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The lagrangian formulation of chiral scalars
THE LAGRANGIAN C. IMBIMBO
23 July 1987
PHYSICS LETTERS B
Volume 193, number 4
FORMULATION
OF CHIRAL SCALARS
’
Centerfor Theoretical Physics, MIT, Cambridge, MA 02139, USA and
A. SCHWIMMER Department of Nuclear Physics, Weizmann Institute of Science, 76100 Rehovot, Israel Received 5 May 1987
The lagrangian proposed by Siegel describing left- or right-moving scalars in two dimensions is analysed. The existence of anomalies is pointed out and a modification of the lagrangian leading to the elimination of anomalies is proposed. The generalization for chiral scalars defined on an arbitrary Riemann surface is discussed.
Chiral scalars are an essential ingredient in the construction of heterotic string theories [ 1] Left- or right-moving scalars are easily described in the hamiltonian language, however, a consistent lagrangian formulation is lacking. Starting with the free lagrangian for the scalar field, the unwanted (leftor right-moving) component should be projected out using a Lagrange multiplier, i.e., a constraint has to be imposed on the system. In order that the multiplier does not become dynamical, the constraint should be first class following from a gauge invariance. An ingenious proposal in this direction was made by Siegel [ 21: The Lagrange multiplier is coupled to the unwanted energy-momentum tensor component and the gauge symmetry is provided by a diagonal subgroup of reparametrizations and Weyl invariance. The action proposed by Siegel is d’x [a+yla_(~+n(&~)*],
The transformations (2) are the infinitesimal sion of reparametrizations x+=x+, ’
+=ed_p,,
The action
iii=-a+t+ca_;l-la-t.
(a-p)’
Iphys) =0 .
(4)
This prescription is inconsistent in this case. After quantization (a _P)~ becomes proportional to the -component of the energy-momentum tensor:
e__(x,)=$:(+a,py:,
(5)
where 7c is the momentum conjugate to (0. As is well known, in the equal-time commutation relations of 0_ _ (the Virasoro algebra) there is a Schwinger term:
(1)
e__(y)]
=iS’(x-y)&_(y)
(1) has the + (i/127c)6”‘(x-y) (2)
03.50 0 Elsevier Science Publishers Physics Publishing Division)
(6)
and as a consequence the constraint (4) cannot be consistently imposed in the quantized theory. The origin of this problem can be identified in the
’ On leave from INM, I- 16 146, Genoa, Italy.
0370-2693/87/$ (North-Holland
(3)
accompanied by a Weyl transformation. A standard way to quantize the action (1) would be to choose a physical gauge. This amounts to putting 1=0 and requiring that the physical states (phys ) satisfy
[e__(x),
where x,=(xo~xl)I~. gauge invariance [ 21
x_ =x’+E(X;,XI)
expan-
B.V.
45.5
Volume
193, number
lagrangian functional: W’(l) =
PHYSICS
4
formalism
by considering
invariance
(2)
(7) generates
the
Ward
(8) For 13= 0 (8) becomes
a+e__=o,
(9)
which is a combination of the Ward identities of reparametrization a%‘,, = 0 and Weyl invariance 0; = 0. As a consequence (9) is necessarily anomalous [ 3,4] ; the correlation function of two 8 _ _ operators, Fourier transformed, has the expression
xm,
I
ewy)
Y= s [a+pa_~+n(a_~)2]d2X (15)
(10)
the coefficient c depending on the fields contributing. Therefore the Ward identity (8) acquires an anomalous contribution cd3k The modified equation (8) can be integrated to give ic
where b, 6 are fermionic variables. The lagrangian LP=~) has classically the Siegel gauge invariance but, by the general argument presented above, the symmetry becomes anomalous. A straightforward calculation shows that the contribution of the ghosts to the anomaly is -261247~. Therefore the combined lagrangian ( 1) + (14) is anomalous. A simple modification of (1) based on the fact that a scalar field can serve as its own Wess-Zumino field, gives an anomaly-free lagrangian. We propose to modify ( 1) to
+
(r(e__e__))=Cq4iq2,
(
23 July 1987
(14)
dg, exp[ iS(p, A)] .
