- Email: [email protected]
Membrane symmetries and anomalies
Nuclear Physics B343 (1990) 398—417 North-Holland
MEMBRAJ~4ESYMMETRIES AND ANOMALIES Itzhak BARS* Department of Physics, UniLersity of Southern California, Los Angeles, CA 90089-0484, USA Received 18 December 1989
The symmetries of the bosonic membrane are analysed in several gauges. We discuss two Lorentz-covariant gauges, and a new light-cone gauge that reduces the membrane hamiltonian to a quadratic form which is diagonalized by harmonic oscillators. We then discuss in detail the Lorentz-covariarit quantization and the algebra of constraints that corresponds to the diffeomorphisms in three dimensions (DiffV 3). We introduce an operator product expansion which allows the careful computation of commutators. We find that the subalgebras corresponding to diffeomorphisms of the two-dimensional genus-g membrane surface (Diff~g)and the area-preserving diffeomorphisms (AP Diff ~ do not have any anomalies, but the full algebra Diff V3 has operator anomalies which render the quantization inconsistent. We thus conclude that purely bosonic membranes are quantum mechanically inconsistent in any number of dimensions d.
1. Introduction
Although membrane, super-membrane, and higher (super) p-brane [1—3],theories (p 1 string, p 2 membrane, p 3 jelly, etc.) have been around for some time, little is known about their quantum properties [4—9].Of paramount importance is to determine whether some of these p ~ 2 theories are quantum mechanically consistent, Lorentz covariant, and ghost free. A perturbative investigation of the spectrum in the ghost-free light-cone gauge has provided hints that bosonic p-branes are quantum inconsistent for any p ~ 2 in any space-time dimension d [10]. On the other hand, among the classically possible 12 super p-branes only the d 10 super-string and d 11 super-membranes pass the consistency checks of a Lorentz-covariant massive spectrum at low mass levels [5,6]. But this is insufficient to determine whether the d 11 supermembrane is fully consistent. In a non-linear theory, it is possible that singular gauge choices as well as operator-ordering ambiguities may lead to different quantum theories in different “gauges” (contrast for example string theory in the light-cone gauge [11] with the time-like gauge [12] or the path integral quantization for d < 26 including a =
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Research supported in part by the U.S. Department of Energy Grant No. DE-FGO3-84ER40168.
0550-3213/90/$03.50 © 1990
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Elsevier Science Publishers B.V. (North-Holland)
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Liouville mode [13]). Accordingly, it is also important to investigate these theories in other gauges, in particular in Lorentz-covariant gauges. Some covariant and non-covariant gauges for membranes are proposed in this paper. In each covariant gauge we indicate some infinite-dimensional sub-groups of the diffeomorphism group that have not been noticed until the present. These generalize the Virasoro symmetry of string theory in the sense described below. In a Lorentz-covariant gauge one has a large Hilbert space that contains ghosts due to the indefinite Lorentz metric. The issue in this case is to show that the algebra of gauge generators (constraints) closes without quantum anomalies or that these anomalies, if present, cancel against BRST ghost anomalies. Then either by imposing the constraints on the states or by using BRST methods one must determine that the physical Hilbert space is ghost free. If these procedures fail the theory would have ghosts in the covariant Hilbert space and therefore would be quantum inconsistent. For p-branes one would like to know if there are any consistent theories in any dimension. In the string case quantum fluctuations induce a central extension (or anomaly) in the conformal algebra. This anomaly is responsible for distinguishing the d 26 string and d 10 super-string (and their compactified versions) as the only consistent quantum string theories. In this paper we develop the methods for performing the analogous computation for the symmetries in membrane theory indicated above and apply it to the full diffeomorphism group in the Lorentzcovariant orthonormal gauge. We find no anomalies in some sub-groups but we do find that the full diffeomorphism group has operator as well as c-number anomalies. The operator anomaly turns out to be proportional to the metric. The operator anomaly found in this paper presents problems for the purely bosonic membrane in any dimension and we conclude that purely bosonic membranes are inconsistent quantum theories in the covariant orthonormal gauge. This confirms the hint provided in refs. [5,6, 10] whose investigations were in the light-cone gauge that is a sub-gauge of the covariant orthonormal gauge. The similar problem for the 11-dimensional super-membrane is more involved and is currently under investigation [14]. Throughout this paper we will investigate a general closed bosonic membrane embedded in flat Minkowski space in d dimensions. It is described by the position vector X~(~’), where ~ (r, o-~,o7), p. 0,1 (d 1). At fixed time T, X~(ff) describes a two-dimensional Riemann surface of genus g. Thus g 0 is a sphere, g 1 is a torus, etc. Any of these membrane topologies may appear in interactions since interactions must include topology changes. The topology implies different periodicity conditions on X~(o-)for different genus g. The complete Hilbert space is the direct sum of the Hilbert spaces of all membrane topologies [5, 15] and therefore all topologies must be investigated (for an analogous situation consider the Neveu—Schwarz—Ramond super-string that has NS and R sectors that have different boundary conditions). Issues concerning massless states and the number =
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of gravitons in the theory cannot be settled before considering the effects of all membrane topologies [15]. For arbitrary genus g the action is ~fdrfd2~
~
—
~),
(1.1)
where y13(r, o), i, j 0, 1,2, is a metric, y det y,~is its determinant, and the d2a integral is performed on the surface ~ of genus g. The membrane X~(r,o’) must satisfy the correct periodicity conditions in o~so that the global properties of the surface are taken into account. One way of insuring this property is to expand =
=
X~(r,ff)
=
LX~’(T)Yj(r),
(1.2)
I
where the functions Y 1(o) form a complete and orthonormal set on the surface ~ described by the metric Wab, a, b 1, 2 and determinant w det(wab) (this metric may always be chosen to be locally flat Wab ~ab for any Riemannn surface, while its global properties may be described by Teichmuller parameters) =
=
=
~w(o)
f d2
Y 1(r)Y~(r) =i~,
(1.3)
and the ‘r~can be thought of as an invertible metric in the space of the discrete set of labels I, s~hJ~qJ~ ö~.For example, for a sphere (g 0) we can use the spherical harmonics Y1(o’) Y~m(CO50, ~), and for a torus (g 1) we have Y1(o’) exp(in o). We may use any complete set of functions constructed from these provided they are periodic on the corresponding surface. We will not specify which convenient set we must use until needed. When the surface has genus g 0, and the X’~are compactified in some directions, then the Y1 may be supplemented by additional functions 1~(o-) o~avaa(o),where the V~a(O~),a 1,2, a 1,2, 2g, are the harmonic vectors on the surface. These 2g vectors are associated with the 2g periodic cycles on There are no such terms for a sphere, while for a flat torus the Vaa may be taken as the diagonal matrix Vaa ôaaRa, where the Ra are the constant radii of the torus. The iç may be used to describe membranes in solitonic sectors in which the coordinates o~,cr~may wrap around the 2g cycles. There are such membrane solutions and they are potentially important in describing compactified membranes [4—6,10] just as in string theory [16]. The entire set of functions YJ,Iç is assumed to be included in the expressions (1.2) and (1.3) through an extension of the sum to include the indices a. The Ya enter only if we are dealing with compactified membrane solutions. In the following we will not consider the case of compactified membranes; the inclusion of this case does not alter the essence of our results. =
=
—‘
=
~
=
=
=
=
. . .
,
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The variation of the action yields the classical metric and the equations of motion
aix. aix,
=
a~(%[y~iaJx~) 0, =
(1.4)
and a vanishing “energy—momentum” tensor on the world-volume
rjj=a,x.3JX—yjj.
(1.5)
We will choose gauges to study this system classically and then analyse the quantum behavior of the symmetries in these gauges.
2. Orthonormal covariant gauge For a world-volume in p an arbitrary metric, öy11
+
1 dimensions, a general covariant transformation on
=
+
Ekf3kyiJ
+
leads to the following trans-
formation rule =
0k(E~Y~)
—
c~
~y1k0
—
~ykJ3c1
(2.1)
We define the orthonormal gauge in any dimension (p + 1) by using the gauge freedom of p + 1 local diffeomorphisms ‘, i 0, 1,. , p to fix p + 1 components of the metric and put it into the form =
~(
0
where hab is the inverse of the p that we can solve ~[1~ h and
..
a, b= l,2,...,p,
hh~)’
x p matrix
hab
and h
=
(2.2)
det(hab). This implies
=
~=
(1
(2.3)
hab)~
There are some simplifications in the case of strings and membranes. For strings (p 1) hh’~’ 1, while for membranes (p 2) hh’~ ~uEt)~ihcd, where ~ab is the two-dimensional Levi—Civita symbol. For strings (p 1) this gauge is the conformal gauge since /~y11 diag(— 1,1), and we know that there are remaining gauge symmetries corresponding to the conformal transformations. This remaining symmetry is crucial since it is responsible for the removal of ghosts from the covariant spectrum of strings. Equivalently, the conformal symmetry may be used to fix the gauge further to a unitary gauge, the light-cone gauge X~ T, in which the absence of ghosts is manifest. Is there an analogous remaining symmetry in p-brane theory? To answer =
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the question, consider the transformation rules of eq. (2.1) and see if there are any non-trivial c’ that keep the form of the gauge-fixed metric (2.2). By specializing (2.1) to i =1 = 0 and i = a, j = 0 and using eq. (2.2), we find that the conditions on ~I are [17] 0ØE = Oa~ 0~ = hh~ThobcO. (2.4) For strings, the solution of these equations corresponds to conformal transformations. More generally for p-branes we propose that the solution space of these equations forms a sub-algebra of the diffeomorphism group that generalizes the conformal symmetry of string theory. We note that there are always solutions to these equations in the form
=
~(hh~0h°)
o~°0a(111~0bE0)= 0.
+Aa(ff),
(2.5)
Thus, c’~is fixed in terms of c° and the i--independent arbitrary function fla(~)~ The form of the second equation for c°is identical to the form (1.4) satisfied by an X’2 (for any p) in the gauge (2.2) —
~
=
0.
(2.6)
Therefore the remaining symmetry has as many degrees of freedom as one component of the vector x~on mass-shell, plus the degrees of freedom in flQ• The transformations associated with such A~correspond to time-independent diffeomorphisms and in the case of membranes they generate the diffeomorphisms of the surface ~ which we will denote as Diff ~g• We may, if we wish, fix a light-cone gauge, X~= i-, by using up the remaining c°,A’~gauge freedom. Then, there still remains the area-preserving diffeomorphisms AP Diff ~ corresponding to OaAa = 0, as is well known. But, in this paper we wish to study membranes in a covariant gauge in which the c°,A~gauge degrees of freedom are not fixed. Therefore, we have to understand the structure of the quantum algebra analogous to the Virasoro algebra of string theory, and obtain its anomaly structure which is of fundamental importance in determining the quantum consistency of the theory. Let us examine the membrane equations of motion (1.4) and (2.6) and constraints (1.5) in the orthonormal covariant gauge. We see that after using the relation hab = OaX OhX, the equation of motion may be derived from the Lorentz-covariant action S
=
fdTf
d2u
[~(0 0X)2
—
~ det(OaX.OhX)].
