: replace

text: by

text: in

text: however

text: we

text: n

text: expect

text: a

text: non-trivial

text: cancellation

text: to

text: occur

text: between

text: the

text: traces

text: of

text: the

text: transverse

text: traceless

text: ﬂuctuations

text: and

text: the

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: I [ ] 18 ( ) − ( ) where the regulator depeInd[ent]f≡un∫cti∞onal n− C (is)g=iv∫en∞by the/ i−ntegral+ ( ′) C z zC z C dzzn 1W z dzzd 2 2 . (7.2) n k 0 0 I z[ C] z / − The ﬁrst term in (7.1) is the classical scaling arising from the dimensionful nature of G . The second term k is a quantum correction pr=opo+rtional to=the momentum integral C indicating that this equation is d 2 1 one loop. A. d 2 (cid:15) and d 2 dimensions, a tale of three beta functions → → Before continuing to the general d case we take a digression into the behaviour of the beta function near two dimensions and try to understand the subtleties of the limit d 2. First we note that in the limit d 2 I [ ]∣ = + we have − / − → 2 C ﬁniteterms (7.3) → d 2 1 d 2 d 2 = + = where the singular term is independent of the regulator function C. However the coeﬃcient of G2 has a ﬁnite universal limit for d 2 owing to the pres=enc+e of the gamma function. This allows one to extract a universal beta function in d 2 (cid:15) dimensions and in the case d 2. = + 1. d 2 (cid:15) dimensions = − For d 2 (cid:15) we ﬁnd the universal beta function 38 β (cid:15)G G2 . (7.4) G 3 = / − = We note that this is in agreement with previous studies [38–40] using the linear parameterisation (2.5) but diﬀers from the result obtained with the exponential parameterisation (2.6) [9, 41] with ρ d 2 1 and = − λ¯ 0 which gives b 50 = β (cid:15)G G2 . (7.5) ∝ G( − ) 3 ( ) The reason for this is that by working oﬀ-shell and setting λ¯ 0 the exponential parameterisation gives a b hessian for the scalar mode φ becomes S 2 d 2√∆ and thus the cancellation between these modes and J does not occur. Instead the hessian foφrφ∝the∫conforma(l mo+des mimi)cs the induced action 0 = φ¯ 1 S d2x g R φ¯∆φ¯ (7.6) anom 4 16 → of the conformal anomaly in d 2 despite being of order (cid:15) [41]. In the equations leading to (7.4) such ﬂuctuations are not present since all conformal ﬂuctuations are cancelled by the functional measure. The = result (7.5) can then be understood as arising from the particular way the limit d 2 is taken where the fact that the hessian for φ¯ vanishes is not taken into account. The limit then paradoxically reproduces the expected result in d 2 in the presence of the conformal anomaly even though the induced action (7.6) for has not been accounted for. In order to help clarify the situation we brieﬂy recall the derivations of (7.4) given in [40] and generalise it to an arbitrary parameterisation. The derivation diﬀers in two respects from the methods used in this paper. Firstly dimensional regularisation is used such that only the logarithmic divergences are retained and secondly the freedom to redeﬁne the back√ground metric is exploited in order to deﬁne a physical beta [ ] = ∫ [− ⋅ + ( − )] function. Thus ﬁrst one obtains the one loop equation 1 38 2d ∂ Γ g d4x gµ(cid:15) R ¯b R λ¯ (7.7) t µν 16π 3 (cid:15) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: I

label: paragraph

text: where

text: the

text: regulator

text: depeInd[ent]f≡un∫cti∞onal

text: n−

text: C

text: (is)g=iv∫en∞by

text: the/

text: i−ntegral+

text: C

text: z

text: zC

text: z

text: C

text: dzzn

text: z

text: dzzd

text: n

text: k

text: I

text: z[

text: C]

text: z

text: The

text: ﬁrst

text: term

text: in

text: is

text: the

text: classical

text: scaling

text: arising

text: from

text: the

text: dimensionful

text: nature

text: of

text: G

text: The

text: second

text: term

text: k

text: is

text: a

text: quantum

text: correction

text: pr=opo+rtional

label: section

text: to=the

text: momentum

text: integral

text: C

text: indicating

text: that

text: this

text: equation

text: is

text: d

text: one

text: loop.

text: A.

text: d

text: and

text: d

text: dimensions,

text: a

text: tale

text: of

text: three

text: beta

text: functions

text: Before

text: continuing

text: to

text: the

text: general

text: d

text: case

text: we

text: take

text: a

text: digression

text: into

text: the

text: behaviour

text: of

text: the

text: beta

text: function

text: near

text: two

text: dimensions

text: and

text: try

text: to

text: understand

text: the

text: subtleties

text: of

text: the

text: limit

text: d

text: First

text: we

text: note

text: that

text: in

text: the

text: limit

text: d

text: I

text: we

text: have

text: C

text: ﬁniteterms

text: d

text: d

text: d

text: where

text: the

text: singular

text: term

text: is

text: independent

text: of

text: the

text: regulator

text: function

text: C.

text: However

text: the

text: coeﬃcient

text: of

text: has

text: a

text: ﬁnite

text: universal

text: limit

text: for

text: d

text: owing

text: to

text: the

text: pres=enc+e

text: of

text: the

text: gamma

text: function.

text: This

text: allows

text: one

text: to

text: extract

text: a

text: universal

text: beta

text: function

text: in

text: d

text: dimensions

text: and

text: in

text: the

text: case

text: d

text: d

text: dimensions

text: For

text: d

text: we

text: ﬁnd

text: the

text: universal

text: beta

text: function

text: β

text: G

text: We

text: note

text: that

text: this

text: is

text: in

text: agreement

text: with

text: previous

text: studies

text: using

text: the

text: linear

text: parameterisation

text: but

text: diﬀers

text: from

text: the

text: result

text: obtained

text: with

text: the

text: exponential

text: parameterisation

text: with

text: ρ

text: d

text: and

text: λ¯

text: which

text: gives

text: b

text: β

text: G(

text: The

text: reason

text: for

text: this

text: is

text: that

text: by

text: working

text: oﬀ-shell

text: and

text: setting

text: λ¯

text: the

text: exponential

text: parameterisation

text: gives

text: a

text: b

text: hessian

text: for

text: the

text: scalar

text: mode

text: φ

text: becomes

text: S

text: d

text: 2√∆

text: and

text: thus

text: the

text: cancellation

text: between

text: these

text: modes

text: and

text: J

text: does

text: not

text: occur.

text: Instead

text: the

text: hessian

text: foφrφ∝the∫conforma(l

text: mo+des

text: mimi)cs

text: the

text: induced

text: action

text: φ¯

text: S

text: g

text: R

text: φ¯∆φ¯

text: anom

text: of

text: the

text: conformal

text: anomaly

text: in

text: d

text: despite

text: being

text: of

text: order

text: In

text: the

text: equations

text: leading

text: to

text: such

text: ﬂuctuations

text: are

text: not

text: present

text: since

text: all

text: conformal

text: ﬂuctuations

text: are

text: cancelled

text: by

text: the

text: functional

text: measure.

text: The

text: result

text: can

text: then

text: be

text: understood

text: as

text: arising

text: from

text: the

text: particular

text: way

text: the

text: limit

text: d

text: is

text: taken

text: where

text: the

text: fact

text: that

text: the

text: hessian

text: for

text: φ¯

text: vanishes

text: is

text: not

text: taken

text: into

text: account.

text: The

text: limit

text: then

text: paradoxically

text: reproduces

text: the

text: expected

text: result

text: in

text: d

text: in

text: the

text: presence

text: of

text: the

text: conformal

text: anomaly

text: even

text: though

text: the

text: induced

text: action

text: for

text: has

text: not

text: been

text: accounted

text: for.

text: In

text: order

text: to

text: help

text: clarify

text: the

text: situation

text: we

text: brieﬂy

text: recall

text: the

text: derivations

text: of

text: given

text: in

text: and

text: generalise

text: it

text: to

text: an

text: arbitrary

text: parameterisation.

text: The

text: derivation

text: diﬀers

text: in

text: two

text: respects

text: from

text: the

text: methods

text: used

text: in

text: this

text: paper.

text: Firstly

text: dimensional

text: regularisation

text: is

text: used

text: such

text: that

text: only

text: the

text: logarithmic

text: divergences

text: are

text: retained

text: and

text: secondly

text: the

text: freedom

text: to

text: redeﬁne

text: the

text: back√ground

text: metric

text: is

text: exploited

text: in

text: order

text: to

text: deﬁne

text: a

text: physical

text: beta

text: function.