w(n) =exp
B
the generating
s
The gauge identity:
LETTERS
d*xd*Y
@n(x)
The additional term in (15) violates explicitly the gauge invariance. When one calculates W(A), however, the explicit violation and the anomalies cancel due to the special value of the coefficient chosen in (15). The quantization follows now the standard procedure [ 51. We define a nilpotent BRS operator Q:
>
,
(11) where
where K(x, Y) fulfills the equation
a+a_K(x,y)+a_(na_K)=S(2)(x-y).
(12)
For a single scalar field c= l/2411. The existence of the anomaly requires a careful analysis of the quantization procedure following refs.
The physical
]3,51. The integration over 1 is replaced by the integration over the gauge group parameter E. We need therefore to include the determinant of the jacobian:
or in the sector without
Mx) -= 86(Y)
-a-:d(x-y)
+a_;l(x)d(x--y)
+ayd(x-y)qy)) . The determinant is replaced a ghost lagrangian: 456
(13) by the contribution
of
state condition
Qlph> =O moduloIph)e(@‘)(+)Iph) =O ,
becomes:
Iph)+Qlph)
,
(18)
ghosts: (19)
where 0(p)(+) is the positive-frequency part of 0(p). Since the positive-frequency commutators do not contain the anomaly, (19) can be consistently imposed and gives ak]ph)=O
fork>O,
where ak is the annihilation k. Therefore the physical
(20) operator for momentum space contains only left-
Volume
193, number
PHYSICS
4
moving scalars. The gauge-invariant operators in the theory are non-local and can be reduced to combinations of
r[x+,x-+jlA-(y,x-)dy].
(21)
For scalars defined on a circle, the relevant case for string theory, a more detailed discussion is needed since zero modes are present. The mode expansion for a compact scalar field q with periodicity a defined on a circle parametrized by x,e(O,l) is p(x,)=q+p’x,
+
a(x,)=p+i
c -aPZ+a’, O#O J2n ; (a?, -a,)
C tI*0
exp(i2nnx,)
exp(i2rrnx,)
,
,
(22) [a,,aL1=6,,,,,
b,41=-i,
(23)
all the other commutators vanishing. In particular@ commutes with all the operators and takes the values ma, m = integer. The commutation relations (23) are realized on a Hilbert space ]p;p’,N,,) where the spectrum of p is: 2zmla+a, m=integer and N,, are occupation numbers of the oscillator n. The condition (19) is fulfilled if N,=O,
n>O,
(24a)
p-p’ co.
(24b)
In order to implement
a2 = (27rIM) N,
(24b)
a has to be quantized
A4, N= integer ,
and then the allowed values of the momenta
p’=p=,/mNm.
Trexp(iti)=
~dy,&exp(i~Ydr,dr,)
of
(28)
with Y given by (15) with periodic boundary conditions x0, xl E (0, 1). Due to the boundary conditions constant J. configurations cannot be gauged away and therefore should be integrated over. Now there are gauge transformations originating in diffeomorphisms of the torus not connected continuously to the unity:
O( =
Xi
l-n
-n
n
nfl
x0 >( x, > .
the transformation
(28) il transforms
A’=A-2n.