(2.7)
This corresponds to a quartic interaction among the membrane coordinates X/L.
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The gauge constraints y00 —h, has the vanishing components =
/
‘YOa
Membrane symmetries
=
403
0 lead to an energy—momentum tensor that
2 + ~-det(0aX~0bX) T00
=
0,
~-(00X)
=
T 0a =00X~0aX= 0.
(2.8)
It can easily be checked that the sub-group of diffeomorphism symmetries defined by (2.4) and (2.5) and ÔX~ ‘01X~ are true symmetries of the action (2.7), without using the constraints (2.8). Furthermore, these diffeomorphisms transform T00, T0a of (2.8) into each other, again without requiring that they be identically zero. This means that, if we work in a quantum Hilbert space by 2(G cr’) (thatdefined includes the covariant quantization rules [X~(ff), P’~(ff’)] ig~’~ indefinite-metric ghosts), in which the constraints are not identically zero, we may =
—
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expect that the symmetry, (2.4) and (2.5), classifies states in its representations. Imposition of the constraints on these states would then select the physical sub-space. All this is analogous to the role played by the conformal symmetry or the Virasoro algebra in string theory. These symmetry transformations form a closed algebra at the classical level. We will call these symmetry transformations conformal diffeomorphisms of the three-volume and denote it by CDiffV 3. Evidently, we have the sub-algebra structure DiffV3 ~ CDiffV3 ~ Diff 2g DAPDiff2g. The last term is the area-preserving diffeomorphisnis of the surface by A~of the form OaAa 0, i.e.
(2.9) ~
generated
=
2g =
c’~0hA(o)+ ~ Aa(~)
(2 10)
where the last term is a sum over harmonic vectors on the surface with constant coefficients A” (see second ref. in [9]). It can easily be determined that the generators of these symmetries are as follow DiffV3:
T(11)
CDiffV3:
1~
=
and
T0a,
E°T00+ E~T0a
Diff~g:
10a~
APDiff.~g:
Q(O~)
=~‘0aT~, and
q~
=
1 d2uvcL~Toa,
(2.11)
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where T~describes the generators including the parameters of transformation that are X~-dependentthrough hab, as seen from eq. (2.4). In the commutation rules of T~we should not forget the X dependence of the parameters c~,c”. Using the classical Poisson-bracket algebra between X~ and its conjugate 00X,~,the classical Poisson brackets for the generators of Diff V3 are com=
puted easily
{ T~(O•), Tob(if))
2(o 0h5
=
0a(~
=
0a~~2( if —
2(if— 00(if’)}
T00( o.’))
if’)T
2((T—
if’)Tob(if)
0a(0’) + 0a~
{TOa(if), T
(T00( if),
—
=
o~’)(T 00(if)+ T0~(o’)) if’)
(hh~( if)
~~0b( if)
+ hh’~’((F’)
TOh( if’))
.
(2.12)
Diff ~ is generated by the first line of eq. (2.12) and AP Diff is obtained from that one by differentiation or integration as prescribed in eq. (2.11) and exhibited in ref. [9]. It is more difficult to write the algebra for CDiffV3 since the parameters c°,c” in eqs. (2.4) and (2.5) are X-dependent and are not explicitly known. It is not necessary to construct this algebra to arrive at the conclusions of this paper. Thus, our system has similar features to string theory. Of great interest is to consider the quantum algebra of these operators and determine the anomaly structure. We shall return to this problem after discussing other new covariant and non-covariant gauge choices.
3. Triple conformal covariant gauge There is a variety of other Lorentz-covariant gauge choices that remain invariant under a different set of sub-groups of the diffeomorphism group than those discussed above. For membranes we would3like to display one that has three copies as explicit symmetries. Instead of the of the ordinary conformal group (DiffS1) (i-, ~ cr 2) coordinates we use a new set of coordinates ~ &~, O~~) that are analogous to light-cone coordinates. This new light-cone basis ê1 is related to the time-like basis by e1 (1, 1,0), e2 (1,0, 1), e3 ~ 1, 1)/(V~ 1), which are light-like vectors and satisfy =
=
ejeJ=
=
0 —1 —1
—1 0 —1
—1 —1 0
—
.
(3.1)
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In the new basis we make the gauge choice 0 A3
~[~i~y1J=
A3A2 0 A1
A2A1 which implies
~
(3.2)
,
0
—2A1A2A3, and
=
A~ —A~A2
y,~=
AA
-A1A2
-A1A3
A~ AA 2 3
—A2A3 2 A3
=0~X01X.
(3.3)
Therefore choosing this gauge is equivalent to imposing the constraints T~ 2=01X~02X+ ~(01X)2
V(o2X2
=
0,
T13 =01X~03X+ V(OIX)2 V03x2
=
0,
T~2=02X03X+ ~(02X)2
=0,
~(03X)2
(3.4)
which reduces the equation of motion (1.4) to the form 0i(V(0iX)202Xr~+ V(02X)20iX~) + cyclic (1,2,3)
=
0.
(3.5)
This equation can be derived from the action S= fdTJ[V(03X)202X.oX+ ~(0ix)2o3x.o2x
+
03X+ ~(oix)2(o2x)2(o3x)2
~(02X)201X.
j
(3.6)
provided we use eq. (3.4). This expression makes it evident that the action is 3 of the manifestly form o~ invariant under the triple conformal transformation (DiffS1) 1’ A1(ê1), ö~ A2(o~2), A3(ö3). Thiso~= should be compared to the 2 ofo~ string theory, A ~(o-~), when written in conformal symmetry (DiffS~) light-cone variables O~~=T ±o~. The triple conformal symmetry is only a small part of the symmetry of this gauge. More generally, as in the previous section, we apply the diffeomorphism =
=
=
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transformations ~X~’ c’0,X~’ and look for the space of straints (3.4) and action (3.6) invariant. We find =
(V02x20i +
V(OIX)202)E3
=
0,
(V0ix)203 +
\/(03X)201)2
=
0.
(V(03x)202 +
‘
that leave the con-
V(02X)203)E1
=
0, (3.7)
Unlike the orthonormal gauge of the previous section in which the action was invariant independently from the constraints, in the present gauge the constraints (3.4) must be used to show that the action of (3.6) is invariant under (3.7). The space of solutions of (3.7) is much larger than the triple conformal symmetry exhibited above explicitly. This forms a subgroup of the diffeomorphism group Diff V3 and provides a generalization of conformal symmetry of string theory in a different sense than the previous section.
4. Linear light-cone gauge In string theory the light-cone gauge completely eliminates the unphysical degrees of freedom and at the same time linearizes the equation of motion satisfied by the physical degrees of freedom. In the light-cone gauge proposed by Hoppe [18] the membrane equations remain non-linear and also there remains the area-preserving diffeomorphisms. Here we propose yet another light-cone gauge with no remaining gauge invariance and with a linear equation for the physical degrees of freedom. From the original action (1.1) for any p-brane we define the canonical momentum P~(if) ~y°’0~X~ =
(4.la)
(if)
and, using the y,~of eq. (1.4), we can verify that it satisfies the gauge-independent constraints 2 + det(OQXOhX) 0. (4.lb) OaX•P= 0, P We now choose the gauge partially with the conditions X~=p~r,P~=p~, where p~ is a i-, if-independent constant. Then analyzing the equations of motion (1.4) =
for p.
we learn that OaY°” 0, which, for membranes, is solved by yOa For membranes we can make one more gauge choice. Hoppe’s light-cone gauge is equivalent to taking 4(r, if) 0 in our formalism. We will propose a different choice which gives a purely quadratic hamiltonian, as in string theory, which in turn corresponds to a linear equation of motion. =
+
=
=
~1bobt(T,if).
=
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Before we make our third gauge choice we note that the hamiltonian density is the canonical conjugate to X~,that is P. It is solved from the constraints in eq. (4.lb) as P= (l/2p~)
+
[(p)2
det(OaX’ObX’)]
+
~0a~30bX’P’,
(4.2)
where the label I 1,2, d 2 runs over the transverse components. The first constraint in eq. (4.lb) determines X in terms of the transverse canonical variables through the differential equation OaX= (1/p~)00X’P’,whose integra=
..
.,
—
bility condition demands the constraint ~0aX’0hP’
=
0.
(4.3)
We see that the equation of motion for 4 generated by the hamiltonian (4.2) is precisely this constraint. The equations of motion that (4.2) generates for X’ are identical to those that follow from (1.4) when restricted to the above gauge. Therefore, we have the correct hamiltonian that generates all the conditions of the original theory. We now make our third gauge choice by restricting Pd_2(i-, ~) by the equation P~_~ + det(OaXOhX+OaXd_2OhXd_2)
=
W~1b0aX•0hX
(4.4)
where the X correspond to the components I 1,2,. d 3 that exclude Xd_2, and ~ab is a background metric appropriate to the shape of the membrane defined on ~ This metric may always be chosen to be locally flat for any Riemann surface -~g~Note that the determinant in eq. (4.4) is the same one that appears in the hamiltonian (4.2) except that the terms involving Xd_2 have been explicitly written out. The constraint of eq. (4.3) may now be rewritten by separating out the (d 2)th components and substituting for ~d-2 from (4.4) =
. . ,
—
—
3dX
—
det(3~X-8,X+ OrXd_23,Xd_2)] l/2
+ ahO~X.3hf
=
0.
E~hOaXd_23h[W8r~. (4.5) This is a highly non-linear differential equation whose solution determines Xd_ 2 as a dependent variable in terms of the physical degrees of freedom X, F, thus solving the constraints completely in this gauge. Then ~ is also determined from eq. (4.4). The remaining degrees of freedom X are now governed by the hamiltonian (4.2) after substituting the above form for Xdl, ~d-I• Noting that the 4, term drops after an integration by parts and using the identically solved constraint (4.3), the
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remainder gives P= (1/2p+)(P2
+
(4.6)
W~’0aX•0bX),
which is purely quadratic in the physical canonical variables. By choosing patches of if coordinates on a Riemann surface that correspond to a locally flat metric ~ the equation of motion generated by this hamiltonian can locally be put into the form of the massless Klein—Gordon equation in 2 + 1 dimensions. Its solutions must be chosen to form a basis that obey the global periodicity conditions appropriate to the surface ~ For example for the torus we can simply take the plane waves exp(iif ~ ±i~/n~ + n~~-),with n
(n1, n2) a set of positive or negative integers. The coefficients of these solutions a~,a~are then quantized as harmonic oscillators in the usual way. Then, the remaining components Xd_2, X are determined as non-linear functions of these harmonic oscillators from the equations given above. Thus, in this gauge all the non-linearities of membrane theory have been shifted to the single equation (4.5) that determines X~2. At present this equation remains unsolved. The closure of the Lorentz algebra at the quantum level cannot be checked before determining Xd_2. Of course, we anticipate that Xd_2 is a rather non-linear function of the harmonic oscillators and that the quantum ordering of these oscillators may prevent the closure of the algebra, thus rendering the purely bosonic membrane inconsistent for any dimension d, as anticipated from a study of the spectrum [10]. Indeed, in the present gauge the spectrum is completely known, and it is evident that with only d 3 harmonic oscillators we cannot obtain a Poincaré covariant spectrum for neither massless nor massive states, in any dimension d. The promising d 11 super-membrane may be discussed in a similar gauge, including the fermions. We will not elaborate here on the super-membrane since Xd_2 remains unsolved in this light-cone gauge and the question of the closure of the super Poincaré algebra cannot be answered at the operator level for the time being. However it is worth noting that the hamiltonian can be brought to a quadratic form involving a free Klein—Gordon equation for X’, I 1,2,. 8 and a free Dirac equation involving the SO(8) spinors 8~,8_. For the flat toroidal membrane this system is manifestly invariant under super Poincaré transformations in 8 dimensions. We may now apply the consistency checks of ref. [5] (which used an S0(8) formalism) to show that the low-lying bosonic and fermionic spectrum (massless states and first massive level) is consistent with the 11 -dimensional super Poincaré symmetry and that this is possible only in 11 dimensions. By starting with the SO(8) representations, it was possible to show that the massless states actually form SO(9) representations, and that the massive states at the first level form SO(10) representations. Furthermore, the degenerate fermions and bosons fit into the correct supermultiplets consistent with supersymmetry in 11 .
