Loop variables and the Virasoro group

NUCLEAR

PH YS I C S B

Nuclear Physics B405 (1993) 367—388 North-Holland

Loop variables and the Virasoro group B. Sathiapalan Physics Department, Penn State University, Ridge View Drive, Dunmore, PA 18512, USA Received 16 September 1992 (Revised 2 February 1993) Accepted for publication 15 March 1993

We derive an expression in closed form for the action of a finite element of the Virasoro group on generalized vertex operators. This complements earlier results giving an algorithm to compute the action of a finite string of generators of the Virasoro algebra on generalized vertex operators. The main new idea is to use a first order formalism to represent the infinitesimal group element as a loop variable. To obtain a finite group element it is necessary to thicken the loop to a band of finite thickness. This technique makes the calculation very simple.

1. Introduction Understanding the Virasoro group is of central importance in string theory since the gauge invariance of the theory is a consequence of the conformal invariance of the two dimensional world sheet field theory and the generators of infinitesimal conformal transformations obey the Virasoro algebra. A lot of work has been done on the representation theory of the Virasoro algebra and applications to various branches of physics [1—9]. For many purposes it is sufficient to consider the algebra rather than the group. In particular in string theory attention has been focussed on infinitesimal gauge transformations and for this the algebra is all one needs to consider. Experience with Yang—Mills theories and gravity teaches us however that it is essential to consider the group in order to appreciate the geometrical basis underlying gauge symmetries. Another motivation for studying the Virasoro group is that the string evolution operator eI~oTis obviously an element of the Virasoro group. It is plausible that a gauge invariant generalization of this would be exp(~~L~t~) which is a general element of the Virasoro group. The Virasoro group also has been studied from different points of view [10—151. Our goal in this paper is to approach these issues using the loop variable formalism. This will improve our understanding of the formalism which we feel will be useful for developing computational techniques for string theory. We derive an expression for the action of a Virasoro group element on generalized vertex operators, i.e. we write down an expression for 0550-3213/93/s 06.00

© 1993—Elsevier

Science Publishers B.V. All rights reserved

B. Sathiapalan / Loop variables and the Virasoro group

368

exp (~~20L~) V(z) operator (gvo). exp(i(koX +

I ), where

f

V(z) is a loop variable or generalized vertex

dtk(t)0~X(z + t)))

=

exp(i~ kfl0nX(z)) n >0

(1.1)

~exp(iknYn)

(a sum over the index n is understood). Here we have used

2+...+k 2+..., (1.2) k(t)=k0+k1/t+k2/t 1t+k2t ax(z+t)=ax+ta2x+t2a3x/2!+...+t~’o°x/(n—l)!+....

(1.3) A loop variable contains in it all the vertex operators in the bosonic string theory [16,171. The Taylor expansion in t is allowed because there are no singularities in t. This will not be the case if we consider correlation functions with other operators located inside the contour. Thus in general we must make a distinction between the LHS of (1.1), which is a loop variable and the RHS, which is a generalized vertex operator. The loop variable is more general than the generalized vertex operator. In ref. [17] we described a simple algorithm for calculating the action of an arbitrary finite sequence of L 0’s on the generalized vertex operator. The result, briefly, is as follow: In order to calculate LnLm~•~e1k~~ 0), start with the ‘dressed’ vertex operator exp[iknYn + ~q~oq~YnY~ +(~o~qnoqmYm+ ikn) [ös(_Osq 0q0t1o~(~ro~wYw+ ~kr))]

xDet[l ~

(1.4)

Perform the repeated transformations:

~ L~m

q~oq~

on

o

0,

(1.5)

q~+ (n*p)qn~pc+p; q_~~=0,

(1.6)

qnoqm+2n~m,

(1.7)

On 0 Om + nmA_(n+m).

(1.8)

k~+ (n ±p)k~+~e±~ k~

k~

—p

°~n~

=

Setting q, 0 = Oat the end and the coefficient of the relevant gauge parameter(s) gives the required result. This is an algorithm, then, for performing the computation with an arbitrary but finite number of Ln’S. Since this method could, at least, in principle be used for an arbitrary large number of La’s it would seem that the ‘dressed’ vertex operator is result of thepaper action a finite 1~-~ on somehow e’1’~’10). the In the present weofconsider group element of the form e~

B. Sathiapalan / Loop variables and the Virasoro group

369

the action of a finite element. However, here instead of an algorithm we have a mathematical formula involving some definite matrices and their products for the action of a.flnite group element. Let us outline the approach using loop variables. The basic idea (as explained in more detail in section 2) is to use a first order formalism to represent the group element exp(f~(t)ax(z

+ t)3X(z + t)dt)

(1.9)

as a loop variable

f

Dk(t)exP(ifk(t)3zX(z+t)dt)exP(fdtk2(t)~_1(t)).

(1.10)

Thus in order to evaluate the action of (1.9) on vertex operators ~ one can use the form (1.10) which is more convenient since the operator product of exp(i j’~ k(t)O~X(z + t) dt)exp (i ~n~O ~ Y~)isvery easy to calculate using (schematically)

Kexp(ifk(t)3zX~+ t)dt)exp =

exp

(if dtk(t)(OX(z

(i~~nvn))

+ t)Yn(z))qn).