(29) as
(30)
The contributions to the generating functional coming from the non-zero modes of IJJsatisfy (30). We should therefore discuss only the contributions coming from p. the zero mode part of ~1: Vo(Xo, XI) =4+Pho
SP’X,
3
(31)
with p&= m,ap’ = m,a. We obtain 2, =
C
exp{ifa2( m, -m,)
ml, m2
x [m, +m2+l(ml
-m2)l}
= 1 B(ma2/n, 2a2d17r) m
(25) will be
(26)
Due to (24b) it is impossible to have non-compact chiral scalars on a circle. If we have more than one compact scalar defined on a torus R”/A where A is a lattice, condition (25) becomes A*+vnA#@)
(25). Condition (25) is related to global properties Siegel’s gauge invariance. One calculates
Under
relations
23 July 1987
B
the periodicity imposed by A exists in the quantum theory, we obtain A 1 A* corresponding to M= 1 in
4
J
with the commutation
LETTERS
(27)
where A* is the lattice dual to A and v is a fixed vector. If in addition we require that an operator with
+ 1 exp{ia’[~(~+l)+m]} m
xB(a2(l+m+$)ln,
2a2Ah) ,
(32)
where 0 is the standard 6’[ fJ function. If one requires periodicity of 2, as a function of a2/2, a2A/2and under the transformation (30), 2 will vanish unless condition (25) is satisfied. We remark that the relation between the existence of gauge-invariant states in the Hilbert space and the periodicity of the generating functional is somewhat 457
Volume
193, number
4
PHYSICS
similar to the one encountered for global anomalies in quantum mechanics [ 61. Finally, we discuss the generalization of the lagrangian (15) to the case when the variables x0, xl are on an arbitrary Riemann surface. Chirality has a meaning only in the tangent space. We need therefore to introduce zweibeins e:: with Minkowski metric on the tangent space indices a = 0, 1. The action we propose is
S=
j
d2xe
(
e~eYd,~d,~++;i(e~ee”a,~a,~)
(33) where er = (eg f eI;)Ifi. The Lagrange multiplier A transforms under a local Lorentz transformation as a chiral tensor with two + indices. The Siegel gauge invariance corresponds to a coordinate transformation accompanied by a transformation in the tangent space in such a way that the metric (zweibeins) is not changed:
LETTERS
B
absorbed in 1. The holomorphic property is preserved by quantization only if the Siegel gauge invariance is not anomalous. The aforementioned analytic continuation gives a meaning to the action (33) on a euclidean torus. For higher genus compact Riemann surfaces a similar procedure is not straightforward. In particular, it is not immediate that (33) together with the gaugeinvariant definition of chiral bosonic operators (2 1) reproduces the recently proposed chiral bosonic operators ( 2 1) reproduces the recently proposed chiral bosonisation formulae [ 71. This and related questions are presently under study. We would like to thank the Laboratoire de Physique Theoriqe de I‘Ecole Not-male Superieure, where this work was begun, for their hospitality. Very useful discussions with S. Elitzur, G. Moore and N. Seiberg are gratefully acknowledged. The work of A.S. was supported by a BSF contract. References [ 1] D.J. Gross, J.A. Harvey, E. Martinet
6~ -a+
f+ fa_bAa_
E ,
(34)
where the a, operators are defined as a, = es a,. A check on the validity of (33) is provided by calculating Tr exp( iHr, + ipr ,) for H, P defined by ( 15 ) . The trace reproduces (33) with x0, xl E (0, 1) and e,“=(l,O),ei=(-r,,ro) which after the analytic continuation to euclidean space, xo=ixZ, 7 = ir2 becomes the lagrangian defined on a torus characterized by the modular parameter 7 = 7, + ir2. Moreover, the lagrangian becomes holomorphic, i.e., dependent only on 7, the T dependence being
458
23 July 1987
and R. Rohm, Phys. Rev. Lett. 54 (1985) 502. [2] W. Siegel, Nucl. Phys. B 238 (1984) 307. [ 31 A.M. Polyakov, Phys; Lett. B 103 (1981) 207. [ 41 0. Alvarez, in: Workshop on Unified string theories (World Scientific, Singapore, 1986) p. 103. [ 5) M. Kato and K. Ogawa, Nucl. Phys. B 2 12 (1983) 443; S. Hwang, Phys. Rev. D 28 (1983) 2614. [6] S. Elitzur, E. Rabinovici, Y. Frishman and A. Schwimmer, Nucl. Phys. B 273 (1986) 93. [7] L. Alvarez-Gaume, J.B. Bost, G. Moore, P. Nelson and C. Vafa, Phys. Lett. B 178 (1986) 41; J.B. Bost and P. Nelson, Phys. Rev. Lett. 57 (1986) 795; T. Eguchi and H. Ooguri, LPTENS preprint LPTENS 86.39 (1986).
