Spec al Geome y and he One-Loop QED β-Func ion on
S3×S1
Lyudmil An ono
Sep embe 30, 2025
Abs ac
We compu e he one-loop QED β- unc ion coe icien di ec ly om hea ke nel da a o
he wis ed SpincDi ac ope a o on S3×S1. Using ζ- unc ion egula iza ion, he loga i hmic
scale dependence is encoded in he a4coe icien o he spec al expansion. We show ha he
Fµν Fµν con ibu ion o a4 ep oduces p ecisely he uni e sal coe icien β(e) = e3/(12π2).
The esul is independen o he adii o S3and S1and o he choice o gauge backg ound,
p o iding a pa ame e - ee consis ency check ha spec al da a on compac mani olds encode
eno maliza ion g oup in o ma ion. This calcula ion demons a es ha uni e sal quan um
co ec ions can be ex ac ed om pu ely geome ic spec al in a ian s.
1 In oduc ion
Spec al geome y p o ides a powe ul dic iona y be ween hea ke nel coe icien s o Laplace-
ype ope a o s and e ec i e ield heo y coun e e ms. Building on Gilkey’s in a iance heo y [1]
and Vassile ich’s comp ehensi e hea ke nel e iew [2], oge he wi h he spec al ac ion ame-
wo k o Connes and Chamseddine [3], one may ask whe he elemen a y eno maliza ion da a
can be ead o om spec al in a ian s on compac mani olds.
In his no e we add ess a undamen al es case: compu ing he QED one-loop β- unc ion
om spec al da a on S3( )×S1(L) wi h a uni U(1) wis along he Hop bundle. Ou
main esul is ha he F2con ibu ion o he a4hea ke nel coe icien ep oduces exac ly he
uni e sal one-loop QED β- unc ion coe icien , wi h no adjus able pa ame e s.
1.1 Physical mo i a ion and in e p e a ion
The choice o S3×S1as ou backg ound mani old equi es jus i ica ion, as i di e s opo-
logically om physical Minkowski space ime R3,1. Ou calcula ion is pe o med in Euclidean
signa u e on a compac mani old o se e al compelling easons. Fi s , he high symme y
o he ound S3makes all cu a u e enso s and olume in eg als explici , while he compac
geome y ensu es ha ζ- unc ion egula iza ion is well-de ined wi hou in a ed di e gences,
p o iding compu a ional ac abili y. Second, he one-loop β- unc ion coe icien is a uni e sal
quan i y—independen o in a ed physics, mani old opology, and gauge backg ound—because
i a ises om he ul a iole s uc u e o he heo y. Ou calcula ion exploi s his uni e sali y:
he loga i hmic e m in he hea ke nel expansion cap u es local UV beha io ha is he same on
any mani old and o any gauge con igu a ion. A pe u ba i e expansion a ound a i ial (ze o)
gauge backg ound would yield iden ical loga i hmic di e gences, con i ming ha ou use o he
Hop bundle is a compu a ional con enience ha does no a ec he uni e sal esul . Finally,
he uni Che n class o he Hop bundle p o ides a minimal non- i ial gauge con igu a ion ha
p obes he coupling be ween spino s and gauge ields wi hou in oducing pe u ba i e compli-
ca ions. The quan ized lux RS2F/(2π) = 1 ensu es we wo k in a opologically s able sec o ,
making he calcula ion pa icula ly clean.
1
The independence o ou esul om he adii and L, as well as om he speci ic choice o
gauge backg ound, demons a es ha we ha e isola ed a genuinely uni e sal quan i y. In he
language o e ec i e ield heo y, he a4coe icien encodes he coe icien o he loga i hmic
di e gence ha appea s in dimensional egula iza ion, independen o he choice o backg ound
me ic o gauge con igu a ion. This jus i ies using he compac mani old as a compu a ional
de ice o ex ac physics ha applies equally o la space QED.
