Relative Topology, Principal Bundle Reduction, and Index Theory on Perforated Information Manifolds: Unified Framework Toward S(U(3)\times U(2)), Three-Generation Index, and Yukawa--Winding

Rela i e Topology, P incipal Bundle Reduc ion, and Index
Theo y on Pe o a ed In o ma ion Mani olds:
Unied F amewo k Towa d
S(U(3) ×U(2))
, Th ee-Gene a ion
Index, and YukawaWinding
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Abs ac
Full- ank densi y ma ix mani old
D ull
N={ρ > 0, ρ= 1}
is open con ex con ac ible,
Uhlmann p incipal bundle admi s global squa e- oo sec ion
w=√ρ
on ull domain, hus
absolu e
in ege - alued opological in a ian s a e absen on ull domain. This pape u ns o
pe o a ed ela i e opology
: In
N= 5
case, emo ing ubula neighbo hood o h ee wo
le el gap closing se
Σ3|2={λ3=λ4}
om
D ull
5
yields pe o a ed domain
Dexc
. On
Dexc
cons uc ank
3/2
subbundles
(E3,E2)
ia
Riesz spec al p ojec ion
, ealizing p incipal
bundle s uc u e g oup educ ion
U(5) →U(3) ×U(2)
; u he u ilizing de e minan balancing
yields
S(U(3) ×U(2))
educ ion. We p o e gene al g oup isomo phism
SU(m)×U(n)∼
=SU(m)×SU(n)×U(1)Zlcm(m,n),
wi h
(m, n) = (3,2)
de i ing
(SU(3) ×SU(2) ×U(1))/Z6
. Th ough
ela i e
K
- heo y
bounda y map
uni ying p ojec ionChe n class wi h massclu ching (
de b
Φ
winding), on
wo-dimensional ans e se
S1
ob ain
Ind(DA+ Φ) = wind de b
Φ = ⟨c1(LΦ),[S1]⟩.
In
CP2
spin
c
/Dolbeaul calib a ion compu e
index = 3
as  h ee-gene a ion p o o ype. This
pape p o ides
comple e p oo s
o all co e p oposi ions and heo ems, wi h wo
p o ocol-
le el
ep oducible expe imen al/nume ical schemes (pu ica ion in e e ence loop and pho onic
Di acmass o ex). Appendices include: unied con ou and global smoo hness, g oup iso-
mo phism gcd/lcm no maliza ion and  oo selec ion, igo ous p oo o ela i e
K
- heo y and
Che n cha ac e commu a i e diag am, F edholm cons uc ion o Callias/AnghelBunke index
heo em, and one-page a i hme ic de i a ion o minimal cha ge
1/6
when
Γ = Z6
.
Keywo ds
: Uhlmann p incipal bundle; pe o a ed ela i e opology; Riesz p ojec ion; p in-
cipal bundle educ ion;
S(U(3)×U(2))
;
Z6
quo ien ; ela i e cohomology/
K
- heo y; Dolbeaul /spin
c
index; Callias/AnghelBunke index; de e minan line bundle; line ope a o spec um; ep o-
ducible expe imen al p o ocol
1 No a ion, Assump ions, and Scope

Mixed s a e mani old
:
D ull
N={ρ∈He m+
N:ρ > 0, ρ= 1}
, his pape xes
N= 5
.

Eigen alue o de
:
λ1≥ ··· ≥ λ5
;
spec al gap unc ion
g(ρ) := λ3−λ4
.
1

Pe o a ed domain
: Take
δ > 0
, dene
Dexc := {ρ∈ D ull
5:g(ρ)≥2δ}
. Bounda y
Y:= ∂Tubε(Σ3|2)
equi alen o
g= 2δ
ubula bounda y.

Riesz p ojec ion
: Fix
unied con ou amily
γ3
(see Lemma 1.2 and Appendix A), le
P3(ρ) = 1
2πi Iγ3
(z−ρ)−1dz, P2=I−P3.

Uhlmann p incipal bundle
:
P={√ρ U :ρ∈ D ull
5, U ∈U(5)}
, igh ac ion
w·V=wV
;
π(w) = ww†
yields
U(5)
-p incipal bundle
P→ D ull
5
.

