Coun e e ms A e No he Answe : Cohe ence, Descen , and Global
Consis ency in Highe -Dimensional Holog aphy
And ei T. Pa ascu
1
1
FAST Founda ion, Des in FL, 32541, USA
email: and ei.pa [email p o ec ed]
Recen p og ess in highe –dimensional holog aphy has demons a ed ha physically ele an
con igu a ions—including bubbling geome ies and de ec backg ounds—o en lie ou side he scope
o any ini e– ield AdS
d+1
unca ion, o cing eno maliza ion o be pe o med di ec ly in en o
ele en dimensions. In such se ings, ini e obse ables a e ob ained only a e in oducing speci ic
cu o p esc ip ions and highe –dimensional bounda y coun e e ms, whose jus i ica ion is ypically
alida ed a pos e io i by ag eemen wi h known ield– heo e ic da a. While success ul as compu a ions,
hese cons uc ions emain in insically p esc ip ion–based: hey p o ide nei he a c i e ion o he
uniqueness o he coun e e ms employed no a p inciple go e ning anspo be ween di e en
egula o choices, and hey lea e he physical meaning o he highe –dimensional cu o su ace i sel
un esol ed.
In his wo k we a gue ha his si ua ion e lec s a missing global consis ency s uc u e a he han
a echnical limi a ion o holog aphic eno maliza ion. We show ha choices o cu o , asymp o ic
p esen a ion, and sub ac ion scheme na u ally assemble in o a con ex g oupoid, whose mo phisms
encode admissible changes o egula o . Wi hin his amewo k, coun e e ms acqui e a p ecise
in e p e a ion as cohe ence (descen ) da a equi ed o glue pa chwise–de ined eno malized unc ionals
in o a global objec . Global consis ency equi es cohe ence unde composi ion o con ex changes,
bu does no equi e s ic i ica ion: a cohe en bu non–s ic i iable assignmen al eady de ines a
well–posed global objec . We p o e ha s ic i ica ion is possible i and only i he cohe ence da a is
i ial; when s ic i ica ion ails, he emaining obs uc ion class is in a ian and is iden i ied wi h
he anomaly.
Applying his pe spec i e o ecen en–dimensional compu a ions o de ec anomalies, we demon-
s a e ha he appea ance o ad hoc highe –dimensional bounda y coun e e ms is nei he acciden al
no a bi a y, bu e lec s he p esence o a genuine obs uc ion o s ic i ica ion a he han a
ailu e o pa ching. Ou esul s supply he o ganizing p inciple missing om pu ely compu a ional
app oaches and p o ide a sha p c i e ion dis inguishing emo able scheme dependence om in insic,
physically meaning ul anomalies in highe –dimensional holog aphy.
I. EXECUTIVE SUMMARY FOR PHYSICISTS
The pu pose o his pape is no o in oduce new compu a ional echniques, bu o isola e a missing
consis ency p inciple ha has become una oidable in highe –dimensional holog aphy. The ecen p oli e -
a ion o p esc ip ion–dependen eno maliza ion schemes is no a empo a y echnical incon enience; i is
a s uc u al signal. This sec ion summa izes he diagnosis and he emedy in he mos economical e ms.
2
•Regula o choices a e no gauge unless unc o ial anspo is speci ied.
Changing
a cu o su ace, coo dina e p esc ip ion, o sub ac ion scheme is no a edundancy by de aul .
Such changes mus be accompanied by a well–de ined anspo ule ela ing eno malized
quan i ies ac oss schemes. Absen such a ule, “scheme independence” is an empi ical coincidence
a he han a s uc u al s a emen .
•Coun e e ms supply gluing da a, no me ely di e gence cancella ion.
In highe –
dimensional holog aphy, coun e e ms a e no ixed solely by local powe coun ing. Thei
essen ial ole is o ela e pa chwise–de ined eno malized unc ionals ac oss di e en egula o
choices. In e p e ed co ec ly, coun e e ms p o ide he cohe ence da a equi ed o glue local
de ini ions in o a global objec .
•Anomalies a e obs uc ions, no le o e s.
Uni e sal ini e e ms ha su i e all admis-
sible sub ac ions do no signal an incomple e eno maliza ion. They signal he ailu e o global
s ic i ica ion: a genuine obs uc ion o de ining a egula o –independen obse able. Anomalies
a e he e o e in a ian obs uc ion classes, no a i ac s o poo scheme choices.
•Highe –dimensional holog aphy o ces his issue in o he open.
When holog aphic
obse ables a e sensi i e o in insically en– o ele en–dimensional s uc u e, no ini e lowe –
dimensional unca ion can supply a uni e sal eno maliza ion dic iona y. The dependence
on cu o geome y and bounda y p esc ip ions is una oidable and mus be o ganized, no
elimina ed.
Taken oge he , hese poin s lead o a simple conclusion: highe –dimensional holog aphy equi es a
global no ion o consis ency ha goes beyond local coun e e m sub ac ion. The absence o such a
no ion explains why ecen compu a ions succeed only a e in oducing appa en ly ad hoc bounda y
e ms, and why he meaning and uniqueness o hese p esc ip ions emain obscu e.
This pape supplies he missing consis ency p inciple.
We show ha egula o choices na u ally
assemble in o a con ex g oupoid, ha coun e e ms unc ion as cohe ence (descen ) da a on his g oupoid,
and ha anomalies a ise p ecisely as obs uc ion classes o global s ic i ica ion. This amewo k con e s
p esc ip ion–dependen eno maliza ion in o a well–posed s uc u al p oblem and sha ply dis inguishes
emo able scheme dependence om in insic physical in a ian s.
II. THE SYMPTOM: WHEN HOLOGRAPHY STOPS BEING UNIVERSAL
The s anda d holog aphic dic iona y is o en p esen ed as i i we e a uni e sal eno maliza ion
machine: choose Fe e man–G aham gauge, in oduce a adial cu o , add local coun e e ms de e mined
by co a iance and powe coun ing, emo e he egula o , and ob ain a ini e gene a ing unc ional whose
ini e pa is unambiguous up o he usual scheme choices [
2
–
4
]. This pic u e is accu a e in he egime o
which i was enginee ed: asymp o ically AdS
d+1
solu ions o a ixed lowe –dimensional e ec i e heo y
(o a consis en unca ion) wi h a con olled asymp o ic expansion. The symp om add essed in his
pape is ha a apidly g owing class o physically ele an holog aphic obse ables li es ou side ha
egime, and he adi ional no ion o “scheme dependence” ceases o be he igh o ganizing concep .
