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Chiral symmetries of the WZNW model by hamiltonian methods
__!!!B
8 June 1995
22%
PHYSICS
ELSEVIER
LETTERS
B
Physics Letters B 352 (1995) 64-68
Chiral symmetries of the WZNW model by hamiltonian methods * B. Sazdovi6 Institute
of Physics,
PO.
Box 57.
II001
Beograd,
Yugoslavia
Received 4 April 1995 Editor: L. Alvarez-Gaumk
Abstract The connection between the Kac-Moody algebras of currents and the chiral symmetries of the two dimensional WZNW model is clarified. It is shown that only the zero modes of the Kac-Moody currents are the first class constraints, and that, consequently, the corresponding gauge symmetries are chiral.
1. Introduction
It is well known that the two dimensional WessZumino-Novikov-Witten (WZNW) model [ l-31 in addition to the conformal invariance possesses an invariance under the chirul gauge transformations go+“-(x+,x-)
=n+(xf)g(x+,x-)~2Z1(x-) (1)
when a+ and Sz- are arbitrary group valued functions of the light cone coordinates Xf_
l -pod).
(2)
It is common in the literature [ 3-51 to connect the Kac-Moody (KM) currents with the symmetry ( 1). The goal of this paper is to clarify this connection from the point of view of Dirac’s method of constraints [ 61. The generators of the symmetry must be the first-class (FC) constraints while “almost all” KM currents are the second-class (SC) constraints. We are going to *Work supported Yugoslavia. 0370-2693/95/$09.50
in part by the Serbian
Research
solve this puzzle by showing that only one Fourier mode of the KM currents is the FC constraint, and this mode is the generator of the chiral symmetry ( 1). This explains why the symmetry is chiral. There is another nonstandard application of this canonical analysis. Usualy we use this approach to obtain local gauge symmetries (the parameters of the transformations depend on all space-time coordinates). In this example we are able to find chid gauge symmetries (the parameters of the transformations depend only on some particular combinations of the coordinates, here X+ and x- ). The FC constraints as the generators of these transformations are the Fourier components of the complete constraints.
2. The two dimensional
We will consider the WZNW model described by the action [ 31
W
Foundation,
Here @
1995
SSDI0370-2693(95)00453-X
WZNW model
Elsevier Science B.V. All rights reserved
= So(g)
+
Wg).
(34
65
B. SazdouiC/Physics Letters B 352 (1995) 64-68
So(g) =
&
J(*v-,0-)
Since the three form v3 is a closed form on the three dimensional manifold B, we have locally
T.
is the action of the non-linear a-model on the field g : 2 4 G where Z is two dimensional space-time and G is a semisimple Lie group. The expression (X, Y) is the Cartan-Killing form, invariant under the adjoint action of the group element fI E G (RXir’JlYW’) v- = g-‘dg,
= (X,Y) v+ = gdg-’
(5) = -dgg-’
~(UA,
[UA,UAI)
(10)
where ?-Ais a 2-form TA(pP) = $~Aa&‘ndqPP.