=
—
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dimensions. This happens because of remarkable properties of certain representations of SO(8), SO(9), S0(10), SO(16) and SO(32) and does not happen for any of the groups appropriate to other dimensions, as seen in ref. [5]. Note also that in this gauge there is no evidence of a continuous mass spectrum bordering at zero mass. This possibility arose originally in perturbative approaches [5,6], but there it was argued that higher-order corrections would remove the difficulty. More recently, on the basis of toy models, it was speculated that this possibility may arise in the super-membrane non-perturbatively and that it may be a blow to the theory [19], but we see no basis for its correctness in our formalism (see also the arguments in ref. [15]). 5. Normal ordering, operator products and anomalies Failure of the bosonic membrane theory in the light-cone gauge of the previous section, or in the perturbative analysis [10] in Hoppe’s light-cone gauge [17], may perhaps be inconclusive for dismissing bosonic membranes. It is possible that even though various gauges are equivalent at the classical level they may be inequivalent at the quantum level because of operator-ordering ambiguities; after all the gauge transformations that connect various gauges are rather non-linear transformations and they may not do what is expected from them when operator ordering is taken into account. We also emphasize again that the light-cone gauge may be singular even at the classical level, as in the example of strings in which certain longitudinal motions of strings simply cannot be described even classically in the light-cone gauge [12]. Basically what happens in strings is that there are certain classical solutions for which the momentum density P~(o-)vanishes for certain points 0~ that correspond to folds or end-points that move with the speed of light. When we choose the light-cone gauge we require a constant p~(~.) ~ which never vanishes for any point if, thus banishing such motions altogether. In the absence of these motions the string still manages to be quantum consistent, but only in 26 dimensions. It may not have been quantum consistent at all in any dimension, since certain degrees of freedom were eliminated outright. In view of the remarks in ref. [12], the fact that strings are consistent even in a specific dimension in the light-cone gauge is more a surprise than a sure outcome. According to the hints of ref. [5], it may be that 11-dimensional super-membranes also offer the same kind of surprise, and this would make this special case very interesting and perhaps unique, in the sense of being quantum consistent in the light-cone gauge. However, more generally, we hold the view that a choice of gauge is part of the definition of the quantum theory, and that we may have different quantum theories in different gauges. For strings, this certainly appears to be the case when we compare the light-cone gauge to the time-like gauge X°(’ra-)=p°’r which contains longitudinal degrees of freedom [12], or the Polyakov quantization in d < 26 which also contains longitudinal modes in the form of the Liouville degree
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of freedom. Accordingly, we find it necessary to analyze the consistency of bosonic membranes in a covariant gauge before dismissing them as inconsistent theories. This is the purpose of this section. What we would like to determine is whether the algebra of constraints can consistently be applied to the states, and whether this can insure the elimination of ghosts from the covariant spectrum. This could be done successfully in the case of strings, in which the constraints satisfy the Virasoro algebra with an anomaly in the form of a central extension. 5.1. OSCILLATORS AND NORMAL ORDERING
In this section we analyze the quantum bosonic membrane theory in the covariant orthonormal gauge. Our aim is to calculate the quantum commutator algebra associated with the diffeomorphism group DiffV3 and its subgroups and identify any anomalies that would modify the algebra of (2.12) or its subalgebras listed in (2.11). We would like to define an operator-product expansion in order to evaluate commutators carefully. We shall do this below by defining a Fock space in which operators can be ordered unambiguously. In this space, we will be able to use Wick’s theorem (even though we will not be defining perturbation theory) in order to obtain a definition of operator products. It may be that there are other ways of achieving the same result without our Fock space method, but we think that our approach is convenient and well defined. We recall the expansion X~(r,if) = EX~(T)Y’(if),
P~’(r,if) ~P~(T)Y’(if), =
I
(5.1)
I
where PL~ X’-’, and which satisfy the canonical commutation rules =
[x~’,P;]
=ig’~qjj.
(5.2)
We can always express the commutation rules in terms of a set of harmonic oscillators since we have the freedom to write the position and momentum variables in the form X~=
+
s~
,
PIL
=
—i1~a~ + ~
(5.3)
so that eq. (5.2) is equivalent to [a~t,a~”I =g’~lij
(5.4)
for any value of the frequencies o~.Thus we will allow these frequencies to be
I. Bars
/
Membrane symmetries
411
momentarily free parameters and think of them as a kind of variational parameters of the vacuum annihilated by a~ a~0),~=0.
(5.5)
The Fock space built on this “vacuum” is a complete Hilbert space. The oscillator vacuum defined by eq. (5.5) is not necessarily the lowest-energy eigenstate of the hamiltonian, but it can serve to resolve the quantum-ordering problem and give meaning to operator products by a procedure of normal ordering with respect to this vacuum. In principle we may evaluate expectation values of operators in any state of the theory, and the state of (5.5) is a particularly convenient one. Thus, the basis of operators (a,,a~.) provides a well-defined quantum theory in which we can answer the question of anomalies. The anomalies in the quantum version of the algebra (2.12) will be a function of the parameters w”. Our goal is to first compute the anomaly and then examine its structure, allowing ourselves the ability to vary the frequencies. We emphasize that our Hilbert space should not be confused with the usual perturbative Hilbert space. If we were to choose the frequencies to diagonalize the quadratic piece of the hamiltonian, then our Hilbert space and its vacuum state would coincide with their perturbative counterparts. So, it is clear that our proposal is not the evaluation of the anomaly just in perturbation theory, it is more general and non-perturbative, while we keep our option of evaluating the perturbative anomaly by simply choosing the frequencies appropriately. It is generally believed that anomalies are a property of the theory that may be evaluated even in low orders of perturbation theory. If this is true in membrane theory, then our procedure should have no trouble finding the anomaly even if we were to choose the perturbative frequencies. Therefore, we expect that if there is a true anomaly that there would be no reasonable set of frequencies that would cancel it. Using the Hilbert space defined above we define normal ordering. We will always use Euclidean times, and we will use the shorthand notation X(1) =
X(ii-
1,
o~),X(2)
=
X(ii-2,
if2),
etc. Then
X~’(1)X~(2) :X~(1)X~(2):+ KX~’(1)X~(2)), =
(5.6)
where the vacuum expectation value and the normal ordering are performed according to the vacuum (5.5). For convenience we define the symbol (X~(1)X~(2)~ =g~ii~2.
(5.7)
We do not claim that this is the full propagator of the theory, as the full propagator is the one evaluated in the true vacuum of the theory. However, the behavior of this ~ at short distances will give information about anomalies in the
412
1. Bars
/
Membrane symmetries
algebra of constraints, since the vacuum we have defined allows a consistent ordering of operators and our 4 is closely related to this ordering. In arriving at the form of the anomaly, we will need only general properties rather than detailed information about 4. Only at the end we need to discuss details such as the behavior of the unknown frequencies. The final form of the anomalies indicate that only the short-distance behavior of 4 is relevant. In this case the propagator ~12 depends only on the local properties of the surface and cannot be sensitive to the global properties of ~ therefore it can be approximated by the propagator in an infinite-dimensional sheet that is translationally and rotationally invariant and takes the form (5.8) where
~12
=
if
and i-12 According to these definitions, and P~ 0XiL/0(ii~),we have 1
—
if2
=
—
=
(P~(1)P~(2))
K0a~’(1)0hX~(2))
~g~S12,
=
1
=
2)~ ~
(5.9)
=
K0aX~’(1)P~(2)~ = ~~ig1L~0~z 12,
where the derivatives on the right-hand side are with respect to
i-~ 2, ~ 12~
In the limit
i-12
~12
0 we note the general properties
=
=
~(o~
—
if2),
4(
T12
=
0, ~12)
=
~(~12
=
0,
—
if12)
=
0,if12)
=
zI(r12
=
0,
—
~12),
(5.10)
which follow from eqs. (5.6) and (5.7) and the equal time commutation rules X~(1), P ~(2)]~~~ ig~~~(ifi if 2), [ X”(l), X v(2)] ~2 0 and [P~(1), P”(2)]~~. 0. Thus, the properties (5.10) are independent of any specific assumptions on the i--dependence of the propagator, and these will be sufficient to derive the general form of the anomaly in terms of the propagator. =
—
=
=
5.2. OPERATOR PRODUCTS AND COMMUTATORS
We may now define the operator version of the energy—momentum tensor T00, T0a by applying eq. (5.6) to the expressions (2.8) Toa
=
T00
=
:OaXP:, 2: + ~E”’~’:0aX~0bX0cX~0dX:+ ~
~A” :OaX
OhX’~,
(5.11)
:P
where we have dropped a constant from both expressions since they are irrelevant in the commutators. We see that rewriting our expression in normal-ordered form has produced a quadratic term with the coefficient A~’.This last modification of the energy—momentum tensor is reasonable since normal ordering produces the
I. Bars / Membrane symmetries
413
quadratic terms but, because we have no a priori way of determining the order of operators prior to normal ordering, the value of A” is unknown. From covariance on the surface we expect that A” Aw~’,which locally may be taken proportional to ~ab and we will assume that this is the case. Thus, we will keep Aab as a (possibly cutoff-dependent) parameter and check whether closure of the commutation rules requires a specific value for this parameter. Let us now consider composite operators such as those in (5.11) constructed from the canonical variables X, P, or equivalently from the oscillators, A(X(1),P(1))—~A(T1,if1)and B(X(2),P(2))’—B(i-2,o2), which are assumed to be =
normal ordered according to the vacuum (5.5). We may define their operator products by using Wick’s theorem
:X( 1) X(2) X(3) . . : :X( 1’) X(2’) X(3’) .
=4~.:X(2)X(3)
:X(3)
+ ~1I’~22’
X(2’)X(3’)
...