A. Failu e o e ec i e AdSd+1 unca ions
The b eakdown is no p ima ily compu a ional; i is concep ual. Many o he mos in o ma i e p obes
o AdS/CFT—de ec s, hea y ope a o s, and bubbling geome ies—a e no na u ally desc ibed as solu ions
o a ini e– ield AdS
d+1
bulk heo y. Ra he , hey a e in insically en– o ele en–dimensional supe g a i y
solu ions whose asymp o ics app oach AdS
d+1 ×Mq
bu whose de ining da a and global s uc u e do no
educe o a lowe –dimensional unca ion [
1
,
5
]. In such backg ounds, he pa o he on–shell ac ion ha
is equi ed o ce ain obse ables is no con olled by he uni e sal nea –bounda y expansion o a small
se o ields. I is sensi i e o genuinely highe –dimensional in o ma ion: global opology, in e nal lux
da a, and he de ailed way he geome y “ ills in” away om he asymp o ic egion.
This ma e s because many obse ables o in e es a e ini e obse ables. They a e no me ely a ios
in which di e gences cancel, no a e hey p o ec ed by a educ ion o local bounda y da a. They p obe
3
he eno malized on–shell ac ion (o i s loga i hmic e ms) i sel , and he e o e hey p obe he global
consis ency o he eno maliza ion p esc ip ion, no jus he cancella ion o powe di e gences. In o he
wo ds: he egime ha is adi ionally called “holog aphic eno maliza ion” implici ly assumes ha he
ele an obse able is al eady exp essible in he language o a uni e sal AdS
d+1
e ec i e heo y. Once
ha assump ion ails, uni e sali y is no longe gua an eed by he o malism.
B. The ise o p esc ip ion–based eno maliza ion
A clea mani es a ion o he new egime is he appea ance o eno maliza ion p ocedu es ha succeed
only as p esc ip ions. A ep esen a i e example is he ecen en–dimensional compu a ion o he Eule
con o mal anomaly coe icien o hal –BPS su ace ope a o s in
N
= 4 SYM, ob ained by e alua ing he
on–shell Type IIB supe g a i y ac ion on he ele an bubbling geome y while in oducing a speci ic
highe –dimensional bounda y coun e e m and cu o p esc ip ion [
1
]. This wo k is echnically success ul
and concep ually e ealing: i ep oduces an obse able known exac ly in ield heo y and highligh s ha
he co esponding in o ma ion is no accessible wi hin any ini e– ield AdS5 unca ion [1].
A he same ime, he logic o such cons uc ions is i educibly p ocedu al. The eno malized answe
depends on a chain o choices:
•a choice o cu o su ace (in he highe –dimensional geome y),
•
a choice o asymp o ic p esen a ion/coo dina es (which de e mines wha coun s as “ adial” and
wha da a a e held ixed),
•
a choice o highe –dimensional bounda y coun e e ms (no ixed by he usual AdS
d+1
powe –
coun ing logic),
•
and a no maliza ion/sub ac ion con en ion (o en calib a ed by acuum sub ac ion o ma ching
o a p o ec ed CFT quan i y).
In his egime, “scheme independence” canno be asse ed by ci ing co a iance o he nea –bounda y
expansion alone. I can only be es ed by compa ison wi h independen ield– heo y inpu o by epea ing
he compu a ion unde a di e en se o choices and checking ha he answe s ag ee. Tha is no a
eno maliza ion p inciple; i is an empi ical e i ica ion loop.
This is he p ecise sense in which ecen highe –dimensional holog aphic eno maliza ions a e compu a-
ions wi hou a con olling mechanism: hey show he exis ence o a p esc ip ion ha wo ks, bu hey do
no p o ide (and o en explici ly se aside) a c i e ion ha would decide which p esc ip ions a e admissible,
when hey a e equi alen , and why a gi en coun e e m is compelled a he han me ely con enien . In
pa icula , he physical meaning o he highe –dimensional cu o su ace and he classi ica ion/uniqueness
o allowed bounda y e ms emain opaque in pu ely compu a ional app oaches [1].
C. The key obse a ion
The co e obse a ion is simple and sha p:
The p oblem is no longe ul a iole di e gence; he p oblem is compa ison ac oss schemes.
In he adi ional AdS
d+1
se ing, one may ea he egula o as a echnical sca old because he
eno malized unc ional is con olled by a uni e sal asymp o ic expansion and he admissible coun e e ms
a e classi ied by locali y and co a iance [
2
–
4
]. In highe –dimensional holog aphy beyond unca ions,
by con as , he egula o choice is pa o he de ini ion o he compu a ion unless one also speci ies
a ule o anspo ing eno malized da a be ween di e en choices. Wi hou such anspo , “scheme
dependence” is no a small ambigui y; i is a ailu e o s a e wha is being compa ed.
This pape is buil a ound he ollowing diagnosis. The co ec objec o o ganize highe –dimensional
holog aphic eno maliza ion is no a single p e e ed cu o p esc ip ion, bu he space o p esc ip ions
oge he wi h hei admissible changes. We will o malize his as a con ex g oupoid: objec s a e egula o
con ex s (cu o su ace, asymp o ic p esen a ion, sub ac ion/no maliza ion, allowed bounda y e ms),
and mo phisms a e admissible changes o con ex . In his language, coun e e ms a e no me ely
sub ac ion de ices; hey supply he cohe ence (gluing) da a needed o ela e pa chwise de ini ions
ac oss con ex s. Anomalies appea p ecisely when his cohe ence canno be s ic i ied globally, lea ing
4
an in a ian obs uc ion class. The es o he pape makes his s a emen p ecise and p o es he
co esponding global consis ency c i e ion.