(11)
Note that 7__ = -r+.Using the Stoke’s theorem the Wess-Zumino term I can be locally converted to an integral over the space time B
(6)
are the left (right) invariant Maurer-Cartan forms and *u = *dx+, = ~~,,dxYvp is the dual of v. The second
= -3!dTA((D)
Usp) =
-&
J
r-
term
r(g) =
y&J$v-,iv-.,u-I>
The local expression for the action
is the Wess-Zumino topological term where the integration is carried over a three dimensional manifold B, whose boundary is 8 (Z = 6’B). Instead of working with the field g(x) we will benefit from taking local coordinates on the group manifold @(x) as canonical coordinates. Let t, be the generators of the Lie algebra of G satisfying [tartb] = fabCtc,and let Yab= (to, tb) = facdfbdcbe the group metric. The coordinate indices cy,p, y, . . . and the Lie algebra indices a, b, c, . . . run over the same range. We define the vierbeins on G, E,O(cp), as the coeficients of the expansion of v in dpa and t,, VA =
d@EAaa (9) to = drp”EA,
(8)
where A = +, - are light cone indices. There are two veilbeins, E+,” and Eean, as well as two MaurerCartan forms, v+ and v_. The metric on the group manifold Hap(p) = (EAT, EA,B) = YobE.d%pb
(9)
(no sumation over A) does not depend on A (H,. = H- ) as a consequence of the relation v- = -g-‘v+g and the property (5) of the Cartan-Killing form. The action (4a) can be now rewritten in the form
(4b)
s(V) = so(q) + kr(q) =
K
J
d*x[ -&f‘VH,p(4p)
L
+
E+-~~(~P)ia,hWJ
(3b)
writen in the light cone coordinates takes the form s(q) =
-K
Jd2x[H,8(~P)+ c L
(3c) whereKE&, a&= -&(a0f aI)and eol = -1. The action in the form (3~) is very convenient for the canonical analysis. For the final conclusion we will need boundary conditions, as will be discussed later. 3. Constraints We are going to use Dirac’s canonical approach to find the gauge symmetries. The hamiltonian equations of motion describe dinamical evolution of a system by a trajectory in a phase space. The presence of an arbitrary multiplier in the hamiltonian means that for given initial conditions we will not obtain the unique solution, one trajectory, but a whole set of trajectories, describing the same physical state. The FC constraints are generators of the mapping between these trajectories that do not change the physical state. These unphysical transformations are the gauge symmetries of
B. SazdoviC/Physics Letters B 352 (1995) 64-68
66
the hamiltonian equations of motion. The method essentialy depends on the choice of time variable. If we take x0 as a time coordinate and use the bilinear expression in the time derivative (3b), as a action, there are no constraints and consequently we can not find the symmetries this way. The expression (3~) for the action is more promising because it is linear in the a__cpand J+‘p. Taking first x- and then xf as time coordinates we will find some primary constraints which give us a hope to find some symmetries of the system. To treat both cases together we will use the light cone indices A = {+, -} with xA = {x+,x-} and 84 = {a+,a_}. These two cases are independent and there are no summations over repeated indices A. In the first case (A = -) we will take x- = r and x+ = o and in the second one (A = +) we will take xf = r and x- = --(T. (The minus sign we adopt to preserve the orientation between coordinate axis.) The action corresponding to the choice xA = r has the form &((o) = K
I
d2X[ ( -l)A&fi
- 27-,p]@ba@. (12)
The canonical momentum conjugate to the variable @ is ‘7ra=
K[ (-1)A&p
-
2T_,p]Cp@
The action is linear in the time derivative and consequently the canonical hamiltonian density is zero, 7-i, = 0, while the total hamiltonian takes the form HAT =/d&q
t
= {JA~(x),HAT}
TAO
M -(-1)A2Kd~,(x)
M 0.
(17a)
The multipliers in (17a) looks like determined U’A~= 0 which means that JAMshould be SC constraints. More precisely, only the first derivative of the multipliers is determined and the corresponding constraints are the SC. The o-independent part is arbitrary function of r and corresponding constraints are the FC. The same conclusion we can obtain from Eq. (15a). The Schwinger term contains the derivative of the S-function and gives zero on the CTindependent functions. It means that the Schwinger term does not give a contributions for some modes of the currents and these modes are the FC constraints. Let us make a more precise investigation. We assume that space is compact which means that for all variables X = {JA~,uA’,...} we have X(~,U+ Z,) = X( 7, a). We will use the Fourier expansion X(x)
=
;~xn(Tx)e-‘“-, JEZ
JAn = ?r, + 2KT-47”
- ( - 1) AK&~#?
X, (TV) =
(14)
JAo = -EAo~JA~
(18)
L
(13)
We prefer to rewrite these constraints with the Lie algebra indices as
164
The consistency condition for JAMleads to
(1)
and therefore, there are primary constraints:
= JduuAaJA,.
