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Constructing the Lorentz anomaly out of the Virasoro anomaly
Volume 207, number
PHYSICS
2
CONSTRUCTING
LETTERS
B
16 June 1988
THE LORENTZ ANOMALY OUT OF THE VIRASORO ANOMALY
I. KANI and P. MANSFIELD Department of Theoretical Physics, 1, Keble Road, Oxford OX1 3NP. UK Received
4 February
I988
We use a simple argument based on the algebra of cocycles to construct the Lorentz algebra when strings are quantised in the light-cone gauge.
03.50 0 Elsevier Science Publishers Physics Publishing Division )
anomaly
the anomaly
that occurs in
world sheet parameters, (Tand r, and may be decomposed as a sum of left- and right-moving components X~(o,r)=~~(rr+r)+XU(-~+~)asmaytheother degrees of freedom, the Faddeev-Popov ghosts. x(a) is periodic in G and the Virasoro operators act on 2 and the ghosts as generators of diffeomorphisms of the circle parametrised by 0. Their algebra is a central extension of Diff(S’ ) where the c-number term arises through the need to regularise products of operators,
In the various approaches to the quantisation of strings the critical dimension of space-time emerges from the requirement that some anomaly should vanish. The form of the anomaly is very different in each approach for example in covariant canonical quantisation it is the central term in the Virasoro algebra whereas in canonical quantisation in the lightcone gauge it is a breakdown in the target-space Lorentz algebra, however it is clear that they must be physically equivalent, in some sense, because they lead to the same restriction on the dimension, d, of target space. Originally these anomalies were discovered [ 1,2 ] as a result of detailed calculations which involved careful regularisation of products of operators but obscured the relationship amongst them. However, the inter-relation of the various anomalies that occur in non-abelian gauge theories has been understood more recently by the use of general ideas based on the local cohomology of Lie algebras [ 3 1. In refs. [ 41 these methods were applied to string theories to relate the Virasoro anomaly to the Weyl anomaly of Polyakov’s functional approach and the failure of nilpotency of the BRST charge. In this letter we will show how the breakdown of Lorentz invariance in the light-cone gauge can be simply understood using the algebra of cocycles and without resorting to detailed manipulations of normal-ordered operators. In the covariant canonical quantisation of the bosonic string a central role is played by the Virasoro operators for the full theory, i.e. including reparametrisation ghosts. The open string target space coordinate is a standing wave which depends on two 0370-2693/88/$ ( North-Holland
out of the Virasoro
IL,,
Ll=
+&(d-26)
(n-m) L+, (n3-n)
it,_,.
(1)
The form of the c-number term is strongly constrained by the Jacobi identity for the Virasoro operators. A convenient language for analysing the content of this condition is provided by the local cohomology of Lie algebras. Let p denote a field (or fields) on which acts a set of infinitesimal transformations &yl parametrised by variables 0, with Lie algebra commutation relations [8,, 8,
I = - b,,g1.
(2)
An n-cochain, f,, is defined as a function of p which is both multilinear and skew symmetric in n of the 6,. The coboundary operator d maps n-cochains to (n + 1 )-cochains as (&l)
(ul; 01, ***> &+,)
(3) B.V.
13.5
Volume 207, number
PHYSICS
2
where square brackets around indices denote antisymmetrisation of all the enclosed indices. A is defined in this way so as to be nilpotent, A*=O. Alternatively, we can view the 0, as one-forms and ncochains as n-forms. A can then be defined by its action on a, and the 0, Ayl=&p,
A0,=;[0,
e,].
Its actions on n-cochains then follows by requiring it to satisfy a graded Leibnitz rule. An n-cocycle is defined to be an n-cochain annihilated by A. Because A is nilpotent it is always possible to construct trivial IIcocycles as Afn_, with arbitrary fn_,. The Virasoro anomaly is characterised by being a 2-cocycle for Diff( S’ ) which cannot be obtained by A acting on a local cochain. Diffeomorphisms are parametrised by vector fields. Let v(a) be the component of a vector field on S’ with respect to the basis a/&r, 0~ (T< 2n. The generator of the corresponding diffeomorphism is constructed out of the Virasoro operators and the Fourier modes of v as L(v) = -i
v(a) = 1 v, exp(incs). n
C v_,L,, n
(4)
so that 6,_Y(a)=v(a) 6,X(0,
?(a)=
T)=v(o+r)
[L(v),R(a)], 3+X($
T)
+v(-OO+T)a-x(0, z), 2at = *a,+a,,
(5)
IL %I = +4”,, [u, v] (a)=u(a)
v’(a)-v(a)
and the Virasoro ofL(v) as
algebra may be rewritten
[L(u),
(6) in terms
[u, VI 1+ctu,
s
i(d-26) 24~
da[u”‘(rr)
v(a)+u’(a)
v(a)].