III. WHAT A COMPUTATION ALONE CANNOT TELL YOU
The highe –dimensional esul s mo i a ing his pape a e no “w ong,” no a e hey o be dismissed as
echnical exe cises. On he con a y, hey deli e an impo an ac : ce ain obse ables can be compu ed
holog aphically only by lea ing he com o zone o ini e- ield AdS
d+1
unca ions and wo king di ec ly
in en o ele en dimensions. A ep esen a i e example is he en–dimensional e alua ion o he Type IIB
on–shell ac ion on a bubbling geome y dual o a hal –BPS su ace ope a o , which ep oduces he Eule
anomaly coe icien by supplemen ing he ac ion wi h a speci ic highe –dimensional bounda y e m and a
ca e ully chosen cu o p esc ip ion [1]. This is a genuine success.
The poin o he p esen sec ion is su gical: o sepa a e wha such a compu a ion es ablishes om wha
i canno , by i sel , es ablish. The dis inc ion is no he o ical. I is he di e ence be ween a esul ha is
a one–o calib a ion and a esul ha is con olled by an in e nal p inciple.
A. Exis ence is no mechanism
A compu a ion can es ablish exis ence:
The e exis s a p esc ip ion (cu o choice, coo dina e p esen a ion, bounda y e m) ha yields
a ini e answe ma ching he expec ed ield– heo y quan i y.
This exis ence s a emen is al eady non i ial in highe –dimensional holog aphy beyond unca ions, and
i is exac ly wha is achie ed in [1].
Howe e , exis ence is no a mechanism. I does no ell you whe he he p esc ip ion is compelled o
acciden al, canonical o enginee ed, s able o agile. In he adi ional AdS
d+1
se ing, his gap is la gely
closed because he admissible coun e e ms a e classi ied by locali y and co a iance and he asymp o ic
expansion p o ides a uni e sal con ol pa ame e [
2
–
4
]. Ou side ha egime, he gap eappea s, and i is
p ecisely he gap ha p oduces he sensa ion o “ad hoc coun e e ms.”
B. Fou ques ions a compu a ion canno answe
Once one lea es a uni e sal unca ion, a eno malized answe comes wi h a hidden bu den: one mus
con ol he space o pe missible egula o choices and hei ela ions. A compu a ion, by i sel , canno
p o ide ha con ol. Conc e ely, i canno answe he ollowing ou ques ions.
(i) Why is his p esc ip ion equi ed? A wo king p esc ip ion may be one poin in a la ge space o
possible p esc ip ions. Wi hou a p inciple, he e is no dis inc ion be ween a coun e e m ha is o ced
by consis ency and one ha is me ely su icien o ma ch a known da um. In [
1
] he highe –dimensional
bounda y e m is in oduced because i makes he compu a ion wo k; he deepe eason i should be
p esen (and in wha p ecise sense i is he igh objec ) is no supplied by he compu a ion i sel [1].
(ii) Is he p esc ip ion unique? E en i one accep s he need o addi ional highe –dimensional bounda y
e ms, one mus s ill decide whe he he choice is unique unde a clea equi alence ela ion. In he absence
o a classi ica ion o admissible bounda y e ms in he highe –dimensional se ing, “uniqueness” educes
o a p ac ical ques ion: can one modi y he p esc ip ion wi hou changing he esul ? Pu e compu a ion
answe s his only by epea ed ial. The concep ual ask is o de ine he equi alence ela ion and show
whe he he answe is in a ian unde i . Tha ask is no pa o he compu a ional pipeline.
(iii) How does i gene alize? A p esc ip ion calib a ed on one class o backg ounds does no au oma i-
cally anspo o ano he . In pa icula , once he ele an obse able depends on global bulk s uc u e,
he ques ion o gene aliza ion becomes a ques ion o anspo : how do choices made in one con ex map o
ano he con ex , and wha da a mus be ca ied along? The en–dimensional compu a ion o [
1
] exhibi s
he need o such anspo p ecisely by con on ing he misma ch be ween he ou –dimensional locus
suppo ing he de ec CFT da a and he highe –dimensional cu o su ace used in he eno maliza ion
[1]. Bu again: exhibi ing he need is no he same as p o iding he anspo law.
5
(i ) Wha in a ian is being measu ed? Ma ching a p o ec ed ield– heo y quan i y iden i ies a numbe ,
bu i does no iden i y he s uc u al s a us o ha numbe . Is i an in insic in a ian o he holog aphic
se up, o an a i ac o he chosen egula o con ex ? Is i s able unde admissible changes, o only unde
he pa icula changes ha we e ied? In he adi ional amewo k, uni e sal e ms (no ably anomalies)
a e unde s ood as in a ian da a because he space o admissible local coun e e ms is classi ied and
he emaining ambigui y is con olled [
2
,
4
]. In he highe –dimensional egime highligh ed by [
1
], he
compu a ion ep oduces he anomaly coe icien , bu he compu a ion alone canno supply he s uc u al
a gumen ha his coe icien is he in a ian emainde o a well–posed compa ison p oblem a he han
he esidue o a con enien sub ac ion scheme [1].
C. The clinical conclusion
The p eceding ou ques ions a e no op ional philosophical upg ades. They a e he minimal condi ions
unde which “scheme dependence” becomes a con olled s a emen a he han a slogan. The ecen
highe –dimensional compu a ions a e he e o e bes unde s ood as high–quali y symp oms: hey show
clea ly whe e he adi ional uni e sali y s o y ails, and hey pinpoin he exac loca ion whe e addi ional
s uc u e mus en e .
Wha is missing is a global no ion o consis ency.
IV. CONTEXTS ARE NOT GAUGE: THE MINIMAL STRUCTURE
The p esc ip ion–dependence encoun e ed in highe –dimensional holog aphy is o en discussed in he
loose language o “scheme choices.” Tha language is oo weak o he egime o in e es . A scheme
is usually ea ed as an auxilia y choice ha can be changed wi hou concep ual consequence. In he
p esen se ing, by con as , he egula o choice becomes pa o he de ini ion o he compu a ion unless
one also speci ies how esul s a e anspo ed be ween choices. The minimal s uc u e equi ed o make
his p ecise is small: one needs only a no ion o con ex and a no ion o admissible change o con ex .
