Conformal mappings and the three-string bosonic vertex

Volume 179 number 3

PHYSICS LETTERS B

23 October 1986

C O N F O R M A L M A P P I N G S A N D THE T H R E E - S T R I N G B O S O N I C VERTEX A NEVEU CERN, CH-1211 Geneva 23, Swttzerland

and P WEST Mathemattcs Department, King's College, Strand, London WC2 2RU, UK Received 31 July 1986 A new approach to the three-string bosomc vertex is proposed and a more powerful m e t h o d of d e n v a t m n of its gauge properties Is given

Recently there has been considerable progress in the development of a gauge covarlant theory of strings In our opinion, the best (up to now) formulation of the free open bosonlc string can be found in refs [1,2] , i It has a particularly simple form, namely × Q × The Interacting open bosonlc string was found In two different formulatlons, one by the authors of this paper in ref [4] and ref [5],hereafter referred to as I and II, and by the K y o t o group in ref [6], and the other by Wltten in ref [2] The relation between these formulations IS unknown although they both recover many of the known results o f the dual model The closed bosonlc string was found in II and ref [6] This paper is one of the series of papers which wtll find the gauge covariant Interactions of superstrlng theories The scattering of three Neveu-Schwarz superstrlngs wtll be given in the next paper [7] In this paper we develop some new techniques for deriving and finding the properties of the well known three-string vertex for open bosonlC strings In the past literature this vertex was found [8,9] within the context o f the dual model by factorizing the npoint scattering amplitude for tachyons The relation o f this vertex to the three-string vertex which occurs in the gauge covarlant action and transformation rules was discussed in II It was found that they were related by the conformal mapping wluch takes the upper half plane to the Mandelstam strip In this paper we work within the context o f the dual model, that is the upper half plane, and show how to derive the three-string vertex We also estabhsh the gauge identities for this vertex, that is how objects such as L n and C~n~are transmitted and reflected on the three legs One advantage of a further understanding o f the properties o f vertices IS that It will help In putting a name to the local gauge group, given m II, which underhes string theories It would seem likely that for the closed bosonlc string this is associated in some way with the unique lorentzian self-dual lattice m 26 dimensions Clarification o f this point will also help understand the remarkable results which relate vertex operators to group generators [ 1O, 11] In fact, the group generating vertex operators of refs [11,12] can be recovered as special cases of the s p a c e tune dependent closed string theory gauge transformation laws gwen m II It wtll be useful to first summarize some o f the hterature on the three-string scattering vertex which was found by factorizing the n-point scattering amphtude for tachyon scattering to be 4:1 For a comp lete set of references to this subject see ref [3]

0370-2693/86/$ 03 50 © Elsevier Science Publishers B V (North-Holland Physics Publishing Division)

235

Volume 179 number 3

PHYSICS LETTERS B

VCSV = (01'2'3' e x p ( - ~

r=i

FO ~x"r°~ur+l (-1)ml-'(n) = n=l -v,o+l, rtl+n-,,,j

23 October 1986

Q

(1)

We can rewrite this vertex in the form 3

r~=l = f

Vcsv = (01,2,31 exp ( 1 ~

zdZr - - pr(zr)Qr+l(Zr+l) )

zr=0

r

where

QU+(z)_ ~

1 aUz_ n

n=l n

n

'

pU(z ) ~

~ n = _e,o

aUz_ n n

'

Zr+l

=~ 1 1 z

(01231 = 1(012<013(01

(2)

r

An operator R(z) is said to have conformal dtmenslon if

g R(z) g -1 = R ( z l ) I j 1 dzXl d -dY

'

(3)

where g IS a conformal transformation and so is of the form g = exp(Zn=_ ~ anLn) In fact the transformation z -+ z 1 = zl(z) is the same as the one found by nawely applylngg when L n is represented by

L n = z n+l d/dz

(4)

For an mfimteslmal transformation this reduces to the usual definmon [ 12,13 ]

[Ln ,R(z)] = zn(z d/dz + nd)R(z),

(5)

we recall that PU(z), L(z) = Zn=_ ~ Ln z-n , and

QU=n ~=_~ l°t~n - PU lnz + q u, n~O where [q u, a~ ] = rluv, have conformal &menslons + 1, +2 and 0, respectively It will be useful to recover the usual result for the scattering of two strings and the tachyonlc state This involves multiplying one leg by the "twist" factor = e-L1 ( - 1 ) R ,

