× Physics Letters B

(1)

Contents lists available atScienceDirect

Physics Letters B

journalhomepage:www.elsevier.com/locate/physletb

On-shell action for type IIB supergravity and superstrings on AdS 5 × S 5

Subhroneel Chakrabarti

^a^,∗

, Divyanshu Gupta

^b

, Arkajyoti Manna

^c

aFZU- InstituteofPhysicsoftheCzechAcademyofSciences&CEICO,NaSlovance2,18221Prague8,CzechRepublic bInstituteofPhysics,UniversityofAmsterdam,SciencePark904,1098XHAmsterdam,theNetherlands

cCenterforHighEnergyPhysics,IndianInstituteofScience,C.V.RamanAvenue,Bangalore560012,India

a rt i c l e i n f o a b s t r a c t

Articlehistory:

Received11November2022

Receivedinrevisedform15November2022 Accepted16November2022

Availableonline18November2022 Editor: N.Lambert

AdS/CFTpredictsthatthevalueoftheon-shellactionfortypeIIBSupergravity(SUGRA)on AdS5×^S⁵ backgroundmustbeanon-zeronumbercompletelydeterminedfromtheboundarytheory.Weexamine this statement within Sen’s formalism for type IIB SUGRA and ﬁnd that consistency with AdS/CFT requiresus toaddaspeciﬁcboundarytermtothe action.Wecontrast ourresolutionwithtwo other resolutionsrecentlyproposedintheliteratureinthecontextofdifferentapproachestotypeIIBSUGRA.

We explain how our resolution presents a strong benchmark for the possible boundary term ofthe completespacetimeactionfortypeIIBsuperstringandhowitmaypossiblyleadtoapieceofevidence forthestrongestformofAdS/CFTconjectureinAdS5×^S⁵^.^Weâlso^commentôn^the^fateôf^theôn-shell actionforgeneralself-dualp-formfieldsinSen’sformalisminanycurvedbackgrounds.

©²⁰²²^TheÂuthor(s).^Published^byÊlsevier^B.V.^Thisîsânôpenâccessârticleûnder^the^CC^BY^license (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

ofthethird,thecelebratedAdS5×^S⁵ ^[1,2].^Theconstructionofa LorentzcovariantactionfortypeIIBSUGRAhasbeenhistoricallya challengeduetothepresenceofaself-dualRR5-formﬂux.Onthe otherhand,theequationsofmotionofthetypeIIBSUGRAonany backgroundareunambiguously known.Ergo, on AdS5×^S⁵ ^these EOMcanbedimensionallyreducedtoeffective5dequations(along the AdS5), whichcan be interpreted asEuler-Lagrangeequations ofa5deffectiveLagrangian.Thiseffectiveactionobtainedwithout invoking a parent action in 10d is also consistent with AdS/CFT expectations.

https://doi.org/10.1016/j.physletb.2022.137578

0370-2693/©²⁰²²^TheÂuthor(s).^Published^byÊlsevier^B.V.^Thisîsânôpenâccessârticleûnder^the^CC^BY^license(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

(2)

Theﬁeldcontent(bosonic)¹oftypeIIBSUGRAisasfollows[3–

5]

•^The^spacetime^metric ^gab(a,b=⁰,1,· · ·,9).

•^The^dilatonφ.

•^TheKalb-Ramond2-formB⁽²⁾whoseﬁeldstrengthisdenoted asH⁽³⁾=^dB⁽²⁾^.

•^The Ramond-Ramondfluxes(field strengths)- F⁽¹⁾,F⁽³⁾,F⁽⁵⁾. Amongthesethe5-formisself-dual,i.e.F⁽⁵⁾=gF⁽⁵⁾. ThesolutionsofthesefieldsthatgiveusAdS5×^S⁵^background aregivenas(throughoutthispaper,weuse∼=^to^mean^“equalôn- shell”)

ds²₁₀

=

^gabdx^adx^b

∼ = ρ

²

ds²_AdS

5

+

d

²₅

,

F⁽⁵⁾

∼ =

⁴

ρ

⁻¹

5

+

g

5

B⁽²⁾

∼ =

^F⁽¹⁾

∼ =

^F⁽³⁾

∼ = ∂

a

φ ∼ =

⁰

.

(1) Here,thelineelement of AdS₅ andthe S⁵ aredenoted byds²_AdS

5

and d²₅ respectively,

5 is the epsilon tensor in AdS5,

ρ

is the radius of AdS₅ space given by

ρ

⁴=⁴

π α

²g_sN,

α

is the Regge slope parameter, gs is thestring coupling,and N corresponds to theunitsofRR5-formﬂuxinthebackgroundgeometry.

The5deffectiveactionevaluatedinthisbackgroundgives

S5

∼ =

⁸

ρ

⁴

2

κ

₅²^vol

(

^AdS⁵

) .

⁽²⁾

Here

κ

5 isNewton’s constantin5d. Alternatively,intermsofthe 10dgravitationalconstant,wehave

S5

∼ =

⁴

ρ

⁸

κ

₁₀² ^vol

(

^AdS⁵

) ,

^where

1 2

κ

₅²

=

ρ

⁴vol

S⁵

2

κ

₁₀²

.

⁽³⁾

Herewearedeliberatelyusingthesameconventionforthe5don- shellactionas[6] foreaseofcomparisonlater.Thisisindeedcon- sistent withAdS/CFT, which predicts the5d on-shellaction must begivenbytheconformala-anomalyoftheN=⁴,SU(N)Super- Yang-Millstheorylivingontheboundary S⁴[7–12].