The es is bookkeeping—bu i is he bookkeeping ha u ns a compu a ion in o a p inciple.
A. De ini ion: con ex
Acon ex is he ull se o da a equi ed o s a e a eno malized holog aphic obse able as a well–posed
p oblem. Conc e ely, a con ex consis s o he ollowing componen s:
1. Cu o su ace.
A choice o egula ing hype su ace Σ
in he bulk geome y (o amily he eo )
on which bounda y condi ions a e imposed and on which bounda y e ms a e e alua ed. In highe –
dimensional se ups his su ace may no coincide wi h a canonical “ adial” slice and may depend
non i ially on in e nal coo dina es.
2. Asymp o ic p esen a ion.
A choice o asymp o ic coo dina e sys em and ield pa ame iza ion
used o iden i y he egula ed egion and o de ine wha i means o hold bounda y da a ixed.
In AdS
d+1
his is o en encoded in Fe e man–G aham gauge; beyond unca ions, he ele an
p esen a ion da a can be mo e in ica e.
3. Sub ac ion/no maliza ion scheme.
A choice o e e ence backg ound and no maliza ion con-
en ion (e.g. acuum sub ac ion, choice o coun e e m eno maliza ion scale, o o he calib a ion)
used o de ine he ini e pa .
4. Allowed bounda y e ms.
A speci ica ion o he admissible local (o quasi–local) bounda y
unc ionals ha may be added on Σ
as coun e e ms, including he symme y and co a iance
equi emen s hey mus sa is y.
5. Bounda y condi ions.
A speci ica ion o he bounda y condi ions imposed on Σ
(Di ichle /Neu-
mann/mixed, ensemble choice, and any addi ional cons ain s equi ed o a well–posed a ia ional
p inciple).
6
We deno e a con ex by a symbol such as
c
, wi h he unde s anding ha
c
packages he abo e i e
ing edien s.
Two ema ks a e essen ial. Fi s , he poin is no o p oli e a e da a, bu o acknowledge da a ha
is al eady p esen implici ly in any ac ual compu a ion. Second, in he highe –dimensional egime
exempli ied by [
1
], se e al o hese ing edien s a e no ixed by he lowe –dimensional AdS
d+1
dic iona y;
hey mus be speci ied independen ly. This is p ecisely why one encoun e s appa en ly ad hoc bounda y
coun e e ms and cu o p esc ip ions [1].
B. Allowed changes o con ex
Gi en wo con ex s
c
and
c0
, a change o con ex is an admissible map ha ela es he co esponding
egula ed se ups. The wo d “admissible” is doing he wo k: we only allow changes ha p ese e he
class o p oblems unde conside a ion (symme ies, a ia ional well–posedness, and he no ion o ixed
bounda y da a). The basic mo es a e:
1. Coo dina e and ield ede ini ions.
Changes o asymp o ic coo dina es and ield a iables
ha p ese e he asymp o ic class o he solu ion and he iden i ica ion o sou ces/VEVs. In he
AdS
d+1
se ing hese include he amilia PBH ans o ma ions; in highe dimensions hey include
epa ame iza ions mixing “ adial” and in e nal di ec ions, p o ided hey p ese e he admissibili y
c i e ia.
2. Cu o de o ma ions.
De o ma ions o he egula ing su ace Σ
7→
Σ
0
wi hin he admissible
amily. Physically, hese a e changes o how one slices he bulk nea he bounda y (o nea a
degene a e bounda y), and hey gene ally induce changes in he induced me ic and o he bounda y
ields on he cu o su ace.
3. Induced coun e e m shi s.
Since he cu o su ace and p esen a ion da a change, he allowed
bounda y e ms mus ans o m acco dingly. E en when he space o admissible coun e e ms is
ixed abs ac ly, a change o con ex gene ically induces a shi in he ep esen a i e coun e e m
unc ional used o de ine he ini e pa .
A change o con ex he e o e does no mean “pick a di e en egula o and hope no hing changes.” I
means: speci y a map be ween egula o s and ca y along he induced ans o ma ion o all con ex ual
da a. In pa icula , i a compu a ion yields he same ini e answe unde wo di e en con ex s, his is
meaning ul only i he con ex s a e connec ed by an admissible change and i he co esponding anspo
is p ope ly accoun ed o .
C. Con ex s o m a g oupoid
The collec ion o con ex s oge he wi h admissible changes o con ex na u ally o ms a g oupoid. We
deno e his g oupoid by Cand summa ize i s s uc u e as ollows:
•Objec s:
con ex s
c
(cu o , p esen a ion, sub ac ion, admissible bounda y e ms, bounda y
condi ions).
•Mo phisms: admissible changes o con ex φ:c→c0.
•Composi ion: conca ena ion o admissible changes, ψ◦φ:c→c00.
•In e sion:
each admissible change is e e sible a he le el o con ex ual da a (a leas locally in
he space o con ex s), yielding φ−1:c0→c.
The g oupoid iewpoin is he minimal ma hema ical upg ade equi ed o make “scheme dependence”
a con olled s a emen . I eco ds no only he exis ence o mul iple con ex s, bu also he admissible
ela ions be ween hem. In pa icula , loops in
C
encode he possibili y o e u ning o he “same” con ex
by di e en ou es, a mechanism by which non i ial obs uc ion da a (anomalies) can a ise.
Key message.
I you do no ack mo phisms, you canno alk abou in a iance.
7
A eno malized obse able is no me ely a numbe assigned o one p e e ed egula o choice. I is, a
minimum, an assignmen ha is s able unde he admissible changes o con ex encoded by
C
. The
emainde o he pape makes his s a emen p ecise: coun e e ms will be iden i ied as he cohe ence
da a go e ning anspo along mo phisms, and anomalies will eme ge as he obs uc ion o global
s ic i ica ion o e he con ex g oupoid.