...
X(3’)
..
..
+
.
. .
.
:+
permutations of (123...)
permutations of (123...)
:+permutations of (123...)
~~1I~22’~33
+ ...
.
(5.12)
Then the equal-time commutator [A, B] is carefully defined as [A(1), B(2)]~...T2
=
urn (A(i-, if1)B(0,if2) i—’
—
B(T,if2)A(0,if~)), (5.13)
t)
where r appears only in the first factor. We first evaluate the operator products using Wick’s theorem as above, take the difference between terms, and then go to the i- 0 limit using the general properties of the propagator in eq. (5.10). The terms containing a single propagator in the operator-product expansion will contribute to the commutator the same expression as the naive classical Poisson bracket, while those containing two or more propagators may lead to anomalous terms in the commutation rules. We apply this procedure to compute the quantum algebra of the diffeomorphism group and its sub-groups. We will do explicitly the simpler case of Diff ~ generated by T0a and quote the result for the rest. We first evaluate the operator product =
Toa(TI,ifI)Tob(T2,if2)
=
:OaXP: (1) :0~X~P: (2) 2)
+ 2L112 :0~X(1).
=
+
OhX(
+~~0b~I2 :OaX(1) ~P(2):+ 4”~12
~ a b’’12
:0~X~P(1) 00XP(2): ~~0a0b4I2 :P(1) P(2):
—
2~a~l2
~ ~ A ~y”ea~12”h~12
:P(1) .OX(2):
1. Bars / Membrane symmetries
414
and then apply eqs. (5.5) and (5.10) to obtain the commutator 2(ifi
[TOa(if !), T
0~(if2)] =
~
if2)
—
TOa(if2) +
~0a~
—
if 2)Toh(ifl)
,
(5.15)
which is identical to the classical Poisson bracket result (2.12). Note that no detailed information of the r dependence of ~I2 was needed here, and only the generally correct information (5.10) was sufficient. Hence there is no anomaly in the subalgebra Diff Furthermore the area-preserving diffeomorphisrns AP Diff ~g listed in eq. (2.11) cannot have an anomaly since it is to be obtained directly from eq. (5.15) by differentiation and integration. In ref. [9]it was argued that, on purely algebraic grounds, the area-preserving diffeomorphisms could have a central extension. We have now shown that in bosonic membrane theory this central extension is absent. The quantum algebra for area-preserving diffeomorphisms is then 2(if
[Q(if1),
If the surface
Q(if2)]
_jEtho~~
=
is a torus we expand Q(if)
~g
1
=
—
(5.16)
if2)0~Q(if1).
~J~L~e’~”. Then eq. (5.16) implies
[L~,L~] =n XmLn+m,
(5.17)
without an anomaly. With the same procedure we find that the commutation rules involving T00 do have anomalies. Instead of the Poisson bracket algebra of eq. (2.12) we now have [T00(if1),T00(if2)]
=iacE0a62(ifl_if2)(:T~b(1)[hCd(1)+ACd]:+(1~~~2)),
(5.18) where hbc
=
:ObX 0~X:.If desired we may use Wick’s theorem to rewrite
:TOb( 1) [hCd(1)
ACd] :
+
=
[hCd(1)
+
AcdI TOb(l) ,
(5.19)
where ACd, which represents the deviation from the Poisson bracket algebra of (2.12), is a new constant that depends on the old ALd and the propagator 4. Finally we have the commutator with the worse anomaly 2(ifi
[TOa(ifi), T00(if2)]
=
—
if 2)(T00(if1) + T00(if2))
~0a~ —
j4hc((F
—
if2)hbC(if2) + iB~(if1—
~2)’
(5.20)
I. Bars / Membrane symmetries
415
where the anomalous coefficients are given by A~~C = ~(d
Ba
—
1)cbkcCt[Iim(3a0(k4l20I)~l2)]
l~m(~120~~12
=
—
+ (AbCOa
—
l~(bAc)ko)~2(if
—
(5.21)
Abc0a0bLtl20c~l2)1.
To instill confidence in our expressions, we point out that the same procedure can be applied for strings. The A-anomaly which arose because of the quartic terms in T00 are absent for strings, but the B-anomaly is present. Thus, specializing our expression to one dimension, and taking A 1, we have the string anomaly B ~d(~ 4”)~’ where the prime indicates a if-derivative. Using the string propagator, 4 ln(i-~+ us), this expression becomes proportional to ö”(o-12), which is the correct anomaly in the Virasoro algebra. Up to this point 2(if we have used only the general properties (5.10). We see that lirnT,,O ~12 ö 12) indicates in (5.21) we only need the short-distance behavior of 4. We might reasonably expect a propagator that has a singular behavior near 0~~I2 0 and then the product of two propagators is expected to produce terms that are even more singular. Thus, a cubic derivative of the delta function can easily appear in the coefficient A. This can be seen by taking the simple example of the free particle propagator =
=
—
=
=
=
w(k)
=
1
~k~+k~
~~12
—
(5.22)
.
4ii-~/i-~+
if~2
More generally, we may try propagators of the form 2k 4 12(r12,
~i2I)
f
1
d exp[ —w(k)i-12
2
(2ir)
+
ik
if12]
(5.23)
W~tL)
that satisfy (5.10) for any frequencies w(k) (they are closely related to those of the eq. 4 then (5.3)). If the large-k behavior of the w(k) blows up faster than k A-anomaly can be cancelled by choosing appropriately the unknown coefficient A, while the B-anomaly becomes divergent. However, this behavior seems strange since the propagator becomes non-singular for small T, o~.If the w(k) correspond to a singular behavior of propagators near the origin, then all such w(k) produce cubic derivatives of the delta function in the A-anomaly. With such behavior it appears impossible to cancel terms in the anomalous coefficient A~C for any value of the constant A~~c even if we allow this constant to depend on the cutoff parameter T. In the presence of this anomaly, the algebra (5.20) contains the
416
1. Bars / Membrane symmetries
operator hbC on the right-hand side, indicating that DiffV3, which is equivalent to the algebra of constraints, does not close at the quantum level. Recall that the physical states are defined as those satisfying T00Iphys) 0 =
=
aiphys). Therefore, the commutator of any two constraints is required to annihilate the physical states. If the non-closure were only a c-number central extension as in string theory, it could be handled by requiring that only the positive frequency components of the constraints annihilate the physical states. However, we have now found an operator anomaly. Under such circumstances, in order to construct a consistent quantum theory, we would want to enlarge the algebra of constraints by including the operator ~ in our list of constraints. However, demanding the constraint hat, 0 on the states would require that the entire metric, as given in eq. (2.3), vanishes on the states ~ 0. This reduces the theory to triviality. Therefore, we conclude that the purely bosonic membrane cannot be a consistent theory in any dimension d! This result is in agreement with previous negative results in sect. 4 and refs. [5, 6, 10] based on the Lorentz invariance consistency checks in the light-cone gauge. For the purely bosonic membrane and higher p-branes we had concluded that they were not Lorentz invariant in any dimension d [10]. We point out that the light-cone gauge of Hoppe (which is the one used in the previous consistency checks) is a sub-gauge of the present covariant conformal gauge, as explained in sect. 2. In this light-cone gauge the Lorentz generator performs a naive gauge transformation followed by a diffeornorphism transformation that restores the gauge. It is not surprizing that quantum Lorentz invariance would fail in such a gauge since, as we have now established, the underlying diffeomorphism group algebra fails at the quantum level. The present result distinguishes bosonic strings from all other extended bosonic p-branes as having the unique property of being quantum consistent. The next interesting question is “what about super p-branes”? Unfortunately, we do not yet know how to treat the super p-branes (including the Green—Schwarz super-string) quantum mechanically in covariant gauges. Only light-cone quantization has been developed. Instead, we may apply the present operator-product techniques to compute the anomalies in the super Poincaré algebra in the light-cone gauge. This is considerably more involved algebraically, and is currently under investigation [14]. It is evident that using our method we should be able to determine whether the 11-dimensional super-membrane, that survived the consistency checks, is really quantum consistent or not. =
=
References [11 J. Hughes, J. Liu and J. Polchinski, Phys. Lett. B180 (1986) 370 [2] E. Bergshoeff, E. Sezgin and P.K. Townsend, Phys. Lett. B189 (1987) 75 [3] A. Achbcarro, J.M. Evans, P.K. Townsend and D. Witshire, Phys. Lett. B198 (1987) 441
I. Bars / Membrane symmetries
417
[4] M.J. Duff, T. Inami, C.N. Pope, E. Sezgin and K. Stelle, NucI. Phys. B297 (1988) 515 [5] I. Bars, NucI. Phys. B308 (1988) 462 [6] I. Bars and C.N. Pope, Class, and Quantum Gray. 5 (1988) 1157 [71B. de Wit, J. Hoppe and H. Nicolai, NucI. Phys. B305 (1988) 545 [8] K. Kikkawa and M. Yamasaki, Prog. Theor. Phys. 76 (1986) 379; L. Mezincescu, R. Nepomechie, P. van Nieuwenhuizen, UMTG-139/ITP-SB-84-43; I. Bars, C.N. Pope and E. Sezgin, Phys. Lett, B198 (1987) 455; S. Ghandi and K. Stelle, Class, and Quantum Gray. 5 (1988) 127; C.N. Pope and K. Stelle, Class, and Quantum Gray. (1989) [9] E. Floratos and J. Illiopoulos, Phys. Lett, B201 (1988) 237; I. Bars, C.N. Pope and E. Sezgin, Phys. Lett, B210 (1988) 85; D. Fairlie, P. Fletcher and C. Zachos, Phys. Lett. B218 (1989) 203; C.N. Pope and L. Romans, Class. Quantum Gray. 7 (1990) 97 [10] I. Bars, preprint USC-88/HEPO6, in Proc. of 17th Int. Conf. on Group theoretical methods in physics (Montreal 1988), eds, Y. Saint-Aubin and L. Vinet (World Scientific, Singapore, 1989), p. 693 [11] M. Green, J. Schwarz and E. Witten, Superstring theory (Cambridge, 1987) [12] WA. Bardeen, I. Bars, A.J. Hanson and RD. Peccei, Phys. Rev. D13 (1976) 2364; D14 (1976) 2193 [13] A. Polyakov, Phys. Lett. B103 (1981) 207 [141 I. Bars and K. Sfetsos, to be published [15] I. Bars, Issues of topology and the spectrum of the supermembrane, preprint USC-89/026, Proc. Workshop on membranes and physics in 2 + I dimensions, Trieste (1989) [16] Heterotic superstring in ref. [11], and its 4-dimensional versions [17] I. Bars, Lecture at Aspen workshop, 1987 [18] J. Hoppe, MIT Ph.D. thesis, 1982 [19] B. de Wit, M. Liisher and H. Nicolai, Nucl. Phys. B320 (1989) 135
MEMBRAJ~4ESYMMETRIES AND ANOMALIES Itzhak BARS* Department of Physics, UniLersity of Southern California, Los Angeles, CA 90089-0484, USA Received 18 December 1989
The symmetries of the bosonic membrane are analysed in several gauges. We discuss two Lorentz-covariant gauges, and a new light-cone gauge that reduces the membrane hamiltonian to a quadratic form which is diagonalized by harmonic oscillators. We then discuss in detail the Lorentz-covariarit quantization and the algebra of constraints that corresponds to the diffeomorphisms in three dimensions (DiffV 3). We introduce an operator product expansion which allows the careful computation of commutators. We find that the subalgebras corresponding to diffeomorphisms of the two-dimensional genus-g membrane surface (Diff~g)and the area-preserving diffeomorphisms (AP Diff ~ do not have any anomalies, but the full algebra Diff V3 has operator anomalies which render the quantization inconsistent. We thus conclude that purely bosonic membranes are quantum mechanically inconsistent in any number of dimensions d.