23 July 1987
PHYSICS LETTERS B
Volume 193, number 4
FORMULATION
OF CHIRAL SCALARS
’
Centerfor Theoretical Physics, MIT, Cambridge, MA 02139, USA and
A. SCHWIMMER Department of Nuclear Physics, Weizmann Institute of Science, 76100 Rehovot, Israel Received 5 May 1987
The lagrangian proposed by Siegel describing left- or right-moving scalars in two dimensions is analysed. The existence of anomalies is pointed out and a modification of the lagrangian leading to the elimination of anomalies is proposed. The generalization for chiral scalars defined on an arbitrary Riemann surface is discussed.
Chiral scalars are an essential ingredient in the construction of heterotic string theories [ 1] Left- or right-moving scalars are easily described in the hamiltonian language, however, a consistent lagrangian formulation is lacking. Starting with the free lagrangian for the scalar field, the unwanted (leftor right-moving) component should be projected out using a Lagrange multiplier, i.e., a constraint has to be imposed on the system. In order that the multiplier does not become dynamical, the constraint should be first class following from a gauge invariance. An ingenious proposal in this direction was made by Siegel [ 21: The Lagrange multiplier is coupled to the unwanted energy-momentum tensor component and the gauge symmetry is provided by a diagonal subgroup of reparametrizations and Weyl invariance. The action proposed by Siegel is d’x [a+yla_(~+n(&~)*],
The transformations (2) are the infinitesimal sion of reparametrizations x+=x+, ’
+=ed_p,,
The action
iii=-a+t+ca_;l-la-t.
(a-p)’
Iphys) =0 .
(4)
This prescription is inconsistent in this case. After quantization (a _P)~ becomes proportional to the -component of the energy-momentum tensor:
e__(x,)=$:(+a,py:,
(5)
where 7c is the momentum conjugate to (0. As is well known, in the equal-time commutation relations of 0_ _ (the Virasoro algebra) there is a Schwinger term:
(1)
e__(y)]
=iS’(x-y)&_(y)
(1) has the + (i/127c)6”‘(x-y) (2)
03.50 0 Elsevier Science Publishers Physics Publishing Division)
(6)
and as a consequence the constraint (4) cannot be consistently imposed in the quantized theory. The origin of this problem can be identified in the
’ On leave from INM, I- 16 146, Genoa, Italy.
0370-2693/87/$ (North-Holland
(3)
accompanied by a Weyl transformation. A standard way to quantize the action (1) would be to choose a physical gauge. This amounts to putting 1=0 and requiring that the physical states (phys ) satisfy
[e__(x),
where x,=(xo~xl)I~. gauge invariance [ 21
x_ =x’+E(X;,XI)
expan-
B.V.
45.5
Volume
193, number
lagrangian functional: W’(l) =
PHYSICS
4
formalism
by considering
invariance
(2)
(7) generates
the
Ward
(8) For 13= 0 (8) becomes
a+e__=o,
(9)
which is a combination of the Ward identities of reparametrization a%‘,, = 0 and Weyl invariance 0; = 0. As a consequence (9) is necessarily anomalous [ 3,4] ; the correlation function of two 8 _ _ operators, Fourier transformed, has the expression
xm,
I
ewy)
Y= s [a+pa_~+n(a_~)2]d2X (15)
(10)
the coefficient c depending on the fields contributing. Therefore the Ward identity (8) acquires an anomalous contribution cd3k The modified equation (8) can be integrated to give ic
where b, 6 are fermionic variables. The lagrangian LP=~) has classically the Siegel gauge invariance but, by the general argument presented above, the symmetry becomes anomalous. A straightforward calculation shows that the contribution of the ghosts to the anomaly is -261247~. Therefore the combined lagrangian ( 1) + (14) is anomalous. A simple modification of (1) based on the fact that a scalar field can serve as its own Wess-Zumino field, gives an anomaly-free lagrangian. We propose to modify ( 1) to
+
(r(e__e__))=Cq4iq2,
(
23 July 1987
(14)
dg, exp[ iS(p, A)] .
w(n) =exp
B
the generating
s
The gauge identity:
LETTERS
d*xd*Y
@n(x)
The additional term in (15) violates explicitly the gauge invariance. When one calculates W(A), however, the explicit violation and the anomalies cancel due to the special value of the coefficient chosen in (15). The quantization follows now the standard procedure [ 51. We define a nilpotent BRS operator Q:
>
,
(11) where
where K(x, Y) fulfills the equation
a+a_K(x,y)+a_(na_K)=S(2)(x-y).