(7)
The Jacobi identity for L( vi) leads to the condition on the c-number term that C(V[i,
136
[VI,
V!f]
I)
but C(u, v) is an antisymmetric functional of the transformation parameters u and v and so can be regarded as a 2-cochain for Diff(S’) and if AsI is the coboundary operator for this Lie algebra then this condition is just that AsI C should vanish. This formalism is constructed to be explicitly Lorentz covariant. Under a Lorentz boost parametrised by w the target space coordinates transform as S,x~=u(w)~“x”=
[M(w),
=O,
Xfl],
(9)
where the generator of this transformation, M(w), is a( w),,MuP/2 where Mfi” is the total angular momentum Mfl”=
s
da2XtP(,,
The components Lorentz algebra quantisation,
T)
a,xq0, T).
(10)
of angular momentum commutation relations,
satisfy the even after
[M/I”, Al”“] =~“a~u$_~~~~va+~YP~~a_~v~Mll$.
(11)
In the transition to light-cone gauge [ 1 ] quantisation the freedom to make Diff( S ’ ) transformations is completely used up by choosing world sheet coordinates IT and r such that in one inertial frame the combination of target space coordinates (x0 + Xd- ’ ) should be independent of r~. This combination corcoordinates light-cone using responds to (X’=(~+Xd-1)/$?,X~=X’,...,Xd-2).WithX+ fixed X- can be obtained by solving the Virasoro gauge conditions in terms of the CY’,x’, p’, x+, p+ where x+=x++p+q cosnci.
v),
2n x
16 June 1988
B
X’=x’+p’r+i~$O~a~exp(-inr)
L(v) I = -L(
C(u, v)=
u’(c7),
LETTERS
(8)
Since the timelike oscillators associated with X0 have now been eliminated there are no negative norm states in the light-cone gauge. However, the abandonment of explicit Lorentz covariance leads to the sickness of the quantum theory away from the critical value d=26 manifesting itself as a breakdown of Lorentz covariance. The Lorentz generators are obtained from those of the covariant scheme by eliminating X’ in favour of the dynamical degrees of freedom. We will distinguish these operators from those in expression
Volume 207, number
PHYSICS
2
( 10 ) by placing a hat over them. The action of A( w ) on the dynamical degrees of freedom is different from that of (9) because it preserves the gauge condition X+ =x+ +p+r. So in addition to (9) A(w) includes a compensating reparametrisation. If we denote all the dynamical degrees of freedom by 0 then for y any operator in the theory A(w) generates the transformation (12)
&Y= &JY + ~“,$o,W,Y.
The S ’ vector, u, that parametrises the compensating diffeomorphism is fixed by requiring that the new value ofX+, X+ +&X+, be the Lorentz boosted value ofx+++p+7, i.e. x+ +p+7+
6,(x+
z)=a(w)+,X”(q
7)
16 June 1988
then the anomalous piece, K, depends on the operator degrees of freedom 01,in contrast to C( U, v) which was a c-number term in the Virasoro algebra. As we will now show the Jacobi identity once again implies that K is a 2-cocycle for the Lorentz algebra but this time it is operator valued. In an obvious notation we have denoted by [w,, w2] ,_the parameter of the Lorentz boost obtained by commuting infinitesimal boosts themselves parametrised by w, and w2, (17)
mv,,Ll=-~~~L.
The algebra of the A(w)
implies that
[8,, >&, I = -~[WI.W*IL.
[&4,&c,l
+v(o+r)d+x+(a,7)+24--(T+7)a_x+(qT) =u(w)+,(X”+p”T)
B
(18)
Since this result will be essential it is worth trying to understand it from a different perspective. By an argument similar to that which led to equation ( 14) it follows that
+p+t),
of &X+(0,
LETTERS
(13)
~+=[&v,,~w.,l
(X++P+7).