Thus the idea is to calculate the OPE with (1.10) and and then do the (Gaussian) integration over k (t) rather than the other way around. In order to implement the above idea a crucial element is a normal ordering prescription for the 1oop variable. Thus we have to write exP(ifk(t)azx(z

=

+ t)dt)

: exP(ifk(t)OzX(z + t) dt) : N[k(t)],

where N is the normal ordering constant (this is analogous to to writing ~ = 212 in a for the tachyon vertex operator, where a is an ultraviolet ek cutoff). In section 2 we will present a simple prescription. It gives N[k(t)] = exp(~nmknOnmkm) for some matrix 0. However, it will turn out on checking group composition that this prescription is too naive and works only to quadratic order. The correct prescription, it turns out, is obtained by a simple extension of the naive prescription and gives a somewhat more complicated 0 matrix. We give here the final result. It is proved in section 3 using an induction technique. In section 4 we show how to derive this result by modifying the loop to an annulus of finite thickness. This dramatically simplifies the derivation. In addition it

B. Sathiapalan / Loop variables and the Virasoro group

370

seems to provide a prescription for regulating ultraviolet divergences for the loop variable. This is an issue that one faces in string interactions. However in the context of interactions it remains to find a suitable interpretation of the ~ n > 0 variables. The formula is as follows: We define =

=

00102

).m,n(a)

N0(a1

—

=

~‘n+m,

0

zs~a

~ l;n,m e 7/,

a2) + NTO(a2_ a1);0(x)

=

~ 1, x>0 , (~0, x <0

(1.11) (1.12)

where N is an infinite dimensional matrix acting on infinite dimensional vectors of the form y,~,n E Z. If we consider the subspace (~), n > 0 it has the form N = (~ ~) When n = 0, N = (?~).We make a distinction between .v+~and Y—o and set the former equal to zero i.e.y~0= 0. Finally, define the (infinite) column vector Y = (~3”~ ) where .

I

—ink~n

Y_n ~

>

0

1~—iko n=O

and Y+n

Y,~~0~X/(n

—

1)!, n

>

Oand Y+n ~O,n

=

0.

In terms of these variables, exp (~~~nLn) : exp (i>~nYn) exP(_~YTPAY)exP(i~ ~nYn) : Det[l

_~j-D/2,

(1.13)

}~(a2).

(1.14)

n~0

with pA

[daj~

1

J

The object in curly brackets is a matrix with continuous indices a1, a2 and multiplies a column vector ,~ (a2). Thus a2 is summed over. Inside the curly bracket there is no sum on a1. pA can be represented as an infinite series in ~ by expanding the expression in curly brackets:

Idai I~3~t ~(a1) +

=

+

/~i

Jda2

fda22(ai)0(aia2)~(a2) 1a2)~22a3)~(a3) +

=A + (l/2)~(N + NT)~+ (l/3!)[ANAN,~ +~NT~NT~

B. Sathiapalan / Loop variables and the Virasoro group + 2ANANTA + 2ANTANA] +

371

(1.15)

...

Thus exp ( ~ yTpA Y) can be regarded as a representation of the group element There is a well defined group composition law pA e”1~-’~. 0pii = pp(A~u) When A and ~i are infinitesimal this law satisfies the Virasoro algebra. Equations (1. 1 3), (1.14) summarize the main result of this paper. They describe the action of an element of the Virasoro group on an arbitrary vertex operator. Comparing this result with (1.1 3) we also see the similarity of the two expressions. In this paper we have not tried to relate the two expressions. This paper is organized as follows: In section 2 we motivate the result in an intuitive way. In section 3 we give a more complete proof. In section 4 we discuss the group composition property. This can be used to construct an alternate proof of the result. A sample calculation is also given. In section 5 we give some concluding remarks. 2. Loop variable representation of group element Consider the integral

f

Vk(t)exP(ifk(t)0zX(z

+ t)dt) ~[k(t)I,

(2.1)

2 + •.. which is a loop variable representation of with k(t) = k0 + fields k1/t +ofk2/t (D-dimensional) the bosonic string as considered in ref. [16]. (Later we will generalize to include positive powers of t also). If we choose ~[k(t)]

=

exp(_~fdtk2(t)A1(t))

(2.2)

and perform the functional integral over k (t) we get, upto an overall normalization factor,

f

exp (—~ dtaX(z + t)DX(z + t)A(t)) [DetA]D/2.

(2.3)

The exponent is nothing but the energy momentum tensor multiplied by a gauge parameter. If we choose 2 + 2 3 + ... + A~t~ + ... (2.4) 0t + A1 + A2/t + + A1t 2t then (2.3) becomes exp(~~A~L..~) which is an element of the Virasoro group. A(t)

=

2

The A~in (2.4) is the same as in (1.11). The above discussion has to be modified to include the effects of normal ordering when we make X a two dimensional quantum field. Furthermore k(t) will also include positive powers oft when we consider operator products ofthe loop variable (2.1) with other vertex operators at the point z. Nevertheless the basic idea illustrated above will be used. Namely we will use the first order representation (2.1) instead of (2.3). In order to

B. Sathiapalan / Loop variables and the Virasoro group

372

Fig. 1. Loop variables are defined along circles of different radii.

calculate the action of a group element on a vertex operator we first perform the functional integral over X to get the terms in the operator product expansion of (2.1) with the vertex operator and then do the k (t) integration. Now we have to incorporate quantum mechanics since otherwise the Ln’S in (2.3) will commute. That means we have to include normal ordering effects. We can try the following: write exP(ifk(t)8zX(z+t)dt)

x exp

=:exP(ifk(t)a~x(z+t)dt)

(~f

dtk(t)(OX(z + t)~~~)kn).