J
da,X( x) ei”Wr”
0
where w = F, and the a-function representation
where EA~Ois the inverse of EAT”. The Poisson brackets (PB) between these constraints define two independent KM algebras (A = + or -) :
(19) The PB between modes ~~~~( T) are
{JA&h -
JAb(y)) (
= fobCJAc(~)&flx
-1)A2K~&5’(ff,
- CTy).
-
ay)
(15a)
{ JAG~D JAbnt}
= fabC J~c(n+m) - t - 1 )A4~iK%b&n.0 (15b)
The expression (14) is a convenient linear combination of the previous constraints. We adapt a correspondence between Jh and E& in order to obtain the algebra ( Ha).
while plus and minus constraints commute. This expression does not depend on the period L. In terms of modes the total hamiltonian becomes
67
B. SazdoviC/ Physics Letters B 352 (1995) 64-68
=-
c
1 L
obtain the gauge generators and the corresponding gauge transformation&The gauge parameters WA’( 7) will depend only on the time coordinate 7, as well as the FC constraints JAB and the arbitrary multipliers
(16b)
UAan(7x)JAa(-n)(Tx)*
KZ
Now it is easy to separate the FC and the SC constraints, and check the consistency condition for the primary Constraints JAon( 7) :
UA'O.
All the FC constraints JAM are primary in this case, and the expression for the gauge symmetry generator is GA=wA’(T)JA~(T)
(A=+,-).
(21)
The gauge transformations produced by GA are
(17b) For n = 0 the multipliers u,@(r) are arbitrary functions of time and JA&(7) constraints are FC. For n # 0 we can determined the multipliers UAon (7) = 0 (n # 0) which means that JA*,, (n # 0) are SC constraints. The same conclusion follows from ( 15b) because
{J~ao,JAbn) = {J~on,J,tb(--n)}
for
n $0.
1 -UA~O(T)JACIO(~).
L
g-‘&g
= S_cpnE_,at,, = S+SpaE+~ata.
(23)
With the help of Eq. (22) we get
$0
(20)
In any case there are no secondary constraints. The total hamiltonian is now HAT=
(22)
We are interested in the gauge transformations of the field g. Using Eqs. (6) and (8) we can connect the transformations of the fields g and (p
- 6+gg-’
.fabCJAcn = 0, M -(-l)A‘hiKny,b
S,J(~~= {@, GA} = -wA’(T)EA~~.
(16c)
In Section 5 we will verify that expression ( 16~) gives the correct equations of motion. The assumption of the periodicaly boundary conditions is not necessary. The general solution of (17a) is u.&,((T, r) = UAO (7) and substituting this into ( 16a) we obtain (16~) with uA’(r) instead of +uA(la(r) and JAM = s daJAa( (T, 7). The constraints JAM are FC because they correspond to the arbitrary multiplier U,Ja(7). They also weakly vanish with all constraints, which can be easily verified integrating IQ. ( 15a) over ox. For our purpose any boundary condition with finite JAM is a good condition.
s-g = -gw_,
6,g = o+g
(24)
where @A= WAat,l. Because WAis infinitesimal we can write g”- =g+S_g=g(l -w_) age-O-, gO+ =g+6,g=(1+w+)g=eeO+g. Recalling that w_ E o-(x-) we have gn- =gK’(x-),
(25)
and w+ = w+ (x+)
go+ =n+(n+)g
(26)
where fiA (XA) = eoA(“‘). Fqs. (26) are equivalent with the EQ. ( 1) . Therefore we demonstrated that the gauge symmetries generated by the zero modes of the Kh4 currents are exactly the known chiral gauge symmetries.
5. Equations of motion 4. Chid
gauge symmetries
The presence of arbitrary multipliers uA’O(7) in the total hamiltonian means that the theory has a gauge symmetry. We will use Castellani’s method [7] to
For completeness we briefly derive the hamiltonian equations of motion. For the fields (Dowe have +a = {@, HAT} = -UAaOEAoa.
