- Email: [email protected]
The θ-structure in the string theories I. Bosonic strings
Volume 165B, number 4,5,6
PHYSICS LETTERS
26 December 1985
T H E 0 - S T R U C T U R E IN S T R I N G T H E O R I E S I. B O S O N I C S T R I N G S Miao
LI
International Centrefor TheoreticalPhysics, Trieste, Italy and Centrefor Astrophysics, Universityof Science and Technologyof China, Hefei, Anhui, China i Received 5 August 1985
We explored the 8-structure in bosonic string theories which are similar to that in gauge field theories. The 8-structure of strings is due to the multiply connected spatial compact subspace of space-time. We show that there is an energy band E(0) in the string theory and the first exciting states in 26D string theory are one massive vector and d - 1 massive scalars.
Isham et al. [ 1] pointed out that there may exist a set ofinequivalent quantizations in quantum theories providing the topologies of space M and field space Q are nontrivial. If we put q as a field collection, then the propagator from q to q ' is
Ko(q', q) = [~p] O([p])K[p] (q', q) ,
(1)
where [p] denotes the collection o f paths connecting q and q ' which are homotopic to a particular path p. 0 ([p]) is a complex factor depending on [p]. Mathematically rather than physically, one can distinguish the [p] by choosing a base point q0 in Q and joining it to q and q' by 60 and w ~, then [w t - 1p60] forms a loop passing through q0- We write [7] = [60'-1p60], thus eq. (1) can be written as
Ko(q', q) = ~
0(7)KI.rl (q', q ) .
(2)
0(7) can be determined by demanding K 0 changing a complex factor when 60 and 60' are changed, so 0 (7) is a homeomorphic map: HI(Q ) ~ U(1). Hence, the set o f the inequivalent quantizations is Horn
[rh(O), U(1)]
.
(3)
In bosonic string theories, there is only a consistent theory in which the dimension o f space-time is 26, i.e., the Veneziano model [2,3]. A more realistic 1 Permanent address.
theory is the 10D superstring model of Green and Schwarz [4]. To obtain a realistic correspondence in nature, all these theories have to be compactified. The space-time may be of the form M4 × Md_ 4, where M 4 is minkowskian and Mr_ 4 is a compact spatial space. We suppose the internal space M r _ 4 be multiply connected, IIl(Md_4) =/= e. The string is parameterized by XU(a, r), where a and r are spatial and time-like parameters, respectively. The propagator between two configurations XU(o) and XU '(a) is K(XU'(o), XU(o)), the integral in eq. (1) must be summed over all paths XU(o, r)joining XU(a) and XU'(o). Paths are homotopic to each other only if the trajectories of a = 0 o f them are homotopic to each other (due to continuity), therefore all paths joining XU(o) and XU'(o) can be classified by the fundamental group of Md_4, i.e., IIl(Md_4). We f'md finally that the set of quantizations of the string is Horn [III(Md_4), U(1)] .
(4)
For definitiveness, we consider the VM string and suppose that space-time takes the form MD_ d X T a , where MD_ d is the (D - d)-dimensional minkowskian space-time, T d is the d-dimensional toms, hence IIl(Td ) = Z d and Horn [IIl(Td ), U(1)] = U(1) d .
(5)
We can regain eq. (5) as follows: take the vacuum 279
Volume 165B, number 4,5,6
PHYSICS LETTERS
state as [XU(o)) and separate coordinates into two parts, Xi, i = 1 .... , D - d a n d X a , a = D - d + 1, ...,D, then IXU(o)) = IX i, Xa). The radii of the X a are Ra, because of periodicity, IX i, X a + 27rRa) must be different from IX i, X a) by a complex factor, that is,
[Xi, Xa + 2rrRa) = e i°a IX i, x a ) .
(6)
It is obvious that the 0 a are parameters of U(1) d. We can define vacuum states Ina)just as that in gauge theories, by demanding that the end of string Xa(O) takes values in [2rrnaRa, 2~r(na + 1)Ra]. Define the translation operators ira, T a : X a ~ X a + 27rRa, then
Taln a) = Ina + 1).