(19)
We can compute the commutator on the left hand side from the definitions of 6,X and 8X
so [8&, >h* I = -~~w,,wzl~-~u~(o.w,.wz) v(a)=
-ia(w
C ai exp( -ina)/np+. nfo
(14)
When due care is paid to the regularisation of the generators their algebra is obtained as a modification of the Lorentz algebra ( 10) in which the commutator [A@-, A@- ] acquires the additional term [ 1 ]
-6 [e%w,). av).w)I. 6, is an element
of Diff( S’ ) which is generated by the action of 6, on the y, contained in v( p, w) U(P, WI>w2) =2&G,, V(% WZ]),
(20)
so we can rewrite ( 19) as (d-26)
f
(m-l/m)
I
cr!,a$,/6(p+)‘.
(15)
Here, as in the Virasoro algebra ( 1 ), we have set the normal-ordering constant associated with Lo equal to unity. The anomaly in the Lorentz algebra was originally derived by manipulating products of operators in canonical field theory. We now want to describe a simple way to understand this result from a different point of view, that of the algebra of cocycles which in this case amounts to little more than an analysis of the Jacobi identity. If we write the Lorentz algebra in terms of A(w) [@(%),mw)l =-m
[WI, w2lI_)+K(a),
WI, w),
(16)
(8 [WI,W’2IL+hP,WI,W*~ =i!i [W,,WZ]L(x+p7)
1 X+
+
with ~(~,~,,~2)=~(~,~,,~2)+~~(~,~~)~~(~,~2)1.
The content of eq. ( 13 ) is that it uniquely determines the compensating reparametrisation that must be added to a Lorentz boost acting on X+ to preserve the gauge condition, so it follows that u”(P>WI, wz)=av,
[WI, W21L)
(21)
and from this we can deduce ( 18 ). Having established the algebra of the 6, we can construct a coboundary operator dL. The anomaly in the Lorentz algebra ( 16) can be viewed as an opera-
137
Volume 207, number
PHYSICS
2
tor valued 2-cochain (AK)
and applying AL to it we obtain
(Y, w, >wz, ~3) = &,,WP,
WI 3~21)
LETTERS
16 June 1988
B
wI, w2) by substituting C(u, v) R(P> WI> w2)=C(4c4
Now Jacobi’s identity O=[A(w,A
tmw2)d@w3l)ll
=fiw,,,
J%]lLIL)
tw2,
+[mw,,),mP, -KC&
for the A?(w) is
W[l,
w2, W3]>1
[w2,
(22)
w31ld.
The first term on the right-hand side vanishes by virtue of Jacobi’s identity for the Lorentz algebra ( 17 ) whilst the remaining terms are just A,K given that A(w) is the generator of the transformation 6%. So we conclude that K is an operator valued 2-cocycle for AL. So both C( U, u) and K( rp,w,, w2) are cocycles as are most anomalies in quantum field theory. This puts a surprisingly strong constraint on their form. The question we want to ask is that given the ~-COcycle for Diff( S’ ), C( U, v), is it possible to construct a 2-cocycle for the Lorentz algebra? Our answer to this goes some way to explaining the origin of this Lorentz anomaly. Of course our argument will not guarantee that the cocycle we obtain is indeed K(q, w,, w,), although this turns out to be the case, but this is the price to be paid by abandoning detailed calculation for analysis based only on principles of symmetry. The construction of a 2-cocycle for AL from the 2cocycle for AsI, C( u, v), follows from a consideration of eq. (2 1). Writing it out in terms of the u( p, w) it becomes &,,V(P, W,])--f4vJ,
[WI, W2IL)
We can view v(p, w) as an operator valued one cochain for the 6, in which case the left-hand side of this equation is just - ALv. The right-hand side however is -A,, applied to a cochain consisting of a single S’ vector field and then evaluated for the pair V(q, w,), v(p, w2). So (23) tells us that
(4%
WI ), 4%
w2)).
X?
1
(n-l/n)
crt’ar’!,/6(p+)*.
