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Computation of the topological susceptibility for the 2D CP3 model on a spherical lattice
Volume 234, number 3
PHYSICS LETTERS B
11 January 1990
COMPUTATION OF THE TOPOLOGICAl, SUSCEPTIBILITY F O R T H E 2D C P s M O D E L O N A S P H E R I C A L L A T T I C E B. J O Z E F I N I , M. M O L L E R - P R E U S S K E R and N. S C H U L T K A Joint lnstttute.~r Nuclear Research, Dubna, SU- 101 000 Moscow, C'SSR and Sektion Ph.vsik. ttumboldt-[_.'ntversttdt, DDR-1086 Berhn, GDR
Received 1 September 1989
Using Monte Carlo simulations we calculate the topological susceptibility for the CP 3 model on a simplicial lattice approximating the sphere S2. Our data exhibit the righl scaling bchaviour but do not show a suppression of topologically relevant fluctuations in the small-volume limit.
1. During the last years a lot o f effort has gone in the investigations on the topological vacuum structure o f lattice gauge theories [ 1 ]. As an indicator, probing the contribution o f topologically non-trivial excitations to the quantum vacuum usually serves the topological susceptibility
where Q, represents the topological charge of the gauge field and I "¢'n the volume in the d-dimensional spacetime. Various methods have been invented to define Q¢ on the lattice (see ref. [ I ] ). For the quantized fields generated numerically by Monte Carlo ( M C ) simt, lations the',' lead typically to different Q, values event-by-event and to somewhat different Z, estimates at accessible couplings in the scaling region. Surely, the disagreement is duc to short-range fluctuations and possibly to lattice artifacts, which are taken into account or are suppressed in a different m a n n e r [ 2 ]. Howcvcr, a priori, it is very difficult to say in as far as short-range fluctuations with Q,:/: 0 should bc irrelevant in the c o n t i n u u m limit or whether instanton-likc semi-classical background fields alone d e t e r m i n e the topological properties o f the vacuum state. In this letter we are going to discuss this question within the fiamework o f the two-dimensional C P " - ' model, which has a lot o f similarities with the Y a n g Mills theory in four dimensions. The latter fact concerns also the small-volume limit for both theories
tormulated in the c o n t i n u u m on the spheres S 2 and S4. respectively. It has been shown for the Yang-Mills theory by Ltischcr [ 3 ] and afterwards for the C P " - ' model by Schwab [4] that in this limit Zt tends to zero and becomes d o m i n a t e d by the one-instanton contribution, provided the semi-classical approximation makes sense et al. We want to see whether the same will happen within the formulation o f the theory on a latticized sphere, i.e. under the same boundary conditions as in the c o n t i n u u m case. 2. The C P " - ' model we are considering is defined is defined in a curved c o n t i n u u m space by the action S = ~7 d - x x / g g ~ * ' ( D u - " ) ( D " c " ) * " /L. v = I. 2 ,
(2)
where the complex vectors . : " = c " ( x ) , a = 1, 2 ..... n satisfy the normalization condition Y,z"(_-'*)*= 1. ,~'J'~ denotes the metric tensor on S 2 and de,',( x/g represents the i n v a r i a n t - v o l u m c element. The covariant derivative D¢,=~3,,- i.4,, contains the abelian "'gauge" field .-1,, which can be easily expressed by the =-field using the equations o f motion. Here we want to treat it as a dynamical field which allows to read off the topological charge as a sum over winding numbers Q,= ~
Z
dx,,..l~
(3)
t,
329
Volume 234, number 3
PHYSICS LETTERS B
in a straightforward manner. The theory is approximated on a triangular latlicc. In principle, the latter could be a random one, but we prefer to generate an almost regular lattice. It can be obtained by starting from a regular tetraeder in > 3 the corners of which are placed on the sphere S 2. Then successively finer lattices enumerated by N = 1, 2 .... arc provided by dividing all the edges into two parts each and projecting the midpoints onto the sphere. Connecting the newly projected sites with each other by new links gives us the next finer lattice. In this way the lattices produced contain P = 2-~'v+~+ 2 sites and L = 3 ( P - 2) links. Every site has six nearest neighhours except the original corner points of the tetraeder which have three. This lattice construction has the advantage that it is easily to be mapped onto the two-dimensional plane by cutting the lattice along three of the original edges of the tetraeder. We take the simpliccs of the lattice to be plane ones in D 3. Already for N = 2 the area sum of all simplices represents 88% of the area of the sphere. The lattice formulation of the model (2) looks as follows. We approximate the covariant derivative at lattice site x, in the standard way: Du.7" (.v,) --+ ( z~+u U,.t+u - z',~) /l,.,+i, .