(27)
68
B. Sazdouit / Physics Letters B 352 (1995) 64-68
The equations of motion for the field g are obtained from Eqs. (6)and (8) in terms of the equations for (pa g-‘g = @aE_,ata,
- gg-’ = @aE+aata
(28)
and by using Eq. (27) we find g-‘g = --u-o(7),
kg-’
= u+o(7)
(29a)
where u,z,,,= UAa,,to. If we now recall our definition of time r = xA, it follows that g = a-g for A = - and g = d+g for A = +, so that Eqs. (29a) can be rewritten as g-‘&g = --u_e(X-), d+gg_’ = u+@(X+).
(29b)
Therefore we obtain the well known equations of motion,
zero modes are FC constraints and they are the generators of the gauge symmetries. The only arbitrary parameters are zero mode gauge parameters. They depend on the x- in the first case and on the x+ in the second one, and this explains the chirality of the gauge symmetry. The connection between KM algebra and chiral gauge symmetry is now clear. The KM algebra contains infinitely many modes of the SC constraints (it looks like the algebra of SC constraints) and only one FC mode which is the generator of the symmerty ( 1) . This result is obtained by the application of the canonical approach to a new kind of symmetry when the gauge parameter does not depend on all coordinates, but only on some particular combinations of the coordinates [ 81.
References
a+(g-‘a-g)
= 0,
&(d+gg-1)
=0
(30)
which are, as well as the Eqs. (29b), evidently invariant under the chiral gauge transformations ( 1) .
[11 I. Wess and B. Zumino, Phys. Lett. B 37 (1971) 95.
121S.P. Novikov, Sov. Math. Doklady 24 ( 1981) 222; Usp. Math.Nauk. 37 (1982) 3. 131 E. Witten, Nucl. Phys. B 223 (1983) 422; Commun. Math.
Phys. 92 ( 1984) 455.
6. Conclusion The canonical formalism in the light cone coordinates is developed for the two dimensional WZNW model. Using x- and x+ as the evolution parameters we obtain the two commuting KM algebras of constraints. We made a mode expansion in order to separate the FC constraints from the SC ones. Only the
141 V.G. Knizhnik and A.B. Zamolodchikov, Nucl. Phys. B 247
(1984) 83. 151 D. Gepner and E. Witten, Nucl. Phys. B 278 (1986) 493. 161 P.A.M. Dirac, Lectures on quantum mechanics, Belfer
Graduate School of Science (Yeshiva Univ., 1964). [71 L. Castellani, Ann. Phys. (New York) 143 (1982) 357. 181 M. Blagojevic, M. Vasilic and T. VukaSinac, Class. Quan.
Grav.
11 (
1994) 2134.
8 June 1995
22%
PHYSICS
ELSEVIER
LETTERS
B
Physics Letters B 352 (1995) 64-68
Chiral symmetries of the WZNW model by hamiltonian methods * B. Sazdovi6 Institute
of Physics,
PO.
Box 57.
II001
Beograd,
Yugoslavia
Received 4 April 1995 Editor: L. Alvarez-Gaumk
Abstract The connection between the Kac-Moody algebras of currents and the chiral symmetries of the two dimensional WZNW model is clarified. It is shown that only the zero modes of the Kac-Moody currents are the first class constraints, and that, consequently, the corresponding gauge symmetries are chiral.
1. Introduction
It is well known that the two dimensional WessZumino-Novikov-Witten (WZNW) model [ l-31 in addition to the conformal invariance possesses an invariance under the chirul gauge transformations go+“-(x+,x-)
=n+(xf)g(x+,x-)~2Z1(x-) (1)
when a+ and Sz- are arbitrary group valued functions of the light cone coordinates Xf_
l -pod).