(2.5)

In order to avoid the ambiguity at t = t’ we will take one loop to have a slightly larger radius as shown in fig. 1. On the inner ioop X (z + t) can be expanded in a Taylor series in t (i.e. in positive powers oft) since there is no singularity as the radius of the loop shrinks to zero. We have used (X(z)X(w)) = ln(z —w) to get for the second factor exp

(~

mkm ‘k_m~, /

(2.6)

where +

k(t)

=

~

m= -

k,~rm.

(2.7)

B. Sathiapalan / Loop variables and the Virasoro group

373

Thus (2.5) becomes f+oo

exP(ifk(t)a~X(z + t)dt) : exp C

~kn

km0nm)~

(2.8)

—00

where 0n,m

=

mön,.m,m ~ 0

(2.9)

and (2.1) becomes

f

Vk(t) : exP(ifk(t)azX(z + t)dt): exp(~kn[O~,~ A~Jkm), (2.10)

where ~

is the inverse of An,m, the matrix defined in (1.1 1) It satisfies A~ = (A’ )n+m where 2’ (t) = (A)~t°~’.In fact it will turn out that (2.9) is not the correct matrix to be used nor is (2.10) the correct final answer. But it is quite close. Let us try to use (2.10) in calculating the action ofa Virasoro group element on a vertex operator. In the process we will see both the facility of this approach as well as the ordering subtleties that necessitate a further refinement of the steps leading from (2.5) to (2.10). We want to calculate the operator product of (2.10) with a generalized vertex operator :exp (1 ~n~O q~Yb): where anx Y

0(z)

=

(n_l)!~~’0’~’~

(2.11)

Thus we have to evaluate the operator product (OP) fvk(t):exP(ifk(t)azx(z+t)dt)::exP(i~~nYn): (2.12)

xexp(~kn[Onm—2~]km).

Using (we assume that the vertex operator is located at z) ~ö~X(z

+ t)3nX(z))

=

(2.13)

we get for (2.12) fvk(t):exP(ifk(t)azx(z+t)dt)exP(i~~nYn):

xexP(~kn.~nn +ko.~o)exP(~kn[0nm_A~]km).

(2.14)

n>0

Singularities in t arise only when (2.14) is inserted in correlators with other vertex operators at z. As long as we refrain from doing that one can Taylor expand aX(z + t) in (positive) powers of t to get

374

B. Sathiapalan / Loop variables and the Virasoro group

f[dkn1:exP(i~knYfl)exP(i~~nYn): n>0

n?~0

x exp (~kn~nn + ko.~o)exp

~

kn[Onm

n>0

(2.15)

which becomes (on doing the k integration) :exp(~ ~

Yn(0A’)~nYm

~ ~

n,m>0 —

i

~

n ~0,m>0

nqn(0~)I~,_~mqm

n,m?~0

nq0(O

2.

_A~)i~,mYm)

: {Det[(0

ATh’1}~

In (2. 16) to save space we have assumed implicitly that nq~is q 0 when n = 0. If one defines the column vector

/

(2.16)

to be replaced by

y~’\

I~I(y~\ I

(2.17)

\~—inq~J \Y.~J with the understanding that n = 0 is counted as a negative index (i.e. there is no Y~o= X in (2.17)), then (2.15) can be written compactly as 72

(2.18)

:exp(~YT(0~A1)_1Y)exp(i~qflYfl) :[Det(l~AO)]~ n?’O We have dropped an overall factor of DetA’~’2.Using (O_A)~=_[l/(l_A0)]AE_PA, we finally get e~~:exp(i~qn~n n~~0

=:exp(_~YTPAY)exp(i~qflYfl) :[Det(l~AO)]~/2. n>0

(2.19)

This is a simple looking result. Furthermore it required very little effort using the loop variable formalism. Although as pointed out earlier it is not quite correct, the correct result is very similar to this and is also easy to derive and we will do this in the next section. (We need a more precise treatment of the normal ordering.) Let us first see why the present result cannot be correct. This becomes evident when one checks the group composition property by calculating e~’~’~’ e1’~~” x exp (j >.n ~ 0 q~Y~).This requires us to calculate the action of exp ~ ~ on

B. Sathiapalan / Loop variables and the Virasoro group

375

(2.19). This is not very difficult if one realizes that written in the form (2.15), (2.19) is just another generalized vertex operator ~ multiplied by some Y-independent factors. So we first calculate the action of exp (~ /20L~) on : ~ exp ~

: and then perform the integration over k. And of course for we can again use a first order representation: exP(~iinLn) =fvP(t):ex~(ifP(t)a~x(z

+t)dt):

x exp{~p0[(0 —~~)]nmprn}.