1. Introduction
Although membrane, super-membrane, and higher (super) p-brane [1—3],theories (p 1 string, p 2 membrane, p 3 jelly, etc.) have been around for some time, little is known about their quantum properties [4—9].Of paramount importance is to determine whether some of these p ~ 2 theories are quantum mechanically consistent, Lorentz covariant, and ghost free. A perturbative investigation of the spectrum in the ghost-free light-cone gauge has provided hints that bosonic p-branes are quantum inconsistent for any p ~ 2 in any space-time dimension d [10]. On the other hand, among the classically possible 12 super p-branes only the d 10 super-string and d 11 super-membranes pass the consistency checks of a Lorentz-covariant massive spectrum at low mass levels [5,6]. But this is insufficient to determine whether the d 11 supermembrane is fully consistent. In a non-linear theory, it is possible that singular gauge choices as well as operator-ordering ambiguities may lead to different quantum theories in different “gauges” (contrast for example string theory in the light-cone gauge [11] with the time-like gauge [12] or the path integral quantization for d < 26 including a =
=
=
=
=
=
*
Research supported in part by the U.S. Department of Energy Grant No. DE-FGO3-84ER40168.
0550-3213/90/$03.50 © 1990
—
Elsevier Science Publishers B.V. (North-Holland)
I. Bars / Membrane symmetries
399
Liouville mode [13]). Accordingly, it is also important to investigate these theories in other gauges, in particular in Lorentz-covariant gauges. Some covariant and non-covariant gauges for membranes are proposed in this paper. In each covariant gauge we indicate some infinite-dimensional sub-groups of the diffeomorphism group that have not been noticed until the present. These generalize the Virasoro symmetry of string theory in the sense described below. In a Lorentz-covariant gauge one has a large Hilbert space that contains ghosts due to the indefinite Lorentz metric. The issue in this case is to show that the algebra of gauge generators (constraints) closes without quantum anomalies or that these anomalies, if present, cancel against BRST ghost anomalies. Then either by imposing the constraints on the states or by using BRST methods one must determine that the physical Hilbert space is ghost free. If these procedures fail the theory would have ghosts in the covariant Hilbert space and therefore would be quantum inconsistent. For p-branes one would like to know if there are any consistent theories in any dimension. In the string case quantum fluctuations induce a central extension (or anomaly) in the conformal algebra. This anomaly is responsible for distinguishing the d 26 string and d 10 super-string (and their compactified versions) as the only consistent quantum string theories. In this paper we develop the methods for performing the analogous computation for the symmetries in membrane theory indicated above and apply it to the full diffeomorphism group in the Lorentzcovariant orthonormal gauge. We find no anomalies in some sub-groups but we do find that the full diffeomorphism group has operator as well as c-number anomalies. The operator anomaly turns out to be proportional to the metric. The operator anomaly found in this paper presents problems for the purely bosonic membrane in any dimension and we conclude that purely bosonic membranes are inconsistent quantum theories in the covariant orthonormal gauge. This confirms the hint provided in refs. [5,6, 10] whose investigations were in the light-cone gauge that is a sub-gauge of the covariant orthonormal gauge. The similar problem for the 11-dimensional super-membrane is more involved and is currently under investigation [14]. Throughout this paper we will investigate a general closed bosonic membrane embedded in flat Minkowski space in d dimensions. It is described by the position vector X~(~’), where ~ (r, o-~,o7), p. 0,1 (d 1). At fixed time T, X~(ff) describes a two-dimensional Riemann surface of genus g. Thus g 0 is a sphere, g 1 is a torus, etc. Any of these membrane topologies may appear in interactions since interactions must include topology changes. The topology implies different periodicity conditions on X~(o-)for different genus g. The complete Hilbert space is the direct sum of the Hilbert spaces of all membrane topologies [5, 15] and therefore all topologies must be investigated (for an analogous situation consider the Neveu—Schwarz—Ramond super-string that has NS and R sectors that have different boundary conditions). Issues concerning massless states and the number =
=
=
=
—
=
=
400
/
1. Bars
Membrane symmetries
of gravitons in the theory cannot be settled before considering the effects of all membrane topologies [15]. For arbitrary genus g the action is ~fdrfd2~
~
—
~),
(1.1)
where y13(r, o), i, j 0, 1,2, is a metric, y det y,~is its determinant, and the d2a integral is performed on the surface ~ of genus g. The membrane X~(r,o’) must satisfy the correct periodicity conditions in o~so that the global properties of the surface are taken into account. One way of insuring this property is to expand =
=
X~(r,ff)
=
LX~’(T)Yj(r),
(1.2)
I
where the functions Y 1(o) form a complete and orthonormal set on the surface ~ described by the metric Wab, a, b 1, 2 and determinant w det(wab) (this metric may always be chosen to be locally flat Wab ~ab for any Riemannn surface, while its global properties may be described by Teichmuller parameters) =
=
=
~w(o)
f d2
Y 1(r)Y~(r) =i~,
(1.3)
and the ‘r~can be thought of as an invertible metric in the space of the discrete set of labels I, s~hJ~qJ~ ö~.For example, for a sphere (g 0) we can use the spherical harmonics Y1(o’) Y~m(CO50, ~), and for a torus (g 1) we have Y1(o’) exp(in o). We may use any complete set of functions constructed from these provided they are periodic on the corresponding surface. We will not specify which convenient set we must use until needed. When the surface has genus g 0, and the X’~are compactified in some directions, then the Y1 may be supplemented by additional functions 1~(o-) o~avaa(o),where the V~a(O~),a 1,2, a 1,2, 2g, are the harmonic vectors on the surface. These 2g vectors are associated with the 2g periodic cycles on There are no such terms for a sphere, while for a flat torus the Vaa may be taken as the diagonal matrix Vaa ôaaRa, where the Ra are the constant radii of the torus. The iç may be used to describe membranes in solitonic sectors in which the coordinates o~,cr~may wrap around the 2g cycles. There are such membrane solutions and they are potentially important in describing compactified membranes [4—6,10] just as in string theory [16]. The entire set of functions YJ,Iç is assumed to be included in the expressions (1.2) and (1.3) through an extension of the sum to include the indices a. The Ya enter only if we are dealing with compactified membrane solutions. In the following we will not consider the case of compactified membranes; the inclusion of this case does not alter the essence of our results. =
=
—‘
=
~
=
=
=
=
. . .
,
I. Bars
/
Membrane symmetries
401
The variation of the action yields the classical metric and the equations of motion
aix. aix,
=
a~(%[y~iaJx~) 0, =
(1.4)
and a vanishing “energy—momentum” tensor on the world-volume
rjj=a,x.3JX—yjj.
(1.5)
We will choose gauges to study this system classically and then analyse the quantum behavior of the symmetries in these gauges.
2. Orthonormal covariant gauge For a world-volume in p an arbitrary metric, öy11
+
1 dimensions, a general covariant transformation on
=
+
Ekf3kyiJ
+
leads to the following trans-
formation rule =
0k(E~Y~)
—
c~
~y1k0
—
~ykJ3c1
(2.1)
We define the orthonormal gauge in any dimension (p + 1) by using the gauge freedom of p + 1 local diffeomorphisms ‘, i 0, 1,. , p to fix p + 1 components of the metric and put it into the form =
~(
0
where hab is the inverse of the p that we can solve ~[1~ h and
..
a, b= l,2,...,p,
hh~)’
x p matrix
hab
and h
=
(2.2)
det(hab). This implies
=
~=
(1
(2.3)
hab)~
There are some simplifications in the case of strings and membranes. For strings (p 1) hh’~’ 1, while for membranes (p 2) hh’~ ~uEt)~ihcd, where ~ab is the two-dimensional Levi—Civita symbol. For strings (p 1) this gauge is the conformal gauge since /~y11 diag(— 1,1), and we know that there are remaining gauge symmetries corresponding to the conformal transformations. This remaining symmetry is crucial since it is responsible for the removal of ghosts from the covariant spectrum of strings. Equivalently, the conformal symmetry may be used to fix the gauge further to a unitary gauge, the light-cone gauge X~ T, in which the absence of ghosts is manifest. Is there an analogous remaining symmetry in p-brane theory? To answer =
=
=
=
=
=
=
402
1. Bars
/
Membrane symmetries
the question, consider the transformation rules of eq. (2.1) and see if there are any non-trivial c’ that keep the form of the gauge-fixed metric (2.2). By specializing (2.1) to i =1 = 0 and i = a, j = 0 and using eq. (2.2), we find that the conditions on ~I are [17] 0ØE = Oa~ 0~ = hh~ThobcO. (2.4) For strings, the solution of these equations corresponds to conformal transformations. More generally for p-branes we propose that the solution space of these equations forms a sub-algebra of the diffeomorphism group that generalizes the conformal symmetry of string theory. We note that there are always solutions to these equations in the form
=
~(hh~0h°)
o~°0a(111~0bE0)= 0.
+Aa(ff),
(2.5)
Thus, c’~is fixed in terms of c° and the i--independent arbitrary function fla(~)~ The form of the second equation for c°is identical to the form (1.4) satisfied by an X’2 (for any p) in the gauge (2.2) —
~
=
0.
(2.6)
Therefore the remaining symmetry has as many degrees of freedom as one component of the vector x~on mass-shell, plus the degrees of freedom in flQ• The transformations associated with such A~correspond to time-independent diffeomorphisms and in the case of membranes they generate the diffeomorphisms of the surface ~ which we will denote as Diff ~g• We may, if we wish, fix a light-cone gauge, X~= i-, by using up the remaining c°,A’~gauge freedom. Then, there still remains the area-preserving diffeomorphisms AP Diff ~ corresponding to OaAa = 0, as is well known. But, in this paper we wish to study membranes in a covariant gauge in which the c°,A~gauge degrees of freedom are not fixed. Therefore, we have to understand the structure of the quantum algebra analogous to the Virasoro algebra of string theory, and obtain its anomaly structure which is of fundamental importance in determining the quantum consistency of the theory. Let us examine the membrane equations of motion (1.4) and (2.6) and constraints (1.5) in the orthonormal covariant gauge. We see that after using the relation hab = OaX OhX, the equation of motion may be derived from the Lorentz-covariant action S
=
fdTf
d2u
[~(0 0X)2
—
~ det(OaX.OhX)].