-=exp( - i~p,.,+~) eL( 1 ) .
is given by [ 5 ] g " = 4(3,,~)2
330
(,./1
I
(6)
.7""
where sl,al gets contributions of both adjacent simplices ( i , j . k ) and (i,j. k ' ) sl,., ) = z ' : ( U , , I ; , - (.',~ U~.j£2v- g-:,k Uk'j-C2',,)(z',~) * + c . c . - 31;,.
(7)
with the weights "~ - 12j + l,:2 - l,k. 2 Fv _ 15"~- lTk - l2 / 123,:k 12J,:k.
c-Q,,=(lk,'l~:)/12.4,,~,
C-2b(lk,'l~./)/123,,k.
(8)
Our lattice action is manifestly locally U ( I ) gauge invariant. In contrast to the C P " - ~lattice actions invented in earlier papers [ 6 - 8 ] it is a non-local one containing products of two U( 1 ) link variables. In the following we restrict ourselves to the CP 3 model, i.e. t l = 4 .
3. The model is quantized according to the functional integral d2z76
fi
l z,~, ]2_ 1 \c~=l
d(o/k - I ,, .!iCp/k } ) ] . -9£~- e x p [ -- 5 "L ( ,,.-,
(9)
- n D,kl
k
with the simplex area
St=/3 Y. sl,.,i, IS=
X i
The metric tensor at the fiat simplex ( i , j , k )
- (l,:'l,k)ll,,l,k) 1 _/,,,
The lattice action can be cast into a form o f a sum over links joining the nearest neighbours L j
t=l c¢=1
U,., +a - exp [ - i.-l,, (xi)/,.,+,~ ]
1 -- ( l , ' l , k ) l l , , l , k
3,,~.= ~ ( 21~,1~ + 21~12k + 212,~1~ - l,,4 _ l,a.4 _ la,4 ),/2
(4)
w h e r e / , is the length of the link vector I,j connecting the neighbour sites .v, .v/. The link variable is defined by
X
11 January 1990
(5)
The quantum fields z and cp have been generated in the MC simulation according to (9) by the heat-bath and the standard Metropolis algorithms, respectively. We tried as well different Metropolis update codes for the z-fields, but found the configurations to be strongly correlated from iteration to iteration. Our MC runs were carried out for lattices with divisions N = 2 , 3 and 4. Usually we made 400 therrealization sweeps, after which we started measure° ments during 5000 sweeps (except tbr N = 2, f l = 4.75, 5.00 where we have run 50 000 sweeps). We checked our MC code by comparing the numerical results for (z, U,z* > with a corresponding strong coupling expansion up to order f12.
Volume 234. number 3
PHYSICS I.ETFERS B
4. The topological charge on the lattice for the C P " - ~ model is defined b y = Q~
~p,j . 2~
(10)
,, t l, J],~,,
where the first sum runs over all simplices o" and the second one over the corresponding links (the latter one implies to take the angles ~0,,in the right orientation ). The brackets mean the reduction to the interval - zr < I ~ ~0,1 - ~ q~,,+ 2try,, ~< + 7r, uoe~.
(11)
By Monte Carlo simulation wc measure [~-Z, according to eq. (1). where the average lattice scale 1-= X r,,~jl,/L should behave in accordance with thc renormalization group
,ll.f = ( ~ / 3 ~ ) l / 2 e x p ( -
~/3~).
