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An anomalous transformation in string field theory
Nuclear Physics B303 (1988) 305-328 North-Holland, Amsterdam
A N A N O M A L O U S T R A N S F O R M A T I O N IN S T R I N G FIELD THEORY* J i . MAlqES*
Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08544, USA Received 16 November 1987
Conformal field theory techniques are used to analyze a recently proposed transformation of Witten's interaction vertex from center-of-mass to mid-point variables. As presented in the literature the transformed vertex has the surprising property of being non-local in the mid-point coordinates unless an additional non-unitary transformation is performed. This renders the kinetic term non-local and poses additional problems. Here we show that, besides being non-local, the transformed vertex does not satisfy the overlap equations and fails to be BRST invariant. These bad properties are the result of an associativity anomaly of the type considered by Horowitz and Strominger. However, this anomaly is "almost trivial" in the sense that most of its bad effects can be avoided by choosing a proper regularization.
1. Introduction T h e string field in covariant string field theory [1-6] is a f u n c t i o n a l A[X~'(o)] of the c o o r d i n a t e s of the string.* I n the usual f o r m u l a t i o n the configuration space of the string is parametrized by the center-of-mass position q" a n d the n o r m a l mode c o o r d i n a t e s Xn~. This suggests a representation of the string field in the form of f i r s t - q u a n t i z e d creation operators acting on a suitable vacuum, with tensor fields as coefficients a[X"(o)]
=
+
1+
+
y_l
+ .-.
(1.1) I n this expression the space-time a r g u m e n t x is identified with q. This allows a n i n t e r a c t i n g string field theory to be written as an o r d i n a r y field theory with i n f i n i t e l y m a n y c o m p o n e n t fields in interaction. * Research supported in part by the National Science Foundation under grant No. PHY80-19754. t On leave of absence from Departamento de Fisica, Universidad del Pals Vasco, Bilbao, Spain. * We neglect the role of the ghosts until sect. 3. 0550-3213/88/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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q3
0
0
x
"rr
rrllO Fig. 1. Witten's interaction (schematic). The point o = ~2~ris common to the three strings, unlike their centers of mass.
None of the proposed string interactions [4-6] are local in the center-of-mass coordinates. In particular, this is true for Witten's cubic vertex [4], schematically represented in fig. 1. As a consequence, the interaction among the component fields in eq. (1.1) will contain derivatives of arbitrarily high order. This might be avoided in Witten's theory by identifying the space-time argument in eq. (1.1) with the mid-point of the string X(½~r). This is possible because the point which is common to the three strings also carries a common label (o = {~r). This possibility is not open to any of the other theories. There are several cases in which such a "local" formulation of the theory is desirable. When one considers a hamiltonian treatment [6, 7] of Witten's theory, it seems that the "time" coordinate should appear locally in the action. S=
f( A * Q A + ~ A2* A,A).
(1.2)
This appears to be true for the choice t = X°(½~r), but not for t = q0 (see however ref. [8]). Another example is provided by the purely cubic action of Horowitz, Lykken, Rohm and Strominger [9] s =
fA • A • A
(1.3)
Witten's action (1.2) follows from eq. (1.3) by expanding about a particular solution of the classical equation of motion A * A = 0. This particular solution is an open string field description of a closed string field background. Eq. (1.3), conveniently elaborated [10], can be used to incorporate closed strings, although this part of the theory is still far from clear. The important point is that this picture only makes sense if the action (1.3) is truly independent of the closed string background. There are two ways in which a (gravitational) background dependence could enter eq. (1.3). The first is through the integration measure. This carl be avoided by
J.L. Mahes / Stringfield theory
307
defining the string field A to be a density of weight 1 [11]. The second way is through the connection necessary to covariantize the derivatives present in the interaction vertex. As mentioned before, these derivatives appear because the interaction is not local in x = q. One would hope that by taking q = X(½~r) the interaction would become derivative-free, thus avoiding the second source of gravitational background dependence. This was the motivation behind the work in ref. [11], to which we refer the reader for a complete exposition. Here we present a brief description of the transformation proposed there, which will be analyzed in detail in sects. 2 and 3. In the operator formalism of Witten's theory [12,13], the interaction vertex is represented by a state IV) in the tensor product of three single-string Fock spaces, so that [. tA * A * A = (A 1](Azl (A 3I I V).
(1.4)
Here {A I is the bra conjugate to the ket in eq. (1.1). In momentum representation the state IV) can be written as the exponential of a quadratic form in the oscillators a - n acting on the three-strings vacuum I V)
=
exp
r 1 ~Trs
s ,s r s ~ TlVrnnOtr mOt_n-lNdnp ot n q- 51 N ~r s p rp s }lI2),
m, n = 1 , 2 , . . . , o e , r, s = 1,2, 3.
(1.5)
The vacuum 1£2) satisfies q11~2) =q21$2 ) =q31$2 ),
(1.6a)
3
Y'~ pr]~2) = 0,
(1.6b)
n=l
and space-time indices have been suppressed in (1.5),(1.6). The vertex (1.5) is obviously non-local in the c.m. variables. Ref. [11] proposes a change from the conventional parametrization
x(o) =q+ d
x°cosn
(1.7)
n=l
to "mid-point variables" X ( o ) = m + v~ ~ ~ . [ c o s n o - ~o.1, n=l
(1.8)
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J.L.
Ma~es
/
String field theory
where w, - cos(~n ~r). Eqs. (1.7) and (1.8) are related by the following change of variables
m=q+ 7'2 ~ ~%x,= X(½~'), n=l
~,=x,.
(1.9)
The unitary operator associated with this transformation is
U=exp(ip(q-X(½~r))).
(1.10)
This operator can be used to transform the string field and the interaction vertex to the new variables <-4'1 =
(1.11a)
3
I V') = I-[ Url V ) .
(1.11b)
r=l
The operator U needs regularization due to an infinite normal ordering constant. This is achieved by redefining % %=e
I'l%os(½n~r) .
(1.12)
The transformed vertex is given by eq. (1.5) with "primed" Neumann coefficients. The following result is obtained [11] once the limit e - , 0 is taken rs N 't,rs _- N,~,,
(1.13a)
N6',~'= - ~-23~,
(1.13b)
n
trs _
l_~]rs ~
N0~ - 2"00
1
rs
5 N3 ,
(1.13c)
where N = - ~ In 3 + 2 In 2. This results is surprising. The transformation U takes us from the set of variables (q, x , ) to (X(½~r),x,). By the previous discussion, we would expect the transformed vertex IV') to be local in X(½~r), and instead we see an exponential dependence on p and p2. The fact that the momentum dependence in the transformed vertex is diagonal in r and s is used in ref. [11] to define a single-string (non-unitary)
J.L. Ma~es / Stringfield theory
309
similarity transformation S = exp( - ¼Np 2 +
--pa_n n=l
/'/
)
,
(1.14)
which removes all momentum dependence from*
I V'') - S U I V ) .
(1.15)
The new vertex IV") is thus completely independent of the background metric, and the goals of ref. [11] are fulfilled. This, however, is not a good solution from the point of view of a hamiltonian formulation of Witten's theory, since the transformation S (being non-unitary) shifts the non-locality from the interaction vertex to the kinetic term in (1.2). (This fact was already pointed out in ref. [11]). This would make it very difficult, if not impossible, to formulate a hamiltonian treatment of Witten's theory. But even from the point of view of the cubic action, the meaning of the non-unitary transformation S and how to deal with it (do we need additional ghosts?) was not completely clear. We consider the situation sufficiently puzzling to justify a closer look at the problem. This is facilitated by the use of the powerful techniques of conformal field theory presented in refs. [14,15], instead of oscillator methods. We show in sect. 2 that there is a natural way of regulating the transformation U in terms of a point splitting between positive and negative frequency modes. The computation of the vertex is thus reduced to a trivial evaluation of Neumann functions at some given values of their arguments. The resulting vertex differs from the one previously obtained [11], but is still non-local. This motivates the study of the properties of IV'), which is carried out in sect. 3. The fact that the conformal field theory methods used here permit an easy calculation of higher order corrections in e is crucial at this point. We find that the transformed vertex does not satisfy the (transformed) overlap equations and is not annihilated by the BRST charge Q'. These properties are explained in sect. 4 as the result of an associativity anomaly of the type recently considered by Horowitz and Strominger [16]. We also show that, by an appropriate choice of the regularization, it is possible to avoid most of the bad effects of the anomaly and construct a vertex with the correct properties. The anomaly has, however, some residual manifestations which are discussed in detail. A summary of the results of this work is given in sect. 5.
2. The transformed vertex
In this section we fix the notation, introduce a special type of point splitting regularization and use it to compute the vertex IV'). We find a result which is * U acting o n the vertex will always stand for I~3,=1Ur. This is also true for S.
J.L. Ma~es / Stringfield theory
310
different from the one in the original reference, but still non-local, and explain the discrepancy. The position operator
O~n
X(o,r)=q+pr+i
.
Y'~ - - e - ' " ' c o s n a n4:0
(2.1)
/'/
will be written as
x ( o , T) =
X(z) +
(2.2)
where
X ( z ) = q + iplnz + i Y'. Oln,7,n.
(2.3)
n~O tl Eqs. (2.1) and (2.2) are related by a Wick rotation T ~ - i T and a change of variables z = e x p ( - T - io). Note that with this convention, world-sheet time flows toward the center z = 0. The bosonic part of the interaction vertex
IV) = VxVgh]I2)
(2.4)
is given by
Vx
expf~ _
1 t'~
_)_ ( ~ _ _dw O zdX r ( z ) N rzS ( z , w ) O w X S ( w )
7~c 2qri yc 21ri
(2.5)
The Neumann function N~'(z, w) has been taken from ref. [14] and adapted to our conventions. An explicit expression in terms of ~( + z) can be found in appendix A. The function if(z) is given by ~ ( z ) = (1
+
iz) 1/3
(2.6)
and the cuts in ¢0(+z) are chosen from + i to +iac (see fig. 2) so that N~'(z, w) is analytic for [z[ < 1, {w[ < 1. The contour C will always run counterclockwise around the origin, inside the unit circle. If we compare eqs. (2.5) and (1.4) we immediately find the relation
NrS(z, w) =
~
N~r~,z"w".