TheRicciscalarforthe AdS5×^S⁵^metric^can^be^checked^to^be vanishingforthefull10d,i.e., R∼=^{0 (the}^AdS5andS⁵spaceshave exactlyequalandoppositescalarcurvature).Thismeansthatalong with the Einstein-Hilbertterm, ifone writes the usual Maxwell- liketermfortheonlynon-zeroﬁeldstrengthastheactionfortype IIBSUGRA,thenitwilltriviallyvanishonthesolution.

Butthisvanishingdoesnotmake useofthedetailsofthe solution forthe ﬁeld strength.Forany self-dualform, i.e. F=gF, theMaxwell action F∧gF=^F∧^F identicallyvanishes irrespec- tiveoftheexactformoftheﬁeldstrength!Infact,thisisprecisely the obstaclethat preventedthe communityfromwritingdown a manifestlyLorentz covariant actionfortype IIBSUGRA.Themost commonly adoptedworkaround in theliterature hasbeen to in- steadworkwithapseudoactionthatusesaMaxwell-liketermand imposes the self-duality constraint byhand on the equations of motion. While the pseudoaction is perfectly reasonable to work within obtainingthe EOM, it,by construction,is not thecorrect actionforthetypeIIBSUGRA.

1 Throughoutthis paper,wefocusonly onbosonicparts oftheﬁeldcontent.

Thefermionic partsareﬁxedcompletelyfromsupersymmetry oncethe answers areknownforthebosonicpart.

The vanishing ofthe on-shell 10dpseudoaction andthe non- vanishingoftheon-shell5deffectiveactionispresentedasapuz- zlein[6].Itwasarguedtherethat thispuzzlerequiresresolution evenwithin theframework ofpseudoactionsincethe quantityin questionisanon-shellone.

Wecan extend thepuzzle asfollows- any claim forthe cor- rectactionfortypeIIBSUGRAmustgiveanon-zeroon-shellvalue on AdS5×^S⁵^, ^which îs ^consistent ^with ^the ÂdS/CFT predictions oncethespheredirectionsareintegratedout.Infact,assumingthe strongest formof the AdS/CFT conjecture, we can, in fact, claim that the spacetime action for full type IIB strings (i.e. type IIB stringfieldtheoryaction[13])muston-shellgiveanon-zerovalue consistentwithAdS/CFTprediction.

In this paper, we investigate and resolve the aforementioned puzzle by workingwith the action for type IIBSUGRA given by Senin[14].Ourresolutionforthispuzzleechoesthecoreprinci- pleoftheproposalin[6], i.e.thetype IIBSUGRAactionmustbe supplementedwithasuitableboundaryterm,whichwillnotaffect theequationsofmotionbutgiveanon-vanishingvaluetotheon- shell10daction.Infact,whileourproposedboundarytermhasa differentstructureoff-shell,on-shell,itmatchespreciselywiththe correspondinganswerin[6].

Anotherrecentpaper in[15] has givena newaction fortype IIBSUGRA(infact,theproposedactiondescribesbothtypeIIAand type IIBonthesamefooting). It wasbriefly discussed therehow theproposedactionfortypeIISUGRAalreadycontainsatermthat onthe AdS5×^S⁵ ^background^for^typeÎIB^gives^theêxpectedân- swer.Giventheresultofthepapers[6,15],onemaywonder why anyoneshouldtrytoresolvethepuzzleagainusingadifferentac- tion.Wegivethreefollowingreasons.

• ^Sen’s âction îs ^naturally ^related ^to ^the ^spacetime âction ^for superstrings, viz. type IIB SFT action [13]. Therefore we will seetheresolutionofthispuzzleinSen’sformalismprovidesa benchmarkforthestructure ofboundarytermsforfullstring theory.In fact,inlight oftherecentresultpresentedin[16], anyinsightintostructuresofpossibleboundarytermsinstring field theory at this point is a considerable step forward. In contrast,itisnotclearhowtheactionproposedin[15] isem- beddedinto thefull stringtheory while [6] onlyworks with thepseudoaction.

• ^The ^resolution ôf ^the ^puzzle ^should ^not ^be ^dependent ôn â particular approach to describe type IIB SUGRA. While each approach has its relative advantages, all approaches must agreeon every physicalresult. The on-shell action evaluated onthe AdS₅×^S⁵ ^backgroundîsône^such^physical^result,ând itneedstobesolvedwithinSen’sformalismindependently.

• ^Sen’sconstruction can be used to write down the action for anyself-dual2k+^1-form^ﬁeld^strengthsⁱⁿ^4k+2 dimensions [17]. The resultofthis paperalso leads to an understanding ofpossible boundary terms for other chiral p-form theories.

Since Sen’s construction prima facie is a little unusual, it is interesting in its own right to understand the structure of boundarytermswithinthisformalism.

Thestructureofthispaperisasfollows.Insection2wegivea summaryofSen’sformalism foranyself-dualformﬁeld thatwill be needed to follow the main result of this paper. In section 3 wegive theaction fortype IIBSUGRAduetoSenandshow that despitenothavingaMaxwell-likeaction,itstillvanisheson-shell.

Wealsoshowinthissectionthatthisistrueforanychiralp-form describedinthisformalism.Thereforeinsection4,wesupplement Sen’saction by a pure boundary termandshow how it leads to aconsistentresultwithAdS/CFTprediction.Withtheresolutionat handfortypeIIBSUGRA,weoutlinewhatcanwelearnaboutthe

(3)

resolutionofthepuzzleinfulltypeIIBstringtheoryinsection5.