(2.7)
This corresponds to a quartic interaction among the membrane coordinates X/L.
I. Bars
The gauge constraints y00 —h, has the vanishing components =
/
‘YOa
Membrane symmetries
=
403
0 lead to an energy—momentum tensor that
2 + ~-det(0aX~0bX) T00
=
0,
~-(00X)
=
T 0a =00X~0aX= 0.
(2.8)
It can easily be checked that the sub-group of diffeomorphism symmetries defined by (2.4) and (2.5) and ÔX~ ‘01X~ are true symmetries of the action (2.7), without using the constraints (2.8). Furthermore, these diffeomorphisms transform T00, T0a of (2.8) into each other, again without requiring that they be identically zero. This means that, if we work in a quantum Hilbert space by 2(G cr’) (thatdefined includes the covariant quantization rules [X~(ff), P’~(ff’)] ig~’~ indefinite-metric ghosts), in which the constraints are not identically zero, we may =
—
=
expect that the symmetry, (2.4) and (2.5), classifies states in its representations. Imposition of the constraints on these states would then select the physical sub-space. All this is analogous to the role played by the conformal symmetry or the Virasoro algebra in string theory. These symmetry transformations form a closed algebra at the classical level. We will call these symmetry transformations conformal diffeomorphisms of the three-volume and denote it by CDiffV 3. Evidently, we have the sub-algebra structure DiffV3 ~ CDiffV3 ~ Diff 2g DAPDiff2g. The last term is the area-preserving diffeomorphisnis of the surface by A~of the form OaAa 0, i.e.
(2.9) ~
generated
=
2g =
c’~0hA(o)+ ~ Aa(~)
(2 10)
where the last term is a sum over harmonic vectors on the surface with constant coefficients A” (see second ref. in [9]). It can easily be determined that the generators of these symmetries are as follow DiffV3:
T(11)
CDiffV3:
1~
=
and
T0a,
E°T00+ E~T0a
Diff~g:
10a~
APDiff.~g:
Q(O~)
=~‘0aT~, and
q~
=
1 d2uvcL~Toa,
(2.11)
404
1. Bars
/
Membrane symmetries
where T~describes the generators including the parameters of transformation that are X~-dependentthrough hab, as seen from eq. (2.4). In the commutation rules of T~we should not forget the X dependence of the parameters c~,c”. Using the classical Poisson-bracket algebra between X~ and its conjugate 00X,~,the classical Poisson brackets for the generators of Diff V3 are com=
puted easily
{ T~(O•), Tob(if))
2(o 0h5
=
0a(~
=
0a~~2( if —
2(if— 00(if’)}
T00( o.’))
if’)T
2((T—
if’)Tob(if)
0a(0’) + 0a~
{TOa(if), T
(T00( if),
—
=
o~’)(T 00(if)+ T0~(o’)) if’)
(hh~( if)
~~0b( if)
+ hh’~’((F’)
TOh( if’))
.
(2.12)
Diff ~ is generated by the first line of eq. (2.12) and AP Diff is obtained from that one by differentiation or integration as prescribed in eq. (2.11) and exhibited in ref. [9]. It is more difficult to write the algebra for CDiffV3 since the parameters c°,c” in eqs. (2.4) and (2.5) are X-dependent and are not explicitly known. It is not necessary to construct this algebra to arrive at the conclusions of this paper. Thus, our system has similar features to string theory. Of great interest is to consider the quantum algebra of these operators and determine the anomaly structure. We shall return to this problem after discussing other new covariant and non-covariant gauge choices.
3. Triple conformal covariant gauge There is a variety of other Lorentz-covariant gauge choices that remain invariant under a different set of sub-groups of the diffeomorphism group than those discussed above. For membranes we would3like to display one that has three copies as explicit symmetries. Instead of the of the ordinary conformal group (DiffS1) (i-, ~ cr 2) coordinates we use a new set of coordinates ~ &~, O~~) that are analogous to light-cone coordinates. This new light-cone basis ê1 is related to the time-like basis by e1 (1, 1,0), e2 (1,0, 1), e3 ~ 1, 1)/(V~ 1), which are light-like vectors and satisfy =
=
ejeJ=
=
0 —1 —1
—1 0 —1
—1 —1 0
—
.
(3.1)
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/
Membrane symmetries
405
In the new basis we make the gauge choice 0 A3
~[~i~y1J=
A3A2 0 A1
A2A1 which implies
~
(3.2)
,
0
—2A1A2A3, and
=
A~ —A~A2
y,~=
AA
-A1A2
-A1A3
A~ AA 2 3
—A2A3 2 A3
=0~X01X.
(3.3)
Therefore choosing this gauge is equivalent to imposing the constraints T~ 2=01X~02X+ ~(01X)2
V(o2X2
=
0,
T13 =01X~03X+ V(OIX)2 V03x2
=
0,
T~2=02X03X+ ~(02X)2
=0,
~(03X)2
(3.4)
which reduces the equation of motion (1.4) to the form 0i(V(0iX)202Xr~+ V(02X)20iX~) + cyclic (1,2,3)
=
0.
(3.5)
This equation can be derived from the action S= fdTJ[V(03X)202X.oX+ ~(0ix)2o3x.o2x
+
03X+ ~(oix)2(o2x)2(o3x)2
~(02X)201X.
j
(3.6)
provided we use eq. (3.4). This expression makes it evident that the action is 3 of the manifestly form o~ invariant under the triple conformal transformation (DiffS1) 1’ A1(ê1), ö~ A2(o~2), A3(ö3). Thiso~= should be compared to the 2 ofo~ string theory, A ~(o-~), when written in conformal symmetry (DiffS~) light-cone variables O~~=T ±o~. The triple conformal symmetry is only a small part of the symmetry of this gauge. More generally, as in the previous section, we apply the diffeomorphism =
=
=
406
1. Bars
/
Membrane symmetries
transformations ~X~’ c’0,X~’ and look for the space of straints (3.4) and action (3.6) invariant. We find =
(V02x20i +
V(OIX)202)E3
=
0,
(V0ix)203 +
\/(03X)201)2
=
0.
(V(03x)202 +
‘
that leave the con-
V(02X)203)E1
=
0, (3.7)
Unlike the orthonormal gauge of the previous section in which the action was invariant independently from the constraints, in the present gauge the constraints (3.4) must be used to show that the action of (3.6) is invariant under (3.7). The space of solutions of (3.7) is much larger than the triple conformal symmetry exhibited above explicitly. This forms a subgroup of the diffeomorphism group Diff V3 and provides a generalization of conformal symmetry of string theory in a different sense than the previous section.
4. Linear light-cone gauge In string theory the light-cone gauge completely eliminates the unphysical degrees of freedom and at the same time linearizes the equation of motion satisfied by the physical degrees of freedom. In the light-cone gauge proposed by Hoppe [18] the membrane equations remain non-linear and also there remains the area-preserving diffeomorphisms. Here we propose yet another light-cone gauge with no remaining gauge invariance and with a linear equation for the physical degrees of freedom. From the original action (1.1) for any p-brane we define the canonical momentum P~(if) ~y°’0~X~ =
(4.la)
(if)
and, using the y,~of eq. (1.4), we can verify that it satisfies the gauge-independent constraints 2 + det(OQXOhX) 0. (4.lb) OaX•P= 0, P We now choose the gauge partially with the conditions X~=p~r,P~=p~, where p~ is a i-, if-independent constant. Then analyzing the equations of motion (1.4) =
for p.
we learn that OaY°” 0, which, for membranes, is solved by yOa For membranes we can make one more gauge choice. Hoppe’s light-cone gauge is equivalent to taking 4(r, if) 0 in our formalism. We will propose a different choice which gives a purely quadratic hamiltonian, as in string theory, which in turn corresponds to a linear equation of motion. =
+
=
=
~1bobt(T,if).
=
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/
Membrane symmetries
407
Before we make our third gauge choice we note that the hamiltonian density is the canonical conjugate to X~,that is P. It is solved from the constraints in eq. (4.lb) as P= (l/2p~)
+
[(p)2
det(OaX’ObX’)]
+
~0a~30bX’P’,
(4.2)
where the label I 1,2, d 2 runs over the transverse components. The first constraint in eq. (4.lb) determines X in terms of the transverse canonical variables through the differential equation OaX= (1/p~)00X’P’,whose integra=
..
.,
—
bility condition demands the constraint ~0aX’0hP’
=
0.
(4.3)
We see that the equation of motion for 4 generated by the hamiltonian (4.2) is precisely this constraint. The equations of motion that (4.2) generates for X’ are identical to those that follow from (1.4) when restricted to the above gauge. Therefore, we have the correct hamiltonian that generates all the conditions of the original theory. We now make our third gauge choice by restricting Pd_2(i-, ~) by the equation P~_~ + det(OaXOhX+OaXd_2OhXd_2)
=
W~1b0aX•0hX
(4.4)
where the X correspond to the components I 1,2,. d 3 that exclude Xd_2, and ~ab is a background metric appropriate to the shape of the membrane defined on ~ This metric may always be chosen to be locally flat for any Riemann surface -~g~Note that the determinant in eq. (4.4) is the same one that appears in the hamiltonian (4.2) except that the terms involving Xd_2 have been explicitly written out. The constraint of eq. (4.3) may now be rewritten by separating out the (d 2)th components and substituting for ~d-2 from (4.4) =
. . ,
—
—
3dX
—
det(3~X-8,X+ OrXd_23,Xd_2)] l/2
+ ahO~X.3hf
=
0.
E~hOaXd_23h[W8r~. (4.5) This is a highly non-linear differential equation whose solution determines Xd_ 2 as a dependent variable in terms of the physical degrees of freedom X, F, thus solving the constraints completely in this gauge. Then ~ is also determined from eq. (4.4). The remaining degrees of freedom X are now governed by the hamiltonian (4.2) after substituting the above form for Xdl, ~d-I• Noting that the 4, term drops after an integration by parts and using the identically solved constraint (4.3), the
408
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/
Membrane symmetries
remainder gives P= (1/2p+)(P2
+
(4.6)
W~’0aX•0bX),
which is purely quadratic in the physical canonical variables. By choosing patches of if coordinates on a Riemann surface that correspond to a locally flat metric ~ the equation of motion generated by this hamiltonian can locally be put into the form of the massless Klein—Gordon equation in 2 + 1 dimensions. Its solutions must be chosen to form a basis that obey the global periodicity conditions appropriate to the surface ~ For example for the torus we can simply take the plane waves exp(iif ~ ±i~/n~ + n~~-),with n
(n1, n2) a set of positive or negative integers. The coefficients of these solutions a~,a~are then quantized as harmonic oscillators in the usual way. Then, the remaining components Xd_2, X are determined as non-linear functions of these harmonic oscillators from the equations given above. Thus, in this gauge all the non-linearities of membrane theory have been shifted to the single equation (4.5) that determines X~2. At present this equation remains unsolved. The closure of the Lorentz algebra at the quantum level cannot be checked before determining Xd_2. Of course, we anticipate that Xd_2 is a rather non-linear function of the harmonic oscillators and that the quantum ordering of these oscillators may prevent the closure of the algebra, thus rendering the purely bosonic membrane inconsistent for any dimension d, as anticipated from a study of the spectrum [10]. Indeed, in the present gauge the spectrum is completely known, and it is evident that with only d 3 harmonic oscillators we cannot obtain a Poincaré covariant spectrum for neither massless nor massive states, in any dimension d. The promising d 11 super-membrane may be discussed in a similar gauge, including the fermions. We will not elaborate here on the super-membrane since Xd_2 remains unsolved in this light-cone gauge and the question of the closure of the super Poincaré algebra cannot be answered at the operator level for the time being. However it is worth noting that the hamiltonian can be brought to a quadratic form involving a free Klein—Gordon equation for X’, I 1,2,. 8 and a free Dirac equation involving the SO(8) spinors 8~,8_. For the flat toroidal membrane this system is manifestly invariant under super Poincaré transformations in 8 dimensions. We may now apply the consistency checks of ref. [5] (which used an S0(8) formalism) to show that the low-lying bosonic and fermionic spectrum (massless states and first massive level) is consistent with the 11 -dimensional super Poincaré symmetry and that this is possible only in 11 dimensions. By starting with the SO(8) representations, it was possible to show that the massless states actually form SO(9) representations, and that the massive states at the first level form SO(10) representations. Furthermore, the degenerate fermions and bosons fit into the correct supermultiplets consistent with supersymmetry in 11 .
