Topological--Scattering Solution of Strong CP Problem and Axion in Unified Matrix--QCA Universe

TopologicalSca e ing Solu ion o S ong CP
P oblem and Axion in Unied Ma ixQCA Uni e se
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
No embe 19, 2025
Abs ac
Quan um ch omodynamics (QCD) in i s mos gene al o m admi s a opological
θ
- e m
θ g2
s(32π2)−1Ga
µν ˜
Ga,µν
, which iola es
P
,
T
and
CP
. A e chi al eld ede-
ni ions, he physically obse able s ong
CP
angle is
¯
θ=θ−a g de (YuYd)
, whe e
Yu
and
Yd
a e he up- and down- ype Yukawa ma ices. Na u alness sugges s
¯
θ=O(1)
,
while neu on elec ic dipole momen (nEDM) bounds
|dn|≲1.8×10−26 e·cm
imply
|¯
θ|≲10−10
, cons i u ing he s ong
CP
p oblem.
PecceiQuinn (PQ) heo y p omo es
¯
θ
o a dynamical acuum expec a ion alue
o an axion eld. The axion po en ial is de e mined by he QCD opological sus-
cep ibili y
χ op
, wi h
m2
a 2
a=χ op
and
V(a)≃χ op[1 −cos(a/ a−¯
θ0)]
. La ice
QCD and chi al eec i e heo y now de e mine
χ op(T)
wi h high p ecision, xing
he QCD axion masscoupling ela ion. Howe e , in a b oade unied desc ip ion o
he Uni e se, he o igin and obus ness o PQ symme y agains g a i y, ul a iole
physics and global consis ency condi ions emain unclea .
Wi hin he unied ime-scale, bounda y ime geome y, ma ix uni e se THE-
MATRIX and quan um cellula au oma on (QCA) uni e se amewo k, his wo k
gi es a opologicalsca e ing solu ion o he s ong
CP
p oblem. The main ideas
a e:1. In oduce a pa ame e space
X◦
o all low-ene gy couplings (gauge couplings,
θ
-angles, Yukawa phases, ligh scala pa ame e s) and an ex ended space
Y=M×
X◦
, whe e
M
is space ime. F om he amily o sca e ing ma ices
S(ω;λ)
on a
channel Hilbe space, cons uc a de e minan line bundle
Lde →Y
and i s squa e
oo
L1/2
de
. The obs uc ion o a global smoo h squa e oo is encoded in a ela i e
cohomology class
[K]∈H2(Y, ∂Y ;Z2)
, which has he physical meaning o he global
Z2
holonomy o he "squa e- oo sca e ing de e minan "
pde pS
along pa ame e
loops.
2. Using he p e iously de eloped NullModula double co e and es ic ed uni-
a y bundle amewo k, one has an equi alence be ween: (i) local Eins ein equa ions
wi h app op ia e quan um ene gy condi ions, (ii) small causal diamond gene alized
en opy ex emali y and modula ow consis ency, and (iii) anishing o he ob-
s uc ion class,
[K]=0
. Thus, in any Uni e se admi ing a globally consis en
bounda y ime geome y and semiclassical g a i y, allowed physical sec o s mus
sa is y
[K] = 0
.
3. Embed QCD and i s
θ
-angle in o his unied s uc u e by iden i ying he
QCD sec o con ibu ion
[KQCD]
o
[K]
. The physical s ong
CP
angle
¯
θ=θ−
1
a g de (YuYd)
eappea s as he phase holonomy o
pde pSQCD
along loops in
X◦
.
One shows ha
[KQCD] = 0
is equi alen o he i iali y (modulo
2π
) o all such
holonomies; in pa icula , in any physically ealized sec o compa ible wi h
[K]=0
one has
¯
θe ≈0
wi hou equi ing a anishing qua k mass.
4. In he ma ix uni e se ep esen a ion, he global Uni e se is a gigan ic bu
s uc u ed uni a y ma ix whose block-spa se pa e n encodes causal ela ions and
whose spec al da a encode he unied ime-scale. In his pic u e,
¯
θ
is a " opological
phase" o he QCD block o THE-MATRIX, and
[K] = 0
equi es ha he squa e-
oo de e minan has i ial
Z2
holonomy ac oss all coupling loops. S ong
CP
is
hen eph ased as he equi emen ha such opological sca e ing in a ian s anish
globally.