Weconcludewithsomecommentsandasummaryinsection6.

2. AquickreviewofSen’sformalism

In this section, we briefly review the string field theory in- spiredLagrangian descriptionforanyself-dual(2k+¹)-formfield strength in(4k+²)spacetime dimensionsduetoSen[14,17]. In thisformalism,theself-dualityconditionholdsoff-shell,theaction ispolynomial,whilepreservingmanifestLorentz invarianceatthe cost ofintroducing a single additional unphysicalfield that completely decouples from the dynamics. Some recent works using thisformalism invarious dimensionsatboth classical andquan- tumlevelare[18–26].

Theactioninthisformulationcontainsa2k-formﬁeld P,aself- dual(2k+¹)-formﬁeldstrength Q satisfying

Q

=

^Q ⁽⁴⁾

andalinearmapM(Q)thatmapsself-dualformstoanti-self-dual forms.ThatisforallQ =^Q^,^we^have

M

(

Q

) = −

^M

(

Q

) .

(5)

TheHodgestaroperationisdeﬁnedwithrespecttotheﬂatmet- ric and not with the actual background metric. The Hodge dual withrespect to thedynamical metricwill be denotedby g.See [14,17,19] formoredetailsontheexplicitconstructionofthemap M(Q).Wewillnotrequireitsexplicitforminthispaper.

Theactionfortheself-dualﬁeldtakesthefollowingform S

=

¹

2

d P

∧

d P

−

d P

∧

^Q

+

Q

∧

^M

(

Q

) .

(6) Inthecanonicalformalism,itwasshownin[17] thattheHamilto- niansplitsintoasumofafreeHamiltonianwithonlynon-physical degreesoffreedomandaninteractingHamiltoniancontainingonly thephysicaldegreesoffreedom.Thegravitationalcouplingtothe dynamicalmetricentersonlythroughthemapM(Q),andtheex- tra non-physical field does not couple even to gravity. The form field P contributessolelytothenon-physicalfield.

The invariance ofthe action inequation (6) underdiffeomor- phisms is not manifest due to the non-standard couplingto the backgroundmetric.Nonetheless,thediffeomorphismsymmetryof theactionisindeedpreserved,asshownintheoriginalreferences [14,17].

Due tothe unusualself-dualitycondition on Q andthepres- ence of Hodge star in the kinetic term of P, the fields entering the action in equation (6) are not standard differential forms on thebackgroundmanifold.Werefertothemaspseudoformsfollow- ing [19]. Eventhough Q andM(Q) areindividually not physical forms on the manifold, the following specific linear combination is a proper(2k+¹)-formandsatisfies the self-dualityconstraint w.r.t. thebackgroundmetric g

Q

−

^4M

(

Q

) =

g

(

Q

−

^4M

(

Q

)) .

(7) This particular combination, on-shell, matches with the physical self-dualﬁeldstrengthobtainedviaapseudoactionformalism.For ourcase, thisimpliesthephysicalRR5-formﬂux F(5) oftypeIIB SUGRAactionin10dwillbe givenbytheabove combinationon- shell.

We conclude our discussion on Sen’s action by emphasizing that this formalism does not need to evoke anynotion ofgauge potential anywhere. Rather everything (including possible inter- actions with other fields) is completely captured by the field- strength-likefield.Whilethismayseemunusual,aspointedoutin

theoriginalpaper[14],thisisahighly desirablefeaturefromthe stringtheory perspective,where itis knownthat thestrings and theD-branes are onlysensitive to theRR-ﬂuxes andnot theRR- potentials.Wealsoliketoreiteratethattheactioninequation (8) contains, along withthe physical interactingdegreesof freedom, afree,completelydecoupled, unphysicalextraﬁeld.Formorede- tails,we refer the readerto the original papers[14,17] (also see [19,21]).

3. Sen’sactionfortypeIIBSUGRA

Forourpurpose,itisenoughtoworkwiththebosonicpartof thetype IIBSUGRAaction.Furthermore,since we areonly inter- estedintheon-shellvalueoftheactionfor AdS₅×^S⁵ background, itis suﬃcientto lookinto those termsofthe action thatdo not triviallyvanishoncewegoon-shellfollowingequation(1).Wewill befollowingthenormalizationconventionsof[17] alongwiththe notationof[19,21] forthegravitationalcouplingterm.Therelevant termsintheactionare

S

=

^SS D

+

^SE H

=

¹ 2

dP

∧

dP

−

dP

∧

^Q

+

Q

∧

^M

(

Q

) +

^SE H

.

(8)

SE H istheEinstein-Hilbertaction,whichonceagaingiveszero contributionon-shellsince theRicciscalariszeroforthe AdS5× S⁵ background.

Thisactiongivesthefollowingequationsofmotion(wearenot writingEinstein’sﬁeldequationshere)

d

dP

−

dQ

=

0 1

2

(

dP

−

dP

) +

^2M

(

Q

) =

⁰ ⁽⁹⁾

3.1. On-shellactionoftypeIIBSUGRAinSen’sformalism

Wecanmakeuseoftheequationsofmotiontoobtaintheon- shellvalueoftheactiongiveninequation(8).