Regula /o dina y p ocess ( e ica ion checklis )
: Along pa h ull ank, gene a o local
CPTP and
C1
, op ional con inuous pu ica ion gauge, and
a oiding
Σ3|2
(
g≥2δ
). This
checklis only mo i a ional: ull domain lacks in ege global classes; his pape ocuses
ela i e
quan iza ion on pe o a ed domain.

No maliza ion
: de Rham pai ing uni o mly akes
1
2πi
ac o ; on
CP2
hype plane class
H
no malized as
RCP1H= 1
,
RCP2H2= 1
.
2 Main Resul s (S a emen s)
Theo em 1
(A: G oup Isomo phism, gcd/lcm No maliza ion)
.
Le
g= gcd(m, n)
,
ℓ= lcm(m, n) =
mn/g
. Homomo phism
φ:SU(m)×SU(n)×U(1) →S(U(m)×U(n)), φ(A, B, z) = diagzn/gA, z−m/gB
is su jec i e wi h
ke φ≃Zℓ
. Thus
SU(m)×U(n)∼
=SU(m)×SU(n)×U(1)Zℓ.
Special case
(m, n) = (3,2) ⇒ℓ= 6
.
P oposi ion 2
(B: Pa i ion Uniqueness)
.
Unde cons ain simple ac o s exac ly
SU(3)
,
SU(2)
e aining only one
U(1)
, unique easible pa i ion o
U(5)
is
5 = 3 + 2
.
Theo em 3
(C: Rela i e B idging)
.
Assume mass end e m
Φ
in e ible on
Y
, ake uni iza ion
b
Φ : Y→U(N)
. Then ela i e
K
- heo y bounda y image
∂[de b
Φ] ∈K0(X, Y )
equals p ojec ion line
bundle
[de E3]−[de E2]
; on wo-dimensional link
⟨c1(LΦ),[S2]⟩=⟨c1(de E3),[S2]⟩ ∈ Z.
Theo em 4
(D: Callias/AnghelBunke)
.
I ou e egion in e ibili y
Φ2≥cI
,
[∇,Φ] ∈L∞
,
Φ∈
W1,2
loc
e c. hold, hen
Ind(DA+ Φ) = deg b
Φ|Sd−1
∞∈πd−1(U).
By Bo pe iodici y
πk(U) = Z
(
k
odd),
0
(
k
e en), ob ain: index possibly nonze o only when
ans e se dimension
d
is e en
; when
d= 2
Ind = 1
2πi IT (b
Φ−1db
Φ) = wind de b
Φ = ⟨c1(LΦ),[S1]⟩.
2
Theo em 5
(E:
CP2
Index)
.
Td(TCP2) = 1 + 3
2H+H2
,
ch(O(1)) = 1 + H+1
2H2
, hus
index 
DO(1) = 3
.
Co olla y 6
(F: SM Global G oup)
.
By Theo em A, P oposi ion B, ob ain
S(U(3) ×U(2)) ∼
=SU(3) ×SU(2) ×U(1)
Z6
.
Appendix E u he p o ides elec ic/magne ic cha ge la ice and a i hme ic de i a ion o
min-
imal cha ge s ep
1/6
o line ope a o spec um when
Γ = Z6
.
3 Degene a ion Se Geome y and Unied Con ou
P oposi ion 7
(2.1: Codimension 3 and
S2
-Link)
.
In h ee-dimensional ans e se slice main ain-
ing
(λ2, λ5)
gap wi h no addi ional symme y,
Σ3|2={λ3=λ4}
is codimension 3 egula subse , i s
small sphe e bounda y link homo opic o
S2
.
P oo essen ials