(2.20)

Acting on (2.15) gives

f

Vp(t) : exP(if~(t)3zX(z

+ t)dt):

x : exp(i~(kn + qn)Yn + qoYo) n >0 x exp(k.~q0n+ k0qo)

As before nq0 becomes q0 for n to get

=

f Vp(t) : exp(ifP(t)a2x(z

(2.21)

~Pn[(0JL~)]nmPrn}.

xexp{+~kn[(O—A’)]nmkm+

0. We calculate the OPE using (2.13) again

+ t) dt)

x exp(i~(k0 + qn)Yn + qoYo) n>0

x exp{p0. (k~+ qn)n + k_~ q0n +po.qo + k0

.

qo}

xexp{+~kn[(O—A’)]n~k~+ l/2p~[(O—~~)]~mpm}.

We can Taylor expand DX(z + t) as before to get (reinserting

f f [dk~]

d[p~]

f[

(2.22)

dk~]):

: exp{i~(p~+ k~+ q~)Y0+ n>0

x exp{pn. (k0 + q~)n+ k0

.

qnn + Po

.

q0 + k0qo}

xexp{-f-~kn[(0—A’)]nmkm+ 3pn[(O/~)Inrnpm}.

(2.23)

B. Sathiapalan / Loop variables and the Virasoro group

376

This can be written compactly if we define the column vectors P+n

~‘+n

p~

Y_,~

/ Y’\

—inq~

=1

=

\,Y)

~ k..~

Y~

I

(2.24)

—inqn

and the matrix N

=

(N+n~+rn N+n,_m~

(

=

0

O~

(2.25)

\fl~n,m 0)

\,,N_n,+m N_n,_m)

(2.23) becomes

f [dP]e1PTYexp[T((0~)

N1)P]exp(i~qflYfl)

[Note that since Y+o 0, we can also assume P+o p0k0 k~1.Let us define K

/(o-~-’) (

=

\

Then K~

/

I

=

0

(0— ~c’

0

(0-A’))

and M

\I

0

)~

0

\I

=

/I

0 and P—o 0 N\ 0)

T

~N

/I —r~ o \I

=

\,

(0—A~y~)

0

(2.26)

I

(2.27)

.

~—P.

(2.28)

_pA)

Performing the Gaussian integrals we get :exp(_~yT(K+ MY1Y)exp(i~qnYfl)

:Det[K +Ml_D!2.

(2.29)

n?’O

Using (K + M)~

=

(1 _~M)

~

=

P + PMP + PMPMP +

...

(2.30)

this becomes exp{_~YT(PP+ p’~+ PPNPA + pA NT P’1 + P’1 NP~NT ~0 +

pA NTP’1NPA +

..

‘)Y}exp

~

~n~n)

n~0

: Det[K +

(2.31)

If we set ~t = A in (2.31) we should just end up with p2A But one can check that pA as defined in (2.18) and (2.19) does not satisfy this. This shows that

B. Sathiapalan / Loop variables and the Virasoro group

377

this result is not quite correct. It turns out with some modification of 0 the final result (see (1.14)) has the same general form and is not much harder to derive. Except for this modification of the form of 0 (and therefore of ~2) the calculation proceeds exactly as above. The argument that leads to the correct expression for 0 can be motivated by looking at (2.31). Note that 0 in (2.18) is ~ (N + NT). From (2.31) we see that N and NTalways occur either as P’1 NP2 or as PANTPP i.e. they keep track of the order of P’1vis a vis P2. We can generalize this rule as follows. We imagine rewriting exp (>~ A 0L~) as

21

22

I n N —n’\) exp~ ( nN exp~

~

‘

.

( ,~N —n exp~

for some large (not to beexpect confused with N). in Inathe end we 2 = number = AN.NWe would based on the the matrix above that particular set A’ = A sequence of ANANANT A whether we should have an Nor NTis dependent on which factor of exp(A~L.. 0/N)the particular A (adjacent to that N or NT ) came from. Thus the rule is to have A’NA~if i j. Thus we have a counting problem: The coefficient of a term with a particular sequence of N’s and NT ‘s is equal to the number of ways we can arrange the A”s (in accordance T ‘s. with the above order particular of N’s and N Remember that rule) in theinend the to A’sget arethis all set equal tosequence each other. Implementing this rule modifies P2 defined in (2.19) to the one defined in (1.14) with a more elaborate 0 matrix. We elaborate on this in the next section and prove that (1.14) is the right result. ...

. .

3. Proof

We first outline the steps involved in the proof and then fill in the details: Step]. TolinearorderinAexpression (1.13), (1.14) forexp(~~2~L..~) are correct. Step 2. To bilinear order in A j.i, expression (2.31) for exp(~~A~L_~) x exp (~,1 ~u~L_~) is correct and the Virasoro algebra is satisfied.

B. Sathiapalan / Loop variables and the Virasoro group

378

Step 3. To trilinear order in A p, p the expression for exp ~ exp ~ p~L.~)exp (~ p~L~): exp (i ~In>O q~Y~) : is

A~L_~) x

+ P2 + P’1NP2 + PA NT ~/l + PPNP’1 + PPNP2 + P’1 NT PP + P2NT P’1 + P~NP’1NP2 + PPNPA NT~P + PA NT P~NP’1+ ~/I NTP’1 NP2

exp{_~YT(PP+

]3’1

+ PPNPA NT P’1 + pA NT P’1 NT ~P

(3.1)

xexp (i~qnyn~Det[K + M]_D12. \n~0 /

Here K

=

P~, where (p~ 0

0

OP’1O

(3.2)

0 OP2 and 0 M=

NT

NN

0 N

(3.3)

.