V. PATCHWISE DEFINITIONS VERSUS GLOBAL OBJECTS
Wi h he con ex g oupoid
C
in place, one can s a e he cen al issue in one line: highe –dimensional
holog aphic eno maliza ion cu en ly p oduces pa chwise ini e answe s, while physics equi es a global
objec . The gap be ween “pa chwise” and “global” is no seman ics. I is exac ly whe e coun e e ms
acqui e hei eal meaning and whe e anomalies a ise.
Th oughou his sec ion we delibe a ely a oid ca ego ical ja gon. The only concep s used a e he ones
holog aphe s al eady employ in o mally: local de ini ions, compa isons on o e laps, and consis ency unde
epea ed compa ison.
A. Pa chwise eno malized unc ionals
Fix a con ex
c∈Ob
(
C
), i.e. a choice o cu o su ace, asymp o ic p esen a ion, sub ac ion con en ion,
allowed bounda y e ms, and bounda y condi ions. A highe –dimensional holog aphic compu a ion hen
p oduces a egula ed unc ional
S eg
c
(
)( o example, he on–shell ac ion wi h bounda y e ms e alua ed
on he cu o su ace Σ
), and a e adding coun e e ms and sub ac ing e e ence con ibu ions one
ex ac s a ini e quan i y, which we deno e schema ically by
S en
c:= lim
→0S eg
c() + Sc
c()−S e
c().(1)
The de ining ea u e is no he o mula bu he logic: he ini e answe
S en
c
is p oduced ela i e o
c
.
Change
c
and one ypically changes in e media e s eps and, in gene al, may change he ini e emainde
unless addi ional s uc u e is imposed.
This is p ecisely he si ua ion in highe –dimensional compu a ions beyond unca ions: one ob ains a
clean ini e answe in a chosen con ex , bu one has no ye ea ned he igh o call i con ex –independen .
The ou pu o he compu a ion is he e o e bes iewed as a amily o ini e answe s indexed by con ex s,
c7−→ S en
c.(2)
Physics, howe e , does no ask o a amily indexed by egula o choices. I asks o an obse able.
B. O e laps and ansi ion da a
Suppose now ha wo con ex s
c
and
c0
a e ela ed by an admissible change o con ex
φ
:
c→c0
(a
mo phism in
C
). Then
φ
speci ies how cu o su aces, asymp o ic da a, and bounda y condi ions a e
anspo ed. C ucially, i mus also speci y how he eno malized unc ionals a e compa ed.
In gene al,
S en
c
and
S en
c0
a e no equal as aw unc ionals. They may di e by a ini e, local bounda y
e m (o , mo e gene ally, by a con olled unc ional de e mined by he admissibili y c i e ia). We encode
his by in oducing ansi ion da a
T
(
φ
)assigned o each admissible change
φ
, such ha he compa ison
akes he o m
S en
c0=S en
c+T(φ).(3)
Equa ion
(3)
should be ead as a de ini ion o he p oblem, no as an iden i y ha au oma ically holds.
I s a es: o compa e wo con ex s, one mus speci y ex a da a T(φ).
This is he i s incision:
Coun e e ms do no me ely cancel di e gences; hey li e in he ansi ion da a.
The usual iew o coun e e ms as sub ac ion de ices co esponds o ocusing on a single con ex and
en o cing ini eness o
(1)
. Bu once one asks o a con ex –independen objec , coun e e ms acqui e a
8
second ole: hey con ol how ini e pa s shi unde admissible changes o egula o . In o he wo ds,
coun e e ms a e no an a e hough — hey a e he glue ha ela es pa chwise ini e answe s ac oss he
space o con ex s.
In p ac ice, highe –dimensional compu a ions o en ix
T
(
φ
)implici ly by choosing one p e e ed con ex
and calib a ing i agains ield heo y. Tha p oduces one
S en
c
wi h a desi ed alue, bu i does no
p oduce a sys ema ic ule o
T
(
φ
) o all admissible
φ
. The ule is wha u ns he amily
(2)
in o an
obse able.
C. T iple compa isons and consis ency
The decisi e consis ency check appea s as soon as one compa es h ee con ex s. Le
c0,c1,c2
be con ex s
connec ed by admissible changes
φ01 :c0→c1, φ12 :c1→c2, φ02 :c0→c2,
wi h
φ02
compa able o he composed change
φ12 ◦φ01
. I he ansi ion ule is cohe en , hen compa ing
c0 o c2di ec ly o in wo s eps mus ag ee:
T(φ02) = T(φ12 ◦φ01)and hence T(φ12 ◦φ01) = T(φ12) + T(φ01).(4)
Equa ion
(4)
is he minimal consis ency condi ion: anspo in wo s eps mus equal anspo in one
s ep. No hing mo e sophis ica ed is equi ed o see i s o ce. Once
(4)
ails, he compa ison o ini e
answe s becomes pa h–dependen .
The sha pes way o expose he pa hology is o conside a loop in he con ex g oupoid. Le
γ
be a
sequence o admissible changes ha s a s and ends a he same con ex c:
c
φ1
−→ c1
φ2
−→ · · · φn
−−→ c, γ := φn◦ · · · ◦ φ1.
I he eno malized objec we e genuinely con ex –independen , anspo ing a ound he loop would
p oduce no ne shi :
T(γ)!
= 0.(5)
When
(5)
ails, he ailu e is no a compu a ional e o . I is an in a ian esidue o he compa ison
p oblem:
Failu e o close a ound loops is an obs uc ion.
This obs uc ion is p ecisely wha is measu ed by anomalies in he si ua ions we ca e abou . The poin is
s uc u al: anomalies appea when one canno choose coun e e ms (i.e. ansi ion da a) such ha all
admissible compa isons a e simul aneously consis en .
This sec ion has done only h ee hings. Fi s , i has eph ased he ou pu o highe –dimensional
holog aphic eno maliza ion as a amily
(2)
o pa chwise ini e answe s. Second, i has iden i ied
coun e e ms as he ansi ion da a
(3)
needed o compa e hose answe s ac oss con ex s. Thi d, i has
isola ed he minimal consis ency condi ion
(4)
and i s loop es
(5)
, which is whe e obs uc ions li e.
The nex sec ion u ns hese s a emen s in o a p ecise c i e ion: when he ansi ion da a can be chosen
cohe en ly, a global eno malized objec exis s; when i canno , he esidual obs uc ion class is he
anomaly.