: Res ic Hamil onian o nea -degene a e 2-dimensional eigensubspace, ob ain
h=xσx+yσy+zσz
; degene acy condi ion
(x, y, z) = (0,0,0)
yields h ee independen eal con-
s ain s. See Appendix A.3.
Lemma 8
(2.2: Unied Con ou ; Global
C∞
)
.
Fo any compac
K⊂ Dexc
, he e exis
δ > 0
and ni e co e
{Uj}
wi h closed cu e amily
{γj}
such ha :
∀ρ∈Uj
,
γj
has dis ance
≥δ
om
complemen spec um; hus
P3,2
is
C∞
on
Uj
and can be smoo hly pa ched. De ails in Appendix
A.1A.2.
4 P incipal Bundle Reduc ion o
S(U(3) ×U(2))
Theo em 9
(3.1: Reduc ion = Sec ion)
.
Le
P→X
be
U(5)
-p incipal bundle,
G=P×U(5)G 3(C5)
.
Sec ion
σ
om
P3
exis s i and only i
P
admi s
U(3) ×U(2)
- educ ion
PH⊂P
.
P oo
: S anda d p incipal bundle heo y, Appendix B.4.
P oposi ion 10
(3.2: Gauge Na u e o De e minan Balancing)
.
Backg ound i ial bundle
C5
wi h
xed olume o m yields gauge isomo phism
de E3⊗de E2≃C
, educing o
S(U(3) ×U(2))
.
Theo em 11
(3.3: G oup Isomo phism; Theo em A o
m= 3
,
n= 2
)
.
S(U(3) ×U(2)) ∼
=(SU(3) ×SU(2) ×U(1))/Z6.
P oo
: See Appendix B.1; pa icula ly no e  oo selec ion s ep in su jec i i y: gi en
(g3, g2)
,
ake
z∈U(1)
sa is ying
z6= de g3
, le
A=z−2g3∈SU(3)
,
B=z3g2∈SU(2)
. Ke nel isomo phic
o
Z6
.
P oposi ion 12
(3.4: Pa i ion Uniqueness; P oposi ion B)
.
Pa i ion
5 = 3+2
is unique sa is ying
simple ac o s
SU(3)
,
SU(2)
wi h only one
U(1)
;
(4 + 1)
lacks
SU(2)
,
(3 + 1 + 1)
and
(2 + 2 + 1)
bo h e ain wo
U(1)
's. De ails in Appendix B.2.
3
5 Two Cha ac e iza ions o Rela i e Topology and Thei Equi a-
lence (Theo em C)
5.1 Rela i e
K
-Theo y and Bounda y Map
Fo pai
(X, Y )=(Dexc, ∂Tubε)
, long exac sequence
··· → K1(Y)∂
−−→ K0(X, Y )→K0(X)→ ··· .
I
Φ
in e ible on
Y
, hen uni iza ion
b
Φ : Y→U(N)
denes
[b
Φ] ∈K1(Y)
, i s bounda y
∂[b
Φ] ∈K0(X, Y )
.
5.2 Commu a i e Diag am and de Rham Rep esen a i e
Odd Che n cha ac e
ch1:K1(Y)→H1(Y;Q)
wi h 1-dimensional ep esen a i e
ch1([b
Φ]) = 1
2πiT (b
Φ−1db
Φ).
Commu a i e diag am exis s (Appendix C.1):
K1(Y)∂
−→ K0(X, Y )
↓ch1↓ch
H1(Y)∂
−→ H2(X, Y )
Thus
ch∂[b
Φ]=∂h1
2πiT (b
Φ−1db
Φ)i∈H2(X, Y ).
5.3 Equi alence P oposi ion (Theo em C)
Compa ed wi h
E3
,
E2
om Riesz p ojec ion, u ilizing na u ali y and clu chinggluing a gumen
(Appendix C.2), ob ain
∂[de b