(7)
naOa) I-Ia ina),
f drdaj(a(o,r)[na]
,
(8)
(9)
(12)
where the integral is a total divergence, just like a a in gauge theories [5]. Therefore, ( i/3 2n 2), FuvF~v the amplitudes between 0-vacua can be written as
(O'lexp(-Hr)lO)
f d[XUlexp(-S'),
=
Oa S'= S + i ~ ~ f drdaX a a 2rr2R a
(13)
The lorentzian action derived from eq. (13)is S ; = S L - a ~ 2rr2R----~
and eq. (6) can be rewritten as
Ta[O) = exp(-inaO a)[O) .
naOa = Oa 2rr2R a
oa f drdoka
The vacuum state in eq. (6) can be defined as
[0)= ~ e x p ( i ~ a [na]
26 December 1985
(14)
The addition term on the RHS of eq. (13) does not destroy the 2D Lorentz symmetry. If we take the orthogonal gauge, then
SL =
u
,
- X u X'u)drdo
The amplitudes between vacuum states are
Oa
([mallexp(-Hr)l[na]) = fd[xil
elr-n~
dtXal [ma_na I e x p ( - S ) ,
•
i[na])= I-I Ina),
!
XuXu=O,
T----~c,o
(10)
(t
where S is the euclidean action of the string, and d IX a ] [ma_na ] denotes the measure of all configuragions of Xa(o, r) in which the trajectories Xa(o, r) of fixed o in the interval of r ~- [-o% +oo] form loops with winding number m a - n a in the subspace X a. Using eq. (10) we find again a formula similar to eq.
(2): (O'[e'kp(-Hr)[O) = 6 ( 0 ' - 0)I(0),
=1
i
?
kuX, +XuX'~=O.
(15)
In eq. (15), the addition destroys the explicit Lorentz invariance of Mp, the residual invariance is O(D - d - 1, 1) X U(1)a. In the orthogonal gauge, the momenta which are conjugate to XU are
pi = (1 fir) j(i ,
(16a)
pa = (1/rr)X a - 0 a/2rr2R a .
(16b)
In the canonical quantization formalism, the commutators among X u and pu are
[XU(a, r),pV(o
T---~ oo
, v
', r')] =
inuv6(o-a')6(r-r'),
(17)
the expansions of XU and p u are I ( 0 ) = ~ exp [ ] | - i ~ n a O a l \ / [na]
× fd[Xq d[Xa] thai exp(-S). There is a unique integral expression ofnaOa,
X i = Xio + p i t (11)
+E 1 i - - c o s n o [ a n e x p ( - i n O + h.c.] , n~l rt
X a = Xao + (pa + Oa/2rrRa)r + ~ _I c o s n a [ a a e x p ( - i n r ) +h.c.] n>~l rt
280
(18a)
(18b)
Volume 165B, number 4,5,6
PHYSICS LETTERS
Xa=Oa/21rRa, p u = pU fir + ~ cosno[-ia~n e x p ( - - i n r ) + h.c.], (18c) n~l [X~, Pu ] = i~?uv ,
[a~n,am ] = n rluvSn +m,O" (18d, e)
From the well-known constraint integral we derive 1 ,p , + ~ H = Lo = ~p a_na n - a 0 , n >~l
(19a)
L n = - i p ' a n + ~ a_nam+n, m>~l p' = pi ,
(19b)
= pa + Oa/2rrRa .
(19c)
The appearance of a corrected p ' is due to the fact that not X a / n but x a / n - Oa/2ZrRaare conjugate momenta. One can check the commutators o f L m , they are just the same as the original ones:
[Lm , L n ] = (m - n)Lm+ n + ~ D ( m 3 - m)Sm+n, 0.(20) As we previously pointed out, the Lorentz symmetry O(D - 1, 1) must be broken to O(D - d - 1, 1), because of compactification and non-zero 0 (e.g. p ' is not a covariant vector). In light-cone gauge, following ref. [6], we can obtain the generators o f O(D - 1, 1). By now we let A denote all transverse components. The generators M AB, M A + and M + - keep their original forms. But M A - is
M A- = ~ (XAp-
+ p-X A ) - X-pA
i ~ l ( L _ n a ~ n _ aAnLn) + p+ n>l '
26 December 1985
xA=~aA~ a .
(24)
Therefore, w h e n A = i, B =], [ M i - , M / - ] = O, while A or B is a, the commutator is not zero unless 0 a = O. Thus we showed explicitly that O(D - 1, 1) is broken. Now we check that the first-level exciting states are one massive vector and d - 1 massive scalars * 1. The spectrum formula is l(mass) 2 = l ( p a + X a ) 2 + ~ a_na A A n -- 1. n~>l
pa can only take the valuesMa/R a. Starting from states a_A1IP), where pa [p) = 0, the mass of the first exciting states is (mass)2 = ~2 > 0 .