(6)

1. t~2 One may eamly verify that ~ 2 = 1 and It clearly leaves on-shell states lnvanant up to a where R = L 0 - ~ta0] phase factor Consequently, we define g ( 1 , 2 ) = gCSV[0)3 e-Lqe)(--1)R (e)

(7)

Under a conformal transformation one may verify that

eL1Q+(z) e -L1 = Q+(z/(1 - z ) ) - Q+ ( - 1 ) ,

(8)

and [12,13] <0l e+zL1 = (01 exp[+a~QU+(1/z)] Hence we find that

236

(9)

Volume 179, n u m b e r 3

PHYSICS LETTERS B

23 October 1986

,(2) ,(2) V(1,2) = 1(01 2(01 e -'~I ( - 1 ) " X e x p [ - ~1 nl(f

?P'(z) [Q2+(llz) -

z=O

Q2(1)] + f

dz z - o~U31 0 Q+(I/(1 _z)))l

z=O

?p,(z) = l(0l 2(01 exp ( - .~.~lf 1

Q2(1/z)+ot~3Ql+(1) + (O~O u 2 )Q+(I)) 2 /~1 + or0

oo

1 ( 0 1 2 ( 0 1 exp

~ !~/flO~ n n = l 12 t"/

-- Ot0 [Q+I(1) - Q 2 ( 1 ) ]

(10)

After turning 2 around and identifying the 1 and 2 oscillators we recover a well-known result The vertex for an excited level on line 3 can be evaluated in a smallar way Although the factonzatlon of the general n-point vertices was discussed In the hterature [ 11 ], there is httle discussion of the gauge identities they satBfy In a recent paper, II, It was shown by explicit computation that V csv obeyed the identity

vCS~(A(z)elu (1/(1 - z)) + B(z)e2U((z - 1)/z) + C(z)eaU(z)) = 0,

(11)

where A, B and C satisfy the relation

A / ( - z ) +B + C/(z - 1) = 0

(12)

In order that the series involved in checking eq (11) converge, one of the coefficients A, B, C must vanish A more detatled discussion of this point is gwen below We now wish to show that an operator R(z) of conformal dImenslon d satxsfies the following identity

vcS~[A(z)Rl(1/(1 - z ) ) + B(z)R2((z - 1)/z) + C(z)R3(z)l = 0,

(13)

provided

A(z)/(-z) d + B(z) + C(z)/(z - 1) d = 0

(14)

Let us first consider the transformation z ~ zl = 1/(1 - z) which IS generated by gr = elf1 elf-1 Essentially this transformation maps string 3 into string 2, string 2 into string 1 and string 1 into string 3, a e it cycles the amplitude We notice that

g [A(z)R 1(1/(1 - z)) + B(z)R2((z - 1)/z) + C(z)R3(z)lg -1 = (A(z)[1/(-z)]dRl((z-

1)/z)+B(z)(z- 1)dRZ(z)+C(z)[-z/(z-

1)]dR3(1/(1 - z ) ) } ,

(15)

where g = 113=1 gr If we t a k e r =pu use eqs (11) and (12) and the fact that V csv is cychc then

VCSVg -1 [A(z)RI(1/(1 - z)) + B(z)R2((z - 1)/z) + C'(z)R3(z)] = 0,

(16)

where A,/~ and C can be read off from eq (15) and still satisfy eq (12) Consequently V csv and Vcsv g - 1 satisfy the same pu identity However, as we wtll see later this identity straightforwardly determines the vertex umquely and so

VCSVg -1 = VCSV

(17)

This equation may also be verified using identities stmflar to eqs (8) and (9) and the discussion that follows xt We now sketch, using the mactunery set up in ref [ 12], an alternatwe proof of eq (17) We may write V csv in 237

Volume 179, number 3

PHYSICS LETTERS B

23 October 1986

the generic form V csv = (01,2,31 exp - ( a l f z a 2 + a 212ot3 + a3~2a1),

(18)

where ~2 is an lnfimte dmaensional matrix which corresponds to the transformatmn z -+ 1 - z and hence to the matrix