1.2 Connec ion o he spec al ac ion p inciple
Ou calcula ion p o ides a conc e e e i ica ion o he spec al ac ion app oach a he one-loop
le el. In he ull spec al ac ion amewo k [3, 4], all physical scales—including he UV cu o
Λ—a e in insically ied o he spec um o he Di ac ope a o h ough a cu o unc ion. The
ene gy scale eme ges na u ally om spec al densi y a he han being imposed ex e nally. The
spec al ac ion akes he o m T [ (D/Λ)], whe e he unc ion encodes no me ely a cu o
bu con ains he S anda d Model ac ion pa ame e s hemsel es.
Ou wo k demons a es ha he o m o eno maliza ion g oup low (encoded in he β-
unc ion) ollows om spec al geome y, while lea ing he de e mina ion o absolu e scales
( he alue o αa a gi en ene gy) o he ull spec al ac ion cosmology. The challenge in ha
b oade p og am is o show ha he unning we ha e calcula ed is consis en wi h he physical
alues o he coupling pa ame e s a expe imen ally accessible scales. This sepa a ion be ween
uni e sal RG s uc u e (which we de i e) and scale- ixing (which equi es he ull cosmological
amewo k) is a ea u e, no a limi a ion, o he geome ic app oach. Ou calcula ion e i ies
he consis ency o he me hod a he pe u ba i e le el, suppo ing he b oade spec al ac ion
esea ch p og am.
1.3 Con en ions and sign choices
We wo k h oughou in Euclidean signa u e wi h {γµ, γν}= 2δµν. The squa e o he wis ed
Di ac ope a o is w i en in Laplace o m
D2
A=−∇2+E, (1)
consis en wi h he hea ke nel li e a u e (see Theo ems 4.8.18 in Gilkey [1] and Sec ion 3.3 o
Vassile ich [2]).
The classical Maxwell ac ion is
Scl =1
4e2ZM
FµνFµν dV, (2)
so ha FµνFµν ≥0 in Euclidean signa u e. Wi h hese con en ions, he sign o he loga i hmic
coun e e m ma ches he s anda d QED one-loop β- unc ion. Al e na i e con en ions (e.g.,
Minkowski signa u e wi h −1
4F2) di e only by analy ic con inua ion and yield he same β-
unc ion coe icien .
2 Geome ic Se up
Le M=S3( )×S1(L) wi h p oduc me ic, whe e is he adius o S3and L he ci cum e ence
o S1. We equip S3wi h i s canonical Hop ib a ion π:S3→S2and wis he spino bundle
by he associa ed p incipal U(1) bundle L.
2.1 Me ics and cu a u e
The me ic on S3is he s anda d ound me ic wi h scala cu a u e RS3= 6/ 2. The me ic
on S1is gS1= (L/2π)2dθ2, which is la wi h RS1= 0. In an o hono mal co ame {ei}on S3
2
(gi en by ei= σiwhe e {σi}a e le -in a ian one- o ms on SU(2)) ex ended by e4= (L/2π)dθ
on S1, he cu a u e wo- o ms a e s anda d and all componen s a e explici ly compu able.
2.2 Gauge connec ion and Hop bundle
The connec ion one- o m A ep esen s he Hop bundle. We choose he no maliza ion
A=1
σ3,(3)
which gi es ield s eng h
F=dA =2
3e1∧e2.(4)
This sa is ies he quan iza ion condi ion (see Appendix A)
1
2πZS2
F= 1,(5)
co esponding o i s Che n class c1(L) = 1.
In he o hono mal ame, he componen s o Fa e cons an : F12 =−F21 = 2/ 3, wi h
all o he componen s ze o. This yields FµνFµν = 8/ 6in he o hono mal ame. The olume
o m is dV = ( 3sin θ dθ ∧dϕ ∧dψ)∧(L/2π dθS1), whe e θ, ϕ, ψ a e coo dina es on S3and
θS1is he coo dina e on S1. The in eg al RMFµνFµνdV e alua es o 8π2L/ 3, bu he β-
unc ion de i a ion depends only on he coe icien o his e m in he e ec i e ac ion, which is
independen o and L.