It is clear from ( 15 ) that the 2-cocycle iz obtained from our simple abstract argument is identical with the anomaly in the Lorentz algebra ( 16). Our construction is readily extended to the spinning string. The target space coordinates in covariant quantisation are generalised to superfields [ 5 1. For the open string these are again standing waves which can be decomposed into left and right moving pieces, XP(fr, r)=x$(z+a)+x!!(r-a), X$(cr)=P(,)+e+~fl(a), _ where the 8+ are anticommuting variables and the wP the superpartners of &. Diff(S’) transformations become generalised to superconformal transformations with the generators LB/ &,X+(a)=
D] X+(a)
[Va,+f(DV)
=[L,X+(~)l
D=a/ae+
+e+a,,
and V = u. + 8, U, parametrises infinitesimal supercoordinate transformations. The generators L, are constructed from the super stress-energy tensor for the full theory including reparametrisation ghosts and have the anomalous super conformal algebra +cwJJ)sc,
[v,u],,=va,u-ua,v+~(Dv)~(Du), ~2) ).
(24)
Now define the 2-cochain for the Lorentz algebra x( v, 138
ev>
Coboundary operators act on products according to Leibnitz’ rule so from (24) it follows that the action of AL on I?is the same as the action of AsI on C( U, V) followed by the substitution of v( p, w, ), V(p, w2) and V(P, w3) for the Diff( S’ ) transformation parameters. But AS annihilates C( U, V) so R must be a 2-cocycle for AL. If we compute &explicitly using (7) and ( 14) we obtain
[Lv, LJI =LpwJsc
(&~(a), w) ) (WI >~2) = (4~)
41,
where (23)
=I[4P,W),4P,W2)1.
v(p, w,) and v(p, w2) into
C(V, V),,=
‘(‘1;;‘)
j dodB+(Va:DU).
Volume 207, number
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2
The generators of Lorentz transformations now depend on the v/but have the same anomaly-free algebra as in the purely bosonic case. The light-cone gauge now includes a condition on @’ X’(r,
B
aly by Iwasaki methods.
16 June 1988
and
Kikkawa
[6 ] using
operator
References n)=x++p+7,
ly+=o.
As before Lorentz transformations are implemented non-linearly because a compensating superconforma1 transformation is included to preserve the lightcone gauge. The superfield that parametrises this is easily found to be V(0)=-a(w)‘,
[~“(a)-(x”+p”a)+28+~“(~)]
Substituting this into C(u, u),, and using mode expansions for 2 and W”(a)=C
n
o:exp(-ina)
(where for definiteness we work in the Ramond tor) we obtain the Lorentz 2-cocycle &x,
LETTERS
sec-
WI>w2)
d-10 =---a(w,)+ia(w,)+, 4(P’S
+ (4m2-
1) cu[l,c-w$].
This agrees with the calculation
of the Lorentz anom-
[ 1 ] P. Goddard, J. Goldstone, C. Rebbi and C.B. Thorn, Nucl. Phys. B 56 (1973) 109. [2] A.M. Polyakov, Phys. Lett. B 103 (1981) 207. [3] L.D. Faddeev, Phys. Lett. B 145 (1984) 81; B. Zumino, Nucl. Phys. B 253 (1985) 477; R. Stora, in: New developments in quantum field theory and statistical mechanics, eds. M. Levy and P. Mitter (Plenum, New York, 1977) p. 201; LAPP-TH-94 (Cargese Lectures 1983); L. Bonora and P. Cotta-Ramusino, Commun. Math. Phys. 87 (1983) 589; A.J. Niemi and G. Semenoff, Phys. Rev. Lett. 51 (1983) 2077; B. Zumino, Y.S. Wu and A. Zee, Nucl. Phys. B 239 ( 1984) 477; L. Baulieu, Nucl. Phys. B 241 ( 1984) 557; K.C. Chou, H.Y. Guo and K. Wu, Phys. Lett. B 134 ( 1984) 67; J.L. Manes, Nucl. Phys. B 250 ( 1985) 369; W. Bardeen and B. Zumino, Nucl. Phys. B 244 (1984) 42 I; L. Alvarez-Gaume, S. Della Peitra and G. Moore, Harvard preprint HUTP-84/A-28; R. Jackiw, in: Relativity, groups and topology II (Les Houches, 1983), ed. B.S. Dewitt and R. Stora (North-Holland, Amsterdam, 1984) p. 22 1. [4] P. Mansfield, Ann. Phys. (NY) 180 (1987) 330. [ 51 D. Friedan, E. Martinet and S. Shenker, Nucl. Phys. B 271 (1986) 93. [ 61 Y. Iwasaki and K. Kikkawa, Phys. Rev. D 8 (1973) 440.