(2)
It is common in the literature [ 3-51 to connect the Kac-Moody (KM) currents with the symmetry ( 1). The goal of this paper is to clarify this connection from the point of view of Dirac’s method of constraints [ 61. The generators of the symmetry must be the first-class (FC) constraints while “almost all” KM currents are the second-class (SC) constraints. We are going to *Work supported Yugoslavia. 0370-2693/95/$09.50
in part by the Serbian
Research
solve this puzzle by showing that only one Fourier mode of the KM currents is the FC constraint, and this mode is the generator of the chiral symmetry ( 1). This explains why the symmetry is chiral. There is another nonstandard application of this canonical analysis. Usualy we use this approach to obtain local gauge symmetries (the parameters of the transformations depend on all space-time coordinates). In this example we are able to find chid gauge symmetries (the parameters of the transformations depend only on some particular combinations of the coordinates, here X+ and x- ). The FC constraints as the generators of these transformations are the Fourier components of the complete constraints.
2. The two dimensional
We will consider the WZNW model described by the action [ 31
W
Foundation,
Here @
1995
SSDI0370-2693(95)00453-X
WZNW model
Elsevier Science B.V. All rights reserved
= So(g)
+
Wg).
(34
65
B. SazdouiC/Physics Letters B 352 (1995) 64-68
So(g) =
&
J(*v-,0-)
Since the three form v3 is a closed form on the three dimensional manifold B, we have locally
T.
is the action of the non-linear a-model on the field g : 2 4 G where Z is two dimensional space-time and G is a semisimple Lie group. The expression (X, Y) is the Cartan-Killing form, invariant under the adjoint action of the group element fI E G (RXir’JlYW’) v- = g-‘dg,
= (X,Y) v+ = gdg-’
(5) = -dgg-’
~(UA,
[UA,UAI)
(10)
where ?-Ais a 2-form TA(pP) = $~Aa&‘ndqPP.
(11)
Note that 7__ = -r+.Using the Stoke’s theorem the Wess-Zumino term I can be locally converted to an integral over the space time B
(6)
are the left (right) invariant Maurer-Cartan forms and *u = *dx+, = ~~,,dxYvp is the dual of v. The second
= -3!dTA((D)
Usp) =
-&
J
r-
term
r(g) =
y&J$v-,iv-.,u-I>
The local expression for the action
is the Wess-Zumino topological term where the integration is carried over a three dimensional manifold B, whose boundary is 8 (Z = 6’B). Instead of working with the field g(x) we will benefit from taking local coordinates on the group manifold @(x) as canonical coordinates. Let t, be the generators of the Lie algebra of G satisfying [tartb] = fabCtc,and let Yab= (to, tb) = facdfbdcbe the group metric. The coordinate indices cy,p, y, . . . and the Lie algebra indices a, b, c, . . . run over the same range. We define the vierbeins on G, E,O(cp), as the coeficients of the expansion of v in dpa and t,, VA =
d@EAaa (9) to = drp”EA,
(8)
where A = +, - are light cone indices. There are two veilbeins, E+,” and Eean, as well as two MaurerCartan forms, v+ and v_. The metric on the group manifold Hap(p) = (EAT, EA,B) = YobE.d%pb
(9)
(no sumation over A) does not depend on A (H,. = H- ) as a consequence of the relation v- = -g-‘v+g and the property (5) of the Cartan-Killing form. The action (4a) can be now rewritten in the form
(4b)
s(V) = so(q) + kr(q) =
K
J
d*x[ -&f‘VH,p(4p)
L
+
E+-~~(~P)ia,hWJ
(3b)
writen in the light cone coordinates takes the form s(q) =
-K
Jd2x[H,8(~P)+ c L
(3c) whereKE&, a&= -&(a0f aI)and eol = -1. The action in the form (3~) is very convenient for the canonical analysis. For the final conclusion we will need boundary conditions, as will be discussed later. 3. Constraints We are going to use Dirac’s canonical approach to find the gauge symmetries. The hamiltonian equations of motion describe dinamical evolution of a system by a trajectory in a phase space. The presence of an arbitrary multiplier in the hamiltonian means that for given initial conditions we will not obtain the unique solution, one trajectory, but a whole set of trajectories, describing the same physical state. The FC constraints are generators of the mapping between these trajectories that do not change the physical state. These unphysical transformations are the gauge symmetries of