(12)
For a single scalar field c= l/2411. The existence of the anomaly requires a careful analysis of the quantization procedure following refs.
The physical
]3,51. The integration over 1 is replaced by the integration over the gauge group parameter E. We need therefore to include the determinant of the jacobian:
or in the sector without
Mx) -= 86(Y)
-a-:d(x-y)
+a_;l(x)d(x--y)
+ayd(x-y)qy)) . The determinant is replaced a ghost lagrangian: 456
(13) by the contribution
of
state condition
Qlph> =O moduloIph)e(@‘)(+)Iph) =O ,
becomes:
Iph)+Qlph)
,
(18)
ghosts: (19)
where 0(p)(+) is the positive-frequency part of 0(p). Since the positive-frequency commutators do not contain the anomaly, (19) can be consistently imposed and gives ak]ph)=O
fork>O,
where ak is the annihilation k. Therefore the physical
(20) operator for momentum space contains only left-
Volume
193, number
PHYSICS
4
moving scalars. The gauge-invariant operators in the theory are non-local and can be reduced to combinations of
r[x+,x-+jlA-(y,x-)dy].
(21)
For scalars defined on a circle, the relevant case for string theory, a more detailed discussion is needed since zero modes are present. The mode expansion for a compact scalar field q with periodicity a defined on a circle parametrized by x,e(O,l) is p(x,)=q+p’x,
+
a(x,)=p+i
c -aPZ+a’, O#O J2n ; (a?, -a,)
C tI*0
exp(i2nnx,)
exp(i2rrnx,)
,
,
(22) [a,,aL1=6,,,,,
b,41=-i,
(23)
all the other commutators vanishing. In particular@ commutes with all the operators and takes the values ma, m = integer. The commutation relations (23) are realized on a Hilbert space ]p;p’,N,,) where the spectrum of p is: 2zmla+a, m=integer and N,, are occupation numbers of the oscillator n. The condition (19) is fulfilled if N,=O,
n>O,
(24a)
p-p’ co.
(24b)
In order to implement
a2 = (27rIM) N,
(24b)
a has to be quantized
A4, N= integer ,
and then the allowed values of the momenta
p’=p=,/mNm.
Trexp(iti)=
~dy,&exp(i~Ydr,dr,)
of
(28)
with Y given by (15) with periodic boundary conditions x0, xl E (0, 1). Due to the boundary conditions constant J. configurations cannot be gauged away and therefore should be integrated over. Now there are gauge transformations originating in diffeomorphisms of the torus not connected continuously to the unity:
O( =
Xi
l-n
-n
n
nfl
x0 >( x, > .
the transformation
(28) il transforms
A’=A-2n.