(12)
First topological investigations of the lattice CP ~ model (which is identical with the non-linear 0 ( 3 ) remodel ) showed strong scaling violations for l~-z, due to the dominance of dislocations with an action S t ( d i s l ) < z r [7]. It has been argued that scaling should be restored for n>~4. Moreover. l'etcher and I.iJscher [8] have constructed a modified ("ferromagnetic") lattice action for which in the n = 3 case the M(" data were in agreement with scaling. In our case, the CP 3 model, we have made a rough check, whether dislocations are expected to spoil scaling, for a single simplex cr~ with an arbitrary position we have chosen the surrounding links with I{o¢,l=rr/3+e, r<< 1, such that v,,o according to (I 1 ) became just - 1. All other q~'s were put equal to zero and the z's equal to ( 1.0.0, 0). For this kind of dislocation one
are presented in fig. l for all lattice sizes considered. The errors indicated, for simplicity, are the pure statistical ones tbr Q~. The straight line corresponds to the scaling behaviour according to eq. (12). Wc see that all the data points fit very well to this behaviour. At the given accuracy we do not see any scaling violation in the range 3.25
] 3(t"/e
10-1 ,_
"0. 10 -2
finds
~¢t (disl) = 3.29/3 > re/3,
11 January 1990
(13)
after averaging over all positions of the simplex ao. So this kind of exceptional configuration does not cause danger in the continuum limit of the theory. 5. The topological charge has been measured every t e n t h M C sweep. In this way correlations between subsequent mcasurements were undcr control. Our rcsults tbr the topological susceptibility [Z~/2
10-3
f
I
3.
....
r
_.
I
4
_
n
[
5.
/3
Fig. 1. Topological susceptibility as function of/./. Dols, triangles and stars correspond to lattice sizes with N = 4 , 3 and 2, respectively. The dashed line shows the rcnormalization group behaviour equation ( 12 ).
331
Volume 234, number 3
PHYSICS LETTERS B
that thc semi-classical f i n i t e - v o l u m e picture [3.4] c o u l d be established as well on thc lattice. 6. O u r findings very m u c h r e s e m b l e lattice results o b t a i n e d in SU(2) Y a n g - M i l l s theory, w h e r e Q, has been d e t e r m i n e d on M C e q u i l i b r i u m c o n f i g u r a t i o n s by the g e o m e t r i c m e t h o d o f Phillips and S t o n e [9 ]. A beautiful scaling b c h a v i o u r is seen [ 10], w h e r e at the s a m e t i m e s h o r t - r a n g e f l u c t u a t i o n s are present. Finite-size effects are weak c o m p a r e d with those seen after c o o l i n g [ 1 1 ]. O f course, it w o u l d be w o r t h w h i l e to study the f i n i t e - v o l u m e effects in the 4 D Y a n g Mills case on the sphere in the s a m e way as it has been p r e s e n t e d here for the C P 3 m o d e l .
Acknowledgement We w o u l d like to express o u r g r a t i t u d e to the directory o f the L a b o r a t o r y o f T h e o r e t i c a l Physics o f the J o i n t Institute for N u c l e a r Research, D u b n a , for the kind hospitality e x t e n d e d to us. We a c k n o w l e d g e useful discussions with G. S c h i e r h o l z and U. Wiesc.
332
1I Januar3' 1990
References [ 1 ] For recent reviews see A. Kronfcld, Nucl. Phys. B (Proc. Suppl.) 4 (1988) 329: M. G/3ckeler. Fortschr. Phys. 36 ( 1988 ) 801 : J. Hock, Rutherford Laborato~ preprint RAL-88-097, Nucl. Phys. B (Proc. Suppl.), to appear, and references cited therein. [2] D.J.R. Pugh and M. Teper, Phys. Lett. B 218 (1989) 326: Oxford University preprint OUTP-89-10 P ( 1989 ); M. G6ckclcr et al., preprint DESY 89-065 ( 1989 ): V.G. Bornyakov, B. Jozefini, M. Miiller-Preussker and A.M. Zadorozhny, preprint JINR ( 1989 ). [3] M. Liischer, Nucl. Phys. B 205 [FS5] (1982) 483. [4]P. Schwab. Phys. Lell. B 118 (1982) 373: B 126 (1983) 241. [5]M. Bander and C. Itzykson, Nucl. Phys. B 257 [FSI4] (1985) 531. [6] P. Di Vecchia ct al., Nucl. Phys. B 190 [FS3] ( 1981 ) 719: G. Martinelli, R. Petronzio and M.A Virasoro. Nucl. Phys. B205 (1982) 255. [7] B. Berg and M. Liischer, Nucl. Phys. B 190 [FS3] ( 1981 ) 412: B. Berg, Phys. Left. B 104 ( 1981 ) 475; M. Liischer. Nucl. Phys. B 200 [FS4] (1982) 61. [ 8 ] D. Percher and M. Liischer, Nucl. Phys. B 225 [ FS9 ] ( 1983 ) 53. [ 9 ] A. Phillips and D. Stone, Commun. Math. Phys. 103 ( 1986 ) 599. [ 10J M. Kremer et al., Nucl. Phys. B 305 (1988) 109. [ l l ] J . Hock, Rutherford Laboratory preprint RAI,-88-098 (1988).
