Braz. J. Phys. vol.32 número4

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Comments on Resolution of Nonassoiativity in SFT

- An Example from Axioms of BCFT

-Matsuo Yutaka

Department ofPhysis,UniversityofTokyo

Hongo 7-3-1,Bunkyo-ku,Tokyo113-0033,Japan

Reeivedon17April,2002

Itisknownthattheassoiativityofthestarprodutoftheopenstringeldtheorymaybebroken

in the presene of the nontrivial losed string bakground. We give an argument that suh an

anomalymayberesolvedbyinludingChan-Patonfatorsstartingfromtheaxiomsoftherational

onformaleldtheory.

I Introdution

WhenWittenhasdened thebosoniopenstringeld

theory [1℄, hedened the theory in terms of the

non-ommutative but assoiative star produt. The

non-ommutativity is essential feature of the open string

eld theory sine it hastwo ends and it behaves asa

sort of matrix. On the other hand, the assoiativity

is essentialto ensure thegaugeinvariane ofWitten's

ubi ation. These are the key features of the open

stringeld theory whenwewantto havea

orrespon-denewiththenonommutativegeometry.

In the expliit alulation of the open string eld

theory, however,weoftenmeet thenonassoiativityof

the star produt when we want to treat the singular

statesintheopenstringHilbertspae. Suhastruture

appearsinthesimplied versionofthetheory,namely

treatingtheopenstringeldsasmatries[2℄. The

prod-utsofthreeinnitesizematrixmaynotbeassoiative

(AB)C 6= A(BC) sine itinvolves two innite

sums. While eah summay be absolutely onvergent,

the doublesumsometimesbeomesonlyonditionally

onvergentanditbreakstheassoiativity[3,4℄.

This appearane of theassoiativity anomaly may

lookrathersuperial,namelyonemayneedtobemore

arefulwhenwedenethestateandweshouldruleout

the stateswhihbreaks the assoiativity. However,in

some situations, the nonassoiativity arises from the

verynatureof thenontriviallosedstringbakground.

We illustrate this phenomena by using two

exam-ples.

Intheeldtheorylimit,thenonommutative

geom-etryofthestringeldtheoryreduestothatofthe

tar-getspae. Inthesimplesituationssuhastheonstant

B eldintheatspae,thenonommutativegeometry

is desribed by the Moyal produt whih is obviously

itsatisesdB =0,thetargetspae beomes the

Pois-son manifold and one may use the Kontsevih's star

produt to dene the assoiative produt. The

situa-tionhangesdrastiallywhenweproeed todenethe

nonommutative geometry when dB 6= 0. In suh a

situation,CornalbaandShiappa[5℄observedthatthe

naive extension of Kontsevih's star produt beomes

nonassoiative,

f(gh) (fg)h(dB) ijk

i f

j g

k

h: (1)

Atthelevelofthestringeldtheory,Horowitsand

Strominger[3℄observedthatthespae-timetranslation

generatormay be written in terms of the open string

variableasP

L

IwhereP

L

isthemomentumdensity

in-tegratedoverhalfstringandI istheidentityoperator.

Suh generators, however, break the assoiativity

be-ausetheyusuallyshiftthemidpointoftheopenstring.

Ontheotherhand,intheonventionaldenitionofthe

starprodut,thepositionofthemidpointispreserved.

Strominger[6℄lateremphasizedthatsuhan

assoiativ-ityanomaly is theuniversalfeature of theopenstring

algebra if we want to express the deformation of the

losed string bakground in terms of the open string

degreeoffreedom.

The existene of the assoiativity anomaly signals

thatweannotusetheonventionaldesriptionofthe

nonommutativity. Forexample,the vetor bundlein

theommutativegeometryisusuallytranslatedintothe

projetivemoduledesribedbytheprojetionoperator

ofthe(matrix generalizationofthe) algebra. In

physi-alterminology,itisdesribedbythenonommutative

soliton [7℄. However, suh an operator doesnot have

any topologial meaning one the algebra breaks the

assoiativity.