B. SazdoviC/Physics Letters B 352 (1995) 64-68
66
the hamiltonian equations of motion. The method essentialy depends on the choice of time variable. If we take x0 as a time coordinate and use the bilinear expression in the time derivative (3b), as a action, there are no constraints and consequently we can not find the symmetries this way. The expression (3~) for the action is more promising because it is linear in the a__cpand J+‘p. Taking first x- and then xf as time coordinates we will find some primary constraints which give us a hope to find some symmetries of the system. To treat both cases together we will use the light cone indices A = {+, -} with xA = {x+,x-} and 84 = {a+,a_}. These two cases are independent and there are no summations over repeated indices A. In the first case (A = -) we will take x- = r and x+ = o and in the second one (A = +) we will take xf = r and x- = --(T. (The minus sign we adopt to preserve the orientation between coordinate axis.) The action corresponding to the choice xA = r has the form &((o) = K
I
d2X[ ( -l)A&fi
- 27-,p]@ba@. (12)
The canonical momentum conjugate to the variable @ is ‘7ra=
K[ (-1)A&p
-
2T_,p]Cp@
The action is linear in the time derivative and consequently the canonical hamiltonian density is zero, 7-i, = 0, while the total hamiltonian takes the form HAT =/d&q
t
= {JA~(x),HAT}
TAO
M -(-1)A2Kd~,(x)
M 0.
(17a)
The multipliers in (17a) looks like determined U’A~= 0 which means that JAMshould be SC constraints. More precisely, only the first derivative of the multipliers is determined and the corresponding constraints are the SC. The o-independent part is arbitrary function of r and corresponding constraints are the FC. The same conclusion we can obtain from Eq. (15a). The Schwinger term contains the derivative of the S-function and gives zero on the CTindependent functions. It means that the Schwinger term does not give a contributions for some modes of the currents and these modes are the FC constraints. Let us make a more precise investigation. We assume that space is compact which means that for all variables X = {JA~,uA’,...} we have X(~,U+ Z,) = X( 7, a). We will use the Fourier expansion X(x)
=
;~xn(Tx)e-‘“-, JEZ
JAn = ?r, + 2KT-47”
- ( - 1) AK&~#?
X, (TV) =
(14)
JAo = -EAo~JA~
(18)
L
(13)
We prefer to rewrite these constraints with the Lie algebra indices as
164
The consistency condition for JAMleads to
(1)
and therefore, there are primary constraints:
= JduuAaJA,.
J
da,X( x) ei”Wr”
0
where w = F, and the a-function representation
where EA~Ois the inverse of EAT”. The Poisson brackets (PB) between these constraints define two independent KM algebras (A = + or -) :
(19) The PB between modes ~~~~( T) are
{JA&h -
JAb(y)) (
= fobCJAc(~)&flx
-1)A2K~&5’(ff,
- CTy).
-
ay)
(15a)
{ JAG~D JAbnt}
= fabC J~c(n+m) - t - 1 )A4~iK%b&n.0 (15b)
The expression (14) is a convenient linear combination of the previous constraints. We adapt a correspondence between Jh and E& in order to obtain the algebra ( Ha).
while plus and minus constraints commute. This expression does not depend on the period L. In terms of modes the total hamiltonian becomes
67
B. SazdoviC/ Physics Letters B 352 (1995) 64-68
=-
c
1 L
obtain the gauge generators and the corresponding gauge transformation&The gauge parameters WA’( 7) will depend only on the time coordinate 7, as well as the FC constraints JAB and the arbitrary multipliers
(16b)
UAan(7x)JAa(-n)(Tx)*
KZ
Now it is easy to separate the FC and the SC constraints, and check the consistency condition for the primary Constraints JAon( 7) :
UA'O.
All the FC constraints JAM are primary in this case, and the expression for the gauge symmetry generator is GA=wA’(T)JA~(T)
(A=+,-).
(21)
The gauge transformations produced by GA are
(17b) For n = 0 the multipliers u,@(r) are arbitrary functions of time and JA&(7) constraints are FC. For n # 0 we can determined the multipliers UAon (7) = 0 (n # 0) which means that JA*,, (n # 0) are SC constraints. The same conclusion follows from ( 15b) because
{J~ao,JAbn) = {J~on,J,tb(--n)}
for
n $0.