(2.7)
rt, m = O
In what follows it will be necessary to separate the zero modes from the creation and annihilation parts in X(o, T),
X(o, T) = q +pT + Xc(o, T) + Xa(O , r)
(2.8)
J.L. Mahes / StringfieM theory
311
L
Fig. 2. The N e u m a n n functions are analytic inside the unit circle.
and correspondingly
X(z) = q + ip In z + Xc(Z ) + X . ( z ) ,
(2.9)
where [14] dw
X~(z) = ~c~--~8wX(w)ln(1 - w/z),
Iwl < Izl,
dw Xa(Z) = -- ~ c ~ i OwX(w)ln(1 - z / w ) ,
Izl < Iwl.
(2.10a)
(2.10b)
The exact vacuum expectation value
(X,(z)Xc(w)) = -ln(1 - z/w)
(2.11)
implies the OPE
X(z)X(w) - -in(w-z).
(2.12)
We are now ready to study the operator U, which in our notation reads
U = e x p ( i p ( q - X(½~r, 0))) -- e x p ( - i p ( X c ( l ~ r , 0 ) + X,(½~r,O))) = exp(-- ½p2(Xa(lw,O)Xc(½rr,O)))exp(-ipXc(½"r;,O))exp(-ipX,(½~,O)).
(2.13) The normal ordering constant in eq. (2.13) is infinite, and some kind of regularization is required. We cannot simply drop the infinite constant in the usual way, since
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Mahes / String field theory
that would make U non-unitary. We introduce a point splitting between the positive and negative frequency components of X, so that 1 Xa(½~r,O)--* X~(½~r,e)= lX~(ie-*) + ~X~(-ie-~),
(2.14a)
Xc(½w,O) --* X~(½rr, - e ) = ½X~(ie ~) + ½Xc(-ie*).
(2.14b)
We can use eq. (2.11) to determine the value of the normal ordering constant. This yields the following regulated expression for U
U~=exP(¼p21n(1-e-4~))exp(-ipX~(11r, e))exp(-ipXa(½~,-e))
(2.15)
The equivalence between this point splitting regularization and the one based on oscillator methods used in ref. [11] follows from the identity ½((ie-~)l"l +
60. = e - l n l e c o s ( ½ n T r ) =
(--ie-~)lnl)
.
(2.16)
The last ingredient necessary for the computation of I V') is the transformation of
OzX(z ) by Ue This is easily done with the help of eqs. (2.10) and (2.12). The result is ip U~O~X(z)U] = O~X(z) - -- + ipF,(z), Z
e ' < [z[ < e ~,
(2.17)
where
F~(z) -~
Z g 2 + e -2*
Z z 2 + e 2e •
(2.18)
This expression is only valid inside an annulus of radii e -* and e ~, as a consequence of the restrictions in eqs. (2.10) (see fig. 3). This implies that the limit e ~ 0 should always be taken after the contour integrals have been performed, since there is no value of z for which eq. (2.17) is valid at e = 0. Writing
IV,') = Uyi$2 ) = ( UyU] )U~[~)
(2.19)
and using eqs. (2.15), (2.10) and (2.17) one finds
N'r'(z,w)=Nr'(z,w)-Nr'(O,w)-NrS(z,O)+N~'(O,O)
(2.20a)
+1~1v .... tte/"-~,W)+!~r'('21" ~-,ie-~)+(i~-i)
(2.20b)
+¼Nr~(ie -*,ie -*) + ¼Nrs(ie - * , - i e -*) + ( i ~ - i )
(2.20c)
+ {ln(1 + zZe-2~)3~s+ ½1n(1 + w2e-2")3 r"
(2.20d)
+ ½ 1 n ( 1 -- e - 4 * ) ~
(2.20e)
rs .
J.L. Ma~es / Stringfield theory
_
_
313
N
-ie E
T
Fig. 3. U~O~X(z)U~t, as given by eq. (2.17), is defined only for e ~ < Izl < eq
The terms in the first three lines come from the poles z = 0, + ie -~ in eq. (2.18). The terms in eqs. (2.20d) and (2.20e) arise from the creation part X c (1u r , - e ) and the normal ordering constant, respectively. Eqs. (2.20) reduce the computation of IV') to a simple evaluation of functions. From the expression for N~'(z, w) in appendix A and momentum conservation (1.6b) one obtains
1 N ~ ' ( i e - ~ , w ) + ~ 1N
~, ( - i e
-~,
w) = -½1n(1 +
w 2 ) a r ' + o ( e 2/3)
(2.21)
and we see that (2.20d) cancels against (2.20b). We also have 1 ms (le • - - e , i e - ~ ) + x 1N r~(re • ~, - - i e - ~ ) + (i ~ - i ) = - - (~ln3 + ½1n ~ ) ~ r s q _ ~N
O(e2/3) (2.22)
and l l n ( 1 -- e - 4 ~ ) ~ r s = (½1n e + l n 2 ) ~ rs 4- O ( ~ ) .
(2.23)
Combining (2.22) and (2.23) we obtain a finite value at e = 0 ( - ¼1n3 + ln2)8 rs= ½N8 r ' = 1Nr'(0,0)
(2.24)
and the result is
N'~S(z, w) =- ]imN'~'(~ ,z, w) = Nr'(z, w) - Nrs(o, w) - NrS(z,O) + 3Nrs(o,o). ~0
(2.25) The Neumann coefficients N' ' s can be read from this equation with the help of (2.7). Since N'~S(z,O) = N'~S(0, w) = 0, (2.26) U'~S(0,0) = l~vrs/n 0),
(2.27)
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J.L. Mafies / Stringfield theory
we have _
r$
N(~,~~ = O, N~rS=lMrs= ~ ,~ ~1N6rS
(2.28)
.
Unlike ref. [11], we get zero for No'i ' (see eq. (1.13b)). This discrepancy can be traced to the evaluation of eq. (3.29) of ref. [11], where the regulator was erroneously removed before the contour integrals were performed. That left out several contributions which, once included, reproduce our result. Eq. (2.27) shows that the transformed vertex is still non-local in the mid-point coordinates. Since, as explained in the introduction, this poses some puzzling problems, we must check the validity of our result. This is done in the next section, where the properties of the transformed vertex are studied.
3. P r o p e r t i e s o f the t r a n s f o r m e d v e r t e x
3.1. THE OVERLAP EQUATIONS The overlap conditions on the vertex IV} are [12]
[XS(o,O)-X'+l(~-o,O)]lV}
=0,
~-~ < o -.<<,~.
(3.1)
When trying to prove this equality one considers the following expression [14]
[XS(y)+X'(f)-X'+l(-1/y)-X'+X(-1/Y)]lV},
Rey
(3.2)
which for y on the unit circle is obviously equivalent to the 1.h.s. of eq. (3.1) (see fig. 4). In order to evaluate eq. (3.2) we use _
dz
r
[XS(u),V] = V ~ ) c ~ i O z X ( z ) N r ' ( z , w ) .
(3.3)
Since NrS(z, w) is unambiguously defined only for Iu] < 1, we take lY[ < 1 and analytically continue to u = - 1/y. The following analytic continuation of ~ is valid for lyl < l , R e y < 0
q ~ ( - 1 / y ) = e~/2y-1/3,( y ) , q~(l/y) = ei~/6y-X/3dp(y), where the cut in y-1/3 goes from 0 to - i o c along the imaginary axis.
(3.4)
J.L. Ma~es / String field theory
Fig. 4. A s [y [ ~ 0, the p o i n t s y and -
315
1/y correspond to (o, 0) and (~r - o, 0) in the overlap equations.
Eqs. (3.4) combined with (A.1) give Nil(z,
--
l/y)
N12(z, - l / y ) NI3(z,
--
= N13(z,
y) -
ln(z + l / y ) + In y 2/3,
= N n ( z , y) + ln(y - z) + In y-2/3,
l / y ) = N12(z, y) + In y-2/3.
(3.5)
Then it is easy to show that (3.2) vanishes on the unit circle lim [ X ' ( y ) + X'(fi) lyl~l
x~+l(-1/y)
-
X'+I(-1/~)]]
V) = 0,
Rey<0,
]y[ < 1 .
(3.6)
(A complete account of this method can be found in ref. [14].) In our case we have to evaluate eq. (3.2) with IV) and X replaced by IV') and X' respectively. X' is given by
X[(y) =--U~X(y)U~* = X ( y ) - ½[ Xa(ie- Q + Xc(ie ~) + (i ~ - i ) ] -iplnY+½ipln
(y2+e-2") y2+e2 ~
+ipe.
(3.7)
This expression, like eq. (2.17) is valid only for e -~ < [y[ < e ". After some algebra one finds lim lim [ X ' ~ ( y ) + X~"(.P) - X ' s + l ( - 1 / y ) e~0
-X~'S+I(-1/~)]IV ')
]y[ ~ 1
= iN( p" - / + 1 ) 1 V ' ) .
(3.8)
J.L. Mahes / String field theory
316
Note that the limits have to be taken in the order prescribed. Eq. (3.8) can be rewritten as [X'S(o,0) - X'S+l(rr- o,0)]l V') = liN(pS-pS+l)lV' ) ,
(3.9)
where N was defined in (2.24). We conclude that the transformed vertex does not satisfy the overlap equations. The significance of this fact will be discussed in sect. 4.
3.2. BRST I N V A R I A N C E
The total BRST charge Q is given by
Q=
c'(y)Ts(y) + ghost terms,
(3.10)
where the bosonic energy-momentum tensor is
T(Y)=-I:OyX(y)OyX(y) :
1
y2,
(3.11)
The charge Q annihilates the vertex IV)
OVxVghl~2) = 0
(3.12)
t , . dz ,. d w / z ~ ,. ) Vgh=expl~c-~i~)c-~i[:)b (z)O~N~(z,w)c'(w ) .
(3.13)
with Vgh given by [14]
The Neumann function for the ghosts can be found in appendix A. The vacuum 1~2) is defined so that c~lI2) = O,
Vn >~O,
b,l~2) = 0,
Vn > 0.
(3.14)
We can go from eq. (3.13) to the oscillator expression [12]
Vgh= exp{ br ,,,N,~nc!. }
(3.15)
by using the mode expansions c(z)=
~ c,z "+1, --~
b(z)=
~ b,z "-2 --OQ
(3.16)
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317
and we may write
N~*(z, w) = ~ ,vm.z~r"row",
(3.17)
rn, n ~ O
although the actual values of No~'r"and N~ 0-rs are of course irrelevant. F r o m the mode expansions (3.16) we get
(c(z)b(w)) -
z 1
(3.18)
WW--Z
We shall also need the creation part of c(z), which according to eqs. (3.14) and (3.16) is Cc(Z) =
CnZ n + l =
fcdw (-~) c(w)
Iwl < [z[.