(29) as
(30)
The contributions to the generating functional coming from the non-zero modes of IJJsatisfy (30). We should therefore discuss only the contributions coming from p. the zero mode part of ~1: Vo(Xo, XI) =4+Pho
SP’X,
3
(31)
with p&= m,ap’ = m,a. We obtain 2, =
C
exp{ifa2( m, -m,)
ml, m2
x [m, +m2+l(ml
-m2)l}
= 1 B(ma2/n, 2a2d17r) m
(25) will be
(26)
Due to (24b) it is impossible to have non-compact chiral scalars on a circle. If we have more than one compact scalar defined on a torus R”/A where A is a lattice, condition (25) becomes A*+vnA#@)
(25). Condition (25) is related to global properties Siegel’s gauge invariance. One calculates
Under
relations
23 July 1987
B
the periodicity imposed by A exists in the quantum theory, we obtain A 1 A* corresponding to M= 1 in
4
J
with the commutation
LETTERS
(27)
where A* is the lattice dual to A and v is a fixed vector. If in addition we require that an operator with
+ 1 exp{ia’[~(~+l)+m]} m
xB(a2(l+m+$)ln,
2a2Ah) ,
(32)
where 0 is the standard 6’[ fJ function. If one requires periodicity of 2, as a function of a2/2, a2A/2and under the transformation (30), 2 will vanish unless condition (25) is satisfied. We remark that the relation between the existence of gauge-invariant states in the Hilbert space and the periodicity of the generating functional is somewhat 457
Volume
193, number
4
PHYSICS
similar to the one encountered for global anomalies in quantum mechanics [ 61. Finally, we discuss the generalization of the lagrangian (15) to the case when the variables x0, xl are on an arbitrary Riemann surface. Chirality has a meaning only in the tangent space. We need therefore to introduce zweibeins e:: with Minkowski metric on the tangent space indices a = 0, 1. The action we propose is
S=
j
d2xe
(
e~eYd,~d,~++;i(e~ee”a,~a,~)
(33) where er = (eg f eI;)Ifi. The Lagrange multiplier A transforms under a local Lorentz transformation as a chiral tensor with two + indices. The Siegel gauge invariance corresponds to a coordinate transformation accompanied by a transformation in the tangent space in such a way that the metric (zweibeins) is not changed:
LETTERS
B
absorbed in 1. The holomorphic property is preserved by quantization only if the Siegel gauge invariance is not anomalous. The aforementioned analytic continuation gives a meaning to the action (33) on a euclidean torus. For higher genus compact Riemann surfaces a similar procedure is not straightforward. In particular, it is not immediate that (33) together with the gaugeinvariant definition of chiral bosonic operators (2 1) reproduces the recently proposed chiral bosonic operators ( 2 1) reproduces the recently proposed chiral bosonisation formulae [ 71. This and related questions are presently under study. We would like to thank the Laboratoire de Physique Theoriqe de I‘Ecole Not-male Superieure, where this work was begun, for their hospitality. Very useful discussions with S. Elitzur, G. Moore and N. Seiberg are gratefully acknowledged. The work of A.S. was supported by a BSF contract. References [ 1] D.J. Gross, J.A. Harvey, E. Martinet
6~ -a+
f+ fa_bAa_
E ,
(34)
where the a, operators are defined as a, = es a,. A check on the validity of (33) is provided by calculating Tr exp( iHr, + ipr ,) for H, P defined by ( 15 ) . The trace reproduces (33) with x0, xl E (0, 1) and e,“=(l,O),ei=(-r,,ro) which after the analytic continuation to euclidean space, xo=ixZ, 7 = ir2 becomes the lagrangian defined on a torus characterized by the modular parameter 7 = 7, + ir2. Moreover, the lagrangian becomes holomorphic, i.e., dependent only on 7, the T dependence being
458
23 July 1987
and R. Rohm, Phys. Rev. Lett. 54 (1985) 502. [2] W. Siegel, Nucl. Phys. B 238 (1984) 307. [ 31 A.M. Polyakov, Phys; Lett. B 103 (1981) 207. [ 41 0. Alvarez, in: Workshop on Unified string theories (World Scientific, Singapore, 1986) p. 103. [ 5) M. Kato and K. Ogawa, Nucl. Phys. B 2 12 (1983) 443; S. Hwang, Phys. Rev. D 28 (1983) 2614. [6] S. Elitzur, E. Rabinovici, Y. Frishman and A. Schwimmer, Nucl. Phys. B 273 (1986) 93. [7] L. Alvarez-Gaume, J.B. Bost, G. Moore, P. Nelson and C. Vafa, Phys. Lett. B 178 (1986) 41; J.B. Bost and P. Nelson, Phys. Rev. Lett. 57 (1986) 795; T. Eguchi and H. Ooguri, LPTENS preprint LPTENS 86.39 (1986).