139
PHYSICS
2
CONSTRUCTING
LETTERS
B
16 June 1988
THE LORENTZ ANOMALY OUT OF THE VIRASORO ANOMALY
I. KANI and P. MANSFIELD Department of Theoretical Physics, 1, Keble Road, Oxford OX1 3NP. UK Received
4 February
I988
We use a simple argument based on the algebra of cocycles to construct the Lorentz algebra when strings are quantised in the light-cone gauge.
03.50 0 Elsevier Science Publishers Physics Publishing Division )
anomaly
the anomaly
that occurs in
world sheet parameters, (Tand r, and may be decomposed as a sum of left- and right-moving components X~(o,r)=~~(rr+r)+XU(-~+~)asmaytheother degrees of freedom, the Faddeev-Popov ghosts. x(a) is periodic in G and the Virasoro operators act on 2 and the ghosts as generators of diffeomorphisms of the circle parametrised by 0. Their algebra is a central extension of Diff(S’ ) where the c-number term arises through the need to regularise products of operators,
In the various approaches to the quantisation of strings the critical dimension of space-time emerges from the requirement that some anomaly should vanish. The form of the anomaly is very different in each approach for example in covariant canonical quantisation it is the central term in the Virasoro algebra whereas in canonical quantisation in the lightcone gauge it is a breakdown in the target-space Lorentz algebra, however it is clear that they must be physically equivalent, in some sense, because they lead to the same restriction on the dimension, d, of target space. Originally these anomalies were discovered [ 1,2 ] as a result of detailed calculations which involved careful regularisation of products of operators but obscured the relationship amongst them. However, the inter-relation of the various anomalies that occur in non-abelian gauge theories has been understood more recently by the use of general ideas based on the local cohomology of Lie algebras [ 3 1. In refs. [ 41 these methods were applied to string theories to relate the Virasoro anomaly to the Weyl anomaly of Polyakov’s functional approach and the failure of nilpotency of the BRST charge. In this letter we will show how the breakdown of Lorentz invariance in the light-cone gauge can be simply understood using the algebra of cocycles and without resorting to detailed manipulations of normal-ordered operators. In the covariant canonical quantisation of the bosonic string a central role is played by the Virasoro operators for the full theory, i.e. including reparametrisation ghosts. The open string target space coordinate is a standing wave which depends on two 0370-2693/88/$ ( North-Holland
out of the Virasoro
IL,,
Ll=
+&(d-26)
(n-m) L+, (n3-n)
it,_,.
(1)
The form of the c-number term is strongly constrained by the Jacobi identity for the Virasoro operators. A convenient language for analysing the content of this condition is provided by the local cohomology of Lie algebras. Let p denote a field (or fields) on which acts a set of infinitesimal transformations &yl parametrised by variables 0, with Lie algebra commutation relations [8,, 8,
I = - b,,g1.
(2)
An n-cochain, f,, is defined as a function of p which is both multilinear and skew symmetric in n of the 6,. The coboundary operator d maps n-cochains to (n + 1 )-cochains as (&l)
(ul; 01, ***> &+,)
(3) B.V.
13.5
Volume 207, number
PHYSICS
2
where square brackets around indices denote antisymmetrisation of all the enclosed indices. A is defined in this way so as to be nilpotent, A*=O. Alternatively, we can view the 0, as one-forms and ncochains as n-forms. A can then be defined by its action on a, and the 0, Ayl=&p,
A0,=;[0,
e,].
Its actions on n-cochains then follows by requiring it to satisfy a graded Leibnitz rule. An n-cocycle is defined to be an n-cochain annihilated by A. Because A is nilpotent it is always possible to construct trivial IIcocycles as Afn_, with arbitrary fn_,. The Virasoro anomaly is characterised by being a 2-cocycle for Diff( S’ ) which cannot be obtained by A acting on a local cochain. Diffeomorphisms are parametrised by vector fields. Let v(a) be the component of a vector field on S’ with respect to the basis a/&r, 0~ (T< 2n. The generator of the corresponding diffeomorphism is constructed out of the Virasoro operators and the Fourier modes of v as L(v) = -i
v(a) = 1 v, exp(incs). n
C v_,L,, n
(4)
so that 6,_Y(a)=v(a) 6,X(0,
?(a)=
T)=v(o+r)
[L(v),R(a)], 3+X($
T)
+v(-OO+T)a-x(0, z), 2at = *a,+a,,
(5)
IL %I = +4”,, [u, v] (a)=u(a)
v’(a)-v(a)
and the Virasoro ofL(v) as
algebra may be rewritten
[L(u),
(6) in terms
[u, VI 1+ctu,
s
i(d-26) 24~
da[u”‘(rr)
v(a)+u’(a)
v(a)].