Φ] = [de E3]−[de E2]∈K0(X, Y ),
hus on wo-dimensional link
⟨c1(LΦ),[S2]⟩=⟨c1(de E3),[S2]⟩
.
6 Spin
c
/Dolbeaul Index on
CP2
(Theo em E)
Take
H=c1(O(1))
,
RCP2H2= 1
.
Td(TCP2) = 1 + 1
2c1+1
12(c2
1+c2) = 1 + 3
2H+H2,ch(O(1)) = eH= 1 + H+1
2H2.
Top-dimensional coecien
1 + 3
2+1
2= 3
, hus
index DO(1) = 3
. Kodai a anishing ensu es
χ=h0= 3
.
4
7 Callias/AnghelBunke Index = Deg ee; Two-Dimensional Wind-
ing Fo mula (Theo em D)
7.1 F edholm Condi ions
Le
M
comple e, Di ac- ype ope a o
DA
wi h sel -adjoin end e m
Φ
. I he e exis
R
,
c > 0
such
ha on
M BR
,
Φ2≥cI
, wi h
[∇,Φ] ∈L∞
,
Φ∈W1,2
loc
, hen
DA+ Φ
is F edholm (Appendix D.1).
7.2 Index = Deg ee and Pa i y
Bounda y homomo phism and Bo isomo phism yield
Ind(DA+ Φ) = deg b
Φ|Sd−1
∞∈πd−1(U), πk(U) = (Z, k
odd
0, k
e en
.
Fo wo-dimensional ans e se
Ind = 1
2πi IS1
T (b
Φ−1db
Φ) = wind de b
Φ = ⟨c1(LΦ),[S1]⟩,
consis en wi h ze o-mode coun ing. Sign con en ion:
S1
akes coun e clockwise o ien a ion.
8 Alignmen wi h
GSM
Line Ope a o Spec um and Minimal Cha ge
1/6
By Theo em 3.3:
GSM ∼
=(SU(3) ×SU(2) ×U(1))/Z6
. Ke nel gene a o can ake
(ω−1
3I3,−I2, ei2π/6), ω3=ei2π/3.
Ac ion on
( , s, q)
( espec i ely
SU(3)
iali y,
SU(2)
pa i y,
U(1)
in ege cha ge) is
ω−
3·(−1)s·ei2πq/6.
Necessa y and sucien condi ion o descending o quo ien g oup:
q≡2 + 3s(mod 6)
. Thus
no malized hype cha ge
Y=q/6
has minimal ac ional s ep
1/6
. Appendix E p o ides one-page
de i a ion and example able o elec ic/magne ic cha ge la ice, Di ac pai ing in ege ma ix, and
θ
pe iod.
9 P o ocol-Le el Expe imen al and Nume ical Schemes (O e iew)
E1 Pu ica ion In e e ence (Image a ound
Σ3|2
)
: Disc e ize unied con ou , eadou
2πϕ el
,
whe e
ϕ el =RS2
link Fde E3/(2π)∈Z
. Sampling
Nsho s ≳30
, phase noise
δϕ ≲0.25
ad can s ably de-
e mine in ege . Failu e cases: pa h g azes
Σ3|2
, non-smoo h pu ica ion; coun e measu es: enla ge
con ou adius, inc ease pu i y gap and epea sampling.
E2 Pho onic Di acMass Vo ex
: Encode mass phase
eikθ
, ou e egion
|m| → m∞>0
.
Ze o-mode coun
|k|
, nea -eld in ensi y cen aliza ion, band-gap midpoin ene gy o m nge p in .
Robus egion: phase e o
≤10◦
, coupling misma ch
≤5%
. Appendix F p o ides pa ame e able
and passing s anda d.
5