1 -UA~O(T)JACIO(~).
L
g-‘&g
= S_cpnE_,at,, = S+SpaE+~ata.
(23)
With the help of Eq. (22) we get
$0
(20)
In any case there are no secondary constraints. The total hamiltonian is now HAT=
(22)
We are interested in the gauge transformations of the field g. Using Eqs. (6) and (8) we can connect the transformations of the fields g and (p
- 6+gg-’
.fabCJAcn = 0, M -(-l)A‘hiKny,b
S,J(~~= {@, GA} = -wA’(T)EA~~.
(16c)
In Section 5 we will verify that expression ( 16~) gives the correct equations of motion. The assumption of the periodicaly boundary conditions is not necessary. The general solution of (17a) is u.&,((T, r) = UAO (7) and substituting this into ( 16a) we obtain (16~) with uA’(r) instead of +uA(la(r) and JAM = s daJAa( (T, 7). The constraints JAM are FC because they correspond to the arbitrary multiplier U,Ja(7). They also weakly vanish with all constraints, which can be easily verified integrating IQ. ( 15a) over ox. For our purpose any boundary condition with finite JAM is a good condition.
s-g = -gw_,
6,g = o+g
(24)
where @A= WAat,l. Because WAis infinitesimal we can write g”- =g+S_g=g(l -w_) age-O-, gO+ =g+6,g=(1+w+)g=eeO+g. Recalling that w_ E o-(x-) we have gn- =gK’(x-),
(25)
and w+ = w+ (x+)
go+ =n+(n+)g
(26)
where fiA (XA) = eoA(“‘). Fqs. (26) are equivalent with the EQ. ( 1) . Therefore we demonstrated that the gauge symmetries generated by the zero modes of the Kh4 currents are exactly the known chiral gauge symmetries.
5. Equations of motion 4. Chid
gauge symmetries
The presence of arbitrary multipliers uA’O(7) in the total hamiltonian means that the theory has a gauge symmetry. We will use Castellani’s method [7] to
For completeness we briefly derive the hamiltonian equations of motion. For the fields (Dowe have +a = {@, HAT} = -UAaOEAoa.
(27)
68
B. Sazdouit / Physics Letters B 352 (1995) 64-68
The equations of motion for the field g are obtained from Eqs. (6)and (8) in terms of the equations for (pa g-‘g = @aE_,ata,
- gg-’ = @aE+aata
(28)
and by using Eq. (27) we find g-‘g = --u-o(7),
kg-’
= u+o(7)
(29a)
where u,z,,,= UAa,,to. If we now recall our definition of time r = xA, it follows that g = a-g for A = - and g = d+g for A = +, so that Eqs. (29a) can be rewritten as g-‘&g = --u_e(X-), d+gg_’ = u+@(X+).
(29b)
Therefore we obtain the well known equations of motion,
zero modes are FC constraints and they are the generators of the gauge symmetries. The only arbitrary parameters are zero mode gauge parameters. They depend on the x- in the first case and on the x+ in the second one, and this explains the chirality of the gauge symmetry. The connection between KM algebra and chiral gauge symmetry is now clear. The KM algebra contains infinitely many modes of the SC constraints (it looks like the algebra of SC constraints) and only one FC mode which is the generator of the symmerty ( 1) . This result is obtained by the application of the canonical approach to a new kind of symmetry when the gauge parameter does not depend on all coordinates, but only on some particular combinations of the coordinates [ 81.
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a+(g-‘a-g)
= 0,
&(d+gg-1)
=0
(30)
which are, as well as the Eqs. (29b), evidently invariant under the chiral gauge transformations ( 1) .
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6. Conclusion The canonical formalism in the light cone coordinates is developed for the two dimensional WZNW model. Using x- and x+ as the evolution parameters we obtain the two commuting KM algebras of constraints. We made a mode expansion in order to separate the FC constraints from the SC ones. Only the
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