VI. MAIN RESULT: COHERENCE AND OBSTRUCTION
Sec ions IV–V isola e he p oblem wi h su gical cla i y: highe –dimensional holog aphic eno maliza ion
beyond uni e sal unca ions p oduces ini e answe s only ela i e o a con ex , and wi hou a anspo
law be ween con ex s he e is no meaning o “scheme independence.” This sec ion supplies he missing
p inciple. I does no add ano he p esc ip ion. I makes p esc ip ions unnecessa y by u ning he
compa ison p oblem in o a s uc u al c i e ion.
The co e upg ade is his:
9
Coun e e ms a e cohe ence da a on he con ex g oupoid, and anomalies a e he obs uc ion
o s ic i ying ha da a globally.
This is p ecisely wha pu ely compu a ional app oaches canno p o ide: hey can exhibi a wo king
coun e e m in one con ex , bu hey canno decide whe he he coun e e m is o ced, whe he wo
p esc ip ions a e equi alen , o wha in a ian is being measu ed. He e we gi e he mechanism.
A. Coun e e ms as cohe ence da a
Fix he con ex g oupoid
C
om Sec ion IV C. Fo each con ex
c∈Ob
(
C
)we ha e a pa chwise
eno malized unc ional
S en
c
de ined by some egula iza ion, coun e e m sub ac ion, and no maliza ion
con en ion. To compa e
S en
c
ac oss con ex s, we in oduced ansi ion da a
T
(
φ
)assigned o each
mo phism φ:c→c0:
S en
c0=S en
c+T(φ).(6)
The unc ion
φ7→ T
(
φ
)is no deco a ion; i is he minimal da a needed o in e p e he collec ion
{S en
c}
as desc ibing a single objec . In p ac ice,
T
(
φ
)is implemen ed by he ini e pieces o bounda y
coun e e ms and no maliza ion choices ha mus accompany a change o cu o su ace and asymp o ic
p esen a ion.
The essen ial poin is ha T(φ)is cons ained:
1.
I mus be admissible: compa ible wi h he de ini ion o he con ex class (symme ies, locali y/co-
a iance equi emen s, and well–posed bounda y condi ions).
2. I mus be cohe en : consis en unde composi ion o admissible changes o con ex .
The i s cons ain is wha physicis s usually mean by “allowed coun e e ms.” The second cons ain is
wha is missing om pu ely local discussions: i o ces coun e e ms o beha e as gluing da a a he han
as a bi a y sub ac ions.
Conc e ely, cohe ence equi es ha when wo mo phisms compose,
c
φ
−−→ c0ψ
−−→ c00,(7)
he induced compa ison om
c
o
c00
be independen o whe he one compa es in one s ep o wo s eps.
This is he addi i e law
T(ψ◦φ) = T(ψ) + T(φ),(8)
al eady an icipa ed in
(4)
. Equa ion
(8)
is he s ic o m o cohe ence: i says ha
T
is compa ible wi h
composi ion.
Fi s incision (p inciple e sus p esc ip ion).
A pu ely compu a ional cons uc ion can pick one
coun e e m unc ional and e i y ha i cancels di e gences and yields he desi ed ini e pa in one
con ex . I canno decide whe he he induced ansi ion da a sa is ies
(8)
on he ull class o admissible
mo phisms. Tha decision equi es posing eno maliza ion as a global compa ison p oblem. Tha is he
mea supplied he e.
B. The s ic i ica ion p oblem
E en i one has cohe en ansi ion da a
T
(
φ
), he e emains he ques ion o whe he he con ex
dependence can be elimina ed by a ede ini ion o he pa chwise unc ionals
S en
c
. This is he s ic i ica ion
p oblem.
Conside shi ing each pa chwise unc ional by a ini e coun e e m (a “choice o ep esen a i e”)
depending only on he objec c:
S en
c7−→ e
S en
c:= S en
c+B(c),(9)
whe e
B
(
c
)is an admissible ini e bounda y unc ional in he class allowed by he con ex de ini ion.
Unde such a shi , he ansi ion da a changes as
T(φ)7−→ e
T(φ) := T(φ) + B(c0)−B(c), φ :c→c0.(10)
16
you ansi ion da a is cohe en . I i ails, you ha e de ec ed pa h dependence: he “obse able” depends
on how you mo e h ough con ex space. Tha pa h dependence is no a nuisance. I is he anomaly
mechanism.
Ou pu o S ep 4. A yes/no answe :
•YES: Tis cohe en ; p oceed o s ic i ica ion.
•NO: cohe ence ails; ex ac he obs uc ion om loops (S ep 5).
F. S ep 5: I closu e ails, compu e he anomaly as a loop in a ian
I
(16)
ails, do no “ ix” i by adding mo e ad hoc coun e e ms. Ins ead, compu e he obs uc ion by
e alua ing
T
on loops. Pick a se o gene a ing loops
γ
in
C
( ypically a ising om non i ial ela ions
be ween cu o de o ma ions and coo dina e changes) and compu e
A(γ) := T(γ).(17)
By Theo em VI.1,
A
(
γ
)is in a ian unde admissible objec –wise ede ini ions and he e o e de ines a
physical obs uc ion class. This is he ope a ional de ini ion o an anomaly in ou amewo k: i is he
s ic i ica ion–in a ian esidue o a compa ison p oblem, no a sub ac ion ailu e.
Ou pu o S ep 5.
A se o loop in a ian s
A
(
γ
). In de ec holog aphy, he Eule coe icien ex ac ed
in [
1
] is p ecisely o his ype: i is s able no because one ound a lucky sub ac ion, bu because i is he
in a ian emainde o he con ex compa ison p oblem.
G. S ep 6: I closu e holds, s ic i y and ob ain a global obse able
I (16) holds, compu e an objec –wise shi B(c)such ha
e
T(φ) = T(φ) + B(c0)−B(c)=0 o all φ:c→c0.(18)
This is s ic i ica ion. I p oduces a globally de ined eno malized unc ional
e
S en
independen o con ex
wi hin he admissible class. This is he poin a which “scheme independence” becomes a heo em a he
han a claim. No amoun o single–con ex coun e e m enginee ing can subs i u e o his s ep, because
s ic i ica ion is a global s a emen .