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longstothenontrivialelementintheohomologylass

H 3

(X;Z), the gauge transformation generators fg

ij g

beomesnot losedon the intersetionsof three

oor-dinateneighborhoodsU

i \U j \U k , g ij g jk g k i =e i ijk

6=1: (2)

It shows that the vetor bundle beomes twisted and

ill-dened.

When [dB℄ represents a torsion element in

H 3

(X;Z),say Z

n , e

i

ijk

should satisfye in

ijk

=1. In

this speial situation, when we adjust the number of

D-brane to n, theadjointvetorbundle beomes

on-sistent sine thephase fator belongs to the enter of

the gaugegroupU(n) [8℄. Bya generalizationof this

idea,Bouwknegtand Mathai[9℄ havearguedthat one

maydenewell-denedadjointbundleevenforgeneri

elementofH 3

(X;Z)whenwehavetheinnitenumber

ofD-branes.

Alessonfromthesephenomenaforthe

nonommu-tativesideisthatonemayreovertheassoiativityone

we inlude the degree of freedom of D-branes

(Chan-Paton fator) to modify the originally nonassoiative

algebra to an assoiative one. For the generi

bak-ground, we need the innite number of D-branes for

themodiationofthealgebra.

The neessity of the innite number of D-branes

suggestthat weneed asortof thevauum stringeld

theory[10℄. Namely to express the algebra, we need

to employ an innite dimensional matrix

generaliza-tion of Witten's string eld theory. If all of these

D-branes were physial, we have the innite energy to

reate these D-branes. This isnot thedesired feature

ofourse. Byusingthevauumstringeldtheory,one

maypush most of these D-branes in soalled \losed

string vauum". In suh a situation, while they

on-tribute to make the algebra assoiative, they do not

ostanyenergy.

II BCFT Axioms and

Assoia-tivity

IntheaxiomatiapproahtotheBoundaryConformal

FieldTheory(BCFT)[11℄,theboundaryprimaryeld

is haraterized by four labels a;b 2 V, i 2 E, and

=1;;n ab

i

. Here V represents thepossible

bound-ary onditionsfor thegivenlosed stringbakground.

We have two labels a;b beause we have two

bound-ariesonthegivenopenstring. E isthesetofthehiral

primary elds. n ab

i

representsthe numberof hannels

for the primary eld i when the boundaries are

spe-ied by a;b. If we represent Cardy state in terms of

Ishibashistateasjai= P

i2E a

i

Si1

jiii,wehavethe

rela-tionbetweenthem,

n ab i = X S ij S 1j a j b j (3)

whih is an analogueof Verlinde formula. By

hang-ing S

aj

(the modular transformation matries of the

harater) to a

j

, onemay represent CFT for the

o-diagonalmodularinvariant.

We represent the open string Hilbert spae with

given boundariesa;b asH ab

. Wewrite ab

i

(z)for the

boundaryprimaryeldandji;a;b;itheorresponding

highestweightstate. ThenH ab

anberepresentedas

H ab = L n1 L n`

ji;a;b;iji2V;=1;;n ab

i

(4)

Thestar produtwhihwewantto dene isthe

map-ping,

? : H ab H b !H a : (5)

Equivalently,onemaydenethestarprodutbyusing

thetri-linearmapping,

V 3 : H ab H b H a

!C: (6)

In the CFT type denition of the three string vertex

[12℄, the threestringvertexoperator isdened by the

threepointfuntion,

hV

3

jjAijBijCi=hh

1 ÆA;h

2 ÆB;h

3

ÆCi: (7)

Theonformaltransformationsh

i

aredenedby

h

` (z)=e

2 i(` 1)

3

1+iz

1 iz

2=3

; `=1;2;3: (8)

whihmapstheupperhalfplanesintoeahpieewhih

dividestheunitdiskintothree. Wenotethattheright

handof(7)dependsonlyontheonformalpropertyof

three eldsA;B;C. Sinetheonformalweightof the

boundary primaryelds ome only from thelabelfor

theprimaryelds iandnotfromtheboundaryindies

V, onemayalulate the starprodutfrom theindex

i 2 E for the primary eld and the operator produt

expansionforthehiralprimaryelds,

i (z) j (w) X k 2E C k ij (z w) hi+hj h k k (w)+(desendants) : (9)

Wenote that wedonotrestritthe onformal

dimen-sion to be one sine we onsider the o-shell vertex.