Forexample,

−

dP

∧

^Q

= −

¹ 2

(

dP

−

dP

) ∧

^Q

∼ =

²

M

(

Q

) ∧

^Q

= −

²

Q

∧

^M

(

Q

)

=⇒ −

dP

∧

^Q

+

Q

∧

^M

(

Q

) ∼ = −

Q

∧

^M

(

Q

)

(10) Similarly,wecandointegrationbypartsofthekinetictermfor the4-formP andwriteitas

1 2

dP

∧

dP

= −

¹ 2

P

∧

^d

dP

∼ = −

¹₂

P

∧

^dQ

=⇒

¹ 2

dP

∧

dP

∼ =

¹₂

dP

∧

^Q 1

2

dP

∧

dP

∼ =

Q

∧

^M

(

Q

)

(11)

Pluggingin the answer fromequation (11) andequation (10) intotheactionweget

(4)

S_{S D}

=

¹ 2

dP

∧

dP

−

dP

∧

^Q

+

Q

∧

^M

(

Q

)

∼ =

Q

∧

^M

(

Q

) −

Q

∧

^M

(

Q

)

∼ =

⁰

.

(12)

Therefore,we seethat thepartofSen’sactioninvolvingthe self- dualﬁeldevaluatedexplicitlyon-shellisidenticallyzero.

Itiscuriousthateventhoughthisactiondoesnotcontainany Maxwell-like term, it still vanishes on-shell due to the fact that on-shell thecontribution fromthe parts ofthe actiondepending on the ﬁeld Q exactly cancels the contributionfrom thepart of theactionindependentofQ.

TherestofthetermsintypeIIBSUGRAallvanishon-shellsince theyarethesametermsthatappearinthepseudoaction.

Theresultofequation(12) actuallyholdsforanyself-dualform described bySen’saction inanyappropriate dimensions.There is a shorterwayofunderstandingwhythisisso,which weexplain now.

3.2. WhySen’sactionvanisheson-shell?

Consider the following (inﬁnitesimal) transformation of the ﬁeldvariables P andQ

P

→ (

1

+ )

P

,

Q

→ (

1

+ )

Q

; =

const.

, <<

1

.

⁽¹³⁾

Underthistransformation,theactiontransformsas S

[

^P

+

P

,

Q

+

Q

]

=

¹ 2

(

1

+ )

dP

∧

dP

− (

1

+ )

dP

∧

^Q

+ (

1

+ )

Q

∧

^M

(

Q

) +

^O

(

²

)

= (

1

+ )

S

[

P

,

Q

] +

O

(

²

).

(14)

When evaluated on-shell, i.e. P ∼= ^P⁰ ^and ^Q ∼=^Q⁰^, ^the ^LHS ^of equation(14) givesbackjustS[^P0,Q₀]^due^to^thevariationalprin- ciple,andwehave

S

[

^P0

,

Q0

] = (

1

+ )

S

[

^P0

,

Q0

]

=⇒

^S

[

^P0

,

Q0

] =

⁰

.

⁽¹⁵⁾

So, theon-shellaction inSen’sformalism vanishes identicallyfor any solution of any self-dual p-form theory coupled to a non- dynamical curved background. For a dynamically curved background, Sen’s action needs to be supplemented by the Einstein- Hilbert action and the usual Gibbons-Hawking-Yorke boundary term.Theseterms,whiletheydidnotcontributeanythingon-shell fortypeIIBon AdS5×S⁵ maycontribute tootherself-dualtheo- riescoupledtodynamicalgravityinotherdimensions.

4. Theboundaryterm

It is now clear that to obtain a non-zero on-shell action we mustsupplementtheactionwithaboundaryterm.Sinceon-shell, thereisaclearlinkbetweenSen’sactionandtheequationsofmo- tions obtainedfromthepseudoaction,it istempting tothink the proposal given in [6] will work here as well. However, the pro- posedterm in[6] isa boundary termoff-shell,ifandonlyifwe think oftheRR5-formﬂux variableasanexact form(i.e.aﬁeld strengthofagaugepotential).Asmentionedinsection 2thatthis isnot possibleinSen’saction.So weneedto lookforadifferent resolution.

Theboundarytermweproposeis S_b

= κ

d

Q

∧

^P

,

(16)

where

κ

is aconstant whichwill be ﬁxed bydemanding consis- tencywithAdS/CFTprediction.Beingaboundaryterm,clearly,this doesnotaffecttheequationsofmotion.Tocheckhowitindeedre- solvestheapparent mismatch,we needtoﬁndits on-shellvalue.

Itisanecessaryevilofthisformalismthatanypossibleboundary term, even when evaluated on-shell, will contain a contribution from 3 types of terms. Firstly, terms which are purely built out ofphysicalfields.Secondly,termswhich arecompletely builtout ofthe non-physical field. Finally,there can be terms whichhave amixingofphysicalandthenon-physicalfield.However,AdS/CFT correspondenceisonlycognizantofthephysicalfieldandnothing else.Thereforeitstandstoreasonthatonlythecompletelyphysi- calpartoftheon-shellactioninSen’sformalismshouldreproduce ananswerconsistentwithholography.

Recallfromsection2,thaton-shelltheself-dualRR5-formﬂux iscapturedbythefollowingspeciﬁclinearcombination

F⁽⁵⁾

∼ =

^Q

−

4M

(

Q

) ,

dF⁽⁵⁾

∼ =

⁰

.

(17) On-shellourboundaryterm,ergo,reducesto

S_b

∼ = κ

P

∧

^d

dP

+

²

κ

Q

∧

^M

(

Q

) .