0 The rule for writing down the terms is (as mentioned in the last section) that between any two of PP, P’1, P1, we have an N if the sequence is part of p ~uA and NTif the order is interchanged. Step 4. To pth order, if we have NT NT

e~~” ~

eL~... e~L~ e2~_~ exp

~

~n~n) n~’0

the p-linear term in A’A2 A” consists of a sequence of A”s and N or NT with an N between A’ and A~in the form A’NA’ if i > I and NT (i.e. A!NTAJ ) if i < j. This gives the action of (A~L_~)(A~’Ln)... (A~L~)exp (i ~~> . . .

0q~Y~)

10).

Step 5. If we set A~ = A~ = = A~then step 4 gives us the pth order part of exp (~0A~L_0)exp (i>~>0qnYn)upto a factor of l/p!, i.e. it gives ...

(A~L_0)~exp (i~~>0qnYn). Step 6. Using steps 4 and 5 we get a prescription for writing down the pth order term of exp ~ A~L~)exp(j ~n~0 q~Yn) Take all possible orderings A’ 2 = = A” A” all placing N ‘s and NT ‘s as described in step 4. When we set A’ = A terms with a given sequence of N ‘s and NT ‘s become identical and add together. Thus the number of ways we can order the A’ ‘s to give that particular sequence (of N ‘s and NT ‘s ), divided by p!, is the coefficient of of that particular p th order term in ~ .

...

. . .

B. Sathiapalan / Loop variables and the Virasoro group

379

Step 7. The expression (1. 14) implements this rule to any order in A.

Let us fill in the details now: Step 1. + t)a~X(z+ t) :: exp

(~~

~flYfl)

n~0

+ t)O~X(z+ t)exp (i~~nYn) n~’0

(~(qnazx(z +

+ :

t)~)

+

q0a~x(z+

t)~)

n>0

exp

(~~

~n~n)

n~0

(3.4)

y.~n~nm~m 2 L..s n,m

tn+m+

gives the OPE with the energy momentum tensor. Using a~X(z+ t)

=

0~X+ t32X + t2O3X/2! +

...

+ t’~a”X/(n

—

1)! + (3.5)

we get a~X(z+ t)D~X(z+

t)

q~O~X(z + t)nt”1 Multiplying (3.4) by ~ApYnYp_n

f~dtA(t)

+ ~ApnqnYp+n

=

~nqn

=

n~0

(3.6)

ymtmfl2.

(3.7)

f~dtA~t~~’ we get for the RHS

+ ~A_~nq

0.(pn)q~~ n=1

Ym,

~tn+m_2Y~. nm

=

n=1

+ n~>P

(3.8) (as always nqn is replaced by q0 when 2n == A.0).Thus Sincethe ~ term = Am+n this term is exp(.~YTPAY)is exactly the exponent of (1.13) with P correct to linear order in A This concludes step 1. At this juncture we should point out one fact. At higher orders in A there are two kinds of terms. One of them .

is of the form YTANA AY. These are the “non-trivial” terms we are concerned about. Then there are terms of the form (yT 2 y )m /rn! that come from expanding the exponential in (1.13). This term comes from the repeated (rn-fold) action of A~L~on exp (i>~>0qnYn)directly (i.e. not from commutation of the La’s amongst themselves). In the language of the operator product expansion these are terms that do not involve contraction of X’s from different T~~’s with each other. This is a “trivial” higher order A -dependence (i.e. trivial in that it does . .

B. Sathiapalan / Loop variables and the Virasoro group

380

not require any effort to prove that these terms are present in the right way in (1.13)). Step 2. This step is obvious since the action of exp A~L_~) results in a generalized vertex operator just like the one we started with, as can be seen from (2.15). If this is correct to O(A ), and the subsequent action of exp (~n p0L_0)

(s,,

is correct to O( j.i) then the composition exp ~ 1i~L_~) exp ~ AnL_n) X exp (i ~ q0Y~)must be correct to O(j.tA). For completeness we present an explicit verification ofthe Virasoro algebra with central charge, in the appendix. This concludes step 2. Step 3. The end point of step 2 is (2.31). (2.31) is equivalent to (2.23) which is a generalized vertex operator. One can act on this operator by exp p~L~) in exactly the same way as in in the preceding two steps. If we repeat (with three parameters) the steps leading from (2.23) to (2.31) we end up with (3.1). The algebra is extremely straightforward so we will not repeat it here. By the same logic that was used in step 2, this procedure is guaranteed to givethe right answer to (non-trivial) order pjiA. This concludes step 3. Step 4. This is a p-parameter generalization of step 3 and there is nothing new here. This concludes step 4. Step 5 and step 6 are obvious. Step 7. Consider a typical term in (1.14):

(s,,

Idal ~

(3.9)

Pick a particular sequence of N and NT. The 0-functions enforce that a2 a2 > a3 etc. fdai

Ida2.

.IdamA(al)N0(al

—

>

a1,

a2)A(a2)

xNTO(a3_a2)A(a3)...0(ap_,_ap)A(ap).