The desendants parts of OPE an be dedued from

C k

ij

andtheonformalpropertyalone.

Combining these ideas, one arrives at the

onlu-sion that the star produt an be in priniple dened

withoutusingtheindiesoftheboundary. Weusethe

symboltodenotethisdenition,

ji;Iijj;Ji= X

k ;K C

i j k

I J K

jk;Ki; (10)

where i;j;k 2E represent theindies for theprimary

elds and I;J;K are multi-indies for the onformal

(3)

Thepartiular form of Witten's vertexoperatoris

reetedbytheinnitelinearrelationsamongthe

om-ponentsofthestrutureonstantC. Itanbederived

from theWardidentityforthethreestringvertex[13℄

hV

3 j(L

(1)

vn +L

(2)

vn +L

(3)

vn

)=0; (11)

forthevetoreldswhihareholomorphiintheworld

sheet of the three string vertex. This relation anbe

translatedintotherelationsamongC,

Æ

n C

i j k

I J K

=0: (12)

Theseidentities aresupposed tobesuÆientto

deter-mineC

i j k

I J K

fromtheOPEoeÆientC k

ij for

theprimaryelds.

Thefatthatthehiralprimaryeldsenjoythe

non-trivialmonodromyimpliesthattheprodutwhihwe

dened is not assoiative. Inpartiular, the

transfor-mation rule for the hiral blok funtion implies that

thestrutureonstantCsatises,

X

P C

i j p

I J P

C

p k l

P K L

= X

q;Q F

pq

j k

i l

C

j k q

J K Q

C

i q l

I Q L

(13)

TheoeÆientsF isalled 6j-symbolforthehiralprimaryeldsandsatisestheunitarityondition,

F

F =1: (14)

Theonsistenyonditionfor6j-symbolwasdeterminedas,

FF =FFF (15)

whihisalledthepentagonidentity[14℄. Eq.(13)impliestheassoiativityanomalyitimpliesthat

(ji;Iijj;Ji)jk;Ki= X

p;P C

i j p

I J P

jp;Pijk;Ki

= X

p;P;l;L C

i j p

I J P

C

p k l

P K L

jl;Li (16)

ji;Ii(jj;Jijk;Ki)= X

q;Q C

j k q

J K Q

ji;Iijq;Qi

= X

q;Q;l;L C

j k q

J K Q

C

i q l

I Q L

jl;Li (17)

TheexisteneofthemonodromyfatorF thensuggesttheprodutisnotassoiative,

(ji;Iijj;Ji)jk;Ki6=ji;Ii(jj;Jijk;Ki): (18)

Intheaxiomsoftheboundaryonformaleldtheory,suhananomalyisabsorbedintheboundaryChan-Paton

index asfollows. Weinlude theboundary labelsin the denition ofthe stateji;Ii!ji;I;a;bi. Wethendene

the?produtbyusingthe-numberoeÆients(3j-symbol) asfollows,

ji;I;a;bi?jj;J;b;i= X

k ;K (1)

F

bp

a

i j

C

i j k

I J K

jk;K ;a;i (19)

Theboundary3j-symbol (1)

F isrequiredto satisfy,

(1)

F

bp

a

i j

(1)

F

dp

a

l k

= X

F

pq

i j

k l

(1)

F

aq

b d

i l

(1)

F

q

b d

j k

(4)

The star produt beomes assoiative beause of the

unitarityof6j-symbol.

III Some omments

We give some omments and onjetures to be

on-rmedinthefutureworkinthissetion.