(18) Evidently,theﬁrsttermispurely non-physical,andwecandisre- gardit.Letusnowfocusonthesecondterm.Firstofall,notethat wecanreplace Q by F⁽⁵⁾intheintegrand.Next,wenotethaton thebackgroundofinterest,theRR5-formsplitsintoadirectsum F⁽⁵⁾=^FAdS⊕^FS⁵,with F_AdS=gF_S5.Thisallowsustowrite S_b

∼ = κ

P

∧

^d

dP

+

²

κ

F_AdS

∧

^M

(

Q

)

_S5

+

²

κ

F_S5

∧

^M

(

Q

)

_AdS

.

(19) Finally,wecanuseequation(17) andthefactthatF_AdS=gF_S5 to obtain

S_b

∼ = κ

P

∧

^d

dP

+

¹ 2

κ

F_AdS

∧

^QS⁵

+

¹ 2

κ

F_S5

∧

Q_AdS

+

¹ 2

κ

F_AdS

∧

F_S5

.

(20) Asdiscussed,the firsttermispurely builtofthenon-physical field,the second andthirdare ofmixedtype, andthe finalterm isthepurelyphysicalpart,whichmatchespreciselywiththepro- posedtermin[6].Theconstant

κ

cannowbeﬁxedexactlyalong thelineof[6] bycomparingwithequation (2) (andusingtheso- lution for F⁽⁵⁾ from equation (1)). Also, note that we are using anormalizationfor thephysicalform ﬁeld wherea Maxwell-like termwouldbewritten as∼ ₅¹_!^F∧gF.The valueoftheconstant is

κ =

¹

2

(

5

! )

²

κ

₁₀²

.

⁽²¹⁾

Thusweseethattheproposedboundarytermmakestheon-shell actionnon-zero,anditisindeedconsistentwiththeAdS/CFTpre- diction.

Thereare afew commentswewantto makehere. Firstly,on- shell, our proposal coincides precisely with the proposal of [6].

However, the off-shell realization is very different, which is ex- pectedsincethepseudoaction,andSen’sactionisinequivalentoff- shell.Secondly,unlike [6], ourboundary termdoesnot requirea

(5)

decompositionoftheformsintotheelectricorthemagneticpart, nor doesitinvokethe knowledge ofthefactorized natureof the background geometryprior to going on-shell.This featureof our proposedtermisalsosharedbytheformalismintroducedin[15].

Third, in[6] itwasshownthat theirboundarytermfollowsfrom gaugeinvarianceinthePSTformulationoftypeIIBSUGRA[27,28]

andasimilar observationwasechoed in[15].In contrastour approach requiredus to introduce thisterm by hand.This, in fact, isto beexpectedforSen’sformulationgivenits proximity tothe complete spacetime action of type IIB string theory [13,14] (for a review see[29–31]).In thenext section, we willpoint out ex- actlywhat canbesaidabouttheon-shellspacetimeactionofthe full type IIBstring on AdS5×^S⁵ ^fromôur ânalysis. ^Finally, ît îs quitestraightforwardtoseethatourproposedboundarytermwill be equallyvalidforanyother solutionwiththeproduct structure ofspacetimemanifold,exactlylike[6].Onceagain,bothproposals willagreeonthevalueoftheon-shellaction.

5. Theon-shellspacetimeactionoftypeIIBstringtheoryon Ad S5×^S⁵

Recently, ithasbeenshownthat theclosedstringﬁeldtheory action must vanish on-shell up to possible boundary terms[16].

This isconsistent withthe knownresults fortheeffective action inspacetimeobtainedfromworldsheetsigmamodels[32–36] How will theboundaryterms inSFTlook isan open question.On the otherhand,thereareknowncaseswhereon-shellactionsinstring theoryarerelatedtoanon-zeroquantityofphysicalinterest,such asblackholeentropy[37,38],D-branetensions,andpartitionfunc- tionsindualmatrixmodels[39,40].Theseresultsstronglysuggest thattheclosedSFTactionforcertainbackgroundsneedstobesup- plementedwithanappropriateboundaryterm.

GiventhatthetypeIIBSUGRAactionconsideredherewascon- structedwithdirectmotivationfromtheclosedtypeIIBSFTaction ([13])andtheAdS₅×^S⁵^backgroundîsône^for^which^theôn-shell action must be non-zero (assuming the validity of AdS/CFT), the resultobtainedhereprovidesacrucialbenchmarkforthepossible boundarytermthat needsto beaddedto thetype IIBSFT action inthiscase.

AdS5×^S⁵ îs âlsoêxpected^to^be ânêxact ^background^for^the complete type IIBsuperstring. Thereforethe only non-zero back- groundfieldsforAdS₅×^S⁵^solution^for^typeÎIB^stringsâreâlso^the metricandtheRR5-formflux.Therefore,anystringfieldon-shell must reduce to terms containing only thesetwo massless fields.

Therefore, on-shell, even forthe full string theory, the boundary termthat resolves thepuzzle isthesame onewe proposed. This alsosuggeststhatanypossibleboundaryterminthefull typeIIB stringfieldtheoryactionmustreducetoourproposal,uptopossi- blefieldredefinitions,onceitisevaluatedon-shell.Wecanexpress thisas(in thefollowingS˜_b istheboundarytermfortype IIBSFT action)

S

˜

_b

∼ =

^Sb

on-shell

,

(22)

where S_b isourproposalgiveninequation(16).

Additionally, even attheoff-shell level,oncewe integrate out all the massive states following the procedure outlined in [41]

(also see [42]),the SFT boundary termmust reduce to our pro- posedboundaryterm,

S

˜

_b

Wilsonian RG

−−−−−−−−−−→

^Sb

.