(-1011 1

(19)

We now apply g -1 which corresponds to z 1 -~ 1 - 1/z and so to the matrix g-1~(11

-10)=S

(20)

Applying the rules of ref [12] we find that v e s v g - 1 (012,31 exp - ( a l ~ s ¢ ~ 2 + a2"S~Sot 3 + o t 3 ~ S a 1)

(21)

Now one may verify that the matrix corresponding to Sxs given by S---(01

10)S~1(~

10)=S,

(22)

and hence we find ~2 S = ~2,

(23)

which establishes the result In fact, one may show that S = ~2 Since R is constructed from ~ oscillators a httle thought shows that It wdl satisfy an identity such as eq (13) which explains how the R are transmitted and reflected on each leg provided a certain identity holds This identity can be written m the form A / f (z) + B + c/g(z) = 0,

(24)

as we may scale it by an arbitrary functmn of z However from eq (15) .4 = C ( z ) [ - z / ( z

- 1)] d,

B ( z - 1) d = C,

A(z)/(-z)

d =/~

(25)

must saUsfv the same identity as A, B and C since V csv cannot satisfy two identities of this type This determines f ( z ) = ( - z ) d and g(z) = (z - 1) d , estabhshmg the desired result

One could use the above chain of arguments to derive the vertex V csv Any vertex is bound to have a rule of the form of eq (18) for the transmission and reflection of the a n on the three legs If we then assume it satisfies eq (17) we may then derive eqs (11) and (12) which umquely determine the form of V csv It IS often more useful to write identities of the type discussed above m an integral format In carrymg out the following steps it is unportant to take careful account of the conditions for convergence What must be avoided is an mfinlte sum of creation operators with coefflctents which belong to a divergent series Only operators well defreed when acting on any state consisting of a finite number of creation oscdlators on the vacuum should be considered As a result eqs (13) and (14) are only valid for Izl > 1 f f C = 0, Iz - 11 < Izl xfB = 0 and I1 - zl > 1 ifA = 0 Let us consider then dz

f 7-c(z)R3(z), z=0

238

Volume 179,number3

PHYSICSLETTERSB

23 October 1986

where C(z) as arbitrary We can deform this contour which 1s understood to be In the region Izl "~ 1 to be around z = 1 and z = ~o provided we swoop C for B and A around 1 and oo respectively and use eq (14) Carrying out this manoeuvre we find the result VCSV (£ ~a

dz

z=0

-f

-

dz R3(z) + f z(z--- 1) (_z)d-a R2((z (--Z)d-1 z=1

-~- ¢(z)

(z

%

~z)Rt(1/(1-

1)/z)

z))) :0

(26)

Z=OO

Introducing z 1 = 1/(1 - z), z 3 = 1/(1 - z2), z 1 = z we may write the above equation as

~R3(z)¢(z)(Z-l)d--l+f d~2 (p(z) z=O (-----Z)d-1 ~2=0 ~2 (--Z) d - i

VCSV( f

R2(~2) + f d~l q~(z)RI(~I)) = 0, ~1=0 ~1

(27'

where

~1 =l/(1-z)'

~2 = l - 1 / z

(28)

Two examples of the above Identities are It

oo

V csv iv_3n + ~ n p=O P

( _ 1 ) p (~b

~ (-1) p p=o

-

O~p+.

and eo

n+l

VCSV

n

Lo

(n + 1)'

(n + 1 _p)~pl

+L3

-1

0- l) F0(

n+

p

(30)

The first o f these equations easily determines V csv It as straightforward to transform eq (27) from the upper half plane to any other region of the complex plane by a conformal transformation Of particular Interest as the strap using the conformal transformation of ref [ 14], take z 1 - z = 33 then one finds that z 2 = 32 and z 3 = 33, in accord with the fact that z -+ 1/(1 - z) takes the three strings into each other On the strip eq (27) becomes d3r ¢(z)(z - Zant ) d - 1 (R%r () d3 -r1) = o ' -7-

= 3fr V r~_3-1

(31)