=
—
=
=
. . ,
I. Bars
/
Membrane symmetries
409
dimensions. This happens because of remarkable properties of certain representations of SO(8), SO(9), S0(10), SO(16) and SO(32) and does not happen for any of the groups appropriate to other dimensions, as seen in ref. [5]. Note also that in this gauge there is no evidence of a continuous mass spectrum bordering at zero mass. This possibility arose originally in perturbative approaches [5,6], but there it was argued that higher-order corrections would remove the difficulty. More recently, on the basis of toy models, it was speculated that this possibility may arise in the super-membrane non-perturbatively and that it may be a blow to the theory [19], but we see no basis for its correctness in our formalism (see also the arguments in ref. [15]). 5. Normal ordering, operator products and anomalies Failure of the bosonic membrane theory in the light-cone gauge of the previous section, or in the perturbative analysis [10] in Hoppe’s light-cone gauge [17], may perhaps be inconclusive for dismissing bosonic membranes. It is possible that even though various gauges are equivalent at the classical level they may be inequivalent at the quantum level because of operator-ordering ambiguities; after all the gauge transformations that connect various gauges are rather non-linear transformations and they may not do what is expected from them when operator ordering is taken into account. We also emphasize again that the light-cone gauge may be singular even at the classical level, as in the example of strings in which certain longitudinal motions of strings simply cannot be described even classically in the light-cone gauge [12]. Basically what happens in strings is that there are certain classical solutions for which the momentum density P~(o-)vanishes for certain points 0~ that correspond to folds or end-points that move with the speed of light. When we choose the light-cone gauge we require a constant p~(~.) ~ which never vanishes for any point if, thus banishing such motions altogether. In the absence of these motions the string still manages to be quantum consistent, but only in 26 dimensions. It may not have been quantum consistent at all in any dimension, since certain degrees of freedom were eliminated outright. In view of the remarks in ref. [12], the fact that strings are consistent even in a specific dimension in the light-cone gauge is more a surprise than a sure outcome. According to the hints of ref. [5], it may be that 11-dimensional super-membranes also offer the same kind of surprise, and this would make this special case very interesting and perhaps unique, in the sense of being quantum consistent in the light-cone gauge. However, more generally, we hold the view that a choice of gauge is part of the definition of the quantum theory, and that we may have different quantum theories in different gauges. For strings, this certainly appears to be the case when we compare the light-cone gauge to the time-like gauge X°(’ra-)=p°’r which contains longitudinal degrees of freedom [12], or the Polyakov quantization in d < 26 which also contains longitudinal modes in the form of the Liouville degree
410
1. Bars
/
Membrane symmetries
of freedom. Accordingly, we find it necessary to analyze the consistency of bosonic membranes in a covariant gauge before dismissing them as inconsistent theories. This is the purpose of this section. What we would like to determine is whether the algebra of constraints can consistently be applied to the states, and whether this can insure the elimination of ghosts from the covariant spectrum. This could be done successfully in the case of strings, in which the constraints satisfy the Virasoro algebra with an anomaly in the form of a central extension. 5.1. OSCILLATORS AND NORMAL ORDERING
In this section we analyze the quantum bosonic membrane theory in the covariant orthonormal gauge. Our aim is to calculate the quantum commutator algebra associated with the diffeomorphism group DiffV3 and its subgroups and identify any anomalies that would modify the algebra of (2.12) or its subalgebras listed in (2.11). We would like to define an operator-product expansion in order to evaluate commutators carefully. We shall do this below by defining a Fock space in which operators can be ordered unambiguously. In this space, we will be able to use Wick’s theorem (even though we will not be defining perturbation theory) in order to obtain a definition of operator products. It may be that there are other ways of achieving the same result without our Fock space method, but we think that our approach is convenient and well defined. We recall the expansion X~(r,if) = EX~(T)Y’(if),
P~’(r,if) ~P~(T)Y’(if), =
I
(5.1)
I
where PL~ X’-’, and which satisfy the canonical commutation rules =
[x~’,P;]
=ig’~qjj.
(5.2)
We can always express the commutation rules in terms of a set of harmonic oscillators since we have the freedom to write the position and momentum variables in the form X~=
+
s~
,
PIL
=
—i1~a~ + ~
(5.3)
so that eq. (5.2) is equivalent to [a~t,a~”I =g’~lij
(5.4)
for any value of the frequencies o~.Thus we will allow these frequencies to be
I. Bars
/
Membrane symmetries
411
momentarily free parameters and think of them as a kind of variational parameters of the vacuum annihilated by a~ a~0),~=0.
(5.5)
The Fock space built on this “vacuum” is a complete Hilbert space. The oscillator vacuum defined by eq. (5.5) is not necessarily the lowest-energy eigenstate of the hamiltonian, but it can serve to resolve the quantum-ordering problem and give meaning to operator products by a procedure of normal ordering with respect to this vacuum. In principle we may evaluate expectation values of operators in any state of the theory, and the state of (5.5) is a particularly convenient one. Thus, the basis of operators (a,,a~.) provides a well-defined quantum theory in which we can answer the question of anomalies. The anomalies in the quantum version of the algebra (2.12) will be a function of the parameters w”. Our goal is to first compute the anomaly and then examine its structure, allowing ourselves the ability to vary the frequencies. We emphasize that our Hilbert space should not be confused with the usual perturbative Hilbert space. If we were to choose the frequencies to diagonalize the quadratic piece of the hamiltonian, then our Hilbert space and its vacuum state would coincide with their perturbative counterparts. So, it is clear that our proposal is not the evaluation of the anomaly just in perturbation theory, it is more general and non-perturbative, while we keep our option of evaluating the perturbative anomaly by simply choosing the frequencies appropriately. It is generally believed that anomalies are a property of the theory that may be evaluated even in low orders of perturbation theory. If this is true in membrane theory, then our procedure should have no trouble finding the anomaly even if we were to choose the perturbative frequencies. Therefore, we expect that if there is a true anomaly that there would be no reasonable set of frequencies that would cancel it. Using the Hilbert space defined above we define normal ordering. We will always use Euclidean times, and we will use the shorthand notation X(1) =
X(ii-
1,
o~),X(2)
=
X(ii-2,
if2),
etc. Then
X~’(1)X~(2) :X~(1)X~(2):+ KX~’(1)X~(2)), =
(5.6)
where the vacuum expectation value and the normal ordering are performed according to the vacuum (5.5). For convenience we define the symbol (X~(1)X~(2)~ =g~ii~2.
(5.7)
We do not claim that this is the full propagator of the theory, as the full propagator is the one evaluated in the true vacuum of the theory. However, the behavior of this ~ at short distances will give information about anomalies in the
412
1. Bars
/
Membrane symmetries
algebra of constraints, since the vacuum we have defined allows a consistent ordering of operators and our 4 is closely related to this ordering. In arriving at the form of the anomaly, we will need only general properties rather than detailed information about 4. Only at the end we need to discuss details such as the behavior of the unknown frequencies. The final form of the anomalies indicate that only the short-distance behavior of 4 is relevant. In this case the propagator ~12 depends only on the local properties of the surface and cannot be sensitive to the global properties of ~ therefore it can be approximated by the propagator in an infinite-dimensional sheet that is translationally and rotationally invariant and takes the form (5.8) where
~12
=
if
and i-12 According to these definitions, and P~ 0XiL/0(ii~),we have 1
—
if2
=
—
=
(P~(1)P~(2))
K0a~’(1)0hX~(2))
~g~S12,
=
1
=
2)~ ~
(5.9)
=
K0aX~’(1)P~(2)~ = ~~ig1L~0~z 12,
where the derivatives on the right-hand side are with respect to
i-~ 2, ~ 12~
In the limit
i-12
~12
0 we note the general properties
=
=
~(o~
—
if2),
4(
T12
=
0, ~12)
=
~(~12
=
0,
—
if12)
=
0,if12)
=
zI(r12
=
0,
—
~12),
(5.10)
which follow from eqs. (5.6) and (5.7) and the equal time commutation rules X~(1), P ~(2)]~~~ ig~~~(ifi if 2), [ X”(l), X v(2)] ~2 0 and [P~(1), P”(2)]~~. 0. Thus, the properties (5.10) are independent of any specific assumptions on the i--dependence of the propagator, and these will be sufficient to derive the general form of the anomaly in terms of the propagator. =
—
=
=
5.2. OPERATOR PRODUCTS AND COMMUTATORS
We may now define the operator version of the energy—momentum tensor T00, T0a by applying eq. (5.6) to the expressions (2.8) Toa
=
T00
=
:OaXP:, 2: + ~E”’~’:0aX~0bX0cX~0dX:+ ~
~A” :OaX
OhX’~,
(5.11)
:P
where we have dropped a constant from both expressions since they are irrelevant in the commutators. We see that rewriting our expression in normal-ordered form has produced a quadratic term with the coefficient A~’.This last modification of the energy—momentum tensor is reasonable since normal ordering produces the
I. Bars / Membrane symmetries
413
quadratic terms but, because we have no a priori way of determining the order of operators prior to normal ordering, the value of A” is unknown. From covariance on the surface we expect that A” Aw~’,which locally may be taken proportional to ~ab and we will assume that this is the case. Thus, we will keep Aab as a (possibly cutoff-dependent) parameter and check whether closure of the commutation rules requires a specific value for this parameter. Let us now consider composite operators such as those in (5.11) constructed from the canonical variables X, P, or equivalently from the oscillators, A(X(1),P(1))—~A(T1,if1)and B(X(2),P(2))’—B(i-2,o2), which are assumed to be =
normal ordered according to the vacuum (5.5). We may define their operator products by using Wick’s theorem
:X( 1) X(2) X(3) . . : :X( 1’) X(2’) X(3’) .