(7)
The Jacobi identity for L( vi) leads to the condition on the c-number term that C(V[i,
136
[VI,
V!f]
I)
but C(u, v) is an antisymmetric functional of the transformation parameters u and v and so can be regarded as a 2-cochain for Diff(S’) and if AsI is the coboundary operator for this Lie algebra then this condition is just that AsI C should vanish. This formalism is constructed to be explicitly Lorentz covariant. Under a Lorentz boost parametrised by w the target space coordinates transform as S,x~=u(w)~“x”=
[M(w),
=O,
Xfl],
(9)
where the generator of this transformation, M(w), is a( w),,MuP/2 where Mfi” is the total angular momentum Mfl”=
s
da2XtP(,,
The components Lorentz algebra quantisation,
T)
a,xq0, T).
(10)
of angular momentum commutation relations,
satisfy the even after
[M/I”, Al”“] =~“a~u$_~~~~va+~YP~~a_~v~Mll$.
(11)
In the transition to light-cone gauge [ 1 ] quantisation the freedom to make Diff( S ’ ) transformations is completely used up by choosing world sheet coordinates IT and r such that in one inertial frame the combination of target space coordinates (x0 + Xd- ’ ) should be independent of r~. This combination corcoordinates light-cone using responds to (X’=(~+Xd-1)/$?,X~=X’,...,Xd-2).WithX+ fixed X- can be obtained by solving the Virasoro gauge conditions in terms of the CY’,x’, p’, x+, p+ where x+=x++p+q cosnci.
v),
2n x
16 June 1988
B
X’=x’+p’r+i~$O~a~exp(-inr)
L(v) I = -L(
C(u, v)=
u’(c7),
LETTERS
(8)
Since the timelike oscillators associated with X0 have now been eliminated there are no negative norm states in the light-cone gauge. However, the abandonment of explicit Lorentz covariance leads to the sickness of the quantum theory away from the critical value d=26 manifesting itself as a breakdown of Lorentz covariance. The Lorentz generators are obtained from those of the covariant scheme by eliminating X’ in favour of the dynamical degrees of freedom. We will distinguish these operators from those in expression
Volume 207, number
PHYSICS
2
( 10 ) by placing a hat over them. The action of A( w ) on the dynamical degrees of freedom is different from that of (9) because it preserves the gauge condition X+ =x+ +p+r. So in addition to (9) A(w) includes a compensating reparametrisation. If we denote all the dynamical degrees of freedom by 0 then for y any operator in the theory A(w) generates the transformation (12)
&Y= &JY + ~“,$o,W,Y.
The S ’ vector, u, that parametrises the compensating diffeomorphism is fixed by requiring that the new value ofX+, X+ +&X+, be the Lorentz boosted value ofx+++p+7, i.e. x+ +p+7+
6,(x+
z)=a(w)+,X”(q
7)
16 June 1988
then the anomalous piece, K, depends on the operator degrees of freedom 01,in contrast to C( U, v) which was a c-number term in the Virasoro algebra. As we will now show the Jacobi identity once again implies that K is a 2-cocycle for the Lorentz algebra but this time it is operator valued. In an obvious notation we have denoted by [w,, w2] ,_the parameter of the Lorentz boost obtained by commuting infinitesimal boosts themselves parametrised by w, and w2, (17)
mv,,Ll=-~~~L.
The algebra of the A(w)
implies that
[8,, >&, I = -~[WI.W*IL.
[&4,&c,l
+v(o+r)d+x+(a,7)+24--(T+7)a_x+(qT) =u(w)+,(X”+p”T)
B
(18)
Since this result will be essential it is worth trying to understand it from a different perspective. By an argument similar to that which led to equation ( 14) it follows that
+p+t),
of &X+(0,
LETTERS
(13)
~+=[&v,,~w.,l
(X++P+7).