,
~i
n=l
(3.19)
Z--W
Eqs. (3.13)-(3.19) contain all the information we need about the ghost sector. We now turn our attention to the transformed BRST charge Q~'. This is given by eq. (3.10) with T ' ( y ) instead of In order to write an expression for Tj we remember that
T(y).
U,OyX(y)U,t= ( OyX(y)- ~ ) + ipF,(Y),
e ~ < [y[ < e ~,
(3.20)
where only the last term depends on p. Then T ' is given by
-
-
~
o y x ( y )
-
-
/p
OyX(y) -
1 F ~ ( y ) + ~p
2F21yX e t )
e - ~ < [Yl
y2'
(3.21)
We are now ready to compute the action of Q~' on IV'). The amount of work can be minimized by realizing that the transformation U does not affect any contribution which is independent of p. Indeed we see from eq. (3.20) that only the zero mode dependence has been affected by U, and eq. (2.28) tells us the same about This observation, together with the BRST invariance of IB) allows us to write
IV').
Q'~lV')=
c'(y) -ip ~ 3~X~(y)---f - F~(y) +
~p p F~. (y
Vx V~d~), (3.22)
where there is a single sum over the repeated index s. After all the annihilation operators in this last expression have been commuted through the vertex V' - V(Vgh
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J.L. Mafies / Stringfield theory
we get /,
dy
^
^
(3.23)
Q'V' $2) = V'q) Z--TG~(y)BS(y)ll2), .'c zqrl
where dw
d,= ~ c 2 - ~
lyt/ sr,
B~= -ip'F~(y)
Ow ghty, w ) + y---w
1
cr(w)'
Iw[ < lYl,
~iOzXl(z) O y [ N t S ( z , y ) - N " ( O ,
+ 71 pSpSlff'2 *~ (, Y ) ,
e- ~ < lYl < 1,
y)] +6'"
(3.24)
y--z
Izl < lYl
(3.25)
This last expression is not summed over the index s. Eq. (3.23) can be evaluated by using Cauchy's theorem. The result after integrating over y can be written as dw P 1 s s t(~ Q'~V'[~2) = 2p p V "Z---7.c r (w)H;r(w)[~2) C Z.~l
s , r dz e dw - i p V (fi - - ( f i - - O z X ' ( z ) c r ( w ) L S ' r ( z , w)l12), 7"c 2 qri T"¢ 2 ~ri
(3.26)
where LJr(z, w) and HSr(w) are momentum-independent functions which are computed in appendix B. There we show that L J ~ becomes independent of the index s for e ~ 0, and by momentum conservation lim p'Lslr(z, w ) = 0.
(3.27)
e--~0
H s~ is given by
H: ~= -
4-wl(2)-2/3[(e2)rsep'+(w) -
ieS~op;(w)] +
(l 4~'~+
+ O(g 1/3) (3.28)
W2) 2
with qo+(w) defined by eq. (B.12). The matrix e "r is defined in eq. (B.7). The final result is dw (3.29) Q'I V') = ½p'p~V'~ ~------7.c r ( w ) H S r ( w ) l ~ 2 ) C Z~I
and we see that {V) fails to be BRST invariant (by a divergent expression). 4. The anomaly In sect. 2 we computed the transformed vertex, and observed that it had the strange property of being non-local in the mid-point coordinates. That motivated
J.L. Ma~es / Stringfield theory
319
the closer study of sect. 3. Now we know that, besides being non-local, the transformed vertex has some seemingly unacceptable properties, expressed by eqs. (3.9) and (3.29). In this section we will show that all these bad properties are the result of an anomaly, and that this anomaly depends strongly on the regulator. 4.1. A N A S S O C I A T I V I T Y
ANOMALY
Let us consider BRST invariance, since the analysis of the overlap equations is identical. Our result Q'I v'> * 0
(4.1)
seems to be in contradiction with two well established facts, namely the BRST invariance of the untransformed vertex and the unitarity of U. Indeed, one could write
Q'[ V') = UQUt[ V'> = UQUtU[ v ) = UQ[ v ) = 0.
(4.2)
The contradiction disappears when we take into account the regulator. Then, the unitarity of U is expressed by
U~U~t = u~tu~ = 1,
(4.3)
lim UFQU~tU~]V> = lim U~Q[v ) = O,
(4.4)
which implies ~0
e~0
whereas what we have actually computed is
Q'IV') == - limU~QU/lV')= e--~O
lira lim U,2QU~U,,IV) 4:0.
(4.5)
~2--* 0 ei --+0
The non-invariance of the transformed vertex, eq. (3.29), simply means that these two limits are different lim U~QU/Uel v> 4= lim lim U~2QU~.~U~I[ V>. ~0
(4.6)
e l ~ O e2 "-+0
This can be interpreted as an associativity anomaly of the type considered by Horowitz and Strominger [16], and in that spirit eq. (4.6) could be rewritten as
UQ( UtU )I V> 4= ( UQUt )( UI V> ) .
(4.7)
This lack of associativity can be seen even more clearly by considering the action of U t on [V'), i.e. IV"> - u t [ V') -- lim u~t[ V'). e~O
(4.8)
J.L. Mafies / Stringfieldtheory
320
The result is given by eq. (2.5) with
N .... ( z ,
N"~S(z, w) instead of NrS(z, w)
w ) = NrS(z, w ) -k N~ rs
(4.9)
and we have
[V>=(U*U)IV)-~U*(UIV))=U*IV')=exp ½N
( p r ) 2 IV).
(4.10)
This means that U is not even unitary when acting on the vertex! This last equation implies
(A~I(B~[(C~[ IV')4: (AxI(B2[(C3[ I V)
(4.11)
and we see that the transformed vertex does not reproduce the original amplitudes. It is easy to trace the origin of this anomaly to the fact that the Neumann function NrS(z, w) is not analytic at the points _+i (see fig. 2). While computing N'rS(z, w) we observed that (2.20b) cancels against (2.20d), or in other words that
A~(w) - ½[Nrs(ie-% w) + NrS(-ie-% w)] + ½1n(1 + w2e-2~)3rs (4.12) satisfies lim A,(w) = 0
(4.13)
e--~.O
and these terms don't appear in [V'). However, if we compute (U'U)1V), then A~ is evaluated at the poles _+ie -~ and gives a finite contribution at e = 0 lira ½[A~(ie -~) +A~(-ie-~)] = ½N3~.
(4.14)
e-'*O
This, and an identical contribution from A~(z), are missing from U*(UIV)), thus explaining the anomalous result of eq. (4.9). A similar but more involved analysis can be used to explain the other anomalous properties of IV'). In all cases, infinitesimal terms in U~IV) yield non-vanishing contributions due to poles at ± ie-% This is a general phenomenon which is bound to happen for any regulator. However, the actual effects of the anomaly might depend on the regulator used. This possibility is explored in the next subsection. 4.2. ASYMMETRIC POINT SPLITTING
In order to determine the dependence on the regulator we should look for terms which are not well defined when the regulator is removed. Considering N,'(z, w) as given by eq. (2.20), we observe that the only such terms are (2.22) and (2.23). Eq. (2.23) comes from the normal ordering constant in U~
(X~(½~r, e)X~(½~r, - e ) > = - ~ l n ( 1 - e-4~).
(4.15)
J.L Ma~es / Stringfield theory
321
The key point is that eq. (2.22) depends only on the position of X~, whereas (4.15) is affected by the positions of both X~ and X c. This suggests the introduction of an asymmetric point splitting between the positive and negative frequency components of X X~(½~r,O)~ Xc(l~r,-ae),
X,(½~r,O) ~ X~(½~r, e),
(4.16)
where a is a free parameter which measures the asymmetry of the point splitting. Eq. (2.22) is unchanged, but (2.23) now reads -
,
-
= ½(ln e + ln2 + In(1 + a ) ) 8 r'+ O ( e ) .
(4.17)
Then the new value for N'r~(o, 0) is
N'rs(o,o) -~ R(ot)~ rs ,
(4.18)
R(a) = - 31n3 + ½1n2 + ½1n(1 + a ) .
(4.19)
where
Eq. (4.18) shows that the Neumann coefficient N0~rs, responsible for the nonlocality of I V') is in fact regulator dependent. It is easy to write down the dependence on a for the other relevant expressions. Eq. (4.10), which expresses the non-unitarity of U when acting on IV> becomes*
ut(uIV>)=exp(R(Ct)s~=l(PS)2)lV>.
(4.20)
The overlap eq. (3.4) now reads
[X'S(o,O)-X's+l(qr-o,O)][V '> = i R ( a ) ( p ~ - p s + l ) [ v '>
(4.21)
and we see that for a = ~v/-J- - 1 , R(a) vanishes and we get a vertex with the following nice properties: (i) Locality in the mid-point variables N0~r~= 0.
(4.22)
(ii) Reproduction of the original amplitudes lim U/[ V'> = IV).
(4.23)
E~0
(iii) The overlap equations are satisfied [ X " ( o , 0 ) - X"+l(~r - o,0)] I V') = 0.
(4.24)
* Our notation is somewhat ambiguous. U~t is obtained from Ut by making the substitutions(4.16) and, for a :# 1, is not equal to (UF)t.
322
J.L. Mahes / String field theory
The only property that remains to be checked is BRST invariance. It is easy to see that with the new regulator, as e ---)0 dw Q ' l V ' ) = ~ l1~. .e. . V,.£ ~c~-7~/criw~HSrtw~,IZ, t ) e t 11 2,
(4.25)
where H sr is now given by nsr=
2(1
(e 2) T'+(w) - ieSrep' (w)]
\ 1 + a ](1
-772)2 + 0(el/3).
(4.26)
Thus Q' does not annihilate IV') for any choice of a. However, this lack of BRST invariance is only formal, since property (ii) guarantees the decoupling of spurious states. Indeed, let [A) and [B) satisfy
QIA) = Q[B) = 0
(4.27)
[C) = Q I D ) .
(4.28)
(AIIU~*(BzlUzt(D31Q3U3tIV') = (AII(BzI(D3[Q3Ut[V') = 0 .