1. We note that the use of Chan-Paton degree of

freedom to desribethe assoiativestar produt

isrelatedtotheglobalanomalyanellationofthe

worldvolumetheoryontheD-brane[8℄. Namely

onlywhenwehavetheappropriatenumberof

D-branes to anel anomaly on the world volume.

It isinterestingto seethat this fatis relatedto

thepurelyalgebraionstraintoftheassoiativity

anomaly.

2. It is interesting to study this anomaly

anella-tion mehanisman beapplied to the

Horowitz-Strominger anomaly [3℄. One possibility to

answer this problem may be the following.

Horowitz-Strominger anomaly arises when we

needtodesribethespae-timereparametrization

(namelythegaugedegreeoffreedomofthelosed

string). In CFT language, it amounts to the

marginal deformation of the losed string

bak-ground. Suhadeformationofthetheoryindues

the redenition of the primary elds and at the

same time their OPE relations. In the

onven-tionaloperatorformalism,thereisnoexpliit

de-pendene of Chan-Paton fators andone arrives

at anomaly beause of nontrivial 6-j symbol in

thisbakground.

3. Asweommentedintheintrodution,the

frame-workofthevauumstringeldtheory(VSFT)is

neessarytomakeourdisussiononsistent. The

VSFTonjeturemaybesummarized asfollows.

When the open string tahyon is ondensed, we

aresupposedtoarriveatthe\losedstring"

va-uum. The desription of suh a state is given

byhangingtheonventionalBRSThargeQto

the bakground independent one Q whih is

de-sribedonlyintermsoftheghosteld. This

oper-atorQshouldnothavephysialspetrumsinein

the\losedstring"vauumthereisnoD-branes.

When D-brane is exited from the vauum, the

stringeldisshifted,

!

0

+Æ (21)

and expansion around this new vauum gives a

newBRSTharge,

Q=e K

(Q+[

0 ; ℄

? )e

K

(22)

withanappropriate\unitarytransformation"K.

is desribed by the projetion operator of ?

produt and Q should have nontrivial physial

stateswhihlivesonthereatedD-brane. Dueto

thenonzeroshift

0

,thevauumenergyisshifted

bythevalueoftheationS(

0

)whihanbe

in-terpretedto givethetensionoftheD-brane.

In our disussion, when the losed string

bak-groundisgeneri,the monodromymatrixF an

be aneled only when we introdue the innite

number of D-branes. If all of these D-branes

shouldbephysial,wesuerfromtheinnite

on-tribution of the D-brane tension. This is

inon-sistent. Soif weonly havethe open stringeld

theoryofphysialD-branes,weannotdenethe

openstringdegreeof freedomatall. However,if

wedeform thelosed stringbakgroundto more

trivialonesuhasatspae,theexisteneofthe

nite number of the D-branes beomes possible

and we need the open string eld theory whih

desribethem. Inthissense,thebehaviorof the

open string eld theory heavily depends on the

losed string bakground whih an be

ontinu-ouslydeformed. Itimpliesthatwehavea

dison-tinuityofthedesriptionoftheopenstringdegree

offreedominthemodulispaeofthelosedstring

bakground.

Ifweuse the VSFT senario, however, one may

keep the innite number of D-branes hidden

in the bakground without onsuming energy.

When bakground hanges, the open string

al-gebrahanges ontinuously and at somespei

points,onemayonstruttheprojetionoperator

of?produt. Thereforein thisframework,there

isnodisontinuityofthedesriptionoftheopen

string. Thedisontinuitydependingonthe

bak-ground omes from the onsisteny of the

on-strutionof theprojetionoperator,whih looks

morenatural to work with. One possible

strat-egy to dene VSFT with Chan-Paton fator is

disussedin[15℄.

Aknowledgement: Theauthor would liketo thank

Prof. Berkovits for inviting him to Brazil, many

dis-ussions and a warm hospitality during his stay. He

is also obligedto the organizer forgiving the

oppotu-nity to presenthis resultin \XXII Enontro Naional

deFisiadePartiulaseCampos".

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