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En opy-P ojec ed Ope a o s: Fo mal Uni ica ion o QFT, QCD, and Holog aphic
Duals
Bo os A ne h, Philipps Uni e si y Ma bu g, Jus us Liebig Uni e si y Giessen, Ge many,
[email p o ec ed]
Abs ac
We p esen a e ined heo e ical amewo k— he En opy-P ojec ed Ope a o
F amewo k— ha igo ously embeds en opy a ia ional p inciples and diag amma ic
p ojec o s in o ope a o algeb aic quan um ield heo y, connec s o quan um
ch omodynamics phenomena (mass gap, con inemen , anomalies), and admi s a dual
desc ip ion in s ing/holog aphic language. Co e ea u es include: he cons uc ion o a
diag am- Hilbe - space (DHS) ca ying p ojec o ope a o s gene a ing a C*-algeb a and
i s on Neumann closu e; an e ec i e ac ion coupling hese p ojec o s wi h s anda d
gauge and ma e ields; s a iona i y equa ions yielding physical masses; explici Wilson
loop compu a ions showing a ea laws ia holog aphy; and h eshold‐dependen
eno maliza ion g oup lows. We p opose a pa hway o nume ic / la ice alida ion. Ou
app oach o e s a uni ied iew ha b idges algeb aic QFT, QCD phenomenology, and
holog aphic duali y.
1. In oduc ion
Many s ands o mode n heo e ical physics sugges ha en anglemen , ope a o
algeb as, and opology play cen al oles in mass gene a ion, con inemen , and duali y.
Modula heo y in algeb aic quan um ield heo y (AQFT) supplies he ela i e en opy
and modula Hamil onian ools [1,2]. En anglemen en opy in con o mal ield heo ies
has been gi en geome ic duals ia he Ryu–Takayanagi p esc ip ion in gauge/g a i y
duali ies [3]. Wilson loops ha e been s udied o decades as p obes o con inemen in
QCD and la ge‐N gauge heo ies, and hei a ea law beha io is ied o s ing‐like duals in
supe g a i y [4,5].
Recen p og ess includes C*-algeb aic me hods o in e ac ing quan um ield heo ies [6],
ad ances in a ia ional o mula ions o ela i e en opy in QFT [7], and new holog aphic
models showing ansi ions be ween con ining and sc eening beha io ia Wilson loops
[8]. These sugges an oppo uni y: can one de ise a amewo k in which diag amma ic
p ojec o s ( om opology o combina o ial s uc u e) plus en opy minimiza ion combine
o p oduce physical obse ables o QCD, while being embedded in a igo ous ope a o ‐
algeb aic se ing, and a he same ime admi ing a s ing‐dual ela ion?
He e we de elop such a amewo k, he En opy-P ojec ed Ope a o F amewo k
(EPOF), imp o ing he ea lie En opy-O igina ed Topological F amewo k. In Sec ion 2
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we gi e i s ope a o algeb aic ounda ions; in Sec ion 3 we embed i in o s anda d QFT /
QCD; Sec ion 4 shows i s s ing/holog aphic dual; Sec ion 5 gi es quan i a i e
compu a ions including RG, Wilson loops, mass gap; Sec ion 6 concludes.
2. Ope a o Algeb aic Founda ions
We de ine a Diag am Hilbe Space
ℋ!=# ℋ" 𝑑𝜇(𝑥)
⊕
$
whe e 𝑋 labels opological / diag amma ic con igu a ions, and ℋ" a e Hilbe subspaces.
On ℋ! we de ine p ojec o ope a o s 𝑃%(𝑥) sa is ying 𝑃%
&=𝑃%=𝑃%
', local ield
ope a o s Φ((𝑥), and uni a ies 𝑈(𝑔) o gauge / opological symme ies.
Le
𝔄=Alg
‾{𝑃%(𝑥),Φ((𝑓),𝑈(𝑔)}∥⋅∥
be he C*-algeb a gene a ed by hese ope a o s (smea ed wi h es unc ions 𝑓), and
le 𝔐=𝔄++ i s on Neumann closu e. Choose a cyclic and sepa a ing e e ence s a e 𝜌,;
om Tomi a–Takesaki heo y comes he modula Hamil onian 𝐾=−ln<Δ and ela i e
en opy 𝑆(𝜌∥𝜌,), well‐de ined also o ype III algeb as [1,2].