(4.29)
and let I C ) be spurious
Then property (ii) implies
This can be checked explicitly by observing that eq. (4.29) holds if
a u * l v') - lim OUe* I v ' ) = 0.
(4.30)
e~0
It is easy to show that this is true, in spite of eq. (4.25). This is another manifestation of the associativity anomaly, and has its origin in the fact that Q~', unlike X~' and I Ve'), is not defined in the limit e ---)0. Remembering eq. (3.21) for T / ( y ) , we can observe that although
lim FeZ(y) = 0
(4.31)
e---~ 0
the Fourier components
of FeZ(y) FeZ(y) = ~_~fny n-2
(4.32)
J.L. Mahes / StringfieM theory
323
are given by
fn:~c~iYl-nFe2(y) =e
I"l~cos(½n~r)[ctnh2e + ½[n[]
(4.33)
and diverge for e ~ 0. By varying the value of a we have been able to "mask" the effects of the anomaly to some extent. However Q~', being divergent as e ~ 0, is sensitive to higher order infinitesimals in the difference between IV') and IV/) (see discussion after eq. (4.11)). 5. Conclusion We began this work by pointing out the difficulties associated with a transformed vertex which is non-local in the mid-point coordinates. These difficulties come from the use of an additional non-unitary transformation to eliminate the residual dependence on the momentum. From the point of view of the purely cubic action, the interpretation of this transformation was unclear, and one might have to introduce additional ghosts. From the point of view of obtaining a hamiltonian formulation, the fact that the non unitary similarity transformation made the kinetic term non local seemed unacceptable. We have shown that the transformation is anomalous and that an appropriate choice of the regularization yields a vertex which is local and has all the right properties. The proper regulator is one among a one-parameter family of asymmetric point splittings. The use of powerful conformal field theory techniques has permitted a study of the properties of the transformed vertex. We have shown that, for a general value of the parameter, the vertex does not satisfy the overlap equations and fails to reproduce the original amplitudes. As a residual manifestation of the anomaly, the vertex is not formally BRST invariant even for the correct value of the parameter. However, spurious states do decouple from physical amplitudes and the interaction is effectively BRST invariant. The fact that the BRST charge is not well defined in the limit e ~ 0 makes this transformation awkward for use in a hamiltonian treatment of Witten's theory. A ~-function regularization of Q' [11] yields a finite result after the regulator is removed due to an automatic subtraction. However, this does not solve the problem, since the resulting Q' is not nilpotent [11]. It might be interesting to try to consider other transformations which move the center of mass to the mid-point but also affect the higher modes [17]. There are an infinite number of such transformations and some of them might have better properties than the one considered here. I would like to thank D. Gross, A. Strominger and E. Witten for very helpful discussions. I have enjoyed talking about these matters with J. Labastida and D.
J.L. Mages / Stringfieldtheory
324
Espriu. I also want to thank M. Vozmediano for valuable suggestions on the final form of this work. Note added in proof T.R. Morris has pointed to us that our asymmetric point splitting regularization renders the transformation formally non-unitary even in the limit e ~ 0, and in this sense is analogous to the use of an additional non-unitary transformation in ref. [11]. Our analysis of the anomaly in sect. 4 can be used to show that the transformed vertex IV") of ref. [11] (eq. (1.15) of this work) satisfies the overlap equations and reproduces the original amplitudes. A consequence of the non-unitarity of the transformation for c~¢ 1 is that it can not be used on the kinetic term, and our analysis proves the trivial character of the anomaly only with respect to the cubic action. However, R. Potting and C. Taylor (Rutgers preprint RU-88-08) have apparently shown, by using zeta function regularization, that the anomaly is in fact trivial with respect to the complete Witten's action. They also propose a sense in which Q' can be considered nilpotent.
Appendix A Here we give explicit formulae for the Neumann functions used in the main text. They are taken from ref. [14] and adapted to our conventions. The bosonic Neumann function is given by N i l ( z , w) = ln( ~b(z)2~b(-- w) 2 - ~b(--z)2~b(w) 2} -- ln{ i(z -- w)},
N'2(z,
w) = In (e-i~/6~a(z)20 ( - w ) 2 + ei~/6O(-z)2q~(w)2},
N13(z, w) = ln{ei~/6eO(z)2ep(-w) 2 +
e-i~/6~ ( - z ) 2 O ( w ) 2 ) .
(A.1)
Sometimes the following equivalent form is more convenient
{ Nil(z,
w) = In
2~(z)ep(-w)+2q~(-z)ep(w) } ep(z)2ep(_w) 2 + eo(z)4)(_z)ep(w)ep(_w) + ep(-z)Z~(w) 2 " (a.2)
The function if(z) has been defined in eq. (2.6) of the main text.
J.L. Maftes / Stringfield theory
325
The Neumann function for the ghosts is
{ eo(z)q~(-w) + ~(-z)eP(w) } N2hl(z, w):ln eo(z)eo(_w)--eo(_z~- ~ + ln{ i(z- w)}, { e i:/3+(z)+(-w) + e+:/3+(-z)+(w) } wt : ln
'
{ ei'~/3+(z)+(-w)+e+'~/3d?(-z)q~(w) } Ng~h3(z, w) = In e ~/%(z)+~---~)--- e ' ~ / - - - ~ ( - z - ~ w )
(1.3) "
Appendix B In this appendix we complete the derivation of the action of Q' on the transformed vertex IV'). After we integrate over y in eq. (3.23) we obtain eq. (3.26), where Ulr(z, w) is given by
wz2- e2~ tz, 8"~8I~ L " " w ) = w(w2+e:~)(z:+e:~ ) +½8"~'w(y_w)Y
Or[
Y)-
(B.la)
N'~(O'
Y)]lv:ie-'.
Z
+ ½~+,
.... ' w(y - z) 0 wlVghty, w)l~=,e ~+ (i ++ - i )
I y Oy[N,S(z ' Y) _ +g7
.
+ (i +->-i)
(B.lb) (B.lc)
N/~( O, y)] OwNg~ ( y ' W)[v=i e ,-[- (i ~ --i) (BAd) Z
+F~(w)Ow[N" (,w) z - Nls(o,w)]~sr+ --FF(z)OwN~hr(z,w)8 L~ (B.le) W
(no sum over s). Unlike y, neither ]wl nor Izl are bounded from below by e -* (see eqs. (3.24), (3.25)). Therefore it makes sense to study the limit e--+ 0 of the integrand in eq. (3.26). Since by eq. (2.18) lim F ( w ) = 0, e~0
(B.2)
J.L. Ma~es / Stringfield theory
326
we see that (B.le) vanishes as e goes to zero. On the other hand
OwNg"~t" hkY, w ) -
8r~ w--y
fory=
+i
(B.3)
and eq. (B.lc) cancels against (B.la) at e = 0. Only (B.lb) and (B.ld) remain. If we substitute eq. (B.3) into (B.ld), this last expression seems to cancel against (B.lb). However, at this point we have not been careful enough. According to eq. (A.1), in the limit e ~ 0
Oy[N,S(z, y)_N,s(O,y)]]y=+_ie _O(e
1/3).
(B.4)
Since this quantity diverges at e = 0, we have to worry about infinitesimal contributions to eq. (B.3). In fact ~rs
"~ = -w-T-i + O(e 1/3) OwNgh(Y'W)ly=±ie-"
(B.5)
and this, combined with (B.4), yields a finite contribution at e = 0. After some algebra one finds
limLtSr=--
~-.o
~
w
~(z
)d(0
)2
1 ~ww
d~(w)[e2-ie]
+(z,w,i~, - z , - w , - i ) ,
(B.6)
where the matrix e is
1(o 1 1)
e=--~
-1
0
1
-1
1
•
(B.7)
0
The important point is that (B.6) is independent of the index s, and by m o m e n t u m conservation we get lim pSL:ir(z, w) = 0. e~0
(B.8)
J.L. Ma~es / Stringfieldtheory
327
W e n o w t u r n our attention to the c o n t r i b u t i o n s q u a d r a t i c in p. The following e x p r e s s i o n is o b t a i n e d for H~ r after integrating over y in eq. (3.23)
H~r(z, w) = lyOwOyN~(y, 1 + ~W c t n h Z e
__~sr
W)[y=i e E 71-(i
~ --i)
OwN~hr(ie ~, w) + (i ~ --i)
e 2e
ctnh2e
( W 2 + e2e) 2
-'~ - -
w 2 + e 2e
(B.9)
This e x p r e s s i o n can be evaluated b y using the a s y m p t o t i c b e h a v i o r of the ghost N e u m a n n f u n c t i o n as we a p p r o a c h the m i d - p o i n t [14]
l[N~hS(ie-e, wl + N~h~(--ie-e,w)] = ½1n(w 2 + e 2~)8r'
-3(½e)l/3[(e2)rscp+(w)-ie'*ep_(w)] +0(e4/3),
(B.10)
w h e r e we h a v e defined
,(w) cO + ( w ) - O ( - w ~
,(-w) +
O(w~--)-
(B.11)
S u b s t i t u t i o n o f (B.10) into (B.9) yields eq. (3.28) of the m a i n text.