(19)
We can compute the commutator on the left hand side from the definitions of 6,X and 8X
so [8&, >h* I = -~~w,,wzl~-~u~(o.w,.wz) v(a)=
-ia(w
C ai exp( -ina)/np+. nfo
(14)
When due care is paid to the regularisation of the generators their algebra is obtained as a modification of the Lorentz algebra ( 10) in which the commutator [A@-, A@- ] acquires the additional term [ 1 ]
-6 [e%w,). av).w)I. 6, is an element
of Diff( S’ ) which is generated by the action of 6, on the y, contained in v( p, w) U(P, WI>w2) =2&G,, V(% WZ]),
(20)
so we can rewrite ( 19) as (d-26)
f
(m-l/m)
I
cr!,a$,/6(p+)‘.
(15)
Here, as in the Virasoro algebra ( 1 ), we have set the normal-ordering constant associated with Lo equal to unity. The anomaly in the Lorentz algebra was originally derived by manipulating products of operators in canonical field theory. We now want to describe a simple way to understand this result from a different point of view, that of the algebra of cocycles which in this case amounts to little more than an analysis of the Jacobi identity. If we write the Lorentz algebra in terms of A(w) [@(%),mw)l =-m
[WI, w2lI_)+K(a),
WI, w),
(16)
(8 [WI,W’2IL+hP,WI,W*~ =i!i [W,,WZ]L(x+p7)
1 X+
+
with ~(~,~,,~2)=~(~,~,,~2)+~~(~,~~)~~(~,~2)1.
The content of eq. ( 13 ) is that it uniquely determines the compensating reparametrisation that must be added to a Lorentz boost acting on X+ to preserve the gauge condition, so it follows that u”(P>WI, wz)=av,
[WI, W21L)
(21)
and from this we can deduce ( 18 ). Having established the algebra of the 6, we can construct a coboundary operator dL. The anomaly in the Lorentz algebra ( 16) can be viewed as an opera-
137
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2
tor valued 2-cochain (AK)
and applying AL to it we obtain
(Y, w, >wz, ~3) = &,,WP,
WI 3~21)
LETTERS
16 June 1988
B
wI, w2) by substituting C(u, v) R(P> WI> w2)=C(4c4
Now Jacobi’s identity O=[A(w,A
tmw2)d@w3l)ll
=fiw,,,
J%]lLIL)
tw2,
+[mw,,),mP, -KC&
for the A?(w) is
W[l,
w2, W3]>1
[w2,
(22)
w31ld.
The first term on the right-hand side vanishes by virtue of Jacobi’s identity for the Lorentz algebra ( 17 ) whilst the remaining terms are just A,K given that A(w) is the generator of the transformation 6%. So we conclude that K is an operator valued 2-cocycle for AL. So both C( U, u) and K( rp,w,, w2) are cocycles as are most anomalies in quantum field theory. This puts a surprisingly strong constraint on their form. The question we want to ask is that given the ~-COcycle for Diff( S’ ), C( U, v), is it possible to construct a 2-cocycle for the Lorentz algebra? Our answer to this goes some way to explaining the origin of this Lorentz anomaly. Of course our argument will not guarantee that the cocycle we obtain is indeed K(q, w,, w,), although this turns out to be the case, but this is the price to be paid by abandoning detailed calculation for analysis based only on principles of symmetry. The construction of a 2-cocycle for AL from the 2cocycle for AsI, C( u, v), follows from a consideration of eq. (2 1). Writing it out in terms of the u( p, w) it becomes &,,V(P, W,])--f4vJ,
[WI, W2IL)
We can view v(p, w) as an operator valued one cochain for the 6, in which case the left-hand side of this equation is just - ALv. The right-hand side however is -A,, applied to a cochain consisting of a single S’ vector field and then evaluated for the pair V(q, w,), v(p, w2). So (23) tells us that
(4%
WI ), 4%
w2)).
X?
1
(n-l/n)
crt’ar’!,/6(p+)*.