(23)

Of course, this resolution assumes the strongest form of AdS/CFTconjecture.Conversely,an explicitconstructionoftheSFT boundaryterm ˜S_b thatsatisﬁesthe conditionsgiveninequations

Fig. 1.Themain ﬁndings ofthe paper determined the greenportionsuch that theresultisconsistentwithAdS/CFT.Theredportionhighlightsthepartswhere AdS/CFTisassumed.

(22) and (23) would provideapiece ofstrongevidenceinfavour of the holographic principle. However, there is a glaring techni- caldiﬃcultythat mustbeovercomebeforesuchaconstructionis attempted.

Thestructureofequation(16) suggeststhat thecorresponding terminSFT shouldalsobe abi-linearinstringfields,andoneof thestring fieldsmustbe theextra stringfield that isneededfor the R-sector [13]. However, all SFT actions are based on homo- topyalgebraicconstruction(forareview,see[30,31],andbi-linear terms for any given pair of spacetime fields look naturally like

∼

d¹⁰x1(x)D2(x), where D is some differential operator. In fact, one can ask the same question with regards to homotopy algebraic formulation of familiar local QFTs (see [43] for a re- viewandreferencestherein).Decipheringhowtorepresentatotal boundarytermintermsofthehigherproducts andthesymplec- ticform (the two essential ingredients used in writing down an actioninhomotopyalgebraicconstruction)willbethecrucialﬁrst stepinacompleteconstructionfortheboundaryterminSFT.

6. Conclusionandoutlook

Themainresultofthispaperishowbyaddingaboundaryterm toSen’sactionfortype IIBSUGRA,we canensure matchingwith theAdS/CFT prediction fortheon-shell action.This, inturn,also givespowerfulinsightinto thepossiblereconciliationofthe full- string theory on-shell action with the AdS/CFT prediction. Fig. 1 givesaqualitativesummaryofthemainresult.

As discussed earlier, our proposed boundary term can also be written for any background geometry which is of the form Mn×^Mc,whereMn isanon-compactmanifold,andMc isacom- pactmanifold.Thesameboundarytermwouldleadtoanon-zero on-shellaction forthe effectivetheory on M_n,andit will match theanswerpresentedin[6].Aninteresting question raised in[6]

wasiftheboundarytermswillhave

α

-correctionsonce westart including the massive states of type IIB string theory. A natural answertothat,atleastforourproposal,willbegivenbythecon- structionofthestringﬁeldtheoryboundaryterm.

From the SUGRA side in Sen’s formalism, there is no reason why the action should be supplemented by thisspeciﬁc boundaryterm.Infact,withoutAdS/CFTtoguideus,thereisnoapriori reason to add any boundary terms. This is in contrast with the approachesin [6,15] where such a boundary term is seen to be expectedfromgauge invariance. Onthe other hand theresultof [16] clearly demonstrates that forthepresent formulationofthe

(6)

SFTactionwemustincludeaboundarytermtoﬁndanagreement withAdS/CFTprediction.ItispossiblethattheSUGRAinSen’sfor- malismmaynotknowaboutanypossibleconstraintthatuniquely ﬁxes theboundaryterm. Nonetheless,assuming thetechnicalob- stacle in constructing the SFT boundary term is overcome, the expectationwouldbe that giventheconditionsin equations(22) and(23),onewouldhopefullyendupwithauniquecandidatefor S˜_binSFT.One wouldexpect thatthefull SFTshouldknowabout AdS/CFT, eventhough thatinformationmight benon-triviallyen- coded. A unique boundary term for type IIB SFT on AdS5×^S⁵ whichmatches withAdS/CFT predictionswouldthereforeprovide crucialevidenceforthestrongestformofAdS/CFTconjecture.

Anotherpossibility isto try andmake explicitconnection be- tween theSen’sformulation[14] andthePSTformulation[27,28]

orthe formulationproposedin [15].Thisexplicitconnection will deﬁnitelyhelpinidentifyingtherequiredconsistencyconditionfor singlingout theboundarytermwe proposed inequation (16). In turn,itwouldalsoshedlightintohowthealternativeformulations areembeddedinthefullstringtheory.

Anotherinterestingdirectiontopursueistheconnectionofthe on-shellspacetimeactionforsuperstringwiththespherepartition functionoftheworldsheettheory.Typically,duetothedivisionby the volume of the P S L(2,C), the zero point function in world- sheet theory is naively thought to be zero. However, to be consistent with the non-vanishing of the on-shell spacetime action, theworldsheetzeropoint functionshould besuitablyregularised suchthatitmatchesthespacetimeanswer.Importantadvancesin this directionhave alreadybeen made in the literature foropen strings[39] andnon-criticalstrings[40].Itislikelythatthecom- pleteconstructionoftheboundaryterminfullSFTwillleadtoan understanding of the zero point function in the worldsheetper- spective,especiallysinceSFTisknowntoactasanaturalregulator fortheworldsheettheory[44].

Declarationofcompetinginterest

Theauthorsdeclarethattheyhavenoknowncompetingﬁnan- cialinterestsorpersonalrelationshipsthatcouldhaveappearedto inﬂuencetheworkreportedinthispaper.

Dataavailability

Nodatawasusedfortheresearchdescribedinthearticle.

Acknowledgements

We thank Madhusudhan Raman for collaborating in the earlier stagesof thiswork.We alsothank RenannLipinski Jusinskas and Ashoke Sen for their valuable comments on an earlier ver- sion of the manuscript and numerous discussions. AM acknowl- edges ﬁnancial support from DST through the SERB core grant CRG/2021/000873.