=0 r where V = V csv e x p ( - E n = 1 anL n) For R = pu we have d = 1 and as explained in II we con easily find the a part o f the vertex o f gauge covarlant string theory since the latter vertex laves on the strap The above derivation also explains an more detail the subtraction procedure advocated in II One could also derive V csv by starting on the strip and assuming the vertex is an overlap 6 function as a result of which eq (31) with d = 1 holds for R = pu Mapping onto the upper half plane we can derive eq (12) for R = pu and hence the form of V csv 1 2 The Fubinl-Venezaanv vertex takes the form of eq (10) when multiplied by z - L ozL o The more usual formulation ls found by turning one line into a ket and identifying the oscillators o f this hne with those o f the other line This vertex is an object o f well defined conformal weight, namely k 2, as well as giving, by contour integration over 3, a physical vertex The latter property means that if one leg lS on-shell the second leg as automatically onshell In order to generalise these properties to the three-point vertex let us consider V=f

z=0

dz V csv(z),

z

(32) 239

Volume 179, number 3

PHYSICS LETTERS B

23 October 1986

where

V csv (z) = V esv (zL~(1/(1 --z))Lg((z - "l)/z)Lg)

(33)

We observe that if any two legs are on-shell then the third leg is automatically projected to be on-shell From eq (30) it follows that

VCSV(z)[Ll_n z - n - ( L 1 + n - 1)z n] IX)21X) 3 = 0

(34)

If [X )2 and IX )3 are on-shell consequently

V[X)2[X)3L_n=O

n>~l,

VIX)21X)3(L0-1)=0

(35)

We may also generallse the property of conformal weight to to yield the equation

V(z) glg2g3

V(z)

The identity of eq (30) can be exponentlated

= V(2),

where g 1 = exp

(36)

oo

~,n=_=anLn

2

2

2

and maps z to 2, 2

r2

r2

r2

2

- z)LOe-Ll(--1)LOz -L°] 2 ~ o ( - 1 ) ~ ° e * ' 1 ( 1 - 2 ) - L ° , 33 ( ) g3 = ([z/(z - 1)]LOe-/'-'z -Lg) exp ~ a n L 3 n (2Lge L3-' [ ( 2 g2 = [(1

3

1)/2'] Lo)

(37)

The reader may verify that g2 and g3 are just such as to mduce the same conformal mapping on z as g In II a new vertex lnvolvmg the ghost oscillators of the form V csv V(3,{3) was found Clearly the above discussion can also be extended to consider V(3, 3) We will report on this and the supersymmetnc vertices elsewhere In II It xs shown exphcltly that

V~

3

Qr_ -

-

r = l otr

- o,

(38)

where Q xs the BRST change [5] However, smce Q is unaffected by the conformal transformation mapping the covanant vertex V to the upper half plane, we must fred that 3 IV esv V(3, 3)] ~ r=l

Or = 0

(39)

It will be interesting to discover If the techniques used to derive V csv can be extended to give a simple derwatlon of the hsgher point vertices which occurred in the hterature [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [ 11 ] [12] [13] [14] 240

A Neveu, H Nlcolaaand P West, Phys Lett B 167 (1986)307 E Wltten, Non-commutatwe geometry and string field theory, Princeton preprmt (1986) P C West, Trieste Lecture Notes, CERN preprmt A Neveu and P West, Phys Lett B 168 (1985)192 A Neveu and P West, Symmetrxes of the interacting gauge covarlant bosonlc string, CERN preprmt TH4358/86, Nucl Phys B, to be pub.hshed H Hata, K Itoh, T Kugo, H Kumtomo and K Ogawa, Phys Lett B 172 (1986) 186, 195, B 175 (1986) 138, Kyoto preprmts A Neveu and P West, Supersymmetnc dual model I, m preparation S Scmto, Lett Nuovo Ctrnento 2 (1969) 411 L Canescha,A Schwmamerand G Venezmno,Phys Lett B 30 (1969)351 I B Frankel and V G Kac, Inv Math 62 (1980) 23 D Ohve and P Goddaxd, Vertex operators m mathematics and physics, eds J Lepowsky, S Mandelstam and I Stager (Springer, Berhn, 1984), V Alessandrml, D Amatl, M Le BeUac and D I Ohve, Phys Rep 1 (1971) 170, and references thereto J Schwarz, Phys Rep 8 (1973) 269, and references thereto S Mandelstam, Nucl Phys B64 (1973)205