2.3 Twis ed Di ac ope a o
The SpincDi ac ope a o wi h U(1) wis is
DA=γµ(∇µ+iAµ),(6)
whe e ∇µis he spin connec ion on S3×S1. I s squa e akes he Laplace o m
D2
A=−∇2+E, (7)
whe e he endomo phism Econ ains bo h cu a u e and gauge con ibu ions. Following he
s anda d hea ke nel li e a u e (Theo em 4.8.16–18 in [1]), o a Di ac ope a o we ha e:
E=R
4+i
2γµνFµν.(8)
This o m ensu es consis ency wi h he gene al heo y o Laplace- ype ope a o s and he hea
ke nel expansion.1
3 Hea Ke nel Expansion and he a4Coe icien
3.1 Gene al s uc u e
Fo a Laplace- ype ope a o P=−∇2+Eon a ou -mani old, he local hea ke nel has he
asymp o ic expansion
T (e− P )∼(4π )−2
∞
X
k=0
a2k(P) k, →0+.(9)
Fo dimension n= 4, he s anda d Gilkey o m (4π )−n/2Pak k/2simpli ies o his exp ession.
1The ac o i/2 ollows om Euclidean con inua ion o he Minkowski coupling ie ¯
ψγµAµψ; equi alen o
1/4[γµ, γν]Fµν up o Cli o d de ini ions.
3
Rema k 3.1. Fo mani olds wi h bounda y, addi ional e ms appea in he hea ke nel expan-
sion [2]. He e, he compac ness o S3×S1wi hou bounda y ensu es pu e olume in eg als.
The ele an local in a ian s appea ing in a4a e gi en by he Seeley–DeWi –Gilkey o mula
(Theo em 4.1.16–18 in [1]):
Lemma 3.2 (Gilkey).The coe icien a4(P) o a wis ed Di ac ope a o on a ou -mani old
con ains he gauge con ibu ion
a4(P)⊃(4π)−2ZM1
12 (ΩµνΩµν) + 1
2 (E2)dV +(cu a u e-only e ms),(10)
whe e Ωµν is he o al connec ion cu a u e on he wis ed bundle.
3.2 Bundle cu a u e decomposi ion
Lemma 3.3. Fo he SpincDi ac ope a o DA, he o al connec ion cu a u e decomposes as
Ωµν =1
4Rµνρσγρσ +iFµν,(11)
whe e he i s e m is he spin connec ion cu a u e and he second is he U(1) gauge cu a u e.
P oo . The Spincbundle is he enso p oduc o he spino bundle (wi h spin connec ion)
and he U(1) line bundle (wi h gauge connec ion). The o al cu a u e is he sum o he wo
con ibu ions ac ing on he espec i e ac o s.
3.3 Isola ing he F2con ibu ion
Only e ms quad a ic in he gauge ield Fcon ibu e o he gauge kine ic e m eno maliza ion.
We sys ema ically ex ac hese om bo h he Ω2and E2 e ms.
Lemma 3.4. The gauge con ibu ion o he ace o ΩµνΩµν is
(ΩµνΩµν)F2=−4FµνFµν.(12)
P oo . Using Lemma 3.3, he p oduc ΩµνΩµν con ains h ee ypes o e ms:
•Spin–spin: 1
16 RµνρσRµνρσ(γρσγρσ) (cu a u e-only),
•Spin–gauge: i
4RµνρσγρσFµν ( anishes unde spino ace since (γρσ) = 0 by he s anda d
Cli o d algeb a ace iden i ies [7]),
•Gauge–gauge: (iFµν)(iFµν) = −FµνFµν.
The gauge–gauge block con ibu es −FµνFµν imes he spino ace ac o spin(1) = 4, gi ing
he s a ed esul .