References [1] W. Siegel, Phys. Lett 151B (1985) 391, 396; W. Siegel and B. Zwiebach, Nucl. Phys, B263 (1986) 105; W. Siegel and B. Zwiebach, University of Maryland preprint 87-ill; H. Aratyn, R. Ingermanson and A. Niemi, Ohio State University preprint DOE/ER/01545-39I [2] T. Banks and M. Peskin, Nucl. Phys. B264 (1986) 513; A. Neveu, H. Nicolai and P.C. West, Phys. Lett. 167B (1986) 307; A. Neveu. J. Schwarz and P.C. West, Phys. Lett. 161B (1985) 51 [3] K. Itoh, T. Kugo, H. Kumitono and H. Ooguri, Prog. Theor. Phys. 75 (1986) 162; M. Kaku, Nucl. Phys. B267 (1986) 125 [4] E. Witten, Nucl. Phys. B268 (1986) 253 [5] H. Hata, K. Itoh, T. Kugo, T. Kumitono and K. Ogawa, Phys. Lett. 172B (1986) 186; A. Neveu, P.C. West, Phys. Lett. 168B (1986) 192 [6] E. Witten, Nucl. Phys. B276 (1986) 291 [7] C. Crnkovid, Nucl. Phys. B288 (1987) 419 [81 G. Siopsis, Phys. Lett. 195B (1987) 541
328
J.L. Ma~es / Stringfield theory
[9] G.T. Horowitz, J. Likken, R. Rohm and A. Strominger, Phys. Rev. Lett. 57 (1986) 283 [10] Z. Qiu and A. Strominger, Phys. Rev. D36 (1987) 1794; A. Strominger, Lectures on closed string field theory, IAS preprint IASSNS-HEP-87/28 and references therein [11] T. Morris, Nucl. Phys. B297 (1988) 141 [12] D. Gross and A. Jevicki, Nucl. Phys. 293 (1987) 29; D. Gross and A. Jevicki, Nucl. Phys. 287 (1987) 225; D. Gross and A. Jevicki, Nucl. Phys. 283 (1987) 1; L. J. Romans, Nucl. Phys. B298 (1988) 369 [13] E. Cremmer, A. Schwimmer and C. Thorn, Phys. Lett. 179B (1986) [14] G.T. Horowitz and S.P. Martin, Nucl. Phys. B296 (1988) 220 [15] A. Belavin, A. Polyakov and A. Zamolodchikov, Nucl. Phys. B241 (1984) 333; D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B271 (1986) 93; M.E. Peskin, Introduction to string and superstring theory II, SLAC preprint SLAC-PUB-4251 [16] G.T. Horowtiz and A. Strominger, Phys. Lett. 185B (1987) 45; A. Strominger, Phys. Lett. 187B (1987) 295 [17] E. Witten, private communication [18] A. Leclair, M.E. Peskin and C.R. Preitschopf, String theory on the conformal plane, I and II, in preparation
A N A N O M A L O U S T R A N S F O R M A T I O N IN S T R I N G FIELD THEORY* J i . MAlqES*
Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08544, USA Received 16 November 1987
Conformal field theory techniques are used to analyze a recently proposed transformation of Witten's interaction vertex from center-of-mass to mid-point variables. As presented in the literature the transformed vertex has the surprising property of being non-local in the mid-point coordinates unless an additional non-unitary transformation is performed. This renders the kinetic term non-local and poses additional problems. Here we show that, besides being non-local, the transformed vertex does not satisfy the overlap equations and fails to be BRST invariant. These bad properties are the result of an associativity anomaly of the type considered by Horowitz and Strominger. However, this anomaly is "almost trivial" in the sense that most of its bad effects can be avoided by choosing a proper regularization.
1. Introduction T h e string field in covariant string field theory [1-6] is a f u n c t i o n a l A[X~'(o)] of the c o o r d i n a t e s of the string.* I n the usual f o r m u l a t i o n the configuration space of the string is parametrized by the center-of-mass position q" a n d the n o r m a l mode c o o r d i n a t e s Xn~. This suggests a representation of the string field in the form of f i r s t - q u a n t i z e d creation operators acting on a suitable vacuum, with tensor fields as coefficients a[X"(o)]
=
+
1+
+
y_l
+ .-.
(1.1) I n this expression the space-time a r g u m e n t x is identified with q. This allows a n i n t e r a c t i n g string field theory to be written as an o r d i n a r y field theory with i n f i n i t e l y m a n y c o m p o n e n t fields in interaction. * Research supported in part by the National Science Foundation under grant No. PHY80-19754. t On leave of absence from Departamento de Fisica, Universidad del Pals Vasco, Bilbao, Spain. * We neglect the role of the ghosts until sect. 3. 0550-3213/88/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
306
J.L. Mafies / StringfieMtheory .vr
q3
0
0
x
"rr
rrllO Fig. 1. Witten's interaction (schematic). The point o = ~2~ris common to the three strings, unlike their centers of mass.
None of the proposed string interactions [4-6] are local in the center-of-mass coordinates. In particular, this is true for Witten's cubic vertex [4], schematically represented in fig. 1. As a consequence, the interaction among the component fields in eq. (1.1) will contain derivatives of arbitrarily high order. This might be avoided in Witten's theory by identifying the space-time argument in eq. (1.1) with the mid-point of the string X(½~r). This is possible because the point which is common to the three strings also carries a common label (o = {~r). This possibility is not open to any of the other theories. There are several cases in which such a "local" formulation of the theory is desirable. When one considers a hamiltonian treatment [6, 7] of Witten's theory, it seems that the "time" coordinate should appear locally in the action. S=
f( A * Q A + ~ A2* A,A).
(1.2)
This appears to be true for the choice t = X°(½~r), but not for t = q0 (see however ref. [8]). Another example is provided by the purely cubic action of Horowitz, Lykken, Rohm and Strominger [9] s =
fA • A • A
(1.3)
Witten's action (1.2) follows from eq. (1.3) by expanding about a particular solution of the classical equation of motion A * A = 0. This particular solution is an open string field description of a closed string field background. Eq. (1.3), conveniently elaborated [10], can be used to incorporate closed strings, although this part of the theory is still far from clear. The important point is that this picture only makes sense if the action (1.3) is truly independent of the closed string background. There are two ways in which a (gravitational) background dependence could enter eq. (1.3). The first is through the integration measure. This carl be avoided by
J.L. Mahes / Stringfield theory
307
defining the string field A to be a density of weight 1 [11]. The second way is through the connection necessary to covariantize the derivatives present in the interaction vertex. As mentioned before, these derivatives appear because the interaction is not local in x = q. One would hope that by taking q = X(½~r) the interaction would become derivative-free, thus avoiding the second source of gravitational background dependence. This was the motivation behind the work in ref. [11], to which we refer the reader for a complete exposition. Here we present a brief description of the transformation proposed there, which will be analyzed in detail in sects. 2 and 3. In the operator formalism of Witten's theory [12,13], the interaction vertex is represented by a state IV) in the tensor product of three single-string Fock spaces, so that [. tA * A * A = (A 1](Azl (A 3I I V).
(1.4)
Here {A I is the bra conjugate to the ket in eq. (1.1). In momentum representation the state IV) can be written as the exponential of a quadratic form in the oscillators a - n acting on the three-strings vacuum I V)
=
exp
r 1 ~Trs
s ,s r s ~ TlVrnnOtr mOt_n-lNdnp ot n q- 51 N ~r s p rp s }lI2),
m, n = 1 , 2 , . . . , o e , r, s = 1,2, 3.
(1.5)
The vacuum 1£2) satisfies q11~2) =q21$2 ) =q31$2 ),
(1.6a)
3
Y'~ pr]~2) = 0,
(1.6b)
n=l
and space-time indices have been suppressed in (1.5),(1.6). The vertex (1.5) is obviously non-local in the c.m. variables. Ref. [11] proposes a change from the conventional parametrization
x(o) =q+ d
x°cosn
(1.7)
n=l
to "mid-point variables" X ( o ) = m + v~ ~ ~ . [ c o s n o - ~o.1, n=l
(1.8)
308
J.L.
Ma~es
/
String field theory
where w, - cos(~n ~r). Eqs. (1.7) and (1.8) are related by the following change of variables
m=q+ 7'2 ~ ~%x,= X(½~'), n=l
~,=x,.
(1.9)
The unitary operator associated with this transformation is
U=exp(ip(q-X(½~r))).
(1.10)
This operator can be used to transform the string field and the interaction vertex to the new variables <-4'1 =
(1.11a)
3
I V') = I-[ Url V ) .
(1.11b)
r=l
The operator U needs regularization due to an infinite normal ordering constant. This is achieved by redefining % %=e
I'l%os(½n~r) .
(1.12)
The transformed vertex is given by eq. (1.5) with "primed" Neumann coefficients. The following result is obtained [11] once the limit e - , 0 is taken rs N 't,rs _- N,~,,
(1.13a)
N6',~'= - ~-23~,
(1.13b)
n
trs _
l_~]rs ~
N0~ - 2"00
1
rs
5 N3 ,
(1.13c)
where N = - ~ In 3 + 2 In 2. This results is surprising. The transformation U takes us from the set of variables (q, x , ) to (X(½~r),x,). By the previous discussion, we would expect the transformed vertex IV') to be local in X(½~r), and instead we see an exponential dependence on p and p2. The fact that the momentum dependence in the transformed vertex is diagonal in r and s is used in ref. [11] to define a single-string (non-unitary)
J.L. Ma~es / Stringfield theory
309
similarity transformation S = exp( - ¼Np 2 +
--pa_n n=l
/'/
)
,
(1.14)
which removes all momentum dependence from*
I V'') - S U I V ) .
(1.15)
The new vertex IV") is thus completely independent of the background metric, and the goals of ref. [11] are fulfilled. This, however, is not a good solution from the point of view of a hamiltonian formulation of Witten's theory, since the transformation S (being non-unitary) shifts the non-locality from the interaction vertex to the kinetic term in (1.2). (This fact was already pointed out in ref. [11]). This would make it very difficult, if not impossible, to formulate a hamiltonian treatment of Witten's theory. But even from the point of view of the cubic action, the meaning of the non-unitary transformation S and how to deal with it (do we need additional ghosts?) was not completely clear. We consider the situation sufficiently puzzling to justify a closer look at the problem. This is facilitated by the use of the powerful techniques of conformal field theory presented in refs. [14,15], instead of oscillator methods. We show in sect. 2 that there is a natural way of regulating the transformation U in terms of a point splitting between positive and negative frequency modes. The computation of the vertex is thus reduced to a trivial evaluation of Neumann functions at some given values of their arguments. The resulting vertex differs from the one previously obtained [11], but is still non-local. This motivates the study of the properties of IV'), which is carried out in sect. 3. The fact that the conformal field theory methods used here permit an easy calculation of higher order corrections in e is crucial at this point. We find that the transformed vertex does not satisfy the (transformed) overlap equations and is not annihilated by the BRST charge Q'. These properties are explained in sect. 4 as the result of an associativity anomaly of the type recently considered by Horowitz and Strominger [16]. We also show that, by an appropriate choice of the regularization, it is possible to avoid most of the bad effects of the anomaly and construct a vertex with the correct properties. The anomaly has, however, some residual manifestations which are discussed in detail. A summary of the results of this work is given in sect. 5.
2. The transformed vertex
In this section we fix the notation, introduce a special type of point splitting regularization and use it to compute the vertex IV'). We find a result which is * U acting o n the vertex will always stand for I~3,=1Ur. This is also true for S.
J.L. Ma~es / Stringfield theory
310
different from the one in the original reference, but still non-local, and explain the discrepancy. The position operator
O~n
X(o,r)=q+pr+i
.