It is clear from ( 15 ) that the 2-cocycle iz obtained from our simple abstract argument is identical with the anomaly in the Lorentz algebra ( 16). Our construction is readily extended to the spinning string. The target space coordinates in covariant quantisation are generalised to superfields [ 5 1. For the open string these are again standing waves which can be decomposed into left and right moving pieces, XP(fr, r)=x$(z+a)+x!!(r-a), X$(cr)=P(,)+e+~fl(a), _ where the 8+ are anticommuting variables and the wP the superpartners of &. Diff(S’) transformations become generalised to superconformal transformations with the generators LB/ &,X+(a)=
D] X+(a)
[Va,+f(DV)
=[L,X+(~)l
D=a/ae+
+e+a,,
and V = u. + 8, U, parametrises infinitesimal supercoordinate transformations. The generators L, are constructed from the super stress-energy tensor for the full theory including reparametrisation ghosts and have the anomalous super conformal algebra +cwJJ)sc,
[v,u],,=va,u-ua,v+~(Dv)~(Du), ~2) ).
(24)
Now define the 2-cochain for the Lorentz algebra x( v, 138
ev>
Coboundary operators act on products according to Leibnitz’ rule so from (24) it follows that the action of AL on I?is the same as the action of AsI on C( U, V) followed by the substitution of v( p, w, ), V(p, w2) and V(P, w3) for the Diff( S’ ) transformation parameters. But AS annihilates C( U, V) so R must be a 2-cocycle for AL. If we compute &explicitly using (7) and ( 14) we obtain
[Lv, LJI =LpwJsc
(&~(a), w) ) (WI >~2) = (4~)
41,
where (23)
=I[4P,W),4P,W2)1.
v(p, w,) and v(p, w2) into
C(V, V),,=
‘(‘1;;‘)
j dodB+(Va:DU).
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2
The generators of Lorentz transformations now depend on the v/but have the same anomaly-free algebra as in the purely bosonic case. The light-cone gauge now includes a condition on @’ X’(r,
B
aly by Iwasaki methods.
16 June 1988
and
Kikkawa
[6 ] using
operator
References n)=x++p+7,
ly+=o.
As before Lorentz transformations are implemented non-linearly because a compensating superconforma1 transformation is included to preserve the lightcone gauge. The superfield that parametrises this is easily found to be V(0)=-a(w)‘,
[~“(a)-(x”+p”a)+28+~“(~)]
Substituting this into C(u, u),, and using mode expansions for 2 and W”(a)=C
n
o:exp(-ina)
(where for definiteness we work in the Ramond tor) we obtain the Lorentz 2-cocycle &x,
LETTERS
sec-
WI>w2)
d-10 =---a(w,)+ia(w,)+, 4(P’S
+ (4m2-
1) cu[l,c-w$].
This agrees with the calculation
of the Lorentz anom-
[ 1 ] P. Goddard, J. Goldstone, C. Rebbi and C.B. Thorn, Nucl. Phys. B 56 (1973) 109. [2] A.M. Polyakov, Phys. Lett. B 103 (1981) 207. [3] L.D. Faddeev, Phys. Lett. B 145 (1984) 81; B. Zumino, Nucl. Phys. B 253 (1985) 477; R. Stora, in: New developments in quantum field theory and statistical mechanics, eds. M. Levy and P. Mitter (Plenum, New York, 1977) p. 201; LAPP-TH-94 (Cargese Lectures 1983); L. Bonora and P. Cotta-Ramusino, Commun. Math. Phys. 87 (1983) 589; A.J. Niemi and G. Semenoff, Phys. Rev. Lett. 51 (1983) 2077; B. Zumino, Y.S. Wu and A. Zee, Nucl. Phys. B 239 ( 1984) 477; L. Baulieu, Nucl. Phys. B 241 ( 1984) 557; K.C. Chou, H.Y. Guo and K. Wu, Phys. Lett. B 134 ( 1984) 67; J.L. Manes, Nucl. Phys. B 250 ( 1985) 369; W. Bardeen and B. Zumino, Nucl. Phys. B 244 (1984) 42 I; L. Alvarez-Gaume, S. Della Peitra and G. Moore, Harvard preprint HUTP-84/A-28; R. Jackiw, in: Relativity, groups and topology II (Les Houches, 1983), ed. B.S. Dewitt and R. Stora (North-Holland, Amsterdam, 1984) p. 22 1. [4] P. Mansfield, Ann. Phys. (NY) 180 (1987) 330. [ 51 D. Friedan, E. Martinet and S. Shenker, Nucl. Phys. B 271 (1986) 93. [ 61 Y. Iwasaki and K. Kikkawa, Phys. Rev. D 8 (1973) 440.
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