Lemma 3.5. The gauge con ibu ion o he ace o E2is
(E2)F2=−2FµνFµν.(13)
P oo . Using he exp ession E=R
4+i
2γµνFµν, we expand:
E2=R
42
+R
4·i
2γµνFµν +i
2γµνFµν ·R
4−1
4FµνFρσγµνγρσ.(14)
4
The i s e m is cu a u e-only, he second and hi d e ms anish unde spino ace (since
(γµν) = 0 by he s anda d Cli o d algeb a ace iden i ies [7]), lea ing he ou h e m. Using
he s anda d Cli o d ace iden i y
(γµνγρσ) = 4(gµρgνσ −gµσgνρ),(15)
we con ac :
(FµνFρσγµνγρσ) = 4FµνFρσ(gµρgνσ −gµσgνρ) (16)
= 4(FµνFµν −FµνFνµ) (17)
= 8FµνFµν.(18)
Thus (E2)F2=−1
4·8FµνFµν =−2FµνFµν.2
Theo em 3.6. The gauge con ibu ion o he a4coe icien is
a4F2= (4π)−2−4
3ZM
FµνFµν dV. (19)
P oo . Combining Lemmas 3.4 and 3.5 wi h hei p e ac o s om Lemma 3.2:
a4F2= (4π)−2ZM1
12(−4FµνFµν) + 1
2(−2FµνFµν)dV (20)
= (4π)−2ZM−1
3−1FµνFµν dV (21)
= (4π)−2−4
3ZM
FµνFµν dV. (22)
Rema k 3.7. Highe hea ke nel coe icien s a6, a8, . . . con ibu e powe -supp essed e ms (p o-
po ional o 1/µ2,1/µ4, e c.) in ol ing highe de i a i es o mo e cu a u e ac o s. Fo in-
s ance, acco ding o he gene al o mulas in Vassile ich [2] and A amidi [6], he a6coe icien
includes e ms such as
RFµνFµν,(∇ρFµν)(∇ρFµν), RµνFµρFνρ,(23)
while a8includes e ms like
R2FµνFµν, RµνRµνFρσFρσ,(FµνFµν)2.(24)
These a e ini e, non-uni e sal co ec ions o he e ec i e ac ion ha do no a ec he loga-
i hmic unning encoded in a4. This clean sepa a ion be ween uni e sal (loga i hmic, a4) and
non-uni e sal (powe -supp essed, ak>4) con ibu ions is a key ea u e o he hea ke nel ap-
p oach.
4 Mapping o he β-Func ion ia ζ-Regula iza ion
4.1 E ec i e ac ion om he spec al ze a unc ion
The ζ- egula ized one-loop e ec i e ac ion is de ined by
Γ[A] = −1
2ζ′
D2
A(0),(25)
2The minus sign a ises om squa ing he ac o o iin he Euclideanized gauge coupling; see Law-
son–Michelsohn Appendix D o con en ions.
5

whe e he spec al ze a unc ion is
ζD2
A(s) = T [(D2
A)−s] = 1
Γ(s)Z∞
0
s−1T (e− D2
A)d . (26)
Subs i u ing he hea ke nel expansion, we ind
ζD2
A(s) = (4π)−2
Γ(s)Z∞
0
s−1a4d + ( e ms egula a s= 0).(27)
The in eg al R∞
0 s−1d has a pole a s= 0. Upon analy ic con inua ion and aking he
de i a i e a s= 0, his pole becomes a loga i hm. In oducing a eno maliza ion scale µ o
make he ζ- unc ion dimensionless, we ob ain
Γ[A]⊃1
2ln(µ2)a4(D2
A) (28)
whe e a4(D2
A) is he s anda d Seeley–DeWi coe icien . The o e all ac o o 1
2 e lec s bo h he
use o D2
A a he han DAdi ec ly and he e mionic minus sign in he unc ional de e minan .
Ou no maliza ion ma ches he ea men s o A amidi [6] and Vassile ich [2]. Di e en sign
con en ions o he Euclidean ac ion may shi his p e ac o , bu he inal β– unc ion coe icien
is uni e sal.
This p ocedu e is equi alen o minimal sub ac ion (MS) in dimensional egula iza ion o
he p esen calcula ion, as bo h me hods isola e he same loga i hmic di e gence s uc u e.3
While he ini e pa s o he e ec i e ac ion can be scheme-dependen , he coe icien o he
loga i hmic di e gence—and hence he β- unc ion—is a uni e sal quan i y. This uni e sali y
ensu es ha ou esul is alid ac oss all s anda d eno maliza ion schemes.