References

[1] J.M.Maldacena, ThelargeNlimit ofsuperconformalﬁeldtheoriesand supergravity,Adv. Theor.Math. Phys. 2(1998) 231, https://doi.org/10.1023/A: 1026654312961,arXiv:hep-th/9711200.

[2] D.E.Berenstein,J.M.Maldacena,H.S.Nastase,Stringsinﬂatspaceandppwaves fromN=4superYang-Mills,J.HighEnergyPhys.04(2002)013,https://doi.org/ 10.1088/1126-6708/2002/04/013,arXiv:hep-th/0202021.

[3] J.H.Schwarz, P.West,Symmetries andtransformations ofchiraln=2,d= 10supergravity,Phys.Lett.B126 (5)(1983)301,https://doi.org/10.1016/0370- 2693(83)90168-5.

[4] P.Howe,P.West,Thecompleten=2,d=10supergravity,Nucl.Phys.B238 (1) (1984)181,https://doi.org/10.1016/0550-3213(84)90472-3.

[5] J.H.Schwarz,Covariantﬁeldequationsofchiraln=2d=10supergravity,Nucl.

Phys.B226 (2)(1983)269,https://doi.org/10.1016/0550-3213(83)90192-X.

[6] S.A. Kurlyand, A.A. Tseytlin, Type IIBsupergravity action on M5×^X5 ^solu- tions,Phys. Rev.D106 (8)(2022)086017, https://doi.org/10.1103/PhysRevD. 106.086017,arXiv:2206.14522.

[7] H.Liu,A.A.Tseytlin,D=4superYang-Mills,D=5gaugedsupergravity,andD= 4conformalsupergravity,Nucl.Phys.B533(1998)88,https://doi.org/10.1016/ S0550-3213(98)00443-X,arXiv:hep-th/9804083.

[8] M.Henningson,K.Skenderis,TheholographicWeylanomaly,J.HighEnergy Phys. 07 (1998) 023, https://doi.org/10.1088/1126-6708/1998/07/023, arXiv:

hep-th/9806087.

[9] S.S. Gubser, Einstein manifolds and conformal ﬁeld theories, Phys. Rev. D 59(1999)025006, https://doi.org/10.1103/PhysRevD.59.025006, arXiv:hep-th/ 9807164.

[10] M.Blau,K.S.Narain,E.Gava,OnsubleadingcontributionstotheAdS/CFTtrace anomaly,J. High Energy Phys. 09 (1999)018, https://doi.org/10.1088/1126- 6708/1999/09/018,arXiv:hep-th/9904179.

[11] C.P.Burgess,N.R.Constable,R.C.Myers,ThefreeenergyofN=4superYang-Mills andtheAdS/CFTcorrespondence,J.HighEnergyPhys.08(1999)017,https://

doi.org/10.1088/1126-6708/1999/08/017,arXiv:hep-th/9907188.

[12] J.G.Russo,K.Zarembo,LargeNlimitofN=2SU(N)gaugetheoriesfromlocaliza- tion,J.HighEnergyPhys.10(2012)082,https://doi.org/10.1007/JHEP10(2012) 082,arXiv:1207.3806.

[13] A.Sen,BVmasteractionforheteroticandtypeIIstringﬁeldtheories,J.High Energy Phys. 02(2016)087,https://doi.org/10.1007/JHEP02(2016)087, arXiv:

1508.05387.

[14] A. Sen,Covariant action for type IIBsupergravity, J. HighEnergy Phys. 07 (2016)017,https://doi.org/10.1007/JHEP07(2016)017,arXiv:1511.08220.

[15]K.Mkrtchyan,F.Valach,ActionsfortypeIIsupergravitiesanddemocracy,arXiv:

2207.00626,2022.

[16] T.Erler,Theclosedstringﬁeldtheoryactionvanishes,J.HighEnergyPhys.10 (2022)055,https://doi.org/10.1007/JHEP10(2022)055,arXiv:2204.12863.

[17] A.Sen,Self-dualforms:action,Hamiltonianandcompactiﬁcation,J.Phys.A 53 (8)(2020)084002,https://doi.org/10.1088/1751-8121/ab5423,arXiv:1903. 12196.

[18] N.Lambert, (2, 0)Lagrangian structures, Phys. Lett.B 798(2019) 134948, https://doi.org/10.1016/j.physletb.2019.134948,arXiv:1908.10752.

[19] E.Andriolo, N.Lambert,C.Papageorgakis,GeometricalaspectsofanAbelian (2, 0) action, J. High Energy Phys. 04 (2020) 200, https://doi.org/10.1007/ JHEP04(2020)200,arXiv:2003.10567.

[20] A. Gustavsson, A nonabelian M5 brane Lagrangian in a supergravity background, J. High Energy Phys. 10 (2020) 001, https://doi.org/10.1007/ JHEP10(2020)001,arXiv:2006.07557.

[21] S.Chakrabarti, D. Gupta, A. Manna,M. Raman, Irrelevant deformations of chiral bosons,J. HighEnergy Phys. 02 (2021) 028, https://doi.org/10.1007/ JHEP02(2021)028,arXiv:2011.06352.

[22] E.Andriolo,N.Lambert,T.Orchard,C.Papageorgakis,Apathintegralforthe chiral-formpartitionfunction,J.HighEnergyPhys.04(2022)115,https://doi. org/10.1007/JHEP04(2022)115,arXiv:2112.00040.

[23] D.Rist,C.Saemann,M.vanderWorp,TowardsanM5-branemodel.PartIII.