text: Thus

text: ﬁrst

text: one

text: obtains

text: the

text: one

text: loop

text: equation

text: Γ

text: g

text: R

text: ¯b

text: R

text: λ¯

text: t

text: µν

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 3 I. INTRODUCTION = In quantum ﬁeld theory it is well known that coupling constants become functions of the energy scales entering the renormalisation process. In turn this implies a modiﬁcation of the classical scaling properties of a theory [1]. Such energy dependence of a coupling a is encoded in its beta functions β µ ∂ a, where µ a ∂µ is the renormalisation scale. In quantum chromodynamics (QCD) this non-trivial scaling can be observed = −( − ) in the one-loop beta function [2, 3], 2 α2 ≤ β 11 N s , (1.1) αs 3 f 2π > where α is the strong coupling and N denotes the number of ﬂavours. For N 16 this equation describes s f f the weakening of the strong force as the energy scale µ is increased. On the other hand if we have N 16 the f coupling constant will diverge at a ﬁnite energy. In the former case the theory is said to be asymptotically free and is well deﬁned for all scales. In the later case the coupling has a Landau pole and the theory can only be considered as an eﬀective one, at energies below the pole. When considering gravity as a quantum ﬁeld theory it is natural to ask whether there exists an analog = to (1.1) for Newton’s constant GN. In four spacetime dimensions Newton’s constant is dimensio−nful which implies that the structure of the beta function diﬀers from that of QCD, where α is dimensionless. In s particular, for d spacetime dimensions one should work with the dimensionless coupling G µd 2G . The = ( − ) − N analog of (1.1) will then take the form β d 2 G bG2 , (1.2) G − for some constant b. Whereas the coeﬃcient of G just reﬂects the dimensionality of G the coeﬃcient b N is a quantum correction 1. Clearly b plays the same role as 11 2N in the QCD; if b is positive then 3 f G will remain ﬁnite for all scales, but if b is negative G will blow up at a ﬁnite energy. Thus, although non-perturbative eﬀects may change the naive conclusion, the sign of b is a strong indiction of whether = = quantum gravity should be replaced by some new theory at a ﬁnite energy scale, or whether gravity may = ( − ) ∗ itself constitute a perfectly valid quantum ﬁeld theory. The latter scenario, dubbed asymptotic safety by ∗ Weinberg [4], would involve a ﬁxed point for G where β 0 for some positive value G G . At the level of G the one-loop beta function (1.2) this is found at G 1 d 2 and its existence only depends on the sign of b. b However, one must go beyond one-loop in order to be sure that such a ﬁxed point exists non-perturbatively. Although β has been calculated previously, in these calculations the value of b has been found to depend G on the gauge [5–8] and parameterisation [8, 9] (an exception are calculations with the geometrical eﬀective action [10], which are gauge independent although a particular parameterisation of the ﬁelds is used). Hence these calculations will generically include unphysical degrees of freedom coming from the gauge ﬁxing action and may not be trust worthy. This issue stems from the fact that calculations have to be performed oﬀ-shell and therefore generally depend on both the gauge and the parameterisation. When working on-shell such dependencies vanish. On the other hand the Einstein equations dictate that the scalar curvature is given = by − 2d R λ¯ , (1.3) d 2 where λ¯ is the cosmological constant. As a consequence one cannot disentangle the renormalisation of Newton’s constant from the vacuum energy, forcing us to work oﬀ-shell to determine b. Thus, it is clear that by working oﬀ-shell, in a gauge dependent setup, no deﬁnitive conclusion can be reached as to the sign of b. On the other hand explicit non-perturbative calculations both in four and higher dimensions indicate ̵ ̵ = = 1 b is proportional to h, although here we work in natural units h 1 c. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: I.

text: INTRODUCTION

text: In

text: quantum

text: ﬁeld

text: theory

text: it

text: is

text: well

text: known

text: that

text: coupling

text: constants

text: become

text: functions

text: of

text: the

text: energy

text: scales

text: entering

text: the

text: renormalisation

text: process.

text: In

text: turn

text: this

text: implies

text: a

text: modiﬁcation

text: of

text: the

text: classical

text: scaling

text: properties

text: of

text: a

text: theory

text: Such

text: energy

text: dependence

text: of

text: a

text: coupling

text: a

text: is

text: encoded

text: in

text: its

text: beta

text: functions

text: β

text: µ

text: a,

text: where

text: µ

text: a

text: ∂µ

text: is

text: the

text: renormalisation

text: scale.

text: In

text: quantum

text: chromodynamics

text: (QCD)

text: this

text: non-trivial

text: scaling

text: can

text: be

text: observed

text: in

text: the

text: one-loop

text: beta

text: function

text: α2

text: β

text: N

text: s

text: αs

text: f

text: 2π

text: where

text: α

text: is

text: the

text: strong

text: coupling

text: and

text: N

text: denotes

text: the

text: number

text: of

text: ﬂavours.

text: For

text: N

text: this

text: equation

text: describes

text: s

text: f

text: f

text: the

text: weakening

text: of

text: the

text: strong

text: force

text: as

text: the

text: energy

text: scale

text: µ

text: is

text: increased.

text: On

text: the

text: other

text: hand

text: if

text: we

text: have

text: N

text: the

text: f

text: coupling

text: constant

text: will

text: diverge

text: at

text: a

text: ﬁnite

text: energy.

text: In

text: the

text: former

text: case

text: the

text: theory

text: is

text: said

text: to

text: be

text: asymptotically

text: free

text: and

text: is

text: well

text: deﬁned

text: for

text: all

text: scales.

text: In

text: the

text: later

text: case

text: the

text: coupling

text: has

text: a

text: Landau

text: pole

text: and

text: the

text: theory

text: can

text: only

text: be

text: considered

text: as

text: an

text: eﬀective

text: one,

text: at

text: energies

text: below

text: the

text: pole.

text: When

text: considering

text: gravity

text: as

text: a

text: quantum

text: ﬁeld

text: theory

text: it

text: is

text: natural

text: to

text: ask

text: whether

text: there

text: exists

text: an

text: analog

text: to

text: for

text: Newton’s

text: constant

text: GN.

text: In

text: four

text: spacetime

text: dimensions

text: Newton’s

text: constant

text: is

text: dimensio−nful

text: which

text: implies

text: that

text: the

text: structure

text: of

text: the

text: beta

text: function

text: diﬀers

text: from

text: that

text: of

text: QCD,

text: where

text: α

text: is

text: dimensionless.

text: In

text: s

text: particular,

text: for

text: d

text: spacetime

text: dimensions

text: one

text: should

text: work

text: with

text: the

text: dimensionless

text: coupling

text: G

text: µd

text: The

text: N

text: analog

text: of

text: will

text: then

text: take

text: the

text: form

text: β

text: d

text: G

text: G

text: for

text: some

text: constant

text: b.

text: Whereas

text: the

text: coeﬃcient

text: of

text: G

text: just

text: reﬂects

text: the

text: dimensionality

text: of

text: G

text: the

text: coeﬃcient

text: b

text: N

text: is

text: a

text: quantum

text: correction

text: Clearly

text: b

text: plays

text: the

text: same

text: role

text: as

text: in

text: the

text: QCD;

text: if

text: b

text: is

text: positive

text: then

text: f

text: G

text: will

text: remain

text: ﬁnite

text: for

text: all

text: scales,

text: but

text: if

text: b

text: is

text: negative

text: G

text: will

text: blow

text: up

text: at

text: a

text: ﬁnite

text: energy.

text: Thus,

text: although

text: non-perturbative

text: eﬀects

text: may

text: change

text: the

text: naive

text: conclusion,

text: the

text: sign

text: of

text: b

text: is

text: a

text: strong

text: indiction

text: of

text: whether

text: quantum

text: gravity

text: should

text: be

text: replaced

text: by

text: some

text: new

text: theory

text: at

text: a

text: ﬁnite

text: energy

text: scale,

text: or

text: whether

text: gravity

text: may

text: itself

text: constitute

text: a

text: perfectly

text: valid

text: quantum

text: ﬁeld

text: theory.

text: The

text: latter

text: scenario,

text: dubbed

text: asymptotic

text: safety

text: by

text: Weinberg

text: would

text: involve

text: a

text: ﬁxed

text: point

text: for

text: G

text: where

text: β

text: for

text: some

text: positive

text: value

text: G

text: G

text: At

text: the

text: level

text: of

text: G

text: the

text: one-loop

text: beta

text: function

text: this

text: is

text: found

text: at

text: G

text: d

text: and

text: its

text: existence

text: only

text: depends

text: on

text: the

text: sign

text: of

text: b.

text: b

text: However,

text: one

text: must

text: go

text: beyond

text: one-loop

text: in

text: order

text: to

text: be

text: sure

text: that

text: such

text: a

text: ﬁxed

text: point

text: exists

text: non-perturbatively.

text: Although

text: β

text: has

text: been

text: calculated

text: previously,

text: in

text: these

text: calculations

text: the

text: value

text: of

text: b

text: has

text: been

text: found

text: to

text: depend

text: G

text: on

text: the

text: gauge

text: and

text: parameterisation

text: (an

text: exception

text: are

text: calculations

text: with

text: the

text: geometrical

text: eﬀective

text: action

text: which

text: are

text: gauge

text: independent

text: although

text: a

text: particular

text: parameterisation

text: of

text: the

text: ﬁelds

text: is

text: used).

text: Hence

text: these

text: calculations

text: will

text: generically

text: include

text: unphysical

text: degrees

text: of

text: freedom

text: coming

text: from

text: the

text: gauge

text: ﬁxing

text: action

text: and

text: may

text: not

text: be

text: trust

text: worthy.

text: This

text: issue

text: stems

text: from

text: the

text: fact

text: that

text: calculations

text: have

text: to

text: be

text: performed

text: oﬀ-shell

text: and

text: therefore

text: generally

text: depend

text: on

text: both

text: the

text: gauge

text: and

text: the

text: parameterisation.

text: When

text: working

text: on-shell

text: such

text: dependencies

text: vanish.

text: On

text: the

text: other

text: hand

text: the

text: Einstein

text: equations

text: dictate

text: that

text: the

text: scalar

text: curvature

text: is

text: given

text: by

text: R

text: λ¯

text: d

text: where

text: λ¯

text: is

text: the

text: cosmological

text: constant.

text: As

text: a

text: consequence

text: one

text: cannot

text: disentangle

text: the

text: renormalisation

text: of

text: Newton’s

text: constant

text: from

text: the

text: vacuum

text: energy,

text: forcing

text: us

text: to

text: work

text: oﬀ-shell

text: to

text: determine

text: b.

text: Thus,

text: it

text: is

text: clear

text: that

text: by

text: working

text: oﬀ-shell,

text: in

text: a

text: gauge

text: dependent

text: setup,

text: no

text: deﬁnitive

text: conclusion

text: can

text: be

text: reached

text: as

text: to

text: the

text: sign

text: of

text: b.

text: On

text: the

text: other

text: hand

text: explicit

text: non-perturbative

text: calculations

text: both

text: in

text: four

text: and

text: higher

text: dimensions

text: indicate

text: b

text: is

text: proportional

text: to

text: h,

text: although

text: here

text: we

text: work

text: in

text: natural

text: units

text: h

text: c.