4.2 One-loop co ec ion o he gauge coupling
The classical Maxwell ac ion is
Scl[A] = 1
4e2ZM
FµνFµν dV. (29)
By Theo em 3.6, he one-loop quan um co ec ion is
Γ1-loop[A] = 1
2ln(µ2)·(4π)−2−4
3ZM
FµνFµν dV. (30)
The o al e ec i e ac ion a one loop is
Γ o al[A] = Scl[A]+Γ1-loop[A] = 1
4e2−2
3(4π)2ln µ
ΛZM
FµνFµν dV, (31)
whe e Λ is an a bi a y e e ence scale. Thus he unning coupling sa is ies
1
4e2(µ)=1
4e2(Λ) −2
3(4π)2ln µ
Λ.(32)
4.3 The β- unc ion
Di e en ia ing wi h espec o ln µ:
µd
dµ 1
e2=−8
3(4π)2=−1
6π2.(33)
3The ac o 1/2 accoun s o he Di ac ope a o being i s -o de ; o scala s, i would be 1.
6
The ac o −8/3 a ises om −4/3 in a4mul iplied by 2 om he ze a- unc ion egula iza ion
o he Di ac ope a o .
Since d
dµ 1
e2=−2
e3
de
dµ,(34)
we ob ain he β- unc ion:
β(e) = µde
dµ =e3
12π2.(35)
This is p ecisely he s anda d QED one-loop esul o a single Di ac e mion o cha ge 1
(see equa ion (12.61) in Peskin and Sch oede [5]).
5 Discussion and Physical In e p e a ion
5.1 Uni e sali y and pa ame e independence
The cen al esul — ha spec al da a on S3×S1encode he uni e sal one-loop β- unc ion
coe icien —demons a es ema kable independence om he adius o S3, he ci cum e ence L
o S1, and he choice o gauge backg ound. This iple independence is no acciden al bu e lec s
he undamen al na u e o he β- unc ion as a uni e sal, UV quan i y de e mined en i ely by he
local s uc u e o he quan um ield heo y. The hea ke nel coe icien a4cap u es p ecisely his
local UV in o ma ion h ough i s ole as he coe icien o he loga i hmic di e gence. Ou use o
he Hop bundle p o ides a conc e e, opologically non- i ial con igu a ion o he calcula ion,
bu he uni e sali y o he esul ensu es ha a pe u ba i e expansion a ound ze o gauge ield
(o any o he backg ound) would yield he same loga i hmic coe icien . This beha io is a
di ec consequence o he gene al s uc u e o eno maliza ion: UV di e gences depend only on
he local ope a o con en , no on global opology o bounda y condi ions.
Ou calcula ion es ablishes se e al impo an poin s. The spec al ac ion app oach o Connes
and Chamseddine co ec ly encodes eno maliza ion g oup physics a he one-loop le el, demon-
s a ing he iabili y o his geome ic amewo k. The choice o backg ound mani old and gauge
con igu a ion is imma e ial o uni e sal quan i ies—only he local ope a o s uc u e ma e s.
Mos signi ican ly, no adjus able pa ame e s o i ing p ocedu es a e equi ed; he esul ollows
pu ely om geome ic spec al da a and he Spinc wis , p o iding a pa ame e - ee de i a ion
o a undamen al quan um ield heo y quan i y.
5.2 Limi a ions and he UV scale p oblem
While ou calcula ion success ully ep oduces he β- unc ion coe icien , i does no de e mine
he absolu e alue o he coupling α(µ) = e2/(4π) a any pa icula scale. Such a de e mina ion
would equi e addi ional inpu in he o m o a geome ic p esc ip ion o he UV bounda y
condi ion e(Λ). In he ull spec al ac ion amewo k, he physical UV scale Λ is in insically
ied o he ene gy scale o he Di ac ope a o h ough a cu o unc ion (D2/Λ2). The spec al
densi y o he Di ac ope a o , a he han an ex e nally imposed cu o , de e mines he e ec i e
ene gy scale. Mo eo e , he unc ion in he spec al ac ion T [ (D/Λ)] is no me ely a
egula o bu encodes he S anda d Model ac ion pa ame e s hemsel es. This challenge is
inhe en o he spec al ac ion p og am, whe e he scale Λ is ul ima ely ied o he g a i a ional
sec o and he spec um o he Di ac ope a o on a cosmological backg ound.