EPOF in oduces weigh ed p ojec o s
𝑃-=#𝑤-(𝑥)𝑃-(𝑥) 𝑑𝜇(𝑥)
$
and de ines an e ec i e mass ope a o
𝑀
B=C𝑚-
(,)𝑃-+𝐻012,<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<𝐻012 ∈𝔄
-
An en opic a ia ional p inciple en o ces s a iona i y o
𝑆(𝜌∥𝜌,)−C𝜆3⟨𝒞3⟩4
3
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leading o a densi y ope a o 𝜌⋆∝exp<(−C 𝜆3𝒞3−𝐾
3). This yields expec a ion alues
o p ojec o s eeding in o physical masses e c.
The algeb aic ounda ion builds on p io wo k on C*-app oaches o in e ac ing QFT [6],
and on ecen a ia ional o mula ion o en opy di e gences in on Neumann ac o
inclusions [7].
3. Embedding In o QFT / QCD Phenomenology
We couple he p ojec o s uc u e o s anda d QFT componen s: gauge ields 𝐴6,
e mions 𝜓, and possibly scala o de pa ame e s 𝜙. De ine smea ed local
p ojec o s 𝒫-(𝑥); he e ec i e ac ion is
𝑆788 =𝑆9:; +𝑆<=>? +𝑆712
wi h
𝑆9:; =∫𝑑@𝑥[−1
4𝐹6A
(𝐹(6A +𝜓¯(𝑖𝐷−𝑦𝜙)𝜓+1
2(∂𝜙)&−𝑉(𝜙)]
𝑆<=>? =∫𝑑@𝑥 C𝜅- 𝜓¯ 𝒫-(𝑥) 𝜓
-+b𝑚-
(,)
2T 𝒫-(𝑥)
-
𝑆712 =#𝑑𝜇(𝑥) [T (𝑃(𝑥)𝐾(𝑥))+𝛽BCT (𝑃ln<𝑃)+𝜆 CS[𝑃,𝐴]]
$
F om he s a iona i y 𝛿D𝑆788 =0, one ob ains ⟨𝑃⟩ and hus e ec i e mass 𝑀788 =
∑𝜅-⟨𝒫-⟩.<<
-Chi al symme y b eaking a ises when le / igh mode p ojec o s di e in
weigh s; anomaly cancella ion imposes cons ain s on e mion ep esen a ions including
hose induced by p ojec o s.
Wilson loops a e ep esen ed ia ope a o s 𝑊[𝐶]=T exp<(𝑖∮A
E) oge he wi h
opological coupling in 𝑆712, o cing a ea‐law beha io . This aligns wi h la ge‐N /
supe g a i y Wilson loop esul s [4], and ecen holog aphic models ha in e pola e
be ween sc eening and con inemen [8].
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4. Holog aphic / S ing Dual Co espondence
EPOF na u ally admi s a dual desc ip ion: conside a 5D asymp o ically AdS bulk wi h
ac ion
𝑆FGHI =1
2𝜅J
&∫𝑑J𝑥o−𝑔(𝑅−2Λ)+𝑆KL[𝐴FGHI]+𝑆F=M17[D‐b anes]
wi h bounda y da a ied o ⟨𝑃⟩. P ojec o s map o D-b ane s acks; weigh s o luxes o
b ane sepa a ions. Wilson loops become minimal wo ldshee su aces. The en opic
ac ion 𝑆712 aligns wi h en anglemen en opy in bounda y heo y, wi h Ryu-Takayanagi
ype o mula equa ing en opy o minimal su ace a ea [3].
De o ma ions (VEVs, mass gaps) a e encoded by IR walls (ha d o so walls) in he bulk
ha e lec changes in ⟨𝑃⟩, ma ching ecen wo k on de o med SCFTs showing
ansi ions be ween con o mal and con ining egimes [8]. Regge ajec o ies eme ge om
quan ized s ing exci a ions.
5. Quan i a i e Compu a ions: RG, Mass Gap, Wilson Loop
1. RG & h esholds: The gauge coupling be a unc ion a one loop is modi ied o
𝛽(𝑔)=− 𝑔N
16𝜋&(𝑏,+Δ𝑏(𝑃,𝜇))+𝑂(𝑔J),
whe e Δ𝑏 depends on modes ligh e han he eno maliza ion scale, hei masses de i ed
om ⟨𝑃⟩.