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text: ≡ 22 = = ( − ) + introducing th→e bare UV scale Λ deﬁned by λk Λ λb we c∗an solve this equation to ﬁnd,≡ λk kd Λd λ λb . = (8.4) In the limit k 0 we remove the IR regulator to=obta+in the observed vacuum energy λ λ . In turn we 0 k 0 may express the bare vacuum energy in terms of λ0 as ∗ λ λ Λdλ . (8.5) b 0 Thus the ‘beta function’ (10.4) is nothing but the statement that we must include a counter term proportional to the number of local degrees of freedom in order to get a ﬁnite renormalised vacuum energy λ . → ∞ 0 IX. CONTINUUM LIMIT < As discussed previously the existence of a continuum limit, Λ , relies on the sign of b, the G2 coeﬃcient of β . To understand this we note that the sign of the bare Newtons constant G must be positive if the G = b functional integral is to make sense. If one is forced to take G 0 to renormalise the theory one must b concede that the theory cannot be fundamen=ta(l and−that t)he c+ontinuum limit does not exist. From the (7.1) one may infer the energy dependence of the Plan−ck mas−s κ 1 , = k∗ 16πGk = ∗ → κk kd 2 Λd 2 κ κb≡, = (9.1) − ∗ − where κ 1 is the regulator dependent value of the ﬁxed point for the=dimensionless quantity κ 16πG kva2ludeκko.f TNaekwintognks con0sotnaenth.aTs thheenrweneohramvaelised P=lanck mas+s M−Pdl 2∗ 16π1GN κk 0 where GN is the measured 1 1 > Λd 2κ . (9.2) 16πG 16πG b N for the functional integral to make sense one must >ha−ve Gb 0 otherwise the kinetic terms will have the wrong sign and the theory will breakdown. In particular f−or the semiclassical theory to be valid we need ∗ ∝ ( − ) 1 Λd 2κ . (9.3) ∗> 16πG N Now we know that κ 18 N and that the proportionality constant is positive from which we obtain g ( ) − < ∣ ∣ the inequality for N 18 g ∗ < Λ d 2 > 16π G . (9.4) M Pl Thus, while for 18 N there is no maximum value for which the UV scale can take, for N 18 (9.4) g g provides a bound. In the latter case we can say that the semiclassical theory predicts its own downfall at a ﬁnite energy scale. In the former case no bound exists and the semiclassical approximation suggests that the theory is asymptotically safe. X. INCLUSION OF MATTER It is rather straight forward to include the eﬀects of matter. This is done by adding a matter action to the bare Lagrangian and adding a corresponding regulator term (see for example [47]) or by computing the = regulated one loop eﬀective action [48]. Doing the former adds new traces to the right hand side of (5.6) proportional to the number of matte=r ﬁeld−s. (In t−urn th+ese give−con)tributions to the beta functions. Here we give the result in d 4 spacetime dimensions, although the result is easily generalised to arbitrary d, 4 2 β 2G 16 2N 4N N G2 , (10.1) G D M s 3 4 Here we assume the ﬁelds are massless and minimally coupled to gravity. The beta function will be modiﬁed if these assumptions are dropped. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: introducing

text: th→e

text: bare

text: UV

text: scale

text: Λ

text: deﬁned

text: by

text: λk

text: Λ

text: λb

text: we

text: c∗an

text: solve

text: this

text: equation

text: to

text: ﬁnd,≡

text: λk

text: kd

text: Λd

text: λ

text: λb

text: In

text: the

text: limit

text: k

text: we

text: remove

text: the

text: IR

text: regulator

text: to=obta+in

text: the

text: observed

text: vacuum

text: energy

text: λ

text: λ

text: In

text: turn

text: we

text: k

text: may

text: express

text: the

text: bare

text: vacuum

text: energy

text: in

text: terms

text: of

text: λ0

text: as

text: λ

text: λ

text: Λdλ

text: b

text: Thus

text: the

text: ‘beta

text: function’

text: is

text: nothing

text: but

text: the

text: statement

text: that

text: we

text: must

text: include

text: a

text: counter

text: term

text: proportional

text: to

text: the

text: number

text: of

text: local

text: degrees

text: of

text: freedom

text: in

text: order

text: to

text: get

text: a

text: ﬁnite

text: renormalised

text: vacuum

text: energy

text: λ

text: IX.

text: CONTINUUM

text: LIMIT

text: As

text: discussed

text: previously

text: the

text: existence

text: of

text: a

text: continuum

text: limit,

text: Λ

text: relies

text: on

text: the

text: sign

text: of

text: b,

text: the

text: coeﬃcient

text: of

text: β

text: To

text: understand

text: this

text: we

text: note

text: that

text: the

text: sign

text: of

text: the

text: bare

text: Newtons

text: constant

text: G

text: must

text: be

text: positive

text: if

text: the

text: G

text: b

text: functional

text: integral

text: is

text: to

text: make

text: sense.

text: If

text: one

text: is

text: forced

text: to

text: take

text: G

text: to

text: renormalise

text: the

text: theory

text: one

text: must

text: b

text: concede

text: that

text: the

text: theory

text: cannot

text: be

text: fundamen=ta(l

text: and−that

text: t)he

text: c+ontinuum

text: limit

text: does

text: not

text: exist.

text: From

text: the

text: one

text: may

text: infer

text: the

text: energy

text: dependence

text: of

text: the

text: Plan−ck

text: mas−s

text: κ

text: k∗

text: κk

text: kd

text: Λd

text: κ

text: κb≡,

text: where

text: κ

text: is

text: the

text: regulator

text: dependent

text: value

text: of

text: the

text: ﬁxed

text: point

text: for

text: the=dimensionless

text: quantity

text: κ

text: TNaekwintognks

text: thheenrweneohramvaelised

text: P=lanck

text: mas+s

text: M−Pdl

text: 2∗

text: κk

text: where

text: GN

text: is

text: the

text: measured

text: Λd

text: 2κ

text: b

text: N

text: for

text: the

text: functional

text: integral

text: to

text: make

text: sense

text: one

text: must

text: >ha−ve

text: Gb

text: otherwise

text: the

text: kinetic

text: terms

text: will

text: have

text: the

text: wrong

text: sign

text: and

text: the

text: theory

text: will

text: breakdown.

text: In

text: particular

text: f−or

text: the

text: semiclassical

text: theory

text: to

text: be

text: valid

text: we

text: need

text: Λd

text: 2κ

text: N

text: Now

text: we

text: know

text: that

text: κ

text: N

text: and

text: that

text: the

text: proportionality

text: constant

text: is

text: positive

text: from

text: which

text: we

text: obtain

text: g

text: the

text: inequality

text: for

text: N

text: g

text: Λ

text: d

text: G

text: M

text: Pl

text: Thus,

text: while

text: for

text: N

text: there

text: is

text: no

text: maximum

text: value

text: for

text: which

text: the

text: UV

text: scale

text: can

text: take,

text: for

text: N

text: g

text: g

text: provides

text: a

text: bound.

text: In

text: the

text: latter

text: case

text: we

text: can

text: say

text: that

text: the

text: semiclassical

text: theory

text: predicts

text: its

text: own

text: downfall

text: at

text: a

text: ﬁnite

text: energy

text: scale.

text: In

text: the

text: former

text: case

text: no

text: bound

text: exists

text: and

text: the

text: semiclassical

text: approximation

text: suggests

text: that

text: the

text: theory

text: is

text: asymptotically

text: safe.

text: X.

text: INCLUSION

text: OF

text: MATTER

text: It

text: is

text: rather

text: straight

text: forward

text: to

text: include

text: the

text: eﬀects

text: of

text: matter.

text: This

text: is

text: done

text: by

text: adding

text: a

text: matter

text: action

text: to

text: the

text: bare

text: Lagrangian

text: and

text: adding

text: a

text: corresponding

text: regulator

text: term

text: (see

text: for

text: example

text: or

text: by

text: computing

text: the

text: regulated

text: one

text: loop

text: eﬀective

text: action

text: Doing

text: the

text: former

text: adds

text: new

text: traces

text: to

text: the

text: right

text: hand

text: side

text: of

text: proportional

text: to

text: the

text: number

text: of

text: matte=r

text: ﬁeld−s.

text: (In

text: t−urn

text: th+ese

text: give−con)tributions

text: to

text: the

text: beta

text: functions.

text: Here

text: we

text: give

text: the

text: result

text: in

text: d

text: spacetime

text: dimensions,

text: although

text: the

text: result

text: is

text: easily

text: generalised

text: to

text: arbitrary

text: d,

text: β

text: N

text: G

text: D

text: M

text: s

text: Here

text: we

text: assume

text: the

text: ﬁelds

text: are

text: massless

text: and

text: minimally

text: coupled

text: to

text: gravity.

text: The

text: beta

text: function

text: will

text: be

text: modiﬁed

text: if

text: these

text: assumptions

text: are

text: dropped.

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text: ##LTLine##

text: ##LTLine##


text: 9 survive in the on-shell semiclassical approximation, and which modes cancel in the functional measure, we understand which ﬁelds are carrying physics, and which ﬁelds are only present due to us necessarily breaking diﬀeomorphism invariance. In order not to be forced to expand around a solution to the bare equations of motion we will now drop = the condition (2.4) and instead simply require that the background is an Einstein space, 1 R g R . (3.1) µν µν d As pointed out in the introduction this is crucial if we are to be able to extract the renormalisation of Newton’s constant. A. Gauge independence Our aim is to preserve the gauge independence of the on-shell functional integral (2.27) such that physically meaningful results may be derived oﬀ-shell. The problem that we face is that by working oﬀ-shell the cancellations between the measure and the 2d unphysical degrees of freedom is not guaranteed. For a generic parameterisation terms proportional to the equation of motion arise in the hessians and in the presence of such terms the cancellations observed in the previous section will not occur. In particular the = + ( − ) hessians have the general form, ( ) ( ) − 2d S 2 S˜ 2 X R λ¯ , (3.2) b d 2 for any arbitrary parameterisation, where X is a mat−rix in ﬁeld space that depends on the parameterisation = − ( ) given by d 2 ∂S 2 ( ) X 2d ∂λ¯ . ( ) (3.3) b ≠ which then implicitly deﬁnes S˜ 2 . While X is dependent on the parameterisation S˜ 2 is parameterisation in(de)pendent. = At this point we make two observations, if X 0 the gauge ﬁxing dependence will remain and furthermore ≠ S(2) will possess negative eigenmodes depending on λb. Thus, only by picking a parametrisation for which X 0 will the gauge ﬁxing dependence vanish and the Gaussian integrals converge ((as)suming positivity of = S 2 for the chosen background). By picking a parameterisation for which X 0 the corresponding eﬀective action will not formally exist for certain values of λ¯ where negative eigenvalues of S 2 are present, as well b as being gauge dependent. We are therefore led to the conclusion that a parameterisation for which X 0 is the key to obtaining a physical meaningful gauge independent one-loop result. In turn this removes any dependence of the hessians on the cosmological constant. 3 To ﬁnd such a parametrisation we ﬁrst note that by using the exponential parameterisation (2.6) the equation of motion will only appear in the hessians for ψ and φ. In fact, if we revert to the the parameteri- sation in terms of ψ¯ and φ¯, (see (2.11)) the equation of motion will only appear in the hessian for φ¯ due to (2.8), this ﬁeld being identiﬁed as the conformal factor of the −metric. Then we may choose to parameterise = ( + ) = + + + this scalar degree of freedom in terms of a ﬁeld σ related to the conformal factor by eφd¯ 1 σ d2 1 σ 2 dσ2 ... (3.4) 2 d 4d2 = ( + ) [ ] = + + + such that γ 1 σ d2 g ehˆ λ g hˆ g σ ..., (3.5) µν 2 µλ ν µν µν µν d 3 We note that the dependence on the cosmological constant can also be removed by use of speciﬁc gauge ﬁxing conditions [8] whereby the conformal ﬂuctuations φ¯ are constrained. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: survive

text: in

text: the

text: on-shell

text: semiclassical

text: approximation,

text: and

text: which

text: modes

text: cancel

text: in

text: the

text: functional

text: measure,

text: we

text: understand

text: which

text: ﬁelds

text: are

text: carrying

text: physics,

text: and

text: which

text: ﬁelds

text: are

text: only

text: present

text: due

text: to

text: us

text: necessarily

text: breaking

text: diﬀeomorphism

text: invariance.