Y'~ - - e - ' " ' c o s n a n4:0
(2.1)
/'/
will be written as
x ( o , T) =
X(z) +
(2.2)
where
X ( z ) = q + iplnz + i Y'. Oln,7,n.
(2.3)
n~O tl Eqs. (2.1) and (2.2) are related by a Wick rotation T ~ - i T and a change of variables z = e x p ( - T - io). Note that with this convention, world-sheet time flows toward the center z = 0. The bosonic part of the interaction vertex
IV) = VxVgh]I2)
(2.4)
is given by
Vx
expf~ _
1 t'~
_)_ ( ~ _ _dw O zdX r ( z ) N rzS ( z , w ) O w X S ( w )
7~c 2qri yc 21ri
(2.5)
The Neumann function N~'(z, w) has been taken from ref. [14] and adapted to our conventions. An explicit expression in terms of ~( + z) can be found in appendix A. The function if(z) is given by ~ ( z ) = (1
+
iz) 1/3
(2.6)
and the cuts in ¢0(+z) are chosen from + i to +iac (see fig. 2) so that N~'(z, w) is analytic for [z[ < 1, {w[ < 1. The contour C will always run counterclockwise around the origin, inside the unit circle. If we compare eqs. (2.5) and (1.4) we immediately find the relation
NrS(z, w) =
~
N~r~,z"w".
(2.7)
rt, m = O
In what follows it will be necessary to separate the zero modes from the creation and annihilation parts in X(o, T),
X(o, T) = q +pT + Xc(o, T) + Xa(O , r)
(2.8)
J.L. Mahes / StringfieM theory
311
L
Fig. 2. The N e u m a n n functions are analytic inside the unit circle.
and correspondingly
X(z) = q + ip In z + Xc(Z ) + X . ( z ) ,
(2.9)
where [14] dw
X~(z) = ~c~--~8wX(w)ln(1 - w/z),
Iwl < Izl,
dw Xa(Z) = -- ~ c ~ i OwX(w)ln(1 - z / w ) ,
Izl < Iwl.
(2.10a)
(2.10b)
The exact vacuum expectation value
(X,(z)Xc(w)) = -ln(1 - z/w)
(2.11)
implies the OPE
X(z)X(w) - -in(w-z).
(2.12)
We are now ready to study the operator U, which in our notation reads
U = e x p ( i p ( q - X(½~r, 0))) -- e x p ( - i p ( X c ( l ~ r , 0 ) + X,(½~r,O))) = exp(-- ½p2(Xa(lw,O)Xc(½rr,O)))exp(-ipXc(½"r;,O))exp(-ipX,(½~,O)).
(2.13) The normal ordering constant in eq. (2.13) is infinite, and some kind of regularization is required. We cannot simply drop the infinite constant in the usual way, since
312
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Mahes / String field theory
that would make U non-unitary. We introduce a point splitting between the positive and negative frequency components of X, so that 1 Xa(½~r,O)--* X~(½~r,e)= lX~(ie-*) + ~X~(-ie-~),
(2.14a)
Xc(½w,O) --* X~(½rr, - e ) = ½X~(ie ~) + ½Xc(-ie*).
(2.14b)
We can use eq. (2.11) to determine the value of the normal ordering constant. This yields the following regulated expression for U
U~=exP(¼p21n(1-e-4~))exp(-ipX~(11r, e))exp(-ipXa(½~,-e))
(2.15)
The equivalence between this point splitting regularization and the one based on oscillator methods used in ref. [11] follows from the identity ½((ie-~)l"l +
60. = e - l n l e c o s ( ½ n T r ) =
(--ie-~)lnl)
.
(2.16)
The last ingredient necessary for the computation of I V') is the transformation of
OzX(z ) by Ue This is easily done with the help of eqs. (2.10) and (2.12). The result is ip U~O~X(z)U] = O~X(z) - -- + ipF,(z), Z
e ' < [z[ < e ~,
(2.17)
where
F~(z) -~
Z g 2 + e -2*
Z z 2 + e 2e •
(2.18)
This expression is only valid inside an annulus of radii e -* and e ~, as a consequence of the restrictions in eqs. (2.10) (see fig. 3). This implies that the limit e ~ 0 should always be taken after the contour integrals have been performed, since there is no value of z for which eq. (2.17) is valid at e = 0. Writing
IV,') = Uyi$2 ) = ( UyU] )U~[~)
(2.19)
and using eqs. (2.15), (2.10) and (2.17) one finds
N'r'(z,w)=Nr'(z,w)-Nr'(O,w)-NrS(z,O)+N~'(O,O)
(2.20a)
+1~1v .... tte/"-~,W)+!~r'('21" ~-,ie-~)+(i~-i)
(2.20b)
+¼Nr~(ie -*,ie -*) + ¼Nrs(ie - * , - i e -*) + ( i ~ - i )
(2.20c)
+ {ln(1 + zZe-2~)3~s+ ½1n(1 + w2e-2")3 r"
(2.20d)
+ ½ 1 n ( 1 -- e - 4 * ) ~
(2.20e)
rs .
J.L. Ma~es / Stringfield theory
_
_
313
N
-ie E
T
Fig. 3. U~O~X(z)U~t, as given by eq. (2.17), is defined only for e ~ < Izl < eq
The terms in the first three lines come from the poles z = 0, + ie -~ in eq. (2.18). The terms in eqs. (2.20d) and (2.20e) arise from the creation part X c (1u r , - e ) and the normal ordering constant, respectively. Eqs. (2.20) reduce the computation of IV') to a simple evaluation of functions. From the expression for N~'(z, w) in appendix A and momentum conservation (1.6b) one obtains
1 N ~ ' ( i e - ~ , w ) + ~ 1N
~, ( - i e
-~,
w) = -½1n(1 +
w 2 ) a r ' + o ( e 2/3)
(2.21)
and we see that (2.20d) cancels against (2.20b). We also have 1 ms (le • - - e , i e - ~ ) + x 1N r~(re • ~, - - i e - ~ ) + (i ~ - i ) = - - (~ln3 + ½1n ~ ) ~ r s q _ ~N
O(e2/3) (2.22)
and l l n ( 1 -- e - 4 ~ ) ~ r s = (½1n e + l n 2 ) ~ rs 4- O ( ~ ) .
(2.23)
Combining (2.22) and (2.23) we obtain a finite value at e = 0 ( - ¼1n3 + ln2)8 rs= ½N8 r ' = 1Nr'(0,0)
(2.24)
and the result is
N'~S(z, w) =- ]imN'~'(~ ,z, w) = Nr'(z, w) - Nrs(o, w) - NrS(z,O) + 3Nrs(o,o). ~0
(2.25) The Neumann coefficients N' ' s can be read from this equation with the help of (2.7). Since N'~S(z,O) = N'~S(0, w) = 0, (2.26) U'~S(0,0) = l~vrs/n 0),
(2.27)
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J.L. Mafies / Stringfield theory
we have _
r$
N(~,~~ = O, N~rS=lMrs= ~ ,~ ~1N6rS
(2.28)
.
Unlike ref. [11], we get zero for No'i ' (see eq. (1.13b)). This discrepancy can be traced to the evaluation of eq. (3.29) of ref. [11], where the regulator was erroneously removed before the contour integrals were performed. That left out several contributions which, once included, reproduce our result. Eq. (2.27) shows that the transformed vertex is still non-local in the mid-point coordinates. Since, as explained in the introduction, this poses some puzzling problems, we must check the validity of our result. This is done in the next section, where the properties of the transformed vertex are studied.
3. P r o p e r t i e s o f the t r a n s f o r m e d v e r t e x
3.1. THE OVERLAP EQUATIONS The overlap conditions on the vertex IV} are [12]
[XS(o,O)-X'+l(~-o,O)]lV}
=0,
~-~ < o -.<<,~.
(3.1)
When trying to prove this equality one considers the following expression [14]
[XS(y)+X'(f)-X'+l(-1/y)-X'+X(-1/Y)]lV},
Rey
(3.2)
which for y on the unit circle is obviously equivalent to the 1.h.s. of eq. (3.1) (see fig. 4). In order to evaluate eq. (3.2) we use _
dz
r
[XS(u),V] = V ~ ) c ~ i O z X ( z ) N r ' ( z , w ) .
(3.3)
Since NrS(z, w) is unambiguously defined only for Iu] < 1, we take lY[ < 1 and analytically continue to u = - 1/y. The following analytic continuation of ~ is valid for lyl < l , R e y < 0
q ~ ( - 1 / y ) = e~/2y-1/3,( y ) , q~(l/y) = ei~/6y-X/3dp(y), where the cut in y-1/3 goes from 0 to - i o c along the imaginary axis.
(3.4)
J.L. Ma~es / String field theory
Fig. 4. A s [y [ ~ 0, the p o i n t s y and -
315
1/y correspond to (o, 0) and (~r - o, 0) in the overlap equations.
Eqs. (3.4) combined with (A.1) give Nil(z,
--
l/y)
N12(z, - l / y ) NI3(z,
--
= N13(z,
y) -
ln(z + l / y ) + In y 2/3,
= N n ( z , y) + ln(y - z) + In y-2/3,
l / y ) = N12(z, y) + In y-2/3.
(3.5)
Then it is easy to show that (3.2) vanishes on the unit circle lim [ X ' ( y ) + X'(fi) lyl~l
x~+l(-1/y)
-
X'+I(-1/~)]]
V) = 0,
Rey<0,
]y[ < 1 .
(3.6)
(A complete account of this method can be found in ref. [14].) In our case we have to evaluate eq. (3.2) with IV) and X replaced by IV') and X' respectively. X' is given by
X[(y) =--U~X(y)U~* = X ( y ) - ½[ Xa(ie- Q + Xc(ie ~) + (i ~ - i ) ] -iplnY+½ipln
(y2+e-2") y2+e2 ~
+ipe.
(3.7)
This expression, like eq. (2.17) is valid only for e -~ < [y[ < e ". After some algebra one finds lim lim [ X ' ~ ( y ) + X~"(.P) - X ' s + l ( - 1 / y ) e~0
-X~'S+I(-1/~)]IV ')
]y[ ~ 1
= iN( p" - / + 1 ) 1 V ' ) .