(3.10)

The region 0 ~ a1, a2,~ a,, ~ 1 can be decomposed into p! regions each with a particular ordering of the a’s: ~

(3.11)

Thus (3.10) can be decomposed into p! integrals. Each of the p! terms can only give one of two possible values. The 0-functions either vanish in the entire range of one of these integrals or equal one in the entire range., for if you have 0(a, a1) the integration region either has a > a1 over the entire range, in which case 0 (a, — a~)= 1 or it has a, < a~over the entire range in which case 0(a, — a1) = 0. If any of the 0-functions are zero the integral vanishes. If all the —

B. Sathiapalan / Loop variables and the Virasoro group

381

0-functions are equal to one we get (note that A does not depend on a)

Idau

fdai2...fdai~(ANANTA...) = ~(ANANTA...) (3.12) Thus whenever the ordering ofthe a’s is such that it gives the particular sequence of N’s and NT ‘s we get a factor of l/p!. The full integral (3.10) thus tries all possible (p!) orderings of a, ‘sand it therefore counts the number of ways it can be done. This concludes step 7 and thus the proof of (1. 13). 4. Group composition We briefly discuss the group composition property and give an example of an

explicit calculation for concreteness. Equation (2.31) contains the basic composition rule: Write exp

(~ (~ /inL_n) exp

A~L_~) = exp

(~

p~L_~)e~’1’A).

refers to an overall normalization that depends on (2.31) and (4.1) we see that

PC

ji, A.

1 + P’1NP2 + P2NT P’1 + pP(P~A)= p~+ P PC = ~DTrln[K + M] — ~DTrlnP~.

This defines p implicitly since (1. 14) gives = fda, 1_~

(

(4.1)

Comparing (1. 1 3), ...,

(4.2)

Pa 2

(4.3)

\l —p010102

Note that if we set ~u = A we get

(4.4)

2A=2P2+PA(N+NT)PA+...

P

(4.4) is a non-linear equation for P2 It can be used to determine P2 recursively. If we set .

P2

=

A + aANA + bANTA + cANANA +

(4.5)

...

one can solve recursively for the coefficients a, b, c, .by substituting (4.5) into (4.4). One can check that the result reproduces (1.14). In fact one can prove that (1. 14) satisfies this equation. We will not do this here but the outline of the proof is as follows: Define A(a) : 0 < a < 2 so that . .

IA, 0
l
One then shows that (4.2) can be written in the form (4.3) except that the range of a is from 0 to 2. Doubling the range of a is equivalent to scaling A 2A. This —~

B. Sathiapalan / Loop variables and the Virasoro group

382

proves that (1.14) satisfies (4.4). This is an alternate proof of the correctness of (1.14). Finally one can also show that setting ji = —A gives P’2 = 0. To lowest order this is obvious since P’1 = ~i, however it is not at all obvious a priori that this is satisfied by the full expression (4.2). We have checked this but will not reproduce the argument here. We conclude this section with an example. 4.1. EXAMPLE

e(A2’~_2~_2’~2) e’1’°2’10)

(4.6)

The exact answer of course is (1. 13). We can check this order by order. Some of the non-zero elements of the matrix An,m are A,,,

=

A

=

0,2 = A2,0 = A,,3 = A3,_, = A4,2 = A_2,4 = A2, A0,_2 = A2,0 = A,,_3 = A_3,,A4,2 = A2,_4 = A2, 1

=

A + ~A(N + NT)A + 0(A3),

(4.8)

P

YTAY

=

(Y~,—ink~) (A+n+m A+n,m~

\, A_n,+m

(4.7)

A_n,m)

(

(4.9)

Ym

\ —imkm)

In this example only k 0 is non-zero. This simplifies (4.9) considerably: All the —m indices in the matrix can be set to zero. We remind the reader of our convention that 0 belongs to the negative index set and + m is assumed to be a positive integer. This fact, along with (4.7), quickly reduces (4.9) to the equation: TAY= —A —~Y 2(Y, Y, + 2Y2(—iko)) (4.10) This is just L_2 eu/~oX 0). At the next order in A we have to calculate YT(IA(N

=

+ NT)A)Y

(A+n,+m A+n,m~ (0

m

\~A_n,+m A_n,_m) ~fl

0

(Y~ ik0) I

I

—

~

~

A+m,_p~

(

~ Am,+p A_m,_p) ~

(4.11)

Y~

_jk0)

As before the —p index is forced to be 0, as is the —n index. Once again using (4.7) we get finally 2 IYTA(N + NT)AY + l(YTAY) = —~A~(2Y,.Y 3 4ik0.Y4) + ~A2A_24k~ + ~ —

2, +

2Y2(—iko))

B. Sathiapalan / Loop variables and the Virasoro group

383

where we have added the ‘trivial’ second order contribution to (4.11). One also has to add a contribution from the determinant at this order. Det[ö0102 =

—

A0 00,02 1—D/2

exp[—~~D Trln( 1

=

—

A0)

I

exp{~D[TrA0+ (l/2)Tr(AOAO)...]}

(4.13)

The trace includes one over the a-index as well as the n-index. TrAO= Trf daifda2[AN0(ai—a2) +ANT0(a2_a,)1~(ai_a2)

=

0

(4.14) since TrAN

=

0. At the next order we have

Trfdaifda2A[N0(a,_a2)

a,) + NTO(a, a2)] TO(a 2trf da, da2ANAN 1 a2)

x A[N0(a2 =

+ NTO(a2_a,)]

—

f

—

—

T =

~2TrANAN

=

Thus the contribution of the determinant is ~DA

(4.15)

2A~.2.