Ou pu o S ep 6.
A global eno malized obse able (a s ic i ied ep esen a i e) oge he wi h a p oo
o i s in a iance unde all admissible con ex changes.
H. The scalpel conclusion
The ecipe can be summa ized in a single line:
Build he g oupoid, compu e he ansi ions, check cohe ence, and ead o ei he a s ic i ied
obse able o an obs uc ion class.
This is how one upg ades highe –dimensional holog aphic eno maliza ion om “ ind a coun e e m ha
wo ks” o “classi y wha is emo able and wha is in a ian .” I is p ecisely wha compu a ion–only
app oaches canno do, and i is exac ly how pas esul s—including [
1
]—can be sys ema ized, sha pened,
and pushed beyond he egime whe e calib a ion by known ield– heo y da a is a ailable.
IX. SCOPE, LIMITS, AND PREDICTIONS
The amewo k de eloped in his pape is delibe a ely na ow in one sense and delibe a ely s ong in
ano he . I is na ow because i does no a emp o eplace explici holog aphic calcula ions. I is s ong
because i decides, be o e and independen ly o de ailed compu a ion, wha kind o answe one should
expec . This sec ion makes ha sepa a ion p ecise and spells ou he p edic i e con en .
17
A. Wha his amewo k does no do
Fi s , a clea bounda y.
•
This amewo k does no compu e on–shell ac ions, co ela ion unc ions, o anomaly coe icien s by
i sel .
•
I does no elimina e he need o de ailed bulk analysis, supe symme y cons ain s, o ca e ul
ea men o bounda y condi ions.
•
I does no eplace he adi ional machine y o holog aphic eno maliza ion whe e ha machine y
is su icien .
Any claim o he con a y would be misguided. Explici calcula ions emain indispensable, and no hing
he e diminishes hei echnical o concep ual alue.
Wha his amewo k eplaces is some hing else: he idea ha eno maliza ion p esc ip ions can be
jus i ied pos hoc by ma ching a known answe . I eplaces calib a ion by p inciple.
B. Wha his amewo k does decide
The decisi e gain is ha he amewo k de e mines wha can be made canonical. Gi en a class
o admissible con ex s and admissible changes be ween hem, Theo em VI.1 answe s a ques ion ha
compu a ion alone canno e en o mula e:
Is he e a globally well–de ined eno malized obse able associa ed o his p oblem, o is any
ini e answe necessa ily he esidue o an obs uc ion?
This dis inc ion has conc e e consequences.
•
I s ic i ica ion is possible, hen any wo admissible p esc ip ions a e equi alen , and di e ences
can be emo ed by objec –wise ede ini ions. In his case “scheme dependence” is spu ious and can
be elimina ed once and o all.
•
I s ic i ica ion ails, hen no amoun o coun e e m enginee ing will p oduce a globally in a ian
objec . The emaining ini e quan i y is physical p ecisely because i is una oidable.
This is no a ma e o as e o con en ion. I is a s uc u al s a emen abou compa ison ac oss
egula o con ex s.
C. P edic ions
Because he amewo k is s uc u al, i s p edic ions a e quali a i e bu sha p. They conce n when
ce ain phenomena mus appea , no hei nume ical alues.
When coun e e ms mus appea . Addi ional coun e e ms beyond he s anda d AdS
d+1
lis a e
una oidable whene e he admissible con ex space is la ge han he space con olled by a uni e sal
nea –bounda y expansion. In p ac ice, his occu s p ecisely when he obse able p obes global bulk da a
ha canno be encoded in a ini e– ield unca ion. In such cases, highe –dimensional bounda y e ms
a e no op ional ixes; hey a e he cohe ence da a equi ed o de ine compa isons be ween con ex s.
When anomalies a e ine i able. Anomalies a e ine i able whene e he ansi ion da a ails he closu e
condi ion unde composi ion. Equi alen ly, i he e exis s a non i ial loop in he con ex g oupoid along
which he ne ansi ion shi does no anish, hen no global obse able exis s. The anomaly is he
s ic i ica ion–in a ian esidue associa ed o ha loop. This p edic s, wi hou compu a ion, ha ce ain
ini e e ms mus su i e all admissible sub ac ions.
When scheme dependence is physical. Scheme dependence is physical p ecisely when i is de ec ed by
loop anspo . I changing schemes along wo di e en admissible pa hs leads o di e en ini e answe s,
and i ha di e ence canno be emo ed by objec –wise ede ini ions, hen he dependence is no a law.
I is he obse able con en . Con e sely, i all such di e ences can be s ic i ied away, hen appa en
scheme dependence is an a i ac o incomple e bookkeeping.
These p edic ions con e ague expec a ions in o es able c i e ia. They explain why ce ain coe icien s
a e obus ac oss wildly di e en compu a ional se ups, while o he s luc ua e wi h egula o choice.
18
D. Connec ions and ex ensions
We b ie ly indica e how he amewo k in e aces wi h se e al ac i e hemes, wi hou de eloping hem
he e.
De ec s. De ec obse ables a e a na u al a ena o his analysis because hey o ce a misma ch be ween
he locus suppo ing he ield– heo y da a and he hype su ace on which holog aphic eno maliza ion is
pe o med. The con ex g oupoid makes his misma ch explici and explains why de ec anomalies o en
eme ge as obs uc ion classes a he han emo able ini e e ms.
Highe – o m symme ies. Highe – o m symme ies enla ge he space o admissible bounda y condi-
ions and he e o e enla ge he con ex space. F om he p esen iewpoin , his inc eases he numbe
o admissible mo phisms and po en ial loops, making obs uc ion phenomena mo e, no less, likely.
The amewo k p edic s ha anomaly ma ching o highe – o m symme ies is na u ally ph ased as a
s ic i ica ion p oblem o e an enla ged con ex g oupoid.