Self-dualityfromadditionaltrivialﬁelds,J.HighEnergyPhys.06(2021)036, https://doi.org/10.1007/JHEP06(2021)036,arXiv:2012.09253.

[24] C.A.Cremonini,P.A.Grassi,Self-dualformsinsupergeometryI:thechiralbo- son,Nucl.Phys.B973(2021)115596,https://doi.org/10.1016/j.nuclphysb.2021.

115596,arXiv:2012.10243.

[25] S.Chakrabarti,A.Manna,M.Raman,RenormalizationinTT-deformednoninte- grabletheories,Phys.Rev.D105 (10)(2022)106025,https://doi.org/10.1103/ PhysRevD.105.106025,arXiv:2204.03385.

[26] L.Andrianopoli,C.A.Cremonini,R.D’Auria,P.A.Grassi,R.Matrecano,R.Noris, L.Ravera,M. Trigiante,M5-brane inthesuperspace approach,Phys. Rev.D 106 (2) (2022) 026010, https://doi.org/10.1103/PhysRevD.106.026010, arXiv:

2206.06388.

[27] P.Pasti,D.P.Sorokin,M.Tonin,OnLorentzinvariantactionsforchiralpforms, Phys.Rev.D55(1997)6292,https://doi.org/10.1103/PhysRevD.55.6292,arXiv:

hep-th/9611100.

[28] G.Dall’Agata,K.Lechner,D.P.Sorokin,Covariantactionsforthebosonicsector ofd=10IIBsupergravity,Class. QuantumGravity14(1997) L195,https://

doi.org/10.1088/0264-9381/14/12/003,arXiv:hep-th/9707044.

[29] C.deLacroix,H.Erbin,S.P.Kashyap,A.Sen,M.Verma,Closedsuperstringﬁeld theoryanditsapplications,Int.J.Mod.Phys.A32 (28n29)(2017)1730021, https://doi.org/10.1142/S0217751X17300216,arXiv:1703.06410.

[30] T.Erler,Fourlecturesonclosedstringﬁeldtheory,Phys.Rep.851(2020)1, https://doi.org/10.1016/j.physrep.2020.01.003,arXiv:1905.06785.

[31]H.Erbin,StringFieldTheory:AModernIntroduction,LectureNotesinPhysics, vol. 980,2021,ISBN978-3-030-65320-0,978-3-030-65321-7.

[32] E.S.Fradkin,A.A.Tseytlin,Effectiveﬁeldtheoryfromquantizedstrings,Phys.

Lett.B158(1985)316,https://doi.org/10.1016/0370-2693(85)91190-6.

[33] E.S.Fradkin,A.A.Tseytlin,Quantumstringtheoryeffectiveaction,Nucl.Phys.

B261(1985)1,https://doi.org/10.1016/0550-3213(85)90559-0;Erratum:Nucl.

Phys.B269(1986)745.

(7)

[34] C.G. CallanJr.,I.R. Klebanov,M.J.Perry,Stringtheoryeffectiveactions,Nucl.

Phys.B278(1986)78,https://doi.org/10.1016/0550-3213(86)90107-0.

[35] A.A.Tseytlin,Mobiusinﬁnitysubtractionandeffectiveactioninσ modelap- proachtoclosedstringtheory,Phys.Lett.B208(1988)221,https://doi.org/10. 1016/0370-2693(88)90421-2.

[36] A.A.Tseytlin,Sigmamodelapproachtostring theory,Int.J.Mod.Phys.A4 (1989)1257,https://doi.org/10.1142/S0217751X8900056X.

[37] L. Susskind,J.Uglum,Blackholeentropyincanonicalquantumgravityand superstring theory, Phys. Rev. D 50 (1994) 2700, https://doi.org/10.1103/ PhysRevD.50.2700,arXiv:hep-th/9401070.

[38]Y.Chen,J.Maldacena,E.Witten,Ontheblackhole/stringtransition,arXiv:2109. 08563,2021.

[39] L. Eberhardt, S. Pal, The disk partition function in string theory, J. High EnergyPhys. 08(2021)026,https://doi.org/10.1007/JHEP08(2021)026,arXiv:

2105.08726.

[40] R.Mahajan,D.Stanford,C.Yan,SphereanddiskpartitionfunctionsinLiouville andinmatrixintegrals,J.HighEnergyPhys.07(2022)132,https://doi.org/10. 1007/JHEP07(2022)132,arXiv:2107.01172.

[41] A.Sen,Wilsonianeffectiveactionofsuperstringtheory,J.HighEnergyPhys.01 (2017)108,https://doi.org/10.1007/JHEP01(2017)108,arXiv:1609.00459.

[42] H.Erbin,C.Maccaferri,M.Schnabl,J.Vošmera,Classicalalgebraicstructures instringtheoryeffectiveactions,J.HighEnergyPhys.11(2020)123,https://

doi.org/10.1007/JHEP11(2020)123,arXiv:2006.16270.

[43] B. Jurˇco, L. Raspollini, C. Sämann, M. Wolf, L∞-algebras of classical ﬁeld theoriesand the Batalin-Vilkoviskyformalism, Fortschr.Phys. 67 (7)(2019) 1900025,https://doi.org/10.1002/prop.201900025,arXiv:1809.09899.

[44] A.Sen,Stringﬁeldtheoryasworld-sheetUVregulator,J.HighEnergyPhys.10 (2019)119,https://doi.org/10.1007/JHEP10(2019)119,arXiv:1902.00263.