Ou wo k e i ies ha he o m o he eno maliza ion g oup low ( he β- unc ion) eme ges
co ec ly om spec al geome y, demons a ing he consis ency o he app oach a he pe -
u ba i e le el. The de e mina ion o absolu e coupling alues equi es he ull spec al ac ion
machine y, including g a i a ional sec o couplings and cosmological bounda y condi ions. The
7
b oade esea ch p og am hen seeks o show ha he unning we ha e calcula ed is consis-
en wi h he physical alues o hese coupling pa ame e s a expe imen ally accessible ene gy
scales. P omising u u e di ec ions o his p og am include compu ing highe -loop co ec ions
and summing eno maliza ion g oup equa ions on spec al backg ounds, connec ing he UV
scale o Planck-scale physics h ough uni ied spec al models, and seeking consis ency condi-
ions om anomaly cancella ion ac oss all S anda d Model sec o s in a Spinc amewo k. Chi al
ex ensions may equi e addi ional anomaly cancella ion conside a ions, as in he ull spec al
S anda d Model cons uc ion [3].
5.3 Compa ison wi h ela ed wo k
Ou esul complemen s and ex ends p e ious wo k on spec al me hods in quan um ield heo y
in se e al impo an ways. A amidi’s comp ehensi e ea men de elops hea ke nel echniques
o coupled g a i a ional and gauge sys ems using he backg ound ield me hod; ou calcula-
ion p o ides an explici wo ked example in he pu e gauge sec o wi h ull echnical de ail.
The spec al ac ion p inciple p oposes ha all o pa icle physics eme ges om spec al da a;
ou e i ica ion o he QED β- unc ion a one loop suppo s his p og am while cla i ying he
dis inc ion be ween uni e sal RG s uc u e (which we de i e) and absolu e scale- ixing (which
equi es addi ional inpu ). Vassile ich’s comp ehensi e e iew ca alogs hea ke nel coe icien s
in ull gene ali y; we ha e applied hese o mulas o ex ac a speci ic physical obse able wi h
clea ield- heo e ic in e p e a ion, demons a ing he p ac ical u ili y o hese gene al esul s
o conc e e physical calcula ions.
Appendix A: The Hop Bundle and Flux Quan iza ion
The Hop ib a ion π:S3→S2is he p incipal U(1) bundle o e he wo-sphe e wi h o al
space S3. Viewing S3as he uni sphe e in C2,
S3={(z1, z2)∈C2:|z1|2+|z2|2= 1},(36)
he Hop map is gi en by
π(z1, z2) = 2z1¯z2,|z1|2− |z2|2∈S2⊂R3.(37)
The connec ion one- o m αon S3sa is ies dα =π∗(ωS2), whe e ωS2is he a ea o m on S2
no malized so ha RS2ωS2= 4π. Wi h his no maliza ion,
1
2πZS2
F= 1,(38)
con i ming ha he U(1) bundle has i s Che n class c1(L) = 1 (see De ini ion II.1.3 and
Rema k II.1.8 in Lawson and Michelsohn [7]).
In ou se up, we ake F=dA wi h A= (1/ )σ3on he ound S3o adius . The ac o
o 1/ ensu es he co ec no maliza ion as a ies. In he o hono mal ame, he non-ze o
componen s a e F12 =−F21 = 2/ 3, gi ing FµνFµν = 8/ 6. The olume elemen is dV =
3sin θ dθ ∧dϕ ∧dψ ∧(L/2π)dθS1, and he in eg al RMFµνFµνdV yields 8π2L/ 3, bu does no
a ec he uni e sal β- unc ion coe icien .
Re e ences
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