(26)
Defining ~_L1 = P+haaa_l and a / 1 = a i l + (1/XQ) × pihaaa_ 1, by use of the following equations:
[MiL, aA1] iP) = [ - ( x i + (ip'A/x/~p+)ai_l]
/x/~P+)aA_I [p),
(27)
together with
h a Xa = - X / ~ ,
(28)
one can prove
[Mi],-ak_ 1 ] Ip) = (i6ikTz/_1 -- i8]k a i l ) l P ) ,
(29a)
[M/L, a/_'l] [p) = { - ( x i / v ~ p + ) a ] l
+ i6i! [(p+pL + 1)/2p+2] dE 1)ip), (21)
(25)
[M ij, fiL1 ]]p) = 0,
(29b) (29c)
[M/L, d_E1] [p) = [-(xi/vr~p+)'dL_l
where - ifi~_l ]]p). P
= 2 p1+ ( p ' A p ' A + 2 n > ~ l aA-n aAn --2aO', 1 .
These transformations show that (dL1 IP>, a/-1 IP>} form a massive vector multiplet, although they are not orthogonal, the latter face needs to be studied further. The remaining d - 1 independent states are scalars. De fine
When a 0 -- 1 and D = 26, a careful calculation shows that
[MA - , M B - ] = ( i p - / p + ) X B XA - (1/2p +2)pA ~ B + 1. ~ 1 [AB(L_n4 p+2 n--~l fi
_ aAnLn)
+ XAaB_naaXa] -- (A <+B), where we have used the following notation
(29d)
(22)
(23)
a a l = a a _ l - Xa ~b gbab_l/~b
Xbg b ,
41 1 would like to thank the referee for his reminding me to do this. 281
Volume 165B, number 4,5,6
PHYSICS LETTERS
where {gb} isa set ofparame.ters satisfying ~aXaga We have
--# O.
where • mm M a1 2 = m i n ( M a +
[M i/, da_l] IP) = 0 ,
(30a)
Jill iL , a-a_1][P) = -(Xi/N/~P +) ffa-1 IP),
{30b)
26 December 1985
Oa/2rO 2
= min(O2/4n 2, (1 - Oa/2rr)2 ) .
(32b)
thus the ~a__1 [p) are scalars. Because o f g a ~ a l = 0, there are d - 1 independent scalars. If we introduce a different set o f g ' , 1.e., ' ga,' then
I am grateful to Professor Abdus Salam, International Atomic Energy Agency, and UNESCO for hospitality at ICTP, Trieste.
~a_i(g' ) = ffal(g )
References
-- Xa ~b g'b~b_l(g)/~b Xbg'b .
(31)
It indicates that-~a_l(g) [P) form a complet set of scalars. In conclusion, we have shown that there may exist a 0-structure in the 26-dimensional bosonic string theory and the Lorentz symmetry 0 ( 2 5 , 1) is broken down to 0 ( 2 5 - d, 1) when s p a c e - t i m e takes the form of M26_ d × T d and the 0 a are nonzero. The first exciting states are one massive vector and d - 1 massive scalars. Finally, we point out that the minimal value M 2 i n of the mass squared in the physical spectrum is •
1 2 _a~ lmmMa ~Mmi n ~- R 2 a
282
t2
1,
(32a)
[1] C.J. Isham, Quantum field theory and spatial topology, in: Conf. on Differential geometric methods in theoretical physics, eds. G. Denardo and H. Doebner (World Scientific, Singapore, 1981); M. Laidlaw and C. De Witt, Phys. Rev. D 3 ( 1971) 1395 ; J.S. Dowker, J. Phys. (NY) A5 (1972) 936. [2] G. Veneziano, Nuovo Cimento 57A (1968) 190. [3] See, e.g., C. Rebbi, Phys. Rep. 12C (1974) 1. [4] M.B. Green and J.H. Schwarz, Nucl. Phys. B181 (1981) 502. [5] R. Jackiw and C. Rebbi, Phys. Rev. Lett. 37 (1976) 172; C. Callan, R. Dashen and D. Gross, Phys. Lett. 63B (1976) 334. [6] P. Goddard, J. Goldstone, C. Rebbi and C.B. Thorn, Nucl. Phys. B56 (1973) 109.