text: In

text: order

text: not

text: to

text: be

text: forced

text: to

text: expand

text: around

text: a

text: solution

text: to

text: the

text: bare

text: equations

text: of

text: motion

text: we

text: will

text: now

text: drop

text: the

text: condition

text: and

text: instead

text: simply

text: require

text: that

text: the

text: background

text: is

text: an

text: Einstein

text: space,

text: R

text: g

text: R

text: µν

text: µν

text: d

text: As

text: pointed

text: out

text: in

text: the

text: introduction

text: this

text: is

text: crucial

text: if

text: we

text: are

text: to

text: be

text: able

text: to

text: extract

text: the

text: renormalisation

text: of

text: Newton’s

text: constant.

text: A.

text: Gauge

text: independence

text: Our

text: aim

text: is

text: to

text: preserve

text: the

text: gauge

text: independence

text: of

text: the

text: on-shell

text: functional

text: integral

text: such

text: that

text: physically

text: meaningful

text: results

text: may

text: be

text: derived

text: oﬀ-shell.

text: The

text: problem

text: that

text: we

text: face

text: is

text: that

text: by

text: working

text: oﬀ-shell

text: the

text: cancellations

text: between

text: the

text: measure

text: and

text: the

text: unphysical

text: degrees

text: of

text: freedom

text: is

text: not

text: guaranteed.

text: For

text: a

text: generic

text: parameterisation

text: terms

text: proportional

text: to

text: the

text: equation

text: of

text: motion

text: arise

text: in

text: the

text: hessians

text: and

text: in

text: the

text: presence

text: of

text: such

text: terms

text: the

text: cancellations

text: observed

text: in

text: the

text: previous

text: section

text: will

text: not

text: occur.

text: In

text: particular

text: the

text: hessians

text: have

text: the

text: general

text: form,

text: S

text: S˜

text: X

text: R

text: λ¯

text: b

text: d

text: for

text: any

text: arbitrary

text: parameterisation,

text: where

text: X

text: is

text: a

text: mat−rix

text: in

text: ﬁeld

text: space

text: that

text: depends

text: on

text: the

text: parameterisation

text: given

text: by

text: d

text: ∂S

text: X

text: ∂λ¯

text: b

text: which

text: then

text: implicitly

text: deﬁnes

text: S˜

text: While

text: X

text: is

text: dependent

text: on

text: the

text: parameterisation

text: S˜

text: is

text: parameterisation

text: in(de)pendent.

text: At

text: this

text: point

text: we

text: make

text: two

text: observations,

text: if

text: X

text: the

text: gauge

text: ﬁxing

text: dependence

text: will

text: remain

text: and

text: furthermore

text: will

text: possess

text: negative

text: eigenmodes

text: depending

text: on

text: λb.

text: Thus,

text: only

text: by

text: picking

text: a

text: parametrisation

text: for

text: which

text: X

text: will

text: the

text: gauge

text: ﬁxing

text: dependence

text: vanish

text: and

text: the

text: Gaussian

text: integrals

text: converge

text: ((as)suming

text: positivity

text: of

text: S

text: for

text: the

text: chosen

text: background).

text: By

text: picking

text: a

text: parameterisation

text: for

text: which

text: X

text: the

text: corresponding

text: eﬀective

text: action

text: will

text: not

text: formally

text: exist

text: for

text: certain

text: values

text: of

text: λ¯

text: where

text: negative

text: eigenvalues

text: of

text: S

text: are

text: present,

text: as

text: well

text: b

text: as

text: being

text: gauge

text: dependent.

text: We

text: are

text: therefore

text: led

text: to

text: the

text: conclusion

text: that

text: a

text: parameterisation

text: for

text: which

text: X

text: is

text: the

text: key

text: to

text: obtaining

text: a

text: physical

text: meaningful

text: gauge

text: independent

text: one-loop

text: result.

text: In

text: turn

text: this

text: removes

text: any

text: dependence

text: of

text: the

text: hessians

text: on

text: the

text: cosmological

text: constant.

text: To

text: ﬁnd

text: such

text: a

text: parametrisation

text: we

text: ﬁrst

text: note

text: that

text: by

text: using

text: the

text: exponential

text: parameterisation

text: the

text: equation

text: of

text: motion

text: will

text: only

text: appear

text: in

text: the

text: hessians

text: for

text: ψ

text: and

text: φ.

text: In

text: fact,

text: if

text: we

text: revert

text: to

text: the

text: the

text: parameteri-

text: sation

text: in

text: terms

text: of

text: ψ¯

text: and

text: φ¯,

text: (see

text: the

text: equation

text: of

text: motion

text: will

text: only

text: appear

text: in

text: the

text: hessian

text: for

text: φ¯

text: due

text: to

text: this

text: ﬁeld

text: being

text: identiﬁed

text: as

text: the

text: conformal

text: factor

text: of

text: the

text: −metric.

text: Then

text: we

text: may

text: choose

text: to

text: parameterise

text: this

text: scalar

text: degree

text: of

text: freedom

text: in

text: terms

text: of

text: a

text: ﬁeld

text: σ

text: related

text: to

text: the

text: conformal

text: factor

text: by

text: eφd¯

text: σ

text: σ

text: d

text: such

text: that

text: γ

text: σ

text: g

text: ehˆ

text: λ

text: g

text: hˆ

text: g

text: σ

text: µν

text: µλ

text: ν

text: µν

text: µν

text: µν

text: d

text: We

text: note

text: that

text: the

text: dependence

text: on

text: the

text: cosmological

text: constant

text: can

text: also

text: be

text: removed

text: by

text: use

text: of

text: speciﬁc

text: gauge

text: ﬁxing

text: conditions

text: whereby

text: the

text: conformal

text: ﬂuctuations

text: φ¯

text: are

text: constrained.

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: MINIMALITY AND MUTATION-EQUIVALENCE OF POLYGONS 11 p #q “ t u “ t u Deﬁnition 3.14. Given a seed B, C with B e , . . . , e and C x , . . . , x , the jth mutation p q p 1 1q 1 “ t 1 11u n 1 “ t 1 11 u n of B, C is the seed B , C , where B e , . . . , e and C x , . . . , x are deﬁned by: 1 n 1 n ´ “ 1 “ e , if k j j e ` p q k e max b , 0 e , otherwźise ź k kj j “ t u where b e , e , kl k l 1 “ ‰ 1 “ ` x x if k j, and x x xbjk xblj. k k j j k l k suchąthat l suchăthat bjk 0 bjl 0 § A Recall from that there is a quiver Q and a cluster algebra associated to a Fano polygon P. p q “ p Bq P P Let SC P n, and ﬁx a numbering of the n primitive T-cones in the spanning fan of P. The ith primitive T-cone in the spanning fan of P corresponds to a vertex v of Q , and thus corresponds p q A i P to a basis element e in B , where B , C is the initial seed for . Let E denote the edge of P i P P P P i determined by the ith primitive T-cone; note that diﬀerent primitive T-cones can determine the same edge. Proposition 3.15 (Mutations of seeds induce mutations of polygons). Let P be a Fano polygon with p Bq P singularity content n, . Denote by w M the primitive inner normal vector to E . Consider Ñ p q “ i p q i A the map π: L M such that π e w for each i. Let B , C be the inital seed for , and “ t u “ t i ui p 1 1q P P p q P write B e , . . . , e , C x , . . . , x . Let B , C be the jth mutation of B , C , and write 1 “ t 1 1 1 u n t p 1 q 1 p 1nqu P P 1 B e , . . . , e . Then π e , . . . , π e are the primitive inner normal vectors to the edges of P , 11 n 1 n 1 where P is the mutation of P determined by w . Furthermore, every mutation P of P arises in this j way. § (cid:3) Proof. This is a straightforward calculation using mutation in M (see ). There is a well-known notion of quiver mutation, going back to Bernstein–Gelfand–Ponomarev [9], Fomin–Zelevinsky [16], and others. p q Deﬁnition 3.16. Given a quiver Q and a vertex v of Q, the mutation of Q at v is the quiver mut Q, v obtained from Q by: Ñ Ñ (i) adding, for each subquiver v v v , an arrow from v to v ; 1 2 1 2 (ii) deleting a maximal set of disjoint two-cycles; (iii) reversing all arrows incident to v. The resulting quiver is well-deﬁned up to isomorphism, regardless of the choice of two-cycles in (ii). Proposition 3.17 (Mutations of polygons induce mutations of quivers). Let P be a Fano polygon, let 1 v be a vertex of Q corresponding to a primitive T-cone in P, and let P be the corresponding mutation P “ p q of P. We have Q 1 mut Q , v . P P Proof. Let E denote the edge of P determined by the primitive T-cone corresponding to v, and let P w M denote the primitive inner normal vector to E. Mutation with respect to w acts on M as a piecewise-linear transformation that is the identity in one half-space, and on the other half-space is ÞÑ ` p ^ q a shear transformation u u w u w. Thus determinants between the pairs of normal vectors change as follows: ´ (i) The inner normal vector w to the mutating edge E becomes w, so that all arrows into v change direction; (ii) For a pair of normal vectors in the same half-space (as deﬁned by w), the determinant does not change;

text: MINIMALITY

text: AND

text: MUTATION-EQUIVALENCE

text: OF

text: POLYGONS

text: p

text: #q

text: t

text: u

text: t

text: u

text: Deﬁnition

text: Given

text: a

text: seed

text: B,

text: C

text: with

text: B

text: e

text: e

text: and

text: C

text: x

text: x

text: the

text: jth

text: mutation

text: p

text: q

text: p

text: t

text: n

text: t

text: u

text: n

text: of

text: B,

text: C

text: is

text: the

text: seed

text: B

text: C

text: where

text: B

text: e

text: e

text: and

text: C

text: x

text: x

text: are

text: deﬁned

text: by:

text: n

text: n

text: e

text: if

text: k

text: j

text: j

text: e

text: p

text: q

text: k

text: e

text: max

text: b

text: e

text: otherwźise

text: ź

text: k

text: kj

text: j

text: t

text: u

text: where

text: b

text: e

text: e

text: kl

text: k

text: l

text: x

text: x

text: if

text: k

text: j,

text: and

text: x

text: x

text: xbjk

text: xblj.