5. In he QCA uni e se laye , one cons uc s an
SU(3)
gauge QCA wi h la ice
opological cha ge
Q∈Z
. The QCD
θ
- e m co esponds o a weigh ac o
exp(iθQ)
in he disc e e pa h-sum. By imposing a " opologicalNullModula consis en
QCA" condi ion ha he o al phase o all closed gauge-congu a ion his o ies
be
2πZ
, one ob ains in he con inuum limi a join cons ain on
¯
θ
and possible
g a i a ional
θG
- e ms, he eby simul aneously supp essing s ong and g a i a ional
CP
iola ion.
6. The PQ axion is ein e p e ed as a ela i e cohomology modulus o
[K]
. The
axion eld
a(x)/ a
pa ame izes local ephasings o
pde pS
along a
U(1)
be
o
L1/2
de
. I s eec i e ac ion ep oduces he s anda d o m
S[a]∼R[1
2 2
a(∂a)2+
χ op(1 −cos(a/ a−¯
θ0))]√−gd4x
, whe e
χ op
is he QCD opological suscep ibili y
de e mined om  s -p inciples QCD. The global condi ion
[K]=0
hen en o ces
⟨a⟩/ a=¯
θ0
and
¯
θe = 0
, gi ing a unied opologicalsca e ing e o mula ion o
he PQ mechanism.
Appendices p esen s anda d QCD de i a ions o
¯
θ
, he p ecise cons uc ion o
[K]
om sca e ing heo y, and explici
SU(3)
gauge QCA models wi h disc e e
opological cha ge and
θ
-phase. The esul ing pic u e ea s s ong
CP
as a con-
sis ency cons ain o he ull ma ixQCA Uni e se a he han an independen
ne- uning o a low-ene gy pa ame e .
Keywo ds:
S ong
CP
p oblem; QCD axion; PecceiQuinn mechanism; Topological
suscep ibili y; Sca e ing de e minan line bundle; Ma ix Uni e se; Quan um Cellula
Au oma a (QCA); NullModula double co e
1 In oduc ion & His o ical Con ex
1.1 The S ong
CP
P oblem
Fou -dimensional
SU(3)
YangMills heo y admi s a opological e m in i s Lag angian:
Lθ=θg2
s
32π2Ga
µν ˜
Ga,µν,˜
Ga,µν =1
2ϵµνρσGa
ρσ.
(1)
The space- ime in eg al o his e m con e s o he p oduc o he opological cha ge
Q∈
Z
and he angle
θ
, explici ly iola ing
P
,
T
, and
CP
. Fo QCD wi h mul iple gene a ions
o qua ks, a e in oducing Yukawa ma ices
Yu, Yd
, conside ing chi al edeni ions and
anomaly eec s, he physically obse able s ong
CP
angle is
¯
θ=θ−a g de (YuYd),
(2)
2
which na u ally akes alues in
[0,2π)
.
Wi hou addi ional symme ies o dynamical mechanisms,
¯
θ
would be expec ed o be
O(1)
. Howe e ,
¯
θ= 0
induces a neu on elec ic dipole momen (nEDM). QCD sum ules
and chi al eec i e heo y gi e
dn≃cn¯
θ e ·cm, cn∼10−16.
(3)
Cu en nEDM expe imen s p o ide an uppe bound
|dn|<1.8×10−26 e·cm
, implying
|¯
θ|≲10−10.
(4)
This is he s ong
CP
p oblem: why an angle ha should na u ally be
O(1)
is ne- uned
o he o de o
10−10
.
1.2 T adi ional Solu ions and he QCD Axion
T adi ional solu ions o he s ong
CP
p oblem gene ally all in o a ew ca ego ies: 1.
**Massless Up Qua k:** I
mu= 0
,
¯
θ
can be emo ed ia chi al o a ion. La ice QCD
has essen ially uled ou his possibili y. 2. **Spon aneous
P
o
CP
B eaking:**
P
o
CP
is a undamen al symme y b oken spon aneously a a high ene gy scale, wi h
he acuum choosing
¯
θ= 0
. Such models need o add ess he nEDM induced a e
spon aneous
CP
b eaking and he cha ac e iza ion o disc e e symme ies in quan um
g a i y. 3. **PecceiQuinn Mechanism:** In oduces a global
U(1)PQ
symme y and
a scala eld. I s anomaly s uc u e makes he QCD acuum ene gy dependen on an
angula eld
a/ a
, au oma ically adjus ing
¯
θe = 0
a he minimum o he acuum ene gy.