PHYSICS LETTERS
26 December 1985
T H E 0 - S T R U C T U R E IN S T R I N G T H E O R I E S I. B O S O N I C S T R I N G S Miao
LI
International Centrefor TheoreticalPhysics, Trieste, Italy and Centrefor Astrophysics, Universityof Science and Technologyof China, Hefei, Anhui, China i Received 5 August 1985
We explored the 8-structure in bosonic string theories which are similar to that in gauge field theories. The 8-structure of strings is due to the multiply connected spatial compact subspace of space-time. We show that there is an energy band E(0) in the string theory and the first exciting states in 26D string theory are one massive vector and d - 1 massive scalars.
Isham et al. [ 1] pointed out that there may exist a set ofinequivalent quantizations in quantum theories providing the topologies of space M and field space Q are nontrivial. If we put q as a field collection, then the propagator from q to q ' is
Ko(q', q) = [~p] O([p])K[p] (q', q) ,
(1)
where [p] denotes the collection o f paths connecting q and q ' which are homotopic to a particular path p. 0 ([p]) is a complex factor depending on [p]. Mathematically rather than physically, one can distinguish the [p] by choosing a base point q0 in Q and joining it to q and q' by 60 and w ~, then [w t - 1p60] forms a loop passing through q0- We write [7] = [60'-1p60], thus eq. (1) can be written as
Ko(q', q) = ~
0(7)KI.rl (q', q ) .
(2)
0(7) can be determined by demanding K 0 changing a complex factor when 60 and 60' are changed, so 0 (7) is a homeomorphic map: HI(Q ) ~ U(1). Hence, the set o f the inequivalent quantizations is Horn
[rh(O), U(1)]
.
(3)
In bosonic string theories, there is only a consistent theory in which the dimension o f space-time is 26, i.e., the Veneziano model [2,3]. A more realistic 1 Permanent address.
theory is the 10D superstring model of Green and Schwarz [4]. To obtain a realistic correspondence in nature, all these theories have to be compactified. The space-time may be of the form M4 × Md_ 4, where M 4 is minkowskian and Mr_ 4 is a compact spatial space. We suppose the internal space M r _ 4 be multiply connected, IIl(Md_4) =/= e. The string is parameterized by XU(a, r), where a and r are spatial and time-like parameters, respectively. The propagator between two configurations XU(o) and XU '(a) is K(XU'(o), XU(o)), the integral in eq. (1) must be summed over all paths XU(o, r)joining XU(a) and XU'(o). Paths are homotopic to each other only if the trajectories of a = 0 o f them are homotopic to each other (due to continuity), therefore all paths joining XU(o) and XU'(o) can be classified by the fundamental group of Md_4, i.e., IIl(Md_4). We f'md finally that the set of quantizations of the string is Horn [III(Md_4), U(1)] .
(4)
For definitiveness, we consider the VM string and suppose that space-time takes the form MD_ d X T a , where MD_ d is the (D - d)-dimensional minkowskian space-time, T d is the d-dimensional toms, hence IIl(Td ) = Z d and Horn [IIl(Td ), U(1)] = U(1) d .
(5)
We can regain eq. (5) as follows: take the vacuum 279
Volume 165B, number 4,5,6
PHYSICS LETTERS
state as [XU(o)) and separate coordinates into two parts, Xi, i = 1 .... , D - d a n d X a , a = D - d + 1, ...,D, then IXU(o)) = IX i, Xa). The radii of the X a are Ra, because of periodicity, IX i, X a + 27rRa) must be different from IX i, X a) by a complex factor, that is,
[Xi, Xa + 2rrRa) = e i°a IX i, x a ) .
(6)
It is obvious that the 0 a are parameters of U(1) d. We can define vacuum states Ina)just as that in gauge theories, by demanding that the end of string Xa(O) takes values in [2rrnaRa, 2~r(na + 1)Ra]. Define the translation operators ira, T a : X a ~ X a + 27rRa, then
Taln a) = Ina + 1).