(3.8)
J.L. Mahes / String field theory
316
Note that the limits have to be taken in the order prescribed. Eq. (3.8) can be rewritten as [X'S(o,0) - X'S+l(rr- o,0)]l V') = liN(pS-pS+l)lV' ) ,
(3.9)
where N was defined in (2.24). We conclude that the transformed vertex does not satisfy the overlap equations. The significance of this fact will be discussed in sect. 4.
3.2. BRST I N V A R I A N C E
The total BRST charge Q is given by
Q=
c'(y)Ts(y) + ghost terms,
(3.10)
where the bosonic energy-momentum tensor is
T(Y)=-I:OyX(y)OyX(y) :
1
y2,
(3.11)
The charge Q annihilates the vertex IV)
OVxVghl~2) = 0
(3.12)
t , . dz ,. d w / z ~ ,. ) Vgh=expl~c-~i~)c-~i[:)b (z)O~N~(z,w)c'(w ) .
(3.13)
with Vgh given by [14]
The Neumann function for the ghosts can be found in appendix A. The vacuum 1~2) is defined so that c~lI2) = O,
Vn >~O,
b,l~2) = 0,
Vn > 0.
(3.14)
We can go from eq. (3.13) to the oscillator expression [12]
Vgh= exp{ br ,,,N,~nc!. }
(3.15)
by using the mode expansions c(z)=
~ c,z "+1, --~
b(z)=
~ b,z "-2 --OQ
(3.16)
J.L. Maaes / StringfieM theory
317
and we may write
N~*(z, w) = ~ ,vm.z~r"row",
(3.17)
rn, n ~ O
although the actual values of No~'r"and N~ 0-rs are of course irrelevant. F r o m the mode expansions (3.16) we get
(c(z)b(w)) -
z 1
(3.18)
WW--Z
We shall also need the creation part of c(z), which according to eqs. (3.14) and (3.16) is Cc(Z) =
CnZ n + l =
fcdw (-~) c(w)
Iwl < [z[.
,
~i
n=l
(3.19)
Z--W
Eqs. (3.13)-(3.19) contain all the information we need about the ghost sector. We now turn our attention to the transformed BRST charge Q~'. This is given by eq. (3.10) with T ' ( y ) instead of In order to write an expression for Tj we remember that
T(y).
U,OyX(y)U,t= ( OyX(y)- ~ ) + ipF,(Y),
e ~ < [y[ < e ~,
(3.20)
where only the last term depends on p. Then T ' is given by
-
-
~
o y x ( y )
-
-
/p
OyX(y) -
1 F ~ ( y ) + ~p
2F21yX e t )
e - ~ < [Yl
y2'
(3.21)
We are now ready to compute the action of Q~' on IV'). The amount of work can be minimized by realizing that the transformation U does not affect any contribution which is independent of p. Indeed we see from eq. (3.20) that only the zero mode dependence has been affected by U, and eq. (2.28) tells us the same about This observation, together with the BRST invariance of IB) allows us to write
IV').
Q'~lV')=
c'(y) -ip ~ 3~X~(y)---f - F~(y) +
~p p F~. (y
Vx V~d~), (3.22)
where there is a single sum over the repeated index s. After all the annihilation operators in this last expression have been commuted through the vertex V' - V(Vgh
318
J.L. Mafies / Stringfield theory
we get /,
dy
^
^
(3.23)
Q'V' $2) = V'q) Z--TG~(y)BS(y)ll2), .'c zqrl
where dw
d,= ~ c 2 - ~
lyt/ sr,
B~= -ip'F~(y)
Ow ghty, w ) + y---w
1
cr(w)'
Iw[ < lYl,
~iOzXl(z) O y [ N t S ( z , y ) - N " ( O ,
+ 71 pSpSlff'2 *~ (, Y ) ,
e- ~ < lYl < 1,
y)] +6'"
(3.24)
y--z
Izl < lYl
(3.25)
This last expression is not summed over the index s. Eq. (3.23) can be evaluated by using Cauchy's theorem. The result after integrating over y can be written as dw P 1 s s t(~ Q'~V'[~2) = 2p p V "Z---7.c r (w)H;r(w)[~2) C Z.~l
s , r dz e dw - i p V (fi - - ( f i - - O z X ' ( z ) c r ( w ) L S ' r ( z , w)l12), 7"c 2 qri T"¢ 2 ~ri
(3.26)
where LJr(z, w) and HSr(w) are momentum-independent functions which are computed in appendix B. There we show that L J ~ becomes independent of the index s for e ~ 0, and by momentum conservation lim p'Lslr(z, w ) = 0.
(3.27)
e--~0
H s~ is given by
H: ~= -
4-wl(2)-2/3[(e2)rsep'+(w) -
ieS~op;(w)] +
(l 4~'~+
+ O(g 1/3) (3.28)
W2) 2
with qo+(w) defined by eq. (B.12). The matrix e "r is defined in eq. (B.7). The final result is dw (3.29) Q'I V') = ½p'p~V'~ ~------7.c r ( w ) H S r ( w ) l ~ 2 ) C Z~I
and we see that {V) fails to be BRST invariant (by a divergent expression). 4. The anomaly In sect. 2 we computed the transformed vertex, and observed that it had the strange property of being non-local in the mid-point coordinates. That motivated
J.L. Ma~es / Stringfield theory
319
the closer study of sect. 3. Now we know that, besides being non-local, the transformed vertex has some seemingly unacceptable properties, expressed by eqs. (3.9) and (3.29). In this section we will show that all these bad properties are the result of an anomaly, and that this anomaly depends strongly on the regulator. 4.1. A N A S S O C I A T I V I T Y
ANOMALY
Let us consider BRST invariance, since the analysis of the overlap equations is identical. Our result Q'I v'> * 0
(4.1)
seems to be in contradiction with two well established facts, namely the BRST invariance of the untransformed vertex and the unitarity of U. Indeed, one could write
Q'[ V') = UQUt[ V'> = UQUtU[ v ) = UQ[ v ) = 0.
(4.2)
The contradiction disappears when we take into account the regulator. Then, the unitarity of U is expressed by
U~U~t = u~tu~ = 1,
(4.3)
lim UFQU~tU~]V> = lim U~Q[v ) = O,
(4.4)
which implies ~0
e~0
whereas what we have actually computed is
Q'IV') == - limU~QU/lV')= e--~O
lira lim U,2QU~U,,IV) 4:0.
(4.5)
~2--* 0 ei --+0
The non-invariance of the transformed vertex, eq. (3.29), simply means that these two limits are different lim U~QU/Uel v> 4= lim lim U~2QU~.~U~I[ V>. ~0
(4.6)
e l ~ O e2 "-+0
This can be interpreted as an associativity anomaly of the type considered by Horowitz and Strominger [16], and in that spirit eq. (4.6) could be rewritten as
UQ( UtU )I V> 4= ( UQUt )( UI V> ) .
(4.7)
This lack of associativity can be seen even more clearly by considering the action of U t on [V'), i.e. IV"> - u t [ V') -- lim u~t[ V'). e~O
(4.8)
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320
The result is given by eq. (2.5) with
N .... ( z ,
N"~S(z, w) instead of NrS(z, w)
w ) = NrS(z, w ) -k N~ rs
(4.9)
and we have
[V>=(U*U)IV)-~U*(UIV))=U*IV')=exp ½N
( p r ) 2 IV).
(4.10)
This means that U is not even unitary when acting on the vertex! This last equation implies
(A~I(B~[(C~[ IV')4: (AxI(B2[(C3[ I V)
(4.11)
and we see that the transformed vertex does not reproduce the original amplitudes. It is easy to trace the origin of this anomaly to the fact that the Neumann function NrS(z, w) is not analytic at the points _+i (see fig. 2). While computing N'rS(z, w) we observed that (2.20b) cancels against (2.20d), or in other words that
A~(w) - ½[Nrs(ie-% w) + NrS(-ie-% w)] + ½1n(1 + w2e-2~)3rs (4.12) satisfies lim A,(w) = 0
(4.13)
e--~.O
and these terms don't appear in [V'). However, if we compute (U'U)1V), then A~ is evaluated at the poles _+ie -~ and gives a finite contribution at e = 0 lira ½[A~(ie -~) +A~(-ie-~)] = ½N3~.
(4.14)
e-'*O
This, and an identical contribution from A~(z), are missing from U*(UIV)), thus explaining the anomalous result of eq. (4.9). A similar but more involved analysis can be used to explain the other anomalous properties of IV'). In all cases, infinitesimal terms in U~IV) yield non-vanishing contributions due to poles at ± ie-% This is a general phenomenon which is bound to happen for any regulator. However, the actual effects of the anomaly might depend on the regulator used. This possibility is explored in the next subsection. 4.2. ASYMMETRIC POINT SPLITTING
In order to determine the dependence on the regulator we should look for terms which are not well defined when the regulator is removed. Considering N,'(z, w) as given by eq. (2.20), we observe that the only such terms are (2.22) and (2.23). Eq. (2.23) comes from the normal ordering constant in U~
(X~(½~r, e)X~(½~r, - e ) > = - ~ l n ( 1 - e-4~).
(4.15)
J.L Ma~es / Stringfield theory
321
The key point is that eq. (2.22) depends only on the position of X~, whereas (4.15) is affected by the positions of both X~ and X c. This suggests the introduction of an asymmetric point splitting between the positive and negative frequency components of X X~(½~r,O)~ Xc(l~r,-ae),
X,(½~r,O) ~ X~(½~r, e),
(4.16)
where a is a free parameter which measures the asymmetry of the point splitting. Eq. (2.22) is unchanged, but (2.23) now reads -
,
-
= ½(ln e + ln2 + In(1 + a ) ) 8 r'+ O ( e ) .
(4.17)
Then the new value for N'r~(o, 0) is
N'rs(o,o) -~ R(ot)~ rs ,
(4.18)
R(a) = - 31n3 + ½1n2 + ½1n(1 + a ) .
(4.19)
where
Eq. (4.18) shows that the Neumann coefficient N0~rs, responsible for the nonlocality of I V') is in fact regulator dependent. It is easy to write down the dependence on a for the other relevant expressions. Eq. (4.10), which expresses the non-unitarity of U when acting on IV> becomes*
ut(uIV>)=exp(R(Ct)s~=l(PS)2)lV>.
(4.20)
The overlap eq. (3.4) now reads
[X'S(o,O)-X's+l(qr-o,O)][V '> = i R ( a ) ( p ~ - p s + l ) [ v '>
(4.21)
and we see that for a = ~v/-J- - 1 , R(a) vanishes and we get a vertex with the following nice properties: (i) Locality in the mid-point variables N0~r~= 0.