The co esponding app oxima e Golds one mode is he QCD axion. While he o iginal
PQ model is uled ou , "in isible axion" models like KSVZ and DFSZ emain mains eam
solu ions.
In he PQ amewo k, he axion eec i e po en ial is de e mined by he QCD opo-
logical suscep ibili y
χ op
:
V(a)≃χ op1−cos(a/ a−¯
θ0), m2
a 2
a=χ op.
(5)
La ice QCD and chi al eec i e heo y ha e p o ided high-p ecision esul s o
χ op(T)
,
p ecisely de e mining he ela ionship be ween QCD axion mass and coupling cons an .
1.3 Unied Uni e se F amewo k and P oblem Res a emen
The abo e solu ions mos ly add ess he s ong
CP
p oblem a he le el o low-ene gy
eec i e eld heo y, wi hou embedding i in o a unied amewo k con aining g a -
i y, causal s uc u e, bounda y ime geome y, and global consis ency o he Uni e se.
Fu he mo e, quan um g a i y and black hole he modynamics sugges ha global sym-
me ies migh be iola ed in quan um g a i y, ques ioning he s abili y o he global
U(1)PQ
in he adi ional PQ mechanism.
P e ious wo ks in oduced he mul i-laye objec s o unied ime scale, bounda y
ime geome y, Ma ix Uni e se THE-MATRIX, and QCA Uni e se, cha ac e izing he
Uni e se as a single s uc u e highly consis en ac oss sca e ing, geome y, modula ow,
gene alized en opy, and causal pa ial o de . Key poin s include:
1. **Unied Time Scale Mo he Fo mula:**
κ(ω) = φ′(ω)
π=ρ el(ω) = (2π)−1 Q(ω),
(6)
3
uni ying sca e ing hemi-phase de i a i e, ela i e densi y o s a es, and Wigne Smi h
g oup delay ma ix ace in o a single ime densi y.
2. **Bounda y Time Geome y and NullModula Double Co e :** Gluing mod-
ula ow pa ame e s and sca e ing phases on he bounda y o each causal diamond,
cons aining geome y, en opy, and sca e ing mu ually.
3. **Ma ix Uni e se and QCA Uni e se:** Viewing he en i e cosmic e olu ion as
a gigan ic bu spa se uni a y ma ix o e e sible QCA, whe e block s uc u e encodes
causal pa ial o de , spec al da a implemen s ime scale, and eedback loops ca y
Z2
opology.
In his amewo k, he s ong
CP
p oblem can be es a ed as: why is he QCD sec o 's
con ibu ion o he opologicalsca e ing in a ian s o he whole Uni e se supp essed o
nea ly ze o? This pape will show ha his supp ession is no longe a "ne- uning o
low-ene gy cons an s" bu a manda o y esul o "unied uni e se opologicalsca e ing
consis ency".
2 Model & Assump ions
This sec ion p esen s he models and basic assump ions o he unied uni e se s uc u e,
pa ame e space, sca e ing de e minan line bundle, and QCD sec o embedding used
in his pape .
2.1 Uni e se Objec and Pa ame e Space
Le he geome ic laye o he Uni e se be desc ibed by a globally hype bolic Lo en zian
mani old
(M, g)
wi h causal pa ial o de and app op ia e bounda y s uc u e (including
pas / u u e inni y and possible black hole ho izons). On his basis, in oduce se e al
laye s o he unied uni e se objec : * Geome ic laye
Ugeo
:
(M, g, ≺)
; * Sca e ing
laye
Usca
: A amily o sca e ing pai s and unied ime scale
κ(ω)
; * Bounda y laye
UBTG
: Bounda y algeb a, modula ow, and B ownYo k quasilocal ene gy; * Ma ix
laye
Uma
: Sca e ing ma ix uni e se THE-MATRIX on channel Hilbe space; * QCA
laye
UQCA
: SU(3) gauge QCA dened on a coun able la ice; * Topological laye
U op
:
Rela i e cohomology class
[K]
and NullModula double co e .