text: k

text: k

text: j

text: j

text: k

text: l

text: k

text: suchąthat

text: l

text: suchăthat

text: bjk

text: bjl

text: A

text: Recall

text: from

text: that

text: there

text: is

text: a

text: quiver

text: Q

text: and

text: a

text: cluster

text: algebra

text: associated

text: to

text: a

text: Fano

text: polygon

text: P.

text: p

text: q

text: p

text: Bq

text: P

text: P

text: Let

text: SC

text: P

text: n,

text: and

text: ﬁx

text: a

text: numbering

text: of

text: the

text: n

text: primitive

text: T-cones

text: in

text: the

text: spanning

text: fan

text: of

text: P.

text: The

text: ith

text: primitive

text: T-cone

text: in

text: the

text: spanning

text: fan

text: of

text: P

text: corresponds

text: to

text: a

text: vertex

text: v

text: of

text: Q

text: and

text: thus

text: corresponds

text: p

text: q

text: A

text: i

text: P

text: to

text: a

text: basis

text: element

text: e

text: in

text: B

text: where

text: B

text: C

text: is

text: the

text: initial

text: seed

text: for

text: Let

text: E

text: denote

text: the

text: edge

text: of

text: P

text: i

text: P

text: P

text: P

text: P

text: i

text: determined

text: by

text: the

text: ith

text: primitive

text: T-cone;

text: note

text: that

text: diﬀerent

text: primitive

text: T-cones

text: can

text: determine

text: the

text: same

text: edge.

text: Proposition

text: (Mutations

text: of

text: seeds

text: induce

text: mutations

text: of

text: polygons).

text: Let

text: P

text: be

text: a

text: Fano

text: polygon

text: with

text: p

text: Bq

text: P

text: singularity

text: content

text: n,

text: Denote

text: by

text: w

text: M

text: the

text: primitive

text: inner

text: normal

text: vector

text: to

text: E

text: Consider

text: Ñ

text: p

text: q

text: i

text: p

text: q

text: i

text: A

text: the

text: map

text: π:

text: L

text: M

text: such

text: that

text: π

text: e

text: w

text: for

text: each

text: i.

text: Let

text: B

text: C

text: be

text: the

text: inital

text: seed

text: for

text: and

text: t

text: u

text: t

text: i

text: ui

text: p

text: P

text: P

text: p

text: q

text: P

text: write

text: B

text: e

text: e

text: C

text: x

text: x

text: Let

text: B

text: C

text: be

text: the

text: jth

text: mutation

text: of

text: B

text: C

text: and

text: write

text: t

text: u

text: n

text: t

text: p

text: q

text: p

text: P

text: P

text: B

text: e

text: e

text: Then

text: π

text: e

text: π

text: e

text: are

text: the

text: primitive

text: inner

text: normal

text: vectors

text: to

text: the

text: edges

text: of

text: P

text: n

text: n

text: where

text: P

text: is

text: the

text: mutation

text: of

text: P

text: determined

text: by

text: w

text: Furthermore,

text: every

text: mutation

text: P

text: of

text: P

text: arises

text: in

text: this

text: j

text: way.

text: Proof.

text: This

text: is

text: a

text: straightforward

text: calculation

text: using

text: mutation

text: in

text: M

text: (see

text: There

text: is

text: a

text: well-known

text: notion

text: of

text: quiver

text: mutation,

text: going

text: back

text: to

text: Bernstein–Gelfand–Ponomarev

text: Fomin–Zelevinsky

text: and

text: others.

text: p

text: q

text: Deﬁnition

text: Given

text: a

text: quiver

text: Q

text: and

text: a

text: vertex

text: v

text: of

text: Q,

text: the

text: mutation

text: of

text: Q

text: at

text: v

text: is

text: the

text: quiver

text: mut

text: Q,

text: v

text: obtained

text: from

text: Q

text: by:

text: Ñ

text: Ñ

text: (i)

text: adding,

text: for

text: each

text: subquiver

text: v

text: v

text: v

text: an

text: arrow

text: from

text: v

text: to

text: v

text: (ii)

text: deleting

text: a

text: maximal

text: set

text: of

text: disjoint

text: two-cycles;

text: (iii)

text: reversing

text: all

text: arrows

text: incident

text: to

text: v.

text: The

text: resulting

text: quiver

text: is

text: well-deﬁned

text: up

text: to

text: isomorphism,

text: regardless

text: of

text: the

text: choice

text: of

text: two-cycles

text: in

text: (ii).

text: Proposition

text: (Mutations

text: of

text: polygons

text: induce

text: mutations

text: of

text: quivers).

text: Let

text: P

text: be

text: a

text: Fano

text: polygon,

text: let

text: v

text: be

text: a

text: vertex

text: of

text: Q

text: corresponding

text: to

text: a

text: primitive

text: T-cone

text: in

text: P,

text: and

text: let

text: P

text: be

text: the

text: corresponding

text: mutation

text: P

text: p

text: q

text: of

text: P.

text: We

text: have

text: Q

text: mut

text: Q

text: v

text: P

text: P

text: Proof.

text: Let

text: E

text: denote

text: the

text: edge

text: of

text: P

text: determined

text: by

text: the

text: primitive

text: T-cone

text: corresponding

text: to

text: v,

text: and

text: let

text: P

text: w

text: M

text: denote

text: the

text: primitive

text: inner

text: normal

text: vector

text: to

text: E.

text: Mutation

text: with

text: respect

text: to

text: w

text: acts

text: on

text: M

text: as

text: a

text: piecewise-linear

text: transformation

text: that

text: is

text: the

text: identity

text: in

text: one

text: half-space,

text: and

text: on

text: the

text: other

text: half-space

text: is

text: ÞÑ

text: p

text: q

text: a

text: shear

text: transformation

text: u

text: u

text: w

text: u

text: w.

text: Thus

text: determinants

text: between

text: the

text: pairs

text: of

text: normal

text: vectors

text: change

text: as

text: follows:

text: (i)

text: The

text: inner

text: normal

text: vector

text: w

text: to

text: the

text: mutating

text: edge

text: E

text: becomes

text: w,

text: so

text: that

text: all

text: arrows

text: into

text: v

text: change

text: direction;

text: (ii)

text: For

text: a

text: pair

text: of

text: normal

text: vectors

text: in

text: the

text: same

text: half-space

text: (as

text: deﬁned

text: by

text: w),

text: the

text: determinant

text: does

text: not

text: change;


text: 12 KASPRZYK, NILL, AND PRINCE (iii) Consider primitive T-cones with inner normal vectors in diﬀerent half-spaces (as deﬁned by w), let the corresponding vertices of Q be v and v , and let the corresponding inner normal P 1 2 ^ ą vectors in M be w and w . Without loss of generality we may assume that w w 0 and ^ ă 1 2 Ñ Ñ 1 w w 0, so that there are arrows v v v in Q . Under mutation, the primitive 2 ÞÑ 11 ÞÑ 12 P 1 “ 1 “ ` p ^ q inner normal vectors change as w w , w w where w w , w w w w w. 1 1 2 2 1 1 2 2 2 Thus: 1 ^ 1 “ ^ ` p ^ qp ^ q w w w w w w w w 1 2 1 2Ñ Ñ 2 1 and so we add an arrow for each path v v v . Cancelling two-cycles results in precisely 1 2 the result of calculating the signed total number of arrows from v to v . 1 2 Observing ﬁnally that if v , v give normal vectors in the same half-space then there are no paths Ñ Ñ Ñ Ñ 1 2 (cid:3) v v v or v v v , we see that this description coincides with that of a quiver mutation. 1 2 2 1 Propositions 3.15 and 3.17 give upper and lower bounds on the mutation graphs of Fano polygons. For example: p Bq Corollary 3.18 (see Example 1.4). If a Fano polygon P has singularity content 2, and is such that the primitive inner normal vectors of the two edges corresponding to the two primitive T-cones form a basis of the dual lattice M, then the mutation-equivalence class of P has at most ﬁve members. A Proof. The quiver associated to P is simply the A quiver. The cluster algebra is therefore well- 2 P known (being the cluster algebra associated to the A quiver) and its cluster exchange graph forms a 2 pentagon. Note however that the quiver mutation graph is trivial, as the A quiver mutates only to 2 itself. Proposition 3.15 implies that the mutation graph of P has at most ﬁve vertices. (Proposition 3.17 does not give a non-trivial lower bound here: indeed polygon 7 in Figure 3 on page 30 gives an example (cid:3) of such a polygon P with trivial mutation graph.) Example 3.19. For the polygons R and R considered in Example 3.12 above the associated 15 16 quivers Q and Q are: R15 R16 ‚ (cid:43)(cid:51) ‚ ‚ (cid:43)(cid:51) ‚ (cid:79)(cid:79) (cid:95)(cid:95) (cid:63)(cid:63) (cid:75)(cid:83) ‚ (cid:107)(cid:115) ‚(cid:10)(cid:20) ‚ (cid:107)(cid:115) ‚(cid:11)(cid:19) Q Q R15 R16 Observe that for Q the number of arrows between any two vertices is even. It is easy to see that R16 this property is preserved under mutation. Therefore the quivers Q and Q are not mutation- R15 R16 equivalent, and so the polygons R and R are not mutation-equivalent. 15 16 This section sketches a more geometric approach to ﬁnding mutation invariants of polygons. A detailed description of this material will be presented in forthcoming work of Prince [33], based on the foundational papers of Gross–Siebert [21] and Kontsevich–Soibelman [28]. In broad terms, one wishes ˚ to generalise the fact that the dual polygon P is the base of a special Lagrangian torus ﬁbration given by the moment map, by allowing more general bases and more general torus ﬁbrations. Speciﬁcally, the base of a special Lagrangian torus ﬁbration carries an aﬃne structure [28, 20]. Deforming the torus ﬁbration then leads to a deformation of the “candidate base” (actually exhibiting a ﬁbration over the deformed base is fraught with diﬃculty); the deformed base does not have the structure of a polygon, but rather is an aﬃne manifold. Deﬁnition 3.20. A two-dimensional integral aﬃne manifold B is a manifold which admits a maximal pZq (cid:111) Z atlas with transition functions in GL 2 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: KASPRZYK,

text: NILL,

text: AND

text: PRINCE

text: (iii)

label: list

text: Consider

text: primitive

text: T-cones

text: with

text: inner

text: normal

text: vectors

text: in

text: diﬀerent

text: half-spaces

text: (as

text: deﬁned

text: by

text: w),

text: let

text: the

text: corresponding

text: vertices

text: of

text: Q

text: be

text: v

text: and

text: v

text: and

text: let

text: the

text: corresponding

text: inner

text: normal

text: P

text: ą

text: vectors

text: in

text: M

text: be

text: w

text: and

text: w

text: Without

text: loss

text: of

text: generality

text: we

text: may

text: assume

text: that

text: w

text: w

text: and

text: ă

text: Ñ

text: Ñ

text: w

text: w

text: so

text: that

text: there

text: are

text: arrows

text: v

text: v

text: v

text: in

text: Q

text: Under

text: mutation,

text: the

text: primitive

text: ÞÑ

text: ÞÑ

text: P

text: p

text: q

text: inner

text: normal

text: vectors

text: change

text: as

text: w

text: w

text: w

text: w

text: where

text: w

text: w

text: w

text: w

text: w

text: w

text: w.