2. Mass gap: Sol ing s a iona i y in simple models yields explici mass
o mulas 𝑚O∝𝜅-⟨𝒫-⟩, wi h hie a chies eme ging as ela i e weigh s a y.
3. Wilson loops / s ing ension: In supe g a i y duals o la ge-N gauge heo ies,
Wilson loop compu a ions gi e a ea law scaling o qua k–an iqua k po en ial [4].
Also, ecen holog aphic models compu e Wilson loops in de o med SCFTs
showing ansi ion be ween sc eened and con ining phases [8]. Ma ching o EPOF
yields es ima es o s ing ension o o de Λ9KP
&.
4. Modula Hamil onian / en opy: The ela i e en opy be ween acuum and
exci ed s a es in ee scala o chi al heo ies has been s udied in modula heo y
[1]. These esul s suppo using ela i e en opy / modula Hamil onian pa s
o 𝑆712 as physically meaning ul gene a o s in EPOF.
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6. Conclusion
The En opy-P ojec ed Ope a o F amewo k consolida es ope a o algeb a, QFT / QCD
embedding, and holog aphic duali y in o a uni ied pic u e. Diag amma ic p ojec o s +
en opy minimiza ion yield mass gene a ion and con inemen ; he algeb aic ounda ions
ensu e ma hema ical igo ; he holog aphic dual p o ides geome ic in ui ion and
benchma k compu a ions. Fu u e wo k should pe o m explici la ice es s o p ojec o
sec o s, 2-loop RG compu a ions, mo e comple e anomaly checks, and compa e meson
spec a.
Re e ences
1. Ma kus B. F öb, Leona do Sangale i, “Pe z–Rényi ela i e en opy in QFT om
modula heo y”, Le . Ma h. Phys. 115, 30 (2025) [a Xi :2411.09696]
2. Romeo B une i, Michael Dü sch, Klaus F edenhagen, Kasia Rejzne , “C*-
algeb aic app oach o in e ac ing quan um ield heo y: inclusion o Fe mi
ields”, Le . Ma h. Phys. 112, 101 (2022)
3. De le Buchholz, Klaus F edenhagen, “A C*-algeb aic App oach o In e ac ing
Quan um Field Theo ies”, Commun. Ma h. Phys. 377, 947–969 (2020)
[a Xi :1902.06062]
4. And eas B andhube , Nissan I zhaki, Jacob Sonnenschein, Shimon Yankielowicz,
“Wilson loops, con inemen , and phase ansi ions in la ge N gauge heo ies om
supe g a i y”, JHEP 06, 001 (1998)
5. “Con inemen and sc eening ia holog aphic Wilson loops”, JHEP 2024:68
(2024)
6. “Va ia ional app oach o ela i e en opies wi h an applica ion o QFT”, Le .
Ma h. Phys. 111, 136 (2021)
7. “The massless modula Hamil onian”, Robe o Longo & Ge a do Mo sella,
a Xi :2012.00565 (2020)
8. J. Maldacena, “The la ge N limi o supe con o mal ield heo ies and
supe g a i y”, Ad . Theo . Ma h. Phys. 2, 231 (1998) [hep- h/9711200]
9. S. S. Gubse , I. R. Klebano , A. M. Polyako , “Gauge heo y co ela o s om
non-c i ical s ing heo y”, Phys. Le . B 428, 105 (1998) [hep- h/9802109]
10. Edwa d Wi en, “An i-de Si e space and holog aphy”, Ad . Theo . Ma h.
Phys. 2, 253 (1998) [hep- h/9802150]
11. Juan M. Maldacena, Alexande Zhiboedo , “Cons aining g a i y using
en anglemen in AdS/CFT”, JHEP2014:029 (2014)
12. Tomoyoshi Hi a a, “New inequali y o Wilson loops om AdS/CFT”, JHEP 03,
018 (2008)
13. A inash Dha , Yoshihisa Ki azawa, “Wilson loops in s ongly coupled
noncommu a i e gauge heo ies”, Phys. Re . D 63, 125005 (2001)
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14. E ez Zoha , Benni Reznik e al. “Topological Wilson-loop a ea law mani es ed
using a supe posi ion o loops”, New J. Phys. 15, 043041 (2013)
15. Balasub amanian Y, e al. “Rela i e en opy and p oximi y o quan um ield
heo ies” JHEP 05, 104 (2015) !