To in oduce coupling pa ame e s, dene an open pa ame e space
X◦
whose coo -
dina es include: 1. Gauge coupling cons an s
gs, g, g′
, e c.; 2. Yukawa ma ices and
CKM/PMNS phase pa ame e iza ions; 3. QCD
θ
-angle, possible g a i a ional
θG
angle,
and o he opological e m pa ame e s; 4. Ligh scala s (including possible axions) and
ex e nal mac oscopic pa ame e s.
Dene he ex ended space
Y=M×X◦,
(7)
whose bounda y
∂Y
includes space ime bounda ies and possible bounda ies o he pa-
ame e space.
2.2 Sca e ing Ma ix Family and De e minan Line Bundle
Unde app op ia e locali y and spec al assump ions, o each pa ame e poin
λ∈X◦
and ene gy
ω
, conside a sca e ing pai
(H0, H(λ))
and i s wa e ope a o s, yielding a
4
sca e ing ma ix on he channel Hilbe space
Hchan(ω)
:
S(ω;λ) : Hchan(ω)→ Hchan(ω).
(8)
Assume
S(ω;λ)−1
is a ace-class ope a o o app op ia e o de , such ha he modied
de e minan
de pS(ω;λ)
is dened and a ies smoo hly wi h
(ω, λ)
(unde spec al gap
and app op ia e eno maliza ion).
F om his amily o uni a y ope a o s on he ec o bundle
Hchan
o e
(ω, λ)∈Y
, one
can cons uc a complex line bundle
Lde →Y
, whose be is gene a ed by he o mal
"de e minan ":
Lde |(ω,λ)∼
=C·de pS(ω;λ).
(9)
Locally, one can choose a loga i hmic phase
ϕ(ω, λ)
such ha
de pS(ω;λ) = expiϕ(ω, λ).
(10)
Dene he squa e oo line bundle
L1/2
de
as o mally sa is ying
(L1/2
de )⊗2≃ Lde .
(11)
Locally, one can choose
pde pS= exp( i
2ϕ)
, bu globally he e may be a sign ambigui y.
The
Z2
wis ing o his ambigui y is cha ac e ized by a ela i e cohomology class
[K]∈H2(Y, ∂Y ;Z2).
(12)
[K]=0
i and only i he e exis s a global squa e oo line bundle i ial on
∂Y
. This
cons uc ion can be seen as a sca e ing e sion gene aliza ion o F eed e al.'s wo k on
de e minan line bundles and cohomological obs uc ions.
2.3 NullModula Double Co e and Consis ency Condi ions
Bounda y ime geome y equips he bounda y
∂D
o each small causal diamond
D⊂M
wi h: 1. Bounda y obse able algeb a
A∂(D)
and i s s a e
ω∂(D)
; 2. Modula ow
pa ame e
s∈R
and modula ope a o
∆is
; 3. Phase
φD(ω)
and ime scale
κD(ω)
gi en
by sca e ing ma ix
SD(ω)
and Wigne Smi h g oup delay
QD(ω)
.
The NullModula double co e glues he modula ow pa ame e and sca e ing
phase ia a
Z2
co e , ensu ing consis ency be ween modula ow di ec ion, ime a ow,
and phase single- aluedness, a oiding "hal -cycle sign e e sal" anomalies. P e ious wo k
showed ha unde assump ions o local Eins ein equa ions, QNEC/QFC, and sca e ing
modula consis ency, he ollowing condi ions a e equi alen : 1. Exis ence o a global
NullModula double co e such ha modula ow and sca e ing phase align smoo hly
on his co e o all causal diamonds; 2. Gene alized en opy ex emali y condi ions and
non-nega i i y o second-o de ela i e en opy on each causal diamond a e equi alen o
local g a i a ional eld equa ions; 3. The ela i e cohomology class o he de e minan
squa e oo line bundle sa ises
[K]=0
.
Thus, o an accep able physical uni e se sec o ,
[K] = 0
is no op ional bu a consis-
ency equi emen .
5

2.4 QCD Sec o and Embedding o
¯
θ
In he QCD sec o , conside
SU(3)
gauge elds
Aa
µ
and six a o s o qua ks
ψ
. The
Lag angian includes
LQCD =−1
4Ga
µνGa,µν +¯
ψ(iγµDµ)ψ−(¯uLYuuR+¯
dLYddR+ h.c.) + θg2
s
32π2Ga
µν ˜
Ga,µν.