PHYSICS LETTERS B
11 January 1990
COMPUTATION OF THE TOPOLOGICAl, SUSCEPTIBILITY F O R T H E 2D C P s M O D E L O N A S P H E R I C A L L A T T I C E B. J O Z E F I N I , M. M O L L E R - P R E U S S K E R and N. S C H U L T K A Joint lnstttute.~r Nuclear Research, Dubna, SU- 101 000 Moscow, C'SSR and Sektion Ph.vsik. ttumboldt-[_.'ntversttdt, DDR-1086 Berhn, GDR
Received 1 September 1989
Using Monte Carlo simulations we calculate the topological susceptibility for the CP 3 model on a simplicial lattice approximating the sphere S2. Our data exhibit the righl scaling bchaviour but do not show a suppression of topologically relevant fluctuations in the small-volume limit.
1. During the last years a lot o f effort has gone in the investigations on the topological vacuum structure o f lattice gauge theories [ 1 ]. As an indicator, probing the contribution o f topologically non-trivial excitations to the quantum vacuum usually serves the topological susceptibility
where Q, represents the topological charge of the gauge field and I "¢'n the volume in the d-dimensional spacetime. Various methods have been invented to define Q¢ on the lattice (see ref. [ I ] ). For the quantized fields generated numerically by Monte Carlo ( M C ) simt, lations the',' lead typically to different Q, values event-by-event and to somewhat different Z, estimates at accessible couplings in the scaling region. Surely, the disagreement is duc to short-range fluctuations and possibly to lattice artifacts, which are taken into account or are suppressed in a different m a n n e r [ 2 ]. Howcvcr, a priori, it is very difficult to say in as far as short-range fluctuations with Q,:/: 0 should bc irrelevant in the c o n t i n u u m limit or whether instanton-likc semi-classical background fields alone d e t e r m i n e the topological properties o f the vacuum state. In this letter we are going to discuss this question within the fiamework o f the two-dimensional C P " - ' model, which has a lot o f similarities with the Y a n g Mills theory in four dimensions. The latter fact concerns also the small-volume limit for both theories
tormulated in the c o n t i n u u m on the spheres S 2 and S4. respectively. It has been shown for the Yang-Mills theory by Ltischcr [ 3 ] and afterwards for the C P " - ' model by Schwab [4] that in this limit Zt tends to zero and becomes d o m i n a t e d by the one-instanton contribution, provided the semi-classical approximation makes sense et al. We want to see whether the same will happen within the formulation o f the theory on a latticized sphere, i.e. under the same boundary conditions as in the c o n t i n u u m case. 2. The C P " - ' model we are considering is defined is defined in a curved c o n t i n u u m space by the action S = ~7 d - x x / g g ~ * ' ( D u - " ) ( D " c " ) * " /L. v = I. 2 ,
(2)
where the complex vectors . : " = c " ( x ) , a = 1, 2 ..... n satisfy the normalization condition Y,z"(_-'*)*= 1. ,~'J'~ denotes the metric tensor on S 2 and de,',( x/g represents the i n v a r i a n t - v o l u m c element. The covariant derivative D¢,=~3,,- i.4,, contains the abelian "'gauge" field .-1,, which can be easily expressed by the =-field using the equations o f motion. Here we want to treat it as a dynamical field which allows to read off the topological charge as a sum over winding numbers Q,= ~
Z
dx,,..l~
(3)
t,
329
Volume 234, number 3
PHYSICS LETTERS B
in a straightforward manner. The theory is approximated on a triangular latlicc. In principle, the latter could be a random one, but we prefer to generate an almost regular lattice. It can be obtained by starting from a regular tetraeder in > 3 the corners of which are placed on the sphere S 2. Then successively finer lattices enumerated by N = 1, 2 .... arc provided by dividing all the edges into two parts each and projecting the midpoints onto the sphere. Connecting the newly projected sites with each other by new links gives us the next finer lattice. In this way the lattices produced contain P = 2-~'v+~+ 2 sites and L = 3 ( P - 2) links. Every site has six nearest neighhours except the original corner points of the tetraeder which have three. This lattice construction has the advantage that it is easily to be mapped onto the two-dimensional plane by cutting the lattice along three of the original edges of the tetraeder. We take the simpliccs of the lattice to be plane ones in D 3. Already for N = 2 the area sum of all simplices represents 88% of the area of the sphere. The lattice formulation of the model (2) looks as follows. We approximate the covariant derivative at lattice site x, in the standard way: Du.7" (.v,) --+ ( z~+u U,.t+u - z',~) /l,.,+i, .