Since L2L2 eu/c0~v 0) (4L0 + D/2) eikox 0)

=

[L2, L_2]

=

(2kg + D/2)

e1k0X

0)

e1k0X

0),

(4.16)

we see that (4.12 ) and (4.15) do indeed give the right answer. 5. Conclusions Let us summarize the results. We have given an expression in closed form for the action of an element of the Virasoro group on generalized vertex operators exp (i ~n>O k~Y~)in the form exp AnL_n) exp (i ~n>O k0Y~)10). The result is given in eq. (1 13). The main idea was to use a first order formalism for the energy momentum tensor which recasts the group element in the form of a loop variable exp(if~k(t)i)~X(z+ t)dt). This makes the calculation very easy. The crucial issue that has to be addressed is that of regularizing the 1oop variable or equivalently taking into account contractions of the X field inside the 1oop

(s,,

.

variable. We sidestepped this question by building up a finite group element as a product of infinitesimal group elements, each represented as a loop variable

for which we do not have self contractions (since we only keep the linear piece of each infinitesimal element) We were able to derive the expression (1.13) in this manner.

384

B. Sathiapalan / Loop variables and the Virasoro group

Fig. 2. The 1oop is thickened to an annulus. Concentric circles in the annulus are parameterized by the variable a (0 ~ a ~ 1) as shown.

The naive approach described in section 2 does not give the (right) answer (1.13) although it gives something very similar looking. It would give some insight into the formalism of loop variables if we could identify a procedure that modifies the naive approach to take into account the self interactions of the X fields in a loop variable in a self consistent way and leads to (1.13) directly. It turns out that this is not very difficult. The basic idea is to thicken the loop into a band or an annulus as shown in fig. 2. Then we repeat the method of section 2 but now applied to a superposition of all the loops making a band, i.e. to variables of the form fda f dtk(t, a)öX(z + t). a is assumed to vary from 0 to 1 and parametrizes the different loops in the annulus. Thus we let a = 0 correspond to the inner boundary of the annulus and a = 1 correspond to the outer boundary. Then (2.5) is modified to exP(ifdafk(t~a)OzX(z

+ t,a)dt)

=:exP(ifdafk(t~a)azx(z+t~a)dt):

x exp(f dtf x (aX(z

dt’f dada’k(t,a)k(t’,a’)

+ t)0X(Z +

t’))).

(5.1)

For a > a’ we take t to be the outer loop and t’ to be the inner loop and vice versa when a < a’. The inner loop can be shrunk to a point since there are no

B. Sathiapalan / Loop variables and the Virasoro group

singularities and we can expand DX (z +

385

in a Taylor series with only positive

t)

powers of t. The result is exP(ifdaJk(t~a)DzX(z

+

t,a)dt)

=:exP(ifdafk(t~a)DzX(Z +

t,a)dt):

exp(f da fda’ ~(k~(a)k~(a’)[n0(a

—

a’)]

n~0

+ kn(a)kn(a’)EnO(a’_a)I)).

(5.2)

This can be written as exp(f daf da’kn(a)km(a’)[NO(a

—

a’) + NT0(a~_a)]nm).

(5.3)

The expression in square brackets is 0 of eq. (1 12). If we include the wave function as before, the complete first order form for the group element is .

f

Vk(t,a) : exP(if daf k(t,a)D~X(z +

t,a)dt):

xexp(fdafda’kn(a)km(a’)[~aa2_daia2A(a1)]nm).

(5.4)

Taking the OP ofthis loop (or ‘band’) variable with a generalized vertex operator exp (i >~n>~ q~Y,,) and doingthe k-integration gives the result (1.13). Intuitively the group element exp (>,, A~L..0)is being written as a product of infinitesimal elements: IAnL_n\ IAnL_n\ fA~L_0\ exp~ N ~exp~ N )...exP~ N )~ ~ (5.5) N

factors

Each loop in the annulus represents one of these factors. The contractions between the X-fields of different loops represent the nontrivial commutation between the L0’s coming from different factors. We can also compare the result obtained here with that in ref. [17]. The result obtained there was that the action of a sequence of La’S on a vertex operator exp (i ~In>O k0 Y0) can be obtained by starting with exp(_~Yq1l0Y+ik1l0qY_~k1l0ek)Det(1_eq)~I2,

(5.6)

B. Sathiapalan / Loop variables and the Virasoro group

386

where k0,qn~and °nm were variables (defined in ref. [17]) with simple transformation properties under the Virasoro algebra. Comparing (5.6) with (1.13) and (2.16) shows that they are very similar looking. It must be true then that qnm, °nm are variables that parametrize the group so that they should correspond more or less to A+n,+m, A_n,...m and A+n,m. While we know the transformation properties of q, 0 we do not know how they parametrize the group. On the other hand in the case of the A variables of this paper, we know how they parametrize the group but we have not worked out the transformation properties. Thus the results obtained here complement those of ref. [171. To make a precise comparison one would have to work out the transformations of the A ‘s and compare with those of the q, 0 variables. This is in progress. Finally, on a more speculative level we might hope that some of the techniques used here might be useful for string interactions. The action of a group element on a vertex operator is related to the kinetic term in string field theory. Thus if we can write the kinetic term as an OP of two loop variables as done here we might hope for a more unified treatment of the kinetic and interaction terms in string field theory. I would like to thank M. Gunaydin for a useful discussion.