(13)
Chi al edeni ion and Fujikawa anomaly analysis gi e he physical angle
¯
θ=θ−a g de (YuYd),
(14)
appea ing in he pa h in eg al in he weigh ac o
exp(i¯
θQ)
.
Viewing he QCD sec o as a block o he Ma ix Uni e se THE-MATRIX, o each
pa ame e
λ
and ene gy
ω
, he e is a sca e ing ma ix
SQCD(ω;λ)
. The de e minan and
squa e oo o his amily on he pa ame e space dene a con ibu ion
[KQCD]
o
[K]
.
I s physical meaning is: along a closed loop in pa ame e space con aining
¯
θ
a ia ion,
does
pde pSQCD
unde go an i emo able sign ip?
2.5 SU(3) Gauge QCA Uni e se Laye
In he QCA laye , cons uc an SU(3) gauge QCA: la ice e ices ca y ma e Hilbe
spaces, edges ca y gauge link a iables
Ux,µ ∈SU(3)
, and disc e e ime e olu ion is
implemen ed by local gauge-co a ian uni a y ga es
UQCA
. The disc e e opological cha ge
Q∈Z
can be dened by summing
T (UP˜
UP)
o e la ice poin s.
In his se ing, he QCD
θ
- e m co esponds o adding a phase ac o in he disc e e
pa h sum:
Y
n
expiθQn,
(15)
o in he Heisenbe g pic u e as
UQCA(θ) = exp(iθˆ
Q)UQCA(0)
. TopologicalNullModula
consis ency equi es ha he o al phase o any closed gauge congu a ion loop be
2πZ
,
which will be used o cons ain
¯
θ
.
3 Main Resul s (Theo ems and Alignmen s)
This sec ion o malizes he main conclusions abou he s ong
CP
p oblem and axion
wi hin he unied Ma ixQCA uni e se amewo k in o se e al heo ems and co olla ies.
P oo s a e in he Appendix.
3.1 Theo em 1 (De e minan Squa e Roo and Rela i e Coho-
mology Class)
Theo em 3.1
(De e minan Squa e Roo and Rela i e Cohomology Class)
.
Unde he
a o emen ioned sca e ing and pa ame e space assump ions, he sca e ing ma ix amily
S(ω;λ)
denes a complex line bundle
Lde →Y
, gene a ed by he modied de e minan
de pS(ω;λ)
. The e exis s a na u al ela i e cohomology class
[K]∈H2(Y, ∂Y ;Z2),
(16)
6
sa is ying: 1.
[K] = 0
i and only i he e exis s a squa e oo line bundle
L1/2
de
i ial on
∂Y
, such ha
(L1/2
de )⊗2∼
=Lde
; 2. Fo any closed pa ame e loop
γ⊂X◦
, he e alua ion
o
[K]
on he mapping o us
M×S1
γ
gi es he
Z2
holonomy o
pde pS
along
γ
: i he
holonomy is
+1
, a smoo h squa e oo can be chosen on he loop; i
−1
, he e is an
i emo able sign ip.
3.2 Theo em 2 (NullModula Consis ency and
[K] = 0
)
Theo em 3.2
(NullModula Consis ency and
[K]=0
)
.
Assume he Uni e se sa is-
es: 1.
M
is globally hype bolic, wi h app op ia e bounda y ime geome y s uc u e and
modula ow; 2. Gene alized en opy ex emali y and non-nega i i y o second-o de el-
a i e en opy on small causal diamonds hold and a e equi alen o local g a i a ional eld
equa ions; 3. Bounda y sca e ing ma ices and modula ows can be locally aligned on
a NullModula double co e .
Then he ollowing p oposi ions a e equi alen : 1. The e exis s a global NullModula
double co e such ha modula ow and sca e ing phase on all causal diamonds align
smoo hly on his co e wi hou
Z2
anomaly; 2. The e exis s a global squa e oo line
bundle
L1/2
de
i ial on
∂Y
; 3.
[K]=0
.
Thus, o any physically accep able uni e se sec o ,
[K] = 0
is a necessa y condi ion.
3.3 Theo em 3 (QCD
¯
θ
as Sca e ingTopological Holonomy)
Theo em 3.3
(QCD
¯
θ
as Sca e ingTopological Holonomy)
.