(7)
naOa) I-Ia ina),
f drdaj(a(o,r)[na]
,
(8)
(9)
(12)
where the integral is a total divergence, just like a a in gauge theories [5]. Therefore, ( i/3 2n 2), FuvF~v the amplitudes between 0-vacua can be written as
(O'lexp(-Hr)lO)
f d[XUlexp(-S'),
=
Oa S'= S + i ~ ~ f drdaX a a 2rr2R a
(13)
The lorentzian action derived from eq. (13)is S ; = S L - a ~ 2rr2R----~
and eq. (6) can be rewritten as
Ta[O) = exp(-inaO a)[O) .
naOa = Oa 2rr2R a
oa f drdoka
The vacuum state in eq. (6) can be defined as
[0)= ~ e x p ( i ~ a [na]
26 December 1985
(14)
The addition term on the RHS of eq. (13) does not destroy the 2D Lorentz symmetry. If we take the orthogonal gauge, then
SL =
u
,
- X u X'u)drdo
The amplitudes between vacuum states are
Oa
([mallexp(-Hr)l[na]) = fd[xil
elr-n~
dtXal [ma_na I e x p ( - S ) ,
•
i[na])= I-I Ina),
!
XuXu=O,
T----~c,o
(10)
(t
where S is the euclidean action of the string, and d IX a ] [ma_na ] denotes the measure of all configuragions of Xa(o, r) in which the trajectories Xa(o, r) of fixed o in the interval of r ~- [-o% +oo] form loops with winding number m a - n a in the subspace X a. Using eq. (10) we find again a formula similar to eq.
(2): (O'[e'kp(-Hr)[O) = 6 ( 0 ' - 0)I(0),
=1
i
?
kuX, +XuX'~=O.
(15)
In eq. (15), the addition destroys the explicit Lorentz invariance of Mp, the residual invariance is O(D - d - 1, 1) X U(1)a. In the orthogonal gauge, the momenta which are conjugate to XU are
pi = (1 fir) j(i ,
(16a)
pa = (1/rr)X a - 0 a/2rr2R a .
(16b)
In the canonical quantization formalism, the commutators among X u and pu are
[XU(a, r),pV(o
T---~ oo
, v
', r')] =
inuv6(o-a')6(r-r'),
(17)
the expansions of XU and p u are I ( 0 ) = ~ exp [ ] | - i ~ n a O a l \ / [na]
× fd[Xq d[Xa] thai exp(-S). There is a unique integral expression ofnaOa,
X i = Xio + p i t (11)
+E 1 i - - c o s n o [ a n e x p ( - i n O + h.c.] , n~l rt
X a = Xao + (pa + Oa/2rrRa)r + ~ _I c o s n a [ a a e x p ( - i n r ) +h.c.] n>~l rt
280
(18a)
(18b)
Volume 165B, number 4,5,6
PHYSICS LETTERS
Xa=Oa/21rRa, p u = pU fir + ~ cosno[-ia~n e x p ( - - i n r ) + h.c.], (18c) n~l [X~, Pu ] = i~?uv ,
[a~n,am ] = n rluvSn +m,O" (18d, e)
From the well-known constraint integral we derive 1 ,p , + ~ H = Lo = ~p a_na n - a 0 , n >~l
(19a)
L n = - i p ' a n + ~ a_nam+n, m>~l p' = pi ,
(19b)
= pa + Oa/2rrRa .
(19c)
The appearance of a corrected p ' is due to the fact that not X a / n but x a / n - Oa/2ZrRaare conjugate momenta. One can check the commutators o f L m , they are just the same as the original ones:
[Lm , L n ] = (m - n)Lm+ n + ~ D ( m 3 - m)Sm+n, 0.(20) As we previously pointed out, the Lorentz symmetry O(D - 1, 1) must be broken to O(D - d - 1, 1), because of compactification and non-zero 0 (e.g. p ' is not a covariant vector). In light-cone gauge, following ref. [6], we can obtain the generators o f O(D - 1, 1). By now we let A denote all transverse components. The generators M AB, M A + and M + - keep their original forms. But M A - is
M A- = ~ (XAp-
+ p-X A ) - X-pA
i ~ l ( L _ n a ~ n _ aAnLn) + p+ n>l '
26 December 1985
xA=~aA~ a .
(24)
Therefore, w h e n A = i, B =], [ M i - , M / - ] = O, while A or B is a, the commutator is not zero unless 0 a = O. Thus we showed explicitly that O(D - 1, 1) is broken. Now we check that the first-level exciting states are one massive vector and d - 1 massive scalars * 1. The spectrum formula is l(mass) 2 = l ( p a + X a ) 2 + ~ a_na A A n -- 1. n~>l
pa can only take the valuesMa/R a. Starting from states a_A1IP), where pa [p) = 0, the mass of the first exciting states is (mass)2 = ~2 > 0 .