text: Thus:

text: p

text: qp

text: q

text: w

text: w

text: w

text: w

text: w

text: w

text: w

text: w

text: 2Ñ

text: Ñ

text: and

text: so

text: we

text: add

text: an

text: arrow

text: for

text: each

text: path

text: v

text: v

text: v

text: Cancelling

text: two-cycles

text: results

text: in

text: precisely

text: the

text: result

text: of

text: calculating

text: the

text: signed

text: total

text: number

text: of

text: arrows

text: from

text: v

text: to

text: v

text: Observing

text: ﬁnally

text: that

text: if

text: v

text: v

text: give

text: normal

text: vectors

text: in

text: the

text: same

text: half-space

text: then

text: there

text: are

text: no

text: paths

text: Ñ

text: Ñ

text: Ñ

text: Ñ

text: v

text: v

text: v

text: or

text: v

text: v

text: v

text: we

text: see

text: that

text: this

text: description

text: coincides

text: with

text: that

text: of

text: a

text: quiver

text: mutation.

text: Propositions

text: and

text: give

text: upper

text: and

text: lower

text: bounds

text: on

text: the

text: mutation

text: graphs

text: of

text: Fano

text: polygons.

text: For

text: example:

text: p

text: Bq

text: Corollary

text: (see

text: Example

text: If

text: a

text: Fano

text: polygon

text: P

text: has

text: singularity

text: content

text: and

text: is

text: such

text: that

text: the

text: primitive

text: inner

text: normal

text: vectors

text: of

text: the

text: two

text: edges

text: corresponding

text: to

text: the

text: two

text: primitive

text: T-cones

text: form

text: a

text: basis

text: of

text: the

text: dual

text: lattice

text: M,

text: then

text: the

text: mutation-equivalence

text: class

text: of

text: P

text: has

text: at

text: most

text: ﬁve

text: members.

text: A

text: Proof.

text: The

text: quiver

text: associated

text: to

text: P

text: is

text: simply

text: the

text: A

text: quiver.

text: The

text: cluster

text: algebra

text: is

text: therefore

text: well-

text: P

text: known

text: (being

text: the

text: cluster

text: algebra

text: associated

text: to

text: the

text: A

text: quiver)

text: and

text: its

text: cluster

text: exchange

text: graph

text: forms

text: a

text: pentagon.

text: Note

text: however

text: that

text: the

text: quiver

text: mutation

text: graph

text: is

text: trivial,

text: as

text: the

text: A

text: quiver

text: mutates

text: only

text: to

text: itself.

text: Proposition

text: implies

text: that

text: the

text: mutation

text: graph

text: of

text: P

text: has

text: at

text: most

text: ﬁve

text: vertices.

text: (Proposition

text: does

text: not

text: give

text: a

text: non-trivial

text: lower

text: bound

text: here:

text: indeed

text: polygon

text: in

text: Figure

text: on

text: page

text: gives

text: an

text: example

text: of

text: such

text: a

text: polygon

text: P

text: with

text: trivial

text: mutation

text: graph.)

text: Example

text: For

text: the

text: polygons

text: R

text: and

text: R

text: considered

text: in

text: Example

text: above

text: the

text: associated

text: quivers

text: Q

text: and

text: Q

text: are:

text: Q

text: Q

text: Observe

text: that

text: for

text: Q

text: the

text: number

text: of

text: arrows

text: between

text: any

text: two

text: vertices

text: is

text: even.

text: It

text: is

text: easy

text: to

text: see

text: that

text: this

text: property

text: is

text: preserved

text: under

text: mutation.

text: Therefore

text: the

text: quivers

text: Q

text: and

text: Q

text: are

text: not

text: mutation-

text: equivalent,

text: and

text: so

text: the

text: polygons

text: R

text: and

text: R

text: are

text: not

text: mutation-equivalent.

text: This

text: section

text: sketches

text: a

text: more

text: geometric

text: approach

text: to

text: ﬁnding

text: mutation

text: invariants

text: of

text: polygons.

text: A

text: detailed

text: description

text: of

text: this

text: material

text: will

text: be

text: presented

text: in

text: forthcoming

text: work

text: of

text: Prince

text: based

text: on

text: the

text: foundational

text: papers

text: of

text: Gross–Siebert

text: and

text: Kontsevich–Soibelman

text: In

text: broad

text: terms,

text: one

text: wishes

text: to

text: generalise

text: the

text: fact

text: that

text: the

text: dual

text: polygon

text: P

text: is

text: the

text: base

text: of

text: a

text: special

text: Lagrangian

text: torus

text: ﬁbration

text: given

text: by

text: the

text: moment

text: map,

text: by

text: allowing

text: more

text: general

text: bases

text: and

text: more

text: general

text: torus

text: ﬁbrations.

text: Speciﬁcally,

text: the

text: base

text: of

text: a

text: special

text: Lagrangian

text: torus

text: ﬁbration

text: carries

text: an

text: aﬃne

text: structure

text: Deforming

text: the

text: torus

text: ﬁbration

text: then

text: leads

text: to

text: a

text: deformation

text: of

text: the

text: “candidate

text: base”

text: (actually

text: exhibiting

text: a

text: ﬁbration

text: over

text: the

text: deformed

text: base

text: is

text: fraught

text: with

text: diﬃculty);

text: the

text: deformed

text: base

text: does

text: not

text: have

text: the

text: structure

text: of

text: a

text: polygon,

text: but

text: rather

text: is

text: an

text: aﬃne

text: manifold.

text: Deﬁnition

text: A

text: two-dimensional

text: integral

text: aﬃne

text: manifold

text: B

text: is

text: a

text: manifold

text: which

text: admits

text: a

text: maximal

text: pZq

text: Z

text: atlas

text: with

text: transition

text: functions

text: in

text: GL

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: MINIMALITY AND MUTATION-EQUIVALENCE OF POLYGONS 3 ^ 1 denotes the determinant, and w and w are the primitive inner normal vectors to the edges E and 1 A E of P. The cluster algebra associated to P is the cluster algebra associated to Q ; we denote P p q P the initial seed of this cluster algebra by B , C . P P p q p 1 1q We show in Proposition 3.15 below that a mutation from a seed B, C to a seed B , C induces a 1 mutation between the corresponding Fano polygons P and P . We then show, in Proposition 3.17, that 1 a mutation from a Fano polygon P to a Fano polygon P induces a mutation between the corresponding quivers Q and Q 1. These correspondences have consequences for mutation-equivalence which are P P not readily apparent from the polygon alone. Ă Example 1.4. Consider a Fano polygon P NQ containing only two primitive T-singularities, and suppose that the corresponding inner normal vectors form a basis for the dual lattice M. Then there pZq are at most ﬁve polygons, up to GL -equivalence, that are mutation-equivalent to P. This follows 2 from the facts that the quiver associated to P has underlying graph A , and that the exchange graph 2 of the A cluster algebra is pentagonal; see Corollary 3.18 below. 2 The relation between the lattice points in a convex body and its geometric shape and volume is a key problem in convex geometry and integer optimisation. These connections have been addressed speciﬁcally for lattice polytopes, independently of their signiﬁcance in toric geometry. Here we focus only on the case of interest in this paper, that of a Fano polygon. A classical result in this area is the following (these statements can be generalised and quantiﬁed, see [22]): pZq Theorem 1.5 ([29, 36]). There are only ﬁnitely many GL -equivalence classes of Fano polygons P Z 2 P with I interior lattice points, for each I ą . 0 Corollary 1.6. There are only a ﬁnite number of possibilities for the area and number of lattice points P Z of a Fano polygon with I interior lattice points, for each I ą . 0 In [1] a new equivalence relation on Fano polytopes was introduced, called mutation-equivalence, pZq that is weaker than GL -equivalence. In particular there exist inﬁnitely many mutation-equivalent 2 pZq Fano polytopes that are not GL -equivalent (see, for example, [3, Example 3.14]) and so their area 2 and number of lattice points cannot be bounded. Mutation-equivalence does, however, preserve the ˚ Ă § Ehrhart series (and hence volume) of the dual polytope P MQ (see ) [1, Proposition 4]. Remark 1.7. The reader should be aware that there is no direct relation between the volume of P ˚ p q ¨ p ˚q and that of P . Whilst the product Vol P Vol P of the (normalised) volume of P and of its dual ˚ polygon P cannot be arbitrarily small [30], both can simultaneously become arbitrarily large. For “ tp q p ´ q p´ qu P Z example, P : conv k, 1 , k, 1 , 1, 0 for k ą . k 0 In the two-dimensional case, given a Fano polygon P there exists an explicit formula for the Ehrhart ˚ § series and volume of the dual polygon P given in terms of the singularity content of P (see ) [4]. A consequence of this formula is that mutation-equivalent Fano polygons cannot have an arbitrarily large number of vertices [4, Lemma 3.8]. P Z Ă Recall that the height r ě of a lattice line segment E NQ is the lattice distance of E from E 0 “ | X | ´ the origin, and the width is given by the positive integer k E N 1. Clearly there exist unique ď ă “ ` non-negative integers n and k , 0 k r , such that k nr k . Suppose that E is the edge 0 0 E E 0 § of a Fano polygon, so that the vertices of E are primitive. As described in [4] (see also ), one can ` “ decompose E into n 1 (or n if k 0) lattice line segments with primitive vertices. Of these, n 0 line segments have their width equal to their height; the cones over these line segments correspond to ‰ ă primitive T-singularities. If k 0 then there is one additional lattice line segment of width k r ; 0 0 E the cone over this line segment corresponds to a residual singularity.