-=exp( - i~p,.,+~) eL( 1 ) .
is given by [ 5 ] g " = 4(3,,~)2
330
(,./1
I
(6)
.7""
where sl,al gets contributions of both adjacent simplices ( i , j . k ) and (i,j. k ' ) sl,., ) = z ' : ( U , , I ; , - (.',~ U~.j£2v- g-:,k Uk'j-C2',,)(z',~) * + c . c . - 31;,.
(7)
with the weights "~ - 12j + l,:2 - l,k. 2 Fv _ 15"~- lTk - l2 / 123,:k 12J,:k.
c-Q,,=(lk,'l~:)/12.4,,~,
C-2b(lk,'l~./)/123,,k.
(8)
Our lattice action is manifestly locally U ( I ) gauge invariant. In contrast to the C P " - ~lattice actions invented in earlier papers [ 6 - 8 ] it is a non-local one containing products of two U( 1 ) link variables. In the following we restrict ourselves to the CP 3 model, i.e. t l = 4 .
3. The model is quantized according to the functional integral d2z76
fi
l z,~, ]2_ 1 \c~=l
d(o/k - I ,, .!iCp/k } ) ] . -9£~- e x p [ -- 5 "L ( ,,.-,
(9)
- n D,kl
k
with the simplex area
St=/3 Y. sl,.,i, IS=
X i
The metric tensor at the fiat simplex ( i , j , k )
- (l,:'l,k)ll,,l,k) 1 _/,,,
The lattice action can be cast into a form o f a sum over links joining the nearest neighbours L j
t=l c¢=1
U,., +a - exp [ - i.-l,, (xi)/,.,+,~ ]
1 -- ( l , ' l , k ) l l , , l , k
3,,~.= ~ ( 21~,1~ + 21~12k + 212,~1~ - l,,4 _ l,a.4 _ la,4 ),/2
(4)
w h e r e / , is the length of the link vector I,j connecting the neighbour sites .v, .v/. The link variable is defined by
X
11 January 1990
(5)
The quantum fields z and cp have been generated in the MC simulation according to (9) by the heat-bath and the standard Metropolis algorithms, respectively. We tried as well different Metropolis update codes for the z-fields, but found the configurations to be strongly correlated from iteration to iteration. Our MC runs were carried out for lattices with divisions N = 2 , 3 and 4. Usually we made 400 therrealization sweeps, after which we started measure° ments during 5000 sweeps (except tbr N = 2, f l = 4.75, 5.00 where we have run 50 000 sweeps). We checked our MC code by comparing the numerical results for (z, U,z* > with a corresponding strong coupling expansion up to order f12.
Volume 234. number 3
PHYSICS I.ETFERS B
4. The topological charge on the lattice for the C P " - ~ model is defined b y = Q~
~p,j . 2~
(10)
,, t l, J],~,,
where the first sum runs over all simplices o" and the second one over the corresponding links (the latter one implies to take the angles ~0,,in the right orientation ). The brackets mean the reduction to the interval - zr < I ~ ~0,1 - ~ q~,,+ 2try,, ~< + 7r, uoe~.
(11)
By Monte Carlo simulation wc measure [~-Z, according to eq. (1). where the average lattice scale 1-= X r,,~jl,/L should behave in accordance with thc renormalization group
,ll.f = ( ~ / 3 ~ ) l / 2 e x p ( -
~/3~).