=4~.:X(2)X(3)
:X(3)
+ ~1I’~22’
X(2’)X(3’)
...
...
X(3’)
..
..
+
.
. .
.
:+
permutations of (123...)
permutations of (123...)
:+permutations of (123...)
~~1I~22’~33
+ ...
.
(5.12)
Then the equal-time commutator [A, B] is carefully defined as [A(1), B(2)]~...T2
=
urn (A(i-, if1)B(0,if2) i—’
—
B(T,if2)A(0,if~)), (5.13)
t)
where r appears only in the first factor. We first evaluate the operator products using Wick’s theorem as above, take the difference between terms, and then go to the i- 0 limit using the general properties of the propagator in eq. (5.10). The terms containing a single propagator in the operator-product expansion will contribute to the commutator the same expression as the naive classical Poisson bracket, while those containing two or more propagators may lead to anomalous terms in the commutation rules. We apply this procedure to compute the quantum algebra of the diffeomorphism group and its sub-groups. We will do explicitly the simpler case of Diff ~ generated by T0a and quote the result for the rest. We first evaluate the operator product =
Toa(TI,ifI)Tob(T2,if2)
=
:OaXP: (1) :0~X~P: (2) 2)
+ 2L112 :0~X(1).
=
+
OhX(
+~~0b~I2 :OaX(1) ~P(2):+ 4”~12
~ a b’’12
:0~X~P(1) 00XP(2): ~~0a0b4I2 :P(1) P(2):
—
2~a~l2
~ ~ A ~y”ea~12”h~12
:P(1) .OX(2):
1. Bars / Membrane symmetries
414
and then apply eqs. (5.5) and (5.10) to obtain the commutator 2(ifi
[TOa(if !), T
0~(if2)] =
~
if2)
—
TOa(if2) +
~0a~
—
if 2)Toh(ifl)
,
(5.15)
which is identical to the classical Poisson bracket result (2.12). Note that no detailed information of the r dependence of ~I2 was needed here, and only the generally correct information (5.10) was sufficient. Hence there is no anomaly in the subalgebra Diff Furthermore the area-preserving diffeomorphisrns AP Diff ~g listed in eq. (2.11) cannot have an anomaly since it is to be obtained directly from eq. (5.15) by differentiation and integration. In ref. [9]it was argued that, on purely algebraic grounds, the area-preserving diffeomorphisms could have a central extension. We have now shown that in bosonic membrane theory this central extension is absent. The quantum algebra for area-preserving diffeomorphisms is then 2(if
[Q(if1),
If the surface
Q(if2)]
_jEtho~~
=
is a torus we expand Q(if)
~g
1
=
—
(5.16)
if2)0~Q(if1).
~J~L~e’~”. Then eq. (5.16) implies
[L~,L~] =n XmLn+m,
(5.17)
without an anomaly. With the same procedure we find that the commutation rules involving T00 do have anomalies. Instead of the Poisson bracket algebra of eq. (2.12) we now have [T00(if1),T00(if2)]
=iacE0a62(ifl_if2)(:T~b(1)[hCd(1)+ACd]:+(1~~~2)),
(5.18) where hbc
=
:ObX 0~X:.If desired we may use Wick’s theorem to rewrite
:TOb( 1) [hCd(1)
ACd] :
+
=
[hCd(1)
+
AcdI TOb(l) ,
(5.19)
where ACd, which represents the deviation from the Poisson bracket algebra of (2.12), is a new constant that depends on the old ALd and the propagator 4. Finally we have the commutator with the worse anomaly 2(ifi
[TOa(ifi), T00(if2)]
=
—
if 2)(T00(if1) + T00(if2))
~0a~ —
j4hc((F
—
if2)hbC(if2) + iB~(if1—
~2)’
(5.20)
I. Bars / Membrane symmetries
415
where the anomalous coefficients are given by A~~C = ~(d
Ba
—
1)cbkcCt[Iim(3a0(k4l20I)~l2)]
l~m(~120~~12
=
—
+ (AbCOa
—
l~(bAc)ko)~2(if
—
(5.21)
Abc0a0bLtl20c~l2)1.
To instill confidence in our expressions, we point out that the same procedure can be applied for strings. The A-anomaly which arose because of the quartic terms in T00 are absent for strings, but the B-anomaly is present. Thus, specializing our expression to one dimension, and taking A 1, we have the string anomaly B ~d(~ 4”)~’ where the prime indicates a if-derivative. Using the string propagator, 4 ln(i-~+ us), this expression becomes proportional to ö”(o-12), which is the correct anomaly in the Virasoro algebra. Up to this point 2(if we have used only the general properties (5.10). We see that lirnT,,O ~12 ö 12) indicates in (5.21) we only need the short-distance behavior of 4. We might reasonably expect a propagator that has a singular behavior near 0~~I2 0 and then the product of two propagators is expected to produce terms that are even more singular. Thus, a cubic derivative of the delta function can easily appear in the coefficient A. This can be seen by taking the simple example of the free particle propagator =
=
—
=
=
=
w(k)
=
1
~k~+k~
~~12
—
(5.22)
.
4ii-~/i-~+
if~2
More generally, we may try propagators of the form 2k 4 12(r12,
~i2I)
f
1
d exp[ —w(k)i-12
2
(2ir)
+
ik
if12]
(5.23)
W~tL)
that satisfy (5.10) for any frequencies w(k) (they are closely related to those of the eq. 4 then (5.3)). If the large-k behavior of the w(k) blows up faster than k A-anomaly can be cancelled by choosing appropriately the unknown coefficient A, while the B-anomaly becomes divergent. However, this behavior seems strange since the propagator becomes non-singular for small T, o~.If the w(k) correspond to a singular behavior of propagators near the origin, then all such w(k) produce cubic derivatives of the delta function in the A-anomaly. With such behavior it appears impossible to cancel terms in the anomalous coefficient A~C for any value of the constant A~~c even if we allow this constant to depend on the cutoff parameter T. In the presence of this anomaly, the algebra (5.20) contains the
416
1. Bars / Membrane symmetries
operator hbC on the right-hand side, indicating that DiffV3, which is equivalent to the algebra of constraints, does not close at the quantum level. Recall that the physical states are defined as those satisfying T00Iphys) 0 =
=
aiphys). Therefore, the commutator of any two constraints is required to annihilate the physical states. If the non-closure were only a c-number central extension as in string theory, it could be handled by requiring that only the positive frequency components of the constraints annihilate the physical states. However, we have now found an operator anomaly. Under such circumstances, in order to construct a consistent quantum theory, we would want to enlarge the algebra of constraints by including the operator ~ in our list of constraints. However, demanding the constraint hat, 0 on the states would require that the entire metric, as given in eq. (2.3), vanishes on the states ~ 0. This reduces the theory to triviality. Therefore, we conclude that the purely bosonic membrane cannot be a consistent theory in any dimension d! This result is in agreement with previous negative results in sect. 4 and refs. [5, 6, 10] based on the Lorentz invariance consistency checks in the light-cone gauge. For the purely bosonic membrane and higher p-branes we had concluded that they were not Lorentz invariant in any dimension d [10]. We point out that the light-cone gauge of Hoppe (which is the one used in the previous consistency checks) is a sub-gauge of the present covariant conformal gauge, as explained in sect. 2. In this light-cone gauge the Lorentz generator performs a naive gauge transformation followed by a diffeornorphism transformation that restores the gauge. It is not surprizing that quantum Lorentz invariance would fail in such a gauge since, as we have now established, the underlying diffeomorphism group algebra fails at the quantum level. The present result distinguishes bosonic strings from all other extended bosonic p-branes as having the unique property of being quantum consistent. The next interesting question is “what about super p-branes”? Unfortunately, we do not yet know how to treat the super p-branes (including the Green—Schwarz super-string) quantum mechanically in covariant gauges. Only light-cone quantization has been developed. Instead, we may apply the present operator-product techniques to compute the anomalies in the super Poincaré algebra in the light-cone gauge. This is considerably more involved algebraically, and is currently under investigation [14]. It is evident that using our method we should be able to determine whether the 11-dimensional super-membrane, that survived the consistency checks, is really quantum consistent or not. =
=
References [11 J. Hughes, J. Liu and J. Polchinski, Phys. Lett. B180 (1986) 370 [2] E. Bergshoeff, E. Sezgin and P.K. Townsend, Phys. Lett. B189 (1987) 75 [3] A. Achbcarro, J.M. Evans, P.K. Townsend and D. Witshire, Phys. Lett. B198 (1987) 441
I. Bars / Membrane symmetries
417
[4] M.J. Duff, T. Inami, C.N. Pope, E. Sezgin and K. Stelle, NucI. Phys. B297 (1988) 515 [5] I. Bars, NucI. Phys. B308 (1988) 462 [6] I. Bars and C.N. Pope, Class, and Quantum Gray. 5 (1988) 1157 [71B. de Wit, J. Hoppe and H. Nicolai, NucI. Phys. B305 (1988) 545 [8] K. Kikkawa and M. Yamasaki, Prog. Theor. Phys. 76 (1986) 379; L. Mezincescu, R. Nepomechie, P. van Nieuwenhuizen, UMTG-139/ITP-SB-84-43; I. Bars, C.N. Pope and E. Sezgin, Phys. Lett, B198 (1987) 455; S. Ghandi and K. Stelle, Class, and Quantum Gray. 5 (1988) 127; C.N. Pope and K. Stelle, Class, and Quantum Gray. (1989) [9] E. Floratos and J. Illiopoulos, Phys. Lett, B201 (1988) 237; I. Bars, C.N. Pope and E. Sezgin, Phys. Lett, B210 (1988) 85; D. Fairlie, P. Fletcher and C. Zachos, Phys. Lett. B218 (1989) 203; C.N. Pope and L. Romans, Class. Quantum Gray. 7 (1990) 97 [10] I. Bars, preprint USC-88/HEPO6, in Proc. of 17th Int. Conf. on Group theoretical methods in physics (Montreal 1988), eds, Y. Saint-Aubin and L. Vinet (World Scientific, Singapore, 1989), p. 693 [11] M. Green, J. Schwarz and E. Witten, Superstring theory (Cambridge, 1987) [12] WA. Bardeen, I. Bars, A.J. Hanson and RD. Peccei, Phys. Rev. D13 (1976) 2364; D14 (1976) 2193 [13] A. Polyakov, Phys. Lett. B103 (1981) 207 [141 I. Bars and K. Sfetsos, to be published [15] I. Bars, Issues of topology and the spectrum of the supermembrane, preprint USC-89/026, Proc. Workshop on membranes and physics in 2 + I dimensions, Trieste (1989) [16] Heterotic superstring in ref. [11], and its 4-dimensional versions [17] I. Bars, Lecture at Aspen workshop, 1987 [18] J. Hoppe, MIT Ph.D. thesis, 1982 [19] B. de Wit, M. Liisher and H. Nicolai, Nucl. Phys. B320 (1989) 135