text: MINIMALITY

text: AND

text: MUTATION-EQUIVALENCE

text: OF

text: POLYGONS

text: denotes

text: the

text: determinant,

text: and

text: w

text: and

text: w

text: are

text: the

text: primitive

text: inner

text: normal

text: vectors

text: to

text: the

text: edges

text: E

text: and

text: A

text: E

text: of

text: P.

text: The

text: cluster

text: algebra

text: associated

text: to

text: P

text: is

text: the

text: cluster

text: algebra

text: associated

text: to

text: Q

text: we

text: denote

text: P

text: p

text: q

text: P

text: the

text: initial

text: seed

text: of

text: this

text: cluster

text: algebra

text: by

text: B

text: C

text: P

text: P

text: p

text: q

text: p

text: We

text: show

text: in

text: Proposition

text: below

text: that

text: a

text: mutation

text: from

text: a

text: seed

text: B,

text: C

text: to

text: a

text: seed

text: B

text: C

text: induces

text: a

text: mutation

text: between

text: the

text: corresponding

text: Fano

text: polygons

text: P

text: and

text: P

text: We

text: then

text: show,

text: in

text: Proposition

text: that

text: a

text: mutation

text: from

text: a

text: Fano

text: polygon

text: P

text: to

text: a

text: Fano

text: polygon

text: P

text: induces

text: a

text: mutation

text: between

text: the

text: corresponding

text: quivers

text: Q

text: and

text: Q

text: These

text: correspondences

text: have

text: consequences

text: for

text: mutation-equivalence

text: which

text: are

text: P

text: P

text: not

text: readily

text: apparent

text: from

text: the

text: polygon

text: alone.

text: Ă

text: Example

text: Consider

text: a

text: Fano

text: polygon

text: P

text: NQ

text: containing

text: only

text: two

text: primitive

text: T-singularities,

text: and

text: suppose

text: that

text: the

text: corresponding

text: inner

text: normal

text: vectors

text: form

text: a

text: basis

text: for

text: the

text: dual

text: lattice

text: M.

text: Then

text: there

text: pZq

text: are

text: at

text: most

text: ﬁve

text: polygons,

text: up

text: to

text: GL

text: -equivalence,

text: that

text: are

text: mutation-equivalent

text: to

text: P.

text: This

text: follows

text: from

text: the

text: facts

text: that

text: the

text: quiver

text: associated

text: to

text: P

text: has

text: underlying

text: graph

text: A

text: and

text: that

text: the

text: exchange

text: graph

text: of

text: the

text: A

text: cluster

text: algebra

text: is

text: pentagonal;

text: see

text: Corollary

text: below.

text: The

text: relation

text: between

text: the

text: lattice

text: points

text: in

text: a

text: convex

text: body

text: and

text: its

text: geometric

text: shape

text: and

text: volume

text: is

text: a

text: key

text: problem

text: in

text: convex

text: geometry

text: and

text: integer

text: optimisation.

text: These

text: connections

text: have

text: been

text: addressed

text: speciﬁcally

text: for

text: lattice

text: polytopes,

text: independently

text: of

text: their

text: signiﬁcance

text: in

text: toric

text: geometry.

text: Here

text: we

text: focus

text: only

text: on

text: the

text: case

text: of

text: interest

text: in

text: this

text: paper,

text: that

text: of

text: a

text: Fano

text: polygon.

text: A

text: classical

text: result

text: in

text: this

text: area

text: is

text: the

text: following

text: (these

text: statements

text: can

text: be

text: generalised

text: and

text: quantiﬁed,

text: see

text: pZq

text: Theorem

text: There

text: are

text: only

text: ﬁnitely

text: many

text: GL

text: -equivalence

text: classes

text: of

text: Fano

text: polygons

text: P

text: Z

text: P

text: with

text: I

text: interior

text: lattice

text: points,

text: for

text: each

text: I

text: ą

text: Corollary

text: There

text: are

text: only

text: a

text: ﬁnite

text: number

text: of

text: possibilities

text: for

text: the

text: area

text: and

text: number

text: of

text: lattice

text: points

text: P

text: Z

text: of

text: a

text: Fano

text: polygon

text: with

text: I

text: interior

text: lattice

text: points,

text: for

text: each

text: I

text: ą

text: In

text: a

text: new

text: equivalence

text: relation

text: on

text: Fano

text: polytopes

text: was

text: introduced,

text: called

text: mutation-equivalence,

text: pZq

text: that

text: is

text: weaker

text: than

text: GL

text: -equivalence.

text: In

text: particular

text: there

text: exist

text: inﬁnitely

text: many

text: mutation-equivalent

text: pZq

text: Fano

text: polytopes

text: that

text: are

text: not

text: GL

text: -equivalent

text: (see,

text: for

text: example,

text: Example

text: and

text: so

text: their

text: area

text: and

text: number

text: of

text: lattice

text: points

text: cannot

text: be

text: bounded.

text: Mutation-equivalence

text: does,

text: however,

text: preserve

text: the

text: Ă

text: Ehrhart

text: series

text: (and

text: hence

text: volume)

text: of

text: the

text: dual

text: polytope

text: P

text: MQ

text: (see

text: Proposition

text: Remark

text: The

text: reader

text: should

text: be

text: aware

text: that

text: there

text: is

text: no

text: direct

text: relation

text: between

text: the

text: volume

text: of

text: P

text: p

text: q

text: p

text: ˚q

text: and

text: that

text: of

text: P

text: Whilst

text: the

text: product

text: Vol

text: P

text: Vol

text: P

text: of

text: the

text: (normalised)

text: volume

text: of

text: P

text: and

text: of

text: its

text: dual

text: polygon

text: P

text: cannot

text: be

text: arbitrarily

text: small

text: both

text: can

text: simultaneously

text: become

text: arbitrarily

text: large.

text: For

text: tp

text: q

text: p

text: q

text: p´

text: qu

text: P

text: Z

text: example,

text: P

text: conv

text: k,

text: k,

text: for

text: k

text: ą

text: k

text: In

text: the

text: two-dimensional

text: case,

text: given

text: a

text: Fano

text: polygon

text: P

text: there

text: exists

text: an

text: explicit

text: formula

text: for

text: the

text: Ehrhart

text: series

text: and

text: volume

text: of

text: the

text: dual

text: polygon

text: P

text: given

text: in

text: terms

text: of

text: the

text: singularity

text: content

text: of

text: P

text: (see

text: A

text: consequence

text: of

text: this

text: formula

text: is

text: that

text: mutation-equivalent

text: Fano

text: polygons

text: cannot

text: have

text: an

text: arbitrarily

text: large

text: number

text: of

text: vertices

text: Lemma

text: P

text: Z

text: Ă

text: Recall

text: that

text: the

text: height

text: r

text: ě

text: of

text: a

text: lattice

text: line

text: segment

text: E

text: NQ

text: is

text: the

text: lattice

text: distance

text: of

text: E

text: from

text: E

text: X

text: the

text: origin,

text: and

text: the

text: width

text: is

text: given

text: by

text: the

text: positive

text: integer

text: k

text: E

text: N

text: Clearly

text: there

text: exist

text: unique

text: ď

text: ă

text: non-negative

text: integers

text: n

text: and

text: k

text: k

text: r

text: such

text: that

text: k

text: nr

text: k

text: Suppose

text: that

text: E

text: is

text: the

text: edge

text: E

text: E

text: of

text: a

text: Fano

text: polygon,

text: so

text: that

text: the

text: vertices

text: of

text: E

text: are

text: primitive.

text: As

text: described

text: in

text: (see

text: also

text: one

text: can

text: decompose

text: E

text: into

text: n

text: (or

text: n

text: if

text: k

text: lattice

text: line

text: segments

text: with

text: primitive

text: vertices.

text: Of

text: these,

text: n

text: line

text: segments

text: have

text: their

text: width

text: equal

text: to

text: their

text: height;

text: the

text: cones

text: over

text: these

text: line

text: segments

text: correspond

text: to

text: ă

text: primitive

text: T-singularities.

text: If

text: k

text: then

text: there

text: is

text: one

text: additional

text: lattice

text: line

text: segment

text: of

text: width

text: k

text: r

text: E

text: the

text: cone

text: over

text: this

text: line

text: segment

text: corresponds

text: to

text: a

text: residual

text: singularity.