In he QCD sec o , dene
he physical angle
¯
θ(λ) = θ(λ)−a g de Yu(λ)Yd(λ),
(17)
and conside a closed loop
γ⊂X◦
in pa ame e space. Then: 1. The e alua ion o
[KQCD]
on
M×S1
γ
equals he sign o
exp(i∆γ¯
θ/2)
, whe e
∆γ¯
θ
is he o al phase change
along
γ
(modulo
2π
); 2. I
∆γ¯
θ≡0 (mod 2π)
o all physically ealizable loops
γ
,
hen
[KQCD]=0
; 3. Con e sely, i he e exis s a loop whe e
∆γ¯
θ≡π(mod 2π)
, hen
[KQCD]= 0
.
Specically, in a local pa ame e neighbo hood,
[KQCD] = 0
equi es ha a gauge can
be chosen such ha
¯
θe ≡0 (mod 2π)
.
3.3.1 Co olla y 3.1 (S ong
CP
Supp ession in Unied Uni e se)
Assume he Uni e se sa ises he assump ions o Theo em 2, and non-QCD sec o con-
ibu ions o
[K]
a e cons ained o ze o by independen consis ency condi ions. Then
[K] = 0 ⇒[KQCD] = 0
. Thus: 1. Physically ealizable uni e se sec o s mus sa -
is y
¯
θe ≡0 (mod 2π)
; 2. The expe imen ally obse ed
|¯
θ|≲10−10
is in e p e ed as
an ex emely sensi i e uppe bound on possible esiduals o
[KQCD]
, while he na u al
expec a ion is s ic ly ze o.
3.4 Theo em 4 (Axion as Rela i e Cohomology Modulus o
[K]
)
Theo em 3.4
(Axion as Rela i e Cohomology Modulus o
[K]
)
.
Assume
[K]
li s o an
in ege coecien class
˜
K∈H2(Y, ∂Y ;Z)
. Le he phase o he axion eld be
ϕ(x) = expia(x)/ a∈U(1),
(18)
7
and embed i in o QCD ia he coupling
LaG ˜
G=a
a
g2
s
32π2Ga
µν ˜
Ga,µν.
(19)
Then: 1. The a ia ion o
ϕ
can be iewed as a local escaling o he squa e oo line
bundle
L1/2
de
, and
˜
K
is he  s Che n class o his
U(1)
line bundle; 2. In low-ene gy
eec i e heo y, he axion po en ial can be w i en as
V(a)≃χ op1−cos(a/ a−¯
θ0)+··· ,
(20)
whe e
χ op
is he QCD opological suscep ibili y, and
¯
θ0
is he ba e angle in he UV ame;
3. I he unied uni e se equi es
[K] = 0
, hen globally one mus ha e
⟨a⟩/ a=¯
θ0
, hus
¯
θe =¯
θ0−⟨a⟩/ a= 0
.
This gi es a ein e p e a ion o he PQ mechanism in he unied opologicalsca e ing
amewo k: he axion is he ela i e cohomology modulus ha uly elimina es
[K]
, and
i s acuum selec ion is en o ced by
[K] = 0
a he han being an ex a eedom.
3.5 Theo em 5 (
θ
Sec o and
[KQCD]
in SU(3) Gauge QCA)
Theo em 3.5
(
θ
Sec o and
[KQCD]
in SU(3) Gauge QCA)
.
In he SU(3) gauge QCA
model, le he e olu ion ope a o o each ime s ep be
UQCA(θ) = expiθˆ
QUQCA(0),
(21)
whe e
ˆ
Q
is he disc e e opological cha ge ope a o . Conside he disc e e pa h space
C
composed o all possible gauge congu a ion pa hs. 1. Fo any closed pa h
C
, he o al
phase
Φ(C) = θX
n∈C
Qn
(22)
has a sign (modulo
2π
) consis en wi h he e alua ion o
[KQCD]
on he co esponding
mapping o us; 2. I one equi es
Φ(C)∈2πZ
o all physically ealizable closed pa hs
C
, hen he eec i e
¯
θe ≡0 (mod 2π)
in he disc e e model; 3. In he con inuum limi ,
his condi ion o ces
[KQCD]=0
in he QCD sec o , compa ible wi h he conclusions o
Theo ems 34.
4 P oo s
This sec ion ou lines he p oo amewo ks o he main heo ems. De ails and igo ous
cons uc ions a e in he Appendix.