(26)
Defining ~_L1 = P+haaa_l and a / 1 = a i l + (1/XQ) × pihaaa_ 1, by use of the following equations:
[MiL, aA1] iP) = [ - ( x i + (ip'A/x/~p+)ai_l]
/x/~P+)aA_I [p),
(27)
together with
h a Xa = - X / ~ ,
(28)
one can prove
[Mi],-ak_ 1 ] Ip) = (i6ikTz/_1 -- i8]k a i l ) l P ) ,
(29a)
[M/L, a/_'l] [p) = { - ( x i / v ~ p + ) a ] l
+ i6i! [(p+pL + 1)/2p+2] dE 1)ip), (21)
(25)
[M ij, fiL1 ]]p) = 0,
(29b) (29c)
[M/L, d_E1] [p) = [-(xi/vr~p+)'dL_l
where - ifi~_l ]]p). P
= 2 p1+ ( p ' A p ' A + 2 n > ~ l aA-n aAn --2aO', 1 .
These transformations show that (dL1 IP>, a/-1 IP>} form a massive vector multiplet, although they are not orthogonal, the latter face needs to be studied further. The remaining d - 1 independent states are scalars. De fine
When a 0 -- 1 and D = 26, a careful calculation shows that
[MA - , M B - ] = ( i p - / p + ) X B XA - (1/2p +2)pA ~ B + 1. ~ 1 [AB(L_n4 p+2 n--~l fi
_ aAnLn)
+ XAaB_naaXa] -- (A <+B), where we have used the following notation
(29d)
(22)
(23)
a a l = a a _ l - Xa ~b gbab_l/~b
Xbg b ,
41 1 would like to thank the referee for his reminding me to do this. 281
Volume 165B, number 4,5,6
PHYSICS LETTERS
where {gb} isa set ofparame.ters satisfying ~aXaga We have
--# O.
where • mm M a1 2 = m i n ( M a +
[M i/, da_l] IP) = 0 ,
(30a)
Jill iL , a-a_1][P) = -(Xi/N/~P +) ffa-1 IP),
{30b)
26 December 1985
Oa/2rO 2
= min(O2/4n 2, (1 - Oa/2rr)2 ) .
(32b)
thus the ~a__1 [p) are scalars. Because o f g a ~ a l = 0, there are d - 1 independent scalars. If we introduce a different set o f g ' , 1.e., ' ga,' then
I am grateful to Professor Abdus Salam, International Atomic Energy Agency, and UNESCO for hospitality at ICTP, Trieste.
~a_i(g' ) = ffal(g )
References
-- Xa ~b g'b~b_l(g)/~b Xbg'b .
(31)
It indicates that-~a_l(g) [P) form a complet set of scalars. In conclusion, we have shown that there may exist a 0-structure in the 26-dimensional bosonic string theory and the Lorentz symmetry 0 ( 2 5 , 1) is broken down to 0 ( 2 5 - d, 1) when s p a c e - t i m e takes the form of M26_ d × T d and the 0 a are nonzero. The first exciting states are one massive vector and d - 1 massive scalars. Finally, we point out that the minimal value M 2 i n of the mass squared in the physical spectrum is •
1 2 _a~ lmmMa ~Mmi n ~- R 2 a
282
t2
1,
(32a)
[1] C.J. Isham, Quantum field theory and spatial topology, in: Conf. on Differential geometric methods in theoretical physics, eds. G. Denardo and H. Doebner (World Scientific, Singapore, 1981); M. Laidlaw and C. De Witt, Phys. Rev. D 3 ( 1971) 1395 ; J.S. Dowker, J. Phys. (NY) A5 (1972) 936. [2] G. Veneziano, Nuovo Cimento 57A (1968) 190. [3] See, e.g., C. Rebbi, Phys. Rep. 12C (1974) 1. [4] M.B. Green and J.H. Schwarz, Nucl. Phys. B181 (1981) 502. [5] R. Jackiw and C. Rebbi, Phys. Rev. Lett. 37 (1976) 172; C. Callan, R. Dashen and D. Gross, Phys. Lett. 63B (1976) 334. [6] P. Goddard, J. Goldstone, C. Rebbi and C.B. Thorn, Nucl. Phys. B56 (1973) 109.