(4.22)
(ii) Reproduction of the original amplitudes lim U/[ V'> = IV).
(4.23)
E~0
(iii) The overlap equations are satisfied [ X " ( o , 0 ) - X"+l(~r - o,0)] I V') = 0.
(4.24)
* Our notation is somewhat ambiguous. U~t is obtained from Ut by making the substitutions(4.16) and, for a :# 1, is not equal to (UF)t.
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J.L. Mahes / String field theory
The only property that remains to be checked is BRST invariance. It is easy to see that with the new regulator, as e ---)0 dw Q ' l V ' ) = ~ l1~. .e. . V,.£ ~c~-7~/criw~HSrtw~,IZ, t ) e t 11 2,
(4.25)
where H sr is now given by nsr=
2(1
(e 2) T'+(w) - ieSrep' (w)]
\ 1 + a ](1
-772)2 + 0(el/3).
(4.26)
Thus Q' does not annihilate IV') for any choice of a. However, this lack of BRST invariance is only formal, since property (ii) guarantees the decoupling of spurious states. Indeed, let [A) and [B) satisfy
QIA) = Q[B) = 0
(4.27)
[C) = Q I D ) .
(4.28)
(AIIU~*(BzlUzt(D31Q3U3tIV') = (AII(BzI(D3[Q3Ut[V') = 0 .
(4.29)
and let I C ) be spurious
Then property (ii) implies
This can be checked explicitly by observing that eq. (4.29) holds if
a u * l v') - lim OUe* I v ' ) = 0.
(4.30)
e~0
It is easy to show that this is true, in spite of eq. (4.25). This is another manifestation of the associativity anomaly, and has its origin in the fact that Q~', unlike X~' and I Ve'), is not defined in the limit e ---)0. Remembering eq. (3.21) for T / ( y ) , we can observe that although
lim FeZ(y) = 0
(4.31)
e---~ 0
the Fourier components
of FeZ(y) FeZ(y) = ~_~fny n-2
(4.32)
J.L. Mahes / StringfieM theory
323
are given by
fn:~c~iYl-nFe2(y) =e
I"l~cos(½n~r)[ctnh2e + ½[n[]
(4.33)
and diverge for e ~ 0. By varying the value of a we have been able to "mask" the effects of the anomaly to some extent. However Q~', being divergent as e ~ 0, is sensitive to higher order infinitesimals in the difference between IV') and IV/) (see discussion after eq. (4.11)). 5. Conclusion We began this work by pointing out the difficulties associated with a transformed vertex which is non-local in the mid-point coordinates. These difficulties come from the use of an additional non-unitary transformation to eliminate the residual dependence on the momentum. From the point of view of the purely cubic action, the interpretation of this transformation was unclear, and one might have to introduce additional ghosts. From the point of view of obtaining a hamiltonian formulation, the fact that the non unitary similarity transformation made the kinetic term non local seemed unacceptable. We have shown that the transformation is anomalous and that an appropriate choice of the regularization yields a vertex which is local and has all the right properties. The proper regulator is one among a one-parameter family of asymmetric point splittings. The use of powerful conformal field theory techniques has permitted a study of the properties of the transformed vertex. We have shown that, for a general value of the parameter, the vertex does not satisfy the overlap equations and fails to reproduce the original amplitudes. As a residual manifestation of the anomaly, the vertex is not formally BRST invariant even for the correct value of the parameter. However, spurious states do decouple from physical amplitudes and the interaction is effectively BRST invariant. The fact that the BRST charge is not well defined in the limit e ~ 0 makes this transformation awkward for use in a hamiltonian treatment of Witten's theory. A ~-function regularization of Q' [11] yields a finite result after the regulator is removed due to an automatic subtraction. However, this does not solve the problem, since the resulting Q' is not nilpotent [11]. It might be interesting to try to consider other transformations which move the center of mass to the mid-point but also affect the higher modes [17]. There are an infinite number of such transformations and some of them might have better properties than the one considered here. I would like to thank D. Gross, A. Strominger and E. Witten for very helpful discussions. I have enjoyed talking about these matters with J. Labastida and D.
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Espriu. I also want to thank M. Vozmediano for valuable suggestions on the final form of this work. Note added in proof T.R. Morris has pointed to us that our asymmetric point splitting regularization renders the transformation formally non-unitary even in the limit e ~ 0, and in this sense is analogous to the use of an additional non-unitary transformation in ref. [11]. Our analysis of the anomaly in sect. 4 can be used to show that the transformed vertex IV") of ref. [11] (eq. (1.15) of this work) satisfies the overlap equations and reproduces the original amplitudes. A consequence of the non-unitarity of the transformation for c~¢ 1 is that it can not be used on the kinetic term, and our analysis proves the trivial character of the anomaly only with respect to the cubic action. However, R. Potting and C. Taylor (Rutgers preprint RU-88-08) have apparently shown, by using zeta function regularization, that the anomaly is in fact trivial with respect to the complete Witten's action. They also propose a sense in which Q' can be considered nilpotent.
Appendix A Here we give explicit formulae for the Neumann functions used in the main text. They are taken from ref. [14] and adapted to our conventions. The bosonic Neumann function is given by N i l ( z , w) = ln( ~b(z)2~b(-- w) 2 - ~b(--z)2~b(w) 2} -- ln{ i(z -- w)},
N'2(z,
w) = In (e-i~/6~a(z)20 ( - w ) 2 + ei~/6O(-z)2q~(w)2},
N13(z, w) = ln{ei~/6eO(z)2ep(-w) 2 +
e-i~/6~ ( - z ) 2 O ( w ) 2 ) .
(A.1)
Sometimes the following equivalent form is more convenient
{ Nil(z,
w) = In
2~(z)ep(-w)+2q~(-z)ep(w) } ep(z)2ep(_w) 2 + eo(z)4)(_z)ep(w)ep(_w) + ep(-z)Z~(w) 2 " (a.2)
The function if(z) has been defined in eq. (2.6) of the main text.
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The Neumann function for the ghosts is
{ eo(z)q~(-w) + ~(-z)eP(w) } N2hl(z, w):ln eo(z)eo(_w)--eo(_z~- ~ + ln{ i(z- w)}, { e i:/3+(z)+(-w) + e+:/3+(-z)+(w) } wt : ln
'
{ ei'~/3+(z)+(-w)+e+'~/3d?(-z)q~(w) } Ng~h3(z, w) = In e ~/%(z)+~---~)--- e ' ~ / - - - ~ ( - z - ~ w )
(1.3) "
Appendix B In this appendix we complete the derivation of the action of Q' on the transformed vertex IV'). After we integrate over y in eq. (3.23) we obtain eq. (3.26), where Ulr(z, w) is given by
wz2- e2~ tz, 8"~8I~ L " " w ) = w(w2+e:~)(z:+e:~ ) +½8"~'w(y_w)Y
Or[
Y)-
(B.la)
N'~(O'
Y)]lv:ie-'.
Z
+ ½~+,
.... ' w(y - z) 0 wlVghty, w)l~=,e ~+ (i ++ - i )
I y Oy[N,S(z ' Y) _ +g7
.
+ (i +->-i)
(B.lb) (B.lc)
N/~( O, y)] OwNg~ ( y ' W)[v=i e ,-[- (i ~ --i) (BAd) Z
+F~(w)Ow[N" (,w) z - Nls(o,w)]~sr+ --FF(z)OwN~hr(z,w)8 L~ (B.le) W
(no sum over s). Unlike y, neither ]wl nor Izl are bounded from below by e -* (see eqs. (3.24), (3.25)). Therefore it makes sense to study the limit e--+ 0 of the integrand in eq. (3.26). Since by eq. (2.18) lim F ( w ) = 0, e~0
(B.2)
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we see that (B.le) vanishes as e goes to zero. On the other hand
OwNg"~t" hkY, w ) -
8r~ w--y
fory=
+i
(B.3)
and eq. (B.lc) cancels against (B.la) at e = 0. Only (B.lb) and (B.ld) remain. If we substitute eq. (B.3) into (B.ld), this last expression seems to cancel against (B.lb). However, at this point we have not been careful enough. According to eq. (A.1), in the limit e ~ 0
Oy[N,S(z, y)_N,s(O,y)]]y=+_ie _O(e
1/3).
(B.4)
Since this quantity diverges at e = 0, we have to worry about infinitesimal contributions to eq. (B.3). In fact ~rs
"~ = -w-T-i + O(e 1/3) OwNgh(Y'W)ly=±ie-"
(B.5)
and this, combined with (B.4), yields a finite contribution at e = 0. After some algebra one finds
limLtSr=--
~-.o
~
w
~(z
)d(0
)2
1 ~ww
d~(w)[e2-ie]
+(z,w,i~, - z , - w , - i ) ,
(B.6)
where the matrix e is
1(o 1 1)
e=--~
-1
0
1
-1
1
•
(B.7)
0
The important point is that (B.6) is independent of the index s, and by m o m e n t u m conservation we get lim pSL:ir(z, w) = 0. e~0
(B.8)
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W e n o w t u r n our attention to the c o n t r i b u t i o n s q u a d r a t i c in p. The following e x p r e s s i o n is o b t a i n e d for H~ r after integrating over y in eq. (3.23)
H~r(z, w) = lyOwOyN~(y, 1 + ~W c t n h Z e
__~sr
W)[y=i e E 71-(i
~ --i)
OwN~hr(ie ~, w) + (i ~ --i)
e 2e
ctnh2e
( W 2 + e2e) 2
-'~ - -
w 2 + e 2e
(B.9)
This e x p r e s s i o n can be evaluated b y using the a s y m p t o t i c b e h a v i o r of the ghost N e u m a n n f u n c t i o n as we a p p r o a c h the m i d - p o i n t [14]
l[N~hS(ie-e, wl + N~h~(--ie-e,w)] = ½1n(w 2 + e 2~)8r'
-3(½e)l/3[(e2)rscp+(w)-ie'*ep_(w)] +0(e4/3),
(B.10)
w h e r e we h a v e defined
,(w) cO + ( w ) - O ( - w ~
,(-w) +
O(w~--)-
(B.11)
S u b s t i t u t i o n o f (B.10) into (B.9) yields eq. (3.28) of the m a i n text.
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