4.1 Ske ch o P oo o Theo em 1
De e minan line bundles and cohomological obs uc ions o squa e oo s a e s anda d
opics. 1. Fo each
(ω, λ)
, x an o hono mal s anda d channel basis. The spec um o
S(ω;λ)
can be w i en as
{eiδj(ω,λ)}j
. Dene local hemi-phase
ϕ(ω, λ) = X
j
δj(ω, λ).
(23)
8
2. On a good open co e
{Uα}
o
Y
, selec con inuous b anches
ϕα
, dening local sec-
ions
sα= exp(iϕα)
as local i ializa ions o
Lde
. 3. On in e sec ions
Uα∩Uβ
, ansi ion
unc ions
gαβ =sα/sβ
ake alues in
U(1)
. Thei p ojec i e squa e oo s
hαβ =√gαβ
o m candida e ansi ion unc ions o
L1/2
de
. 4. Whe he
hαβ
can be chosen o sa is y
he cocycle condi ion
hαβhβγhγα = 1
is de e mined by a
Z2
- alued ech 2-cocycle, co -
esponding o a class in
H2(Y;Z2)
. 5. Adding he bounda y condi ion ha he squa e
oo is i ial on
∂Y
yields he class
[K]
in ela i e cohomology
H2(Y, ∂Y ;Z2)
. I s an-
ishing is necessa y and sucien o he exis ence o a global squa e oo i ial on he
bounda y. Holonomy along a closed loop
γ
is de e mined by he ne change
∆γϕ
, wi h
sign
exp(i∆γϕ/2)
, consis en wi h he e alua ion o
[K]
on
M×S1
γ
.
4.2 Ske ch o P oo o Theo em 2
NullModula double co e links bounda y modula ow pa ame e
s
and sca e ing
phase
ϕ(ω, λ)
such ha o each causal diamond
D
: 1. Time ansla ion gene a ed by
modula ow aligns wi h sca e ing ime scale dened by
κ(ω)
on he bounda y; 2. Second
a ia ion o gene alized en opy and non-nega i i y o second-o de ela i e en opy a e
equi alen o local Eins ein equa ions. I
[K]= 0
, he e exis closed pa ame e geome y
loops whe e he holonomy o
pde pS
is
−1
. This means on he NullModula double
co e , one canno con inuously choose he ela i e sign o sca e ing phase and modula
ow phase, leading o a "sign e e sal" jump when gluing bounda ies o ce ain causal
diamonds. This des oys he unied a ia ional p inciple o gene alized en opyg a i y
equa ions. Con e sely, i
[K]=0
, a smoo h
pde pS
can be chosen o e he en i e
Y
, co esponding one- o-one wi h modula ow pa ame e s on bounda ies, cons uc ing
an anomaly- ee NullModula double co e . Consis ency o modula ow, gene alized
en opy, and sca e ing phase ensu es equi alence be ween he a ia ional p inciple on
small causal diamonds and Eins ein equa ions. This yields he equi alence in Theo em
2.
4.3 Ske ch o P oo o Theo em 3 and Co olla y 3.1
In he QCD sec o , he physical angle
¯
θ=θ−a g de (YuYd)
(24)
appea s in he pa h in eg al weigh
exp(i¯
θQ)
. I s con ibu ion o he sca e ing ma ix
de e minan phase can be iewed as
de pSQCD ∼X
Q
P(Q) exp(i¯
θQ) de SQ,
(25)
whe e
P(Q)
is he opological sec o weigh . Along a closed loop
γ
in pa ame e space,
¯
θ
may wind by in ege mul iples o
2π
. Since opological cha ge
Q∈Z
, he o al phase
change is
∆γΦ = ∆γ¯
θ·⟨Q⟩γ+··· .
(26)
Conside ing modulo
2π
, i
∆γ¯
θ≡0 (mod 2π)
, a con inuous squa e oo can be chosen
on he loop wi hou sign ip. I
∆γ¯
θ≡π(mod 2π)
, he squa e oo mus ip sign once,
co esponding o
[KQCD]= 0
. Thus,
[KQCD]=0
is equi alen o
∆γ¯
θ≡0 (mod 2π)
o
all physically ealizable loops. Assuming non-QCD sec o con ibu ions a e supp essed
9