text: MINIMALITY AND MUTATION-EQUIVALENCE OF POLYGONS 21 p ˚q “ ´ ě ÿ Recall from Proposition 3.8 that Vol P 12 n 1, where n is the total number of primitive T-cones spanned by the edges of P; equivalently, | X | ´ “ F N 1 n . PFp q rF F P Since P must have at least three edges, each of which corresponds to a T-cone, and by construction we have that the top edge decomposes into a single primitive T-cone, and that the right-hand vertical 1 ě P t u edge E decomposes into j 5 primitive T-cones, we conclude that j 5, . . . , 9 . From the inequali- “ ě t u “ ties (5.2), (5.3), (5.5), and m r 2, along with the requirement that gcd b, r 1, we obtain P E p q E ﬁnitely many possibilities for the triple j, r , b , as recorded in Table 3. E p q Table 3. The possible values of j, r , b . E j 5 5 6 6 7 7 8 8 8 9 9 9 r 2 5 2 3 2 3 2 3 4 2 3 4 E b 1 2 1 1 1 1 1 1 1 1 1 1 “ “ Analysis of Table 3: First we consider the cases where r 2 and b 1. Here P is contained in the r´ sˆr´ ` s 2 p E´ q P p 2q rectangle 1, 1 j 2, 2 . Let E be the edge with 0, 1 cone E that contains the lower right p ´ q P 2 p ´ q R “ vertex 1, 2 j E . If 0, 1 P then one immediately ﬁnds that j 5 and P is a triangle. This p ´ q P B ď is the triangle T in Table 1. If 0, 1 P then j 6, and P is either the triangle T , or one of P 8 p ´ q P ˝ 2 5 p ´ q P 26 or P in Table 2. Finally, if 0, 1 P then the height of E must be two. Therefore 0, 2 E , 7 giving either the triangle T , or one of P , P , or P . p q p 2 q p 2 q3 4 “ p ˚q “ p ˚q In the cases 2, 5, 2 , 8, 4, 1 , and 9, 3, 1 , we have equality in (5.4), hence 1 Vol P Vol T “ p q and so P T is uniquely determined. This gives the triangles T , T , and T . In the case 6, 3, 1 the “ 1 4 9 triangle T T . In the remaining cases it is easily veriﬁed that T is not a minimal triangle with only 6 T-singularities; this completes the proof of (iii). p q p q p q “ “ Consider the cases 6, 3, 1 , 7, 3, 1 , and 8, 3, 1 . In all three cases r 3 and b 1, hence P is r´ s ˆ r´ ` s P t u E contained in the rectangle 2, 1 j 3, 3 , j 6, 7, 8 . If we assume that P is not a triangle, p ´ q P ˝ p ´ q P p 2q 2 P Fp q it follows that that 0, 1 P , therefore 0, 1 cone E for an edge E P containing the p ´ q 2 bottom-right vertex 1, 3 j . This implies that the edge E has height two or three. Let us assume 2 the height of E is three. Then it must have width three, and so P is not a triangle there must exist ´ one more vertex with ﬁrst coordinate 2. However, this means that there exists a left vertical edge, 2 p ´ q contradicting minimality. Now assume that the height of E is two. Hence 0, 2 is an interior point 2 2 ´ in E , and since E has width two there is a unique vertex with ﬁrst coordinate 1. Hence there can ´ p´ q be at most one vertex left with ﬁrst coordinate 2, excluding the vertex 2, 3 . Enumerating these possibilities shows that none results in a minimal Fano polygon with only T-cones. p q p ´ q P ˝ 2 P Fp q Finally, consider the case 9, 4, 1 . We see that 0, 2 P , hence the non-vertical edge E P p ´ q “ p´ q containing 1, 5 is of height either three or four. If r 2 3 then 2, 1 is a vertex of P, and if “ p´ ´ q E r 2 4 then 3, 1 is a vertex of P. In either case P fails to have only T-cones, and so (iv) is E (cid:3) complete. Corollary 5.5. There are precisely two mutation-equivalence classes of Fano polygons with singularity p ∅q F content 4, . These classes are given by R and R , corresponding to the toric varieties and P ˆ P 15 16 1 1 1, respectively. p ∅q Proof. By Theorem 5.4 there are only three minimal polygons with singularity content 4, : R , R , 3 15 and R . It is easy to see that R and R are mutation-equivalent. That R and R are distinct up 16 3 16 15 16 (cid:3) to mutation is shown in Example 3.12. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: MINIMALITY

text: AND

text: MUTATION-EQUIVALENCE

text: OF

text: POLYGONS

text: p

text: ˚q

text: ě

text: ÿ

text: Recall

text: from

text: Proposition

text: that

text: Vol

text: P

text: n

text: where

text: n

text: is

text: the

text: total

text: number

text: of

text: primitive

text: T-cones

text: spanned

text: by

text: the

text: edges

text: of

text: P;

text: equivalently,

text: X

text: F

text: N

text: n

text: PFp

text: q

text: rF

text: F

text: P

text: Since

text: P

text: must

text: have

text: at

text: least

text: three

text: edges,

text: each

text: of

text: which

text: corresponds

text: to

text: a

text: T-cone,

text: and

text: by

text: construction

text: we

text: have

text: that

text: the

text: top

text: edge

text: decomposes

text: into

text: a

text: single

text: primitive

text: T-cone,

text: and

text: that

text: the

text: right-hand

text: vertical

text: ě

text: P

text: t

text: u

text: edge

text: E

text: decomposes

text: into

text: j

text: primitive

text: T-cones,

text: we

text: conclude

text: that

text: j

text: From

text: the

text: inequali-

text: ě

text: t

text: u

text: ties

text: and

text: m

text: r

text: along

text: with

text: the

text: requirement

text: that

text: gcd

text: b,

text: r

text: we

text: obtain

text: P

text: E

text: p

text: q

text: E

text: ﬁnitely

text: many

text: possibilities

text: for

text: the

text: triple

text: j,

text: r

text: b

text: as

text: recorded

text: in

text: Table

text: E

text: p

text: q

text: Table

text: The

text: possible

text: values

text: of

text: j,

text: r

text: b

text: E

text: j

text: r

text: E

text: b

text: Analysis

text: of

text: Table

text: First

text: we

text: consider

text: the

text: cases

text: where

text: r

text: and

text: b

text: Here

text: P

text: is

text: contained

text: in

text: the

text: r´

text: sˆr´

text: s

text: p

text: E´

text: q

text: P

text: p

text: rectangle

text: j

text: Let

text: E

text: be

text: the

text: edge

text: with

text: cone

text: E

text: that

text: contains

text: the

text: lower

text: right

text: p

text: q

text: P

text: p

text: q

text: R

text: vertex

text: j

text: E

text: If

text: P

text: then

text: one

text: immediately

text: ﬁnds

text: that

text: j

text: and

text: P

text: is

text: a

text: triangle.

text: This

text: p

text: q

text: P

text: B

text: ď

text: is

text: the

text: triangle

text: T

text: in

text: Table

text: If

text: P

text: then

text: j

text: and

text: P

text: is

text: either

text: the

text: triangle

text: T

text: or

text: one

text: of

text: P

text: p

text: q

text: P

text: p

text: q

text: P

text: or

text: P

text: in

text: Table

text: Finally,

text: if

text: P

text: then

text: the

text: height

text: of

text: E

text: must

text: be

text: two.

text: Therefore

text: E

text: giving

text: either

text: the

text: triangle

text: T

text: or

text: one

text: of

text: P

text: P

text: or

text: P

text: p

text: q

text: p

text: q

text: p

text: p

text: ˚q

text: p

text: ˚q

text: In

text: the

text: cases

text: and

text: we

text: have

text: equality

text: in

text: hence

text: Vol

text: P

text: Vol

text: T

text: p

text: q

text: and

text: so

text: P

text: T

text: is

text: uniquely

text: determined.

text: This

text: gives

text: the

text: triangles

text: T

text: T

text: and

text: T

text: In

text: the

text: case

text: the

text: triangle

text: T

text: T

text: In

text: the

text: remaining

text: cases

text: it

text: is

text: easily

text: veriﬁed

text: that

text: T

text: is

text: not

text: a

text: minimal

text: triangle

text: with

text: only

text: T-singularities;

text: this

text: completes

text: the

text: proof

text: of

text: (iii).

text: p

text: q

text: p

text: q

text: p

text: q

text: Consider

text: the

text: cases

text: and

text: In

text: all

text: three

text: cases

text: r

text: and

text: b

text: hence

text: P

text: is

text: r´

text: s

text: ˆ

text: r´

text: s

text: P

text: t

text: u

text: E

text: contained

text: in

text: the

text: rectangle

text: j

text: j

text: If

text: we

text: assume

text: that

text: P

text: is

text: not

text: a

text: triangle,

text: p

text: q

text: P

text: p

text: q

text: P

text: p

text: P

text: Fp

text: q

text: it

text: follows

text: that

text: that

text: P

text: therefore

text: cone

text: E

text: for

text: an

text: edge

text: E

text: P

text: containing

text: the

text: p

text: q

text: bottom-right

text: vertex

text: j

text: This

text: implies

text: that

text: the

text: edge

text: E

text: has

text: height

text: two

text: or

text: three.

text: Let

text: us

text: assume

text: the

text: height

text: of

text: E

text: is

text: three.

text: Then

text: it

text: must

text: have

text: width

text: three,

text: and

text: so

text: P

text: is

text: not

text: a

text: triangle

text: there

text: must

text: exist

text: one

text: more

text: vertex

text: with

text: ﬁrst

text: coordinate

text: However,

text: this

text: means

text: that

text: there

text: exists

text: a

text: left

text: vertical

text: edge,

text: p

text: q

text: contradicting

text: minimality.

text: Now

text: assume

text: that

text: the

text: height

text: of

text: E

text: is

text: two.

text: Hence

text: is

text: an

text: interior

text: point

text: in

text: E

text: and

text: since

text: E

text: has

text: width

text: two

text: there

text: is

text: a

text: unique

text: vertex

text: with

text: ﬁrst

text: coordinate

text: Hence

text: there

text: can

text: p´

text: q

text: be

text: at

text: most

text: one

text: vertex

text: left

text: with

text: ﬁrst

text: coordinate

text: excluding

text: the

text: vertex

text: Enumerating

text: these

text: possibilities

text: shows

text: that

text: results

text: in

text: a

text: minimal

text: Fano

text: polygon

text: with

text: only

text: T-cones.

text: p

text: q

text: p

text: q

text: P

text: P

text: Fp

text: q

text: Finally,

text: consider

text: the

text: case

text: We

text: see

text: that

text: P

text: hence

text: the

text: non-vertical

text: edge

text: E

text: P

text: p

text: q

text: p´

text: q

text: containing

text: is

text: of

text: height

text: either

text: three

text: or

text: four.

text: If

text: r

text: then

text: is

text: a

text: vertex

text: of

text: P,

text: and

text: if

text: p´

text: q

text: E

text: r

text: then

text: is

text: a

text: vertex

text: of

text: P.

text: In

text: either

text: case

text: P

text: fails

text: to

text: have

text: only

text: T-cones,

text: and

text: so

text: (iv)

text: is

text: E

text: complete.

text: Corollary

text: There

text: are

text: precisely

text: two

text: mutation-equivalence

text: classes

text: of

text: Fano

text: polygons

text: with

text: singularity

text: p

text: ∅q

text: F

text: content

text: These

text: classes

text: are

text: given

text: by

text: R

text: and

text: R

text: corresponding

text: to

text: the

text: toric

text: varieties

text: and

text: P

text: ˆ

text: P

text: respectively.

text: p

text: ∅q

text: Proof.

text: By

text: Theorem

text: there

text: are

text: only

text: three

text: minimal

text: polygons

text: with

text: singularity

text: content

text: R

text: R

text: and

text: R

text: It

text: is

text: easy

text: to

text: see

text: that

text: R

text: and

text: R

text: are

text: mutation-equivalent.

text: That

text: R

text: and

text: R

text: are

text: distinct

text: up

text: to

text: mutation

text: is

text: shown

text: in

text: Example

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##