10 Discussion and Ou look

Rela i e s absolu e
: Full domain con ac ible
→
absolu e in ege class anishes; pe o a ed
→
ela i e class quan iza ion.

Dimensional eec
:
de
only ully de ec s in wo-dimensional ans e se; highe dimensions
equi e s able
U
g oup gene a o s.

G oup heo y b idging
: Spec al spli ing induced
S(U(3) ×U(2))
wo ks syne gis ically
wi h line spec um dic iona y, yielding minimal cha ge s ep
1/6
.

Follow-up
: Mul i-de ec supe posi ion ela i e class addi ion, obus window unde noise
non-equilib ium, sys ema ic gene aliza ion wi h highe -o de (
-block) spli ing.
A Spec al Geome y and Unied Con ou (Co esponding o 2)
A.1 Spec al Gap Lowe Bound and Con ou Selec ion
Le
gap(ρ) = min{λ3−λ4, λ2−λ3, λ4−λ5}
. On
X=Dexc
,
gap >0
con inuous; o any compac
K⊂X
, le
δ= minKgap >0
. Fo each
ρ∈K
ake ci cle
γρ
cen e ed a
λ3+λ4
2
wi h adius
δ/2
, i
encloses uppe spec um clus e wi h dis ance
≥δ/2
om complemen spec um.
A.2 Riesz P ojec ion
C∞
Dependence
By esol en es ima e
|(z−ρ)−1| ≤ 2/δ
and smoo hness o
z7→ (z−ρ)−1
,
P3(ρ) = 1
2πi Hγρ(z−ρ)−1dz
is
C∞
in
ρ
. Using ni e co e
{Uj}
wi h pa i ion o uni y pa ching, ob ain global
C∞
p ojec ion
eld
P3,2
.
A.3 Codimension 3 and
S2
-Link
A
λ3
&
λ4
nea -degene acy, ake
E=E34 ⊕E⊥
, eec i e Hamil onian
h=ασz+ℜβ σx+ℑβ σy
;
degene acy
⇔(α, ℜβ, ℑβ) = (0,0,0)
, h ee independen eal equa ions hus codimension 3. Take
no mal small ball
B3
, i s bounda y
S2
is link.
B G oup Isomo phism and Minimal Pa i ion (Co esponding o
3)
B.1 Comple e P oo o Theo em A
Homomo phism
φ(A, B, z) = diag(zn/gA, z−m/gB), g = gcd(m, n), ℓ =mn
g.
Ke nel
:
φ(A, B, z) = I⇒A=z−n/gIm
,
B=zm/gIn
. By
A∈SU(m)⇒z−nm/g = 1 ⇒zℓ=
1
. Map
κ:µℓ→ke φ, κ(z)=(z−n/gIm, zm/gIn, z)
is g oup isomo phism, hus
ke φ≃Zℓ
.
6
Su jec i i y ( oo selec ion)
: Gi en
(g3, g2)∈S(U(m)×U(n))
(i.e.,
de g3de g2= 1
),
ake
z∈U(1)
sa is ying
zℓ= de g3.
Le
A=z−n/gg3∈SU(m), B =zm/gg2∈SU(n).
Then
de A=z−nm/g de g3=z−ℓde g3= 1,de B=znm/g de g2=zℓde g2= 1,
wi h
φ(A, B, z)=(g3, g2)
. Thus ob ain s a ed isomo phism.
B.2 Pa i ion Uniqueness Table
Pa i ion Simple pa
S
-cons ained
U(1)
coun Conclusion
4+1 SU(4)
1 No
SU(2)
3+1+1 SU(3)
2 Viola es one
U(1)