Appendix A

We show that to linear order in A and ~i (2.31) is consistent with the Virasoro algebra. We reproduce (2.31) for convenience: ex~(i~ ~nYn) Det[l _pO]_DI2

eXP(_~YTPPY)

n~0 =

exp (~A~L_0)exp (i~~nYn)

exp ~

2 + P” NP2 + PINT p~+ P’1 NP2 NT pp

exp

(—

I yT

(P’1 + P

+ PA NT P’1NP2 +

.

.)Y) exp

~

: Det[K +

~

\n~’o

/

(A.l) We need to determine p (pt, A) from P’2 as defined above and then calculate the commutator p~u,A) p(A,u). To second order in p —

PP=p+~p(N+NT)p+...

(A.2)

B. Sathiapalan / Loop variables and the Virasoro group

387

In solving for p to O(pA) we need to keep terms upto second order in P’2 since = p+A+ .~.Theansweris (A.3) p~P’2 PP(N+NT)PP+O(PP3)+... Substituting for P” from (2.31) we get p(p,A)

=

P’1 + P2 + ~ [P’1NP2 + pA NT Pp PP NTP2 P2NP’2 PP(N + NT)PP P1(N + NT)P2], -

-

-

-

(A.4)

which gives p(p,A)

—

p(A,p)

= =

P’1NP2 + P2NT P’1 PP NT pA pNA + ANT/A JLNTA ANp —

—

—

P1NP’1

(A.5)

—

to O(pA ). Let us choose Aa and Pb as the two non-vanishing parameters. This will give US [La,Lbl. From the definition of An,m we have Aa_n,n

=

An,a-~n= Aa/lb_n,n

/An,b—n

=

=

Vn.

Pb,

(A.6)

We then get (AN/1)nq

=

AnmLm)d_m,p/Ap,q

~

m~0 =

~An,an(a n~a

=

n)pfl_a,bfl+a (A.7)

~Aapb(an)~q,b+an, n?~a

=

(ANT/.L)nq

—

n)5q,b+an,

(A.8)

Aa/2,~,(Cl— fl)ôq,b+an,

(A.9)

~Aapb(a

—

n~a (ANT/A

—

ANp)nq

(IANTAPNA)nq

=

~

=

~pbAa(ab)~q,b+an,

(A.l0)

which gives (A.ll)

(P)nq _~Aapb(ab)óq,b+an.

Using the definition .on+m = Pn,m we get Pa+b = Aal.tb (a b). This confirms the Virasoro algebra except for the central charge, to which we now turn. The central charge is ~DTrln(K + M) (p A) which, to this order is —

—

~DTr(P’1NP2NT)

‘~-~

—

(p

~

A),

(A.12)

388

B. Sathiapalan / Loop variables and the Virasoro group

with

Tr(PPNP2NT)

=

mpp_p_mA+m+p,

~

m,p> 0

which gives for (A. 12) ~ (li_p_mA +~n+p_A_m_pp+m+p)rnp. m,p>0

(A.l3)

If we let Aa; a> 1 and /J—a be the two non-vanishing parameters, this becomes: m(a—m)p_aAa

~

=

~(a3—a)Dp_aAa

(A.l4)

m=1

which is the central charge. Thus eq. (2.31) is consistent with the Virasoro algebra with central extension. References [1] V.F. Lazutkin and T.F. Pankratova, Funkt. Anal. Jego. Pritzh 9 (1975) [2] G. Segal, Commun. Math. Phys. 80 (1981) 307 [3] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, J. Stat. Phys. 34 (1984) 763, NucI. Phys. B241 (1984) 333 [41 J.-L. Gervais and A. Neveu, Nucl. Phys. B209 (1982) [5] D. Friedan, Z. Qiu, and S. Shenker, Phys. Rev. Lett. 52 (1984) 1575 [6] C.B. Thorn, NucI. Phys. B248 (1984) 551 [7] A. Rocha-Caridi, in: Vertex operators in mathematics and physics, ed. J. Lepowsky et al. (Springer, Berlin, 1984) [8] V.G. Kac, Lecture Notes in Physics 94 (1979) 441 [91 B.L. Feigin and D.B. Fuchs, Funkt. Anal and Appl. 16 (1982) 114 [10] E. Witten, Commun. Math. Phys. 114 (1988) 1 [11] A. Alekseev and S. Shatashvili, NucI. Phys. B323 (1989) 719 [12] B. Zumino in M. Levy et al. , eds., Particle Physics Cargese (1987) (Plenum, New York, 1988) [13] M.J. Bowick and S.G. Rajeev, Nucl. Phys. B293 (1987) 348 [14] V. Aldaya, J. Navarro-Salas and M. Navarro, Phys. Lett. B260 (1991) 311 [15] W. Taylor IV, UCB-PTH-920/09 (U.C. Berkeley Preprint) [16] B. Sathiapalan, Nuci. Phys. B326 (1989) 376 [171 B. Sathiapalan, NucI. Phys. B376 (1992) 387