Non–in e ible s uc u es. Non–in e ible symme ies p o ide a pa icula ly sha p illus a ion o why
g oup–based in ui ion ails. They na u ally gene a e con ex spaces wi h non i ial composi ion laws and
obs uc ions. F om ou pe spec i e, his is no exo ic beha io bu he gene ic si ua ion once in e ibili y
is d opped: non i ial loop anspo is he ule a he han he excep ion. A de ailed analysis o
non–in e ible s uc u es in holog aphy i s na u ally in o he p esen amewo k and will be pu sued
elsewhe e.
E. Ou look
The uni ying message o his sec ion is simple. This amewo k does no compe e wi h compu a ion;
i go e ns i . I ells you, be o e he i s in eg al is e alua ed, whe he you should expec a canonical
obse able, an una oidable anomaly, o genuine physical scheme dependence. In egimes whe e holog aphy
is no longe uni e sal by cons uc ion, ha dis inc ion is no op ional. I is he di e ence be ween
enginee ing p esc ip ions and unde s anding wha he heo y is ac ually elling you.
X. CONCLUSION
Highe –dimensional holog aphy has en e ed a egime in which success ul compu a ions inc easingly
ely on ca e ully chosen cu o p esc ip ions and highe –dimensional bounda y coun e e ms ha canno
be jus i ied by he adi ional AdS
d+1
eno maliza ion dic iona y alone. This de elopmen is no a
empo a y echnical complica ion; i is a s uc u al signal ha he no ion o uni e sali y inhe i ed om
lowe –dimensional unca ions has eached i s limi .
The cen al esul o his pape is o iden i y wha is missing and o supply i . We ha e shown ha
egula o choices in highe –dimensional holog aphy assemble in o a con ex g oupoid, ha coun e e ms
unc ion as cohe ence da a go e ning anspo be ween con ex s, and ha anomalies a ise p ecisely as
obs uc ion classes o global s ic i ica ion. This e ames eno maliza ion om a p esc ip ion–dependen
p ocedu e in o a well–posed compa ison p oblem wi h a sha p consis ency c i e ion.
F om his pe spec i e, ecen en–dimensional compu a ions ha ep oduce p o ec ed ield– heo y
da a a e nei he mys e ious no ad hoc. They a e success ul p ecisely because hey implici ly choose
cohe ence da a in a egime whe e s ic i ica ion ails, lea ing an in a ian esidue. Wha was p e iously
jus i ied only by calib a ion agains known esul s is now explained as he una oidable ou come o a
global consis ency obs uc ion.
Highe –dimensional holog aphy does no need mo e p esc ip ions; i needs a cohe ence
p inciple.
Appendix A: Op ional ca ego ical ema ks ( o comple eness)
This appendix is in en ionally op ional. No hing in he main ex equi es ca ego ical language beyond
he elemen a y g oupoid bookkeeping in oduced in Sec ion IV. The pu pose he e is only o eco d, in
compac o m, he s anda d s uc u al in e p e a ion o ha bookkeeping, and o indica e how one would
gene alize i i needed.
19
1. The con ex g oupoid as a ca ego y o egula o s
A g oupoid Cis a small ca ego y in which e e y mo phism is in e ible. In he p esen pape :
•
objec s
c∈Ob
(
C
)a e egula o con ex s (cu o su ace, asymp o ic p esen a ion, sub ac ion/no -
maliza ion, allowed bounda y e ms, bounda y condi ions);
•
mo phisms
φ
:
c→c0
a e admissible changes o con ex (coo dina e/p esen a ion changes, cu o
de o ma ions, and he induced adjus men s o bounda y da a and bounda y e ms);
•composi ion is conca ena ion o admissible changes.
The con en o he main ex can be ph ased as ollows: a “ eno malized obse able” should no be
unde s ood as a numbe a ached o a p e e ed objec c, bu as a globally consis en assignmen on C.
2. T ansi ion da a as a 1–cocycle and s ic i ica ion as i ializa ion
The ansi ion e ms
T
(
φ
)o Sec ions V–VI may be iewed as a 1–cocycle on
C
wi h alues in an
abelian g oup Ao admissible ini e bounda y unc ionals (o , a e p ojec ing o he ele an sec o , in
he co esponding coe icien space). Conc e ely, cohe ence unde composi ion,
T(ψ◦φ) = T(ψ) + T(φ),(A1)
is p ecisely he cocycle condi ion.
An objec –wise ede ini ion by B(c)(Sec ion VI B) ac s as a cobounda y shi ,
T(φ)7−→ T(φ) + B(c0)−B(c),(A2)
which is he s anda d no ion o equi alence o cocycles. S ic i ica ion is he s a emen ha he cocycle
class is i ial: one can choose
B
so ha he cocycle anishes iden ically. Non i iali y is de ec ed by loop
anspo , and he esul ing loop in a ian s a e exac ly he obs uc ion da a iden i ied wi h anomalies in
Theo em VI.1.
3. Beyond g oupoids: why one migh wan highe cohe ence (no used he e)
The main ex uses a g oupoid because i is he minimal s uc u e needed o make scheme compa isons
well–posed and o ex ac obs uc ions. In mo e elabo a e si ua ions one may encoun e genuinely highe
cohe ence condi ions, o example when:
• he admissible changes o con ex a e na u ally de ined only up o con olled equi alence,
•
bounda y condi ion da a in ol es non i ial ex ensions (e.g. mixing o ensembles o de ec s wi h
junc ions),
•
o one wishes o o ganize amilies o con ex s pa ame ized by moduli wi h non i ial highe
iden i ica ions.
In such cases i may be na u al o eplace he g oupoid by a highe g oupoid (o by a s acky e inemen )
and eplace he cocycle/ i ializa ion p oblem by i s highe analogue. None o his is equi ed o he
esul s o he p esen pape : he g oupoid–le el c i e ion al eady isola es he obs uc ion mechanism and
p o ides he p ac ical ecipe o Sec ion VIII.
[1]
R. Izquie do Ga cía, “Highe dimensional holog aphy,” a Xi :2512.12696 [hep- h] (2025), DOI:
10.48550/a Xi .2512.12696.
[2]
M. Henningson and K. Skende is, “The Holog aphic Weyl anomaly,” JHEP
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