2+2+1 SU(2) ×SU(2)
2 Same
3+2 SU(3) ×SU(2)
1 Unique sa is ying
B.3 Gene aliza ion
S(U(k)×U(ℓ)) ∼
=(SU(k)×SU(ℓ)×U(1))/Zlcm(k,ℓ)
; explici o m o ke nel gene a o depends on
embedding no maliza ion, bu quo ien g oup isomo phism class in a ian .
C Rela i e
K
-Theo y and Che n Cha ac e (Co esponding o 4)
C.1 Commu a i e Diag am
Fo pai
(X, Y )
, odd Che n cha ac e
ch1:K1(Y)→H1(Y;Q)
yields
ch1([u]) = 1
2πiT (u−1du).
E en Che n cha ac e
ch : K0(X, Y )→He en(X, Y ;Q)
wi h de Rham bounda y ope a o
∂
o m commu a i e diag am
K1(Y)∂
−→ K0(X, Y )
↓ch1↓ch
H1(Y)∂
−→ H2(X, Y )
whose commu a i i y ollows om na u ali y and Maye Vie o is pa ching.
7
C.2 B idging Equali y
Le
b
Φ : Y→U(N)
be uni ized mass,
∂[de b
Φ] ∈K0(X, Y )
. On o he hand, Riesz p ojec ion yields
E3
,
E2
, hus
[de E3]−[de E2]∈K0(X, Y )
. Using homo opy ex ension making
b
Φ
s ably compa ible
wi h spec al spli ing mo phism, h ough commu a i e diag am pai ing in
H2(X, Y )
o link
S2
's
in ege equali y, hus wo ela i e classes equal.
C.3 Explici Pai ing in Two-Dimensional T ans e se
I
Y
's piecewise link is
S1
, hen
IS1
1
2πiT (b
Φ−1db
Φ) = ZS2
Fde E3
2π∈Z.
D Callias/AnghelBunke (Co esponding o 6)
D.1 F edholm Cons uc ion
Take ou e egion cu o
χ
and pa ame ix
Q=χΦ−1
. Ha e
(DA+ Φ)Q=I−K1, Q(DA+ Φ) = I−K2,
whe e
K1,2
ela i ely compac (by
[∇,Φ] ∈L∞
, Rellich compac embedding and ou e egion
in e ibili y). Thus
DA+ Φ
is F edholm.
D.2 Bounda y Map and Deg ee
Homo ope ou e egion o di ec ion-only dependen
Φ∞(θ)
, index equals bounda y map
∂[b
Φ∞]∈
e
K0(Sd)∼
=Z
. Bo isomo phism yields
Ind = deg(b
Φ∞)∈πd−1(U).
D.3 Two-Dimensional Single Vo ex Example
Φ( , θ) = U(θ)H( )
,
U(θ) = diag(eikθ,1,1)
,
H( → ∞)→m0I
. Then
b
Φ = U
,
Ind = k
. Taking
coun e clockwise o ien a ion as posi i e,
k→ −k
index changes sign.
E Line Ope a o Spec um and Minimal Cha ge
1/6
(Co espond-
ing o 7)
E.1 Ke nel Gene a o and Cong uence
By Theo em 3.3,
Γ≃Z6
gene a o can ake
g∗= (ω−1
3I3,−I2, ei2π/6), ω3=ei2π/3.
Ac ion on
( , s, q)
is
ω−
3(−1)sei2πq/6
. Quo ien descen condi ion:
ω−
3(−1)sei2πq/6= 1 ⇐⇒ q≡2 + 3s(mod 6).
Le
Y=q/6⇒Y≡ /3 + s/2 (mod Z)
, hus minimal ac ional uni
1/6
.
8
E.2 Elec ic/Magne ic Cha ge La ice and Di ac Pai ing (Schema ic)
Deno e
(e;m)
as elec ic/magne ic cha ge ec o , cen al gluing yields cong uence cons ain ma ix
C
sa is ying
(e;m)7→ (e;m)+Cn
(
n∈Z
) equi alence. Di ac pai ing in ege ma ix
Ω
well-dened
in eg ali y on quo ien ;
θ
pe iod unde goes equi alence con ac ion a e quo ien g oup iden ica-
ion. Example able: undamen al ep esen a ions
3
and
2
's
( , s)
alues b ing
Y
's ac ional pa s
{1/3,1/2}
, syn hesizing wi h
U(1)
phase yields minimal s ep
1/6
span.
F Expe imen al and Nume ical Checklis  (Co esponding o 8)
F.1 E1 Pu ica ion In e e ence

Inpu
: Loop
C
,
δ
, sampling
Nsho s
,
(T1, T2)
.

S eps
: Pu ica ione olu ionin e e ence eadou phase unw apcon ou in eg al.

Ou pu
:
ϕ el ∈Z
.

Passing s anda d
:
|e (ϕ el)|<0.25
can de e mine in ege ; i ails, enla ge loop adius and
Nsho s
.
F.2 E2 Pho onic Vo ex

Inpu
: A ay size, coupling
J
, mass ampli ude
m∞
, o ex numbe
k
.

S eps
: Phase map encodingexci a ionnea -eld imagingspec al localiza ionze o-mode
coun ing.

Ou pu
: Ze o-mode coun
|k|
.

Passing s anda d
: Band gap
>
noise bandwid h, cen al peak signican wi h ene gy nea
midpoin .
F.3 Nume ical Sc ip Essen ials

G id
(Nθ, N )
ake
Nθ≥64
;

Riesz p ojec ion pe o ms con ou quad a u e along xed adius
δ
ci cle;

Wilson-loop's
c1
consis en wi h
wind de b
Φ
, e o
∼ O(h2)
.
End o Main Tex and Appendices
9