(12)
First topological investigations of the lattice CP ~ model (which is identical with the non-linear 0 ( 3 ) remodel ) showed strong scaling violations for l~-z, due to the dominance of dislocations with an action S t ( d i s l ) < z r [7]. It has been argued that scaling should be restored for n>~4. Moreover. l'etcher and I.iJscher [8] have constructed a modified ("ferromagnetic") lattice action for which in the n = 3 case the M(" data were in agreement with scaling. In our case, the CP 3 model, we have made a rough check, whether dislocations are expected to spoil scaling, for a single simplex cr~ with an arbitrary position we have chosen the surrounding links with I{o¢,l=rr/3+e, r<< 1, such that v,,o according to (I 1 ) became just - 1. All other q~'s were put equal to zero and the z's equal to ( 1.0.0, 0). For this kind of dislocation one
are presented in fig. l for all lattice sizes considered. The errors indicated, for simplicity, are the pure statistical ones tbr Q~. The straight line corresponds to the scaling behaviour according to eq. (12). Wc see that all the data points fit very well to this behaviour. At the given accuracy we do not see any scaling violation in the range 3.25
] 3(t"/e
10-1 ,_
"0. 10 -2
finds
~¢t (disl) = 3.29/3 > re/3,
11 January 1990
(13)
after averaging over all positions of the simplex ao. So this kind of exceptional configuration does not cause danger in the continuum limit of the theory. 5. The topological charge has been measured every t e n t h M C sweep. In this way correlations between subsequent mcasurements were undcr control. Our rcsults tbr the topological susceptibility [Z~/2
10-3
f
I
3.
....
r
_.
I
4
_
n
[
5.
/3
Fig. 1. Topological susceptibility as function of/./. Dols, triangles and stars correspond to lattice sizes with N = 4 , 3 and 2, respectively. The dashed line shows the rcnormalization group behaviour equation ( 12 ).
331
Volume 234, number 3
PHYSICS LETTERS B
that thc semi-classical f i n i t e - v o l u m e picture [3.4] c o u l d be established as well on thc lattice. 6. O u r findings very m u c h r e s e m b l e lattice results o b t a i n e d in SU(2) Y a n g - M i l l s theory, w h e r e Q, has been d e t e r m i n e d on M C e q u i l i b r i u m c o n f i g u r a t i o n s by the g e o m e t r i c m e t h o d o f Phillips and S t o n e [9 ]. A beautiful scaling b c h a v i o u r is seen [ 10], w h e r e at the s a m e t i m e s h o r t - r a n g e f l u c t u a t i o n s are present. Finite-size effects are weak c o m p a r e d with those seen after c o o l i n g [ 1 1 ]. O f course, it w o u l d be w o r t h w h i l e to study the f i n i t e - v o l u m e effects in the 4 D Y a n g Mills case on the sphere in the s a m e way as it has been p r e s e n t e d here for the C P 3 m o d e l .
Acknowledgement We w o u l d like to express o u r g r a t i t u d e to the directory o f the L a b o r a t o r y o f T h e o r e t i c a l Physics o f the J o i n t Institute for N u c l e a r Research, D u b n a , for the kind hospitality e x t e n d e d to us. We a c k n o w l e d g e useful discussions with G. S c h i e r h o l z and U. Wiesc.
332
1I Januar3' 1990
References [ 1 ] For recent reviews see A. Kronfcld, Nucl. Phys. B (Proc. Suppl.) 4 (1988) 329: M. G/3ckeler. Fortschr. Phys. 36 ( 1988 ) 801 : J. Hock, Rutherford Laborato~ preprint RAL-88-097, Nucl. Phys. B (Proc. Suppl.), to appear, and references cited therein. [2] D.J.R. Pugh and M. Teper, Phys. Lett. B 218 (1989) 326: Oxford University preprint OUTP-89-10 P ( 1989 ); M. G6ckclcr et al., preprint DESY 89-065 ( 1989 ): V.G. Bornyakov, B. Jozefini, M. Miiller-Preussker and A.M. Zadorozhny, preprint JINR ( 1989 ). [3] M. Liischer, Nucl. Phys. B 205 [FS5] (1982) 483. [4]P. Schwab. Phys. Lell. B 118 (1982) 373: B 126 (1983) 241. [5]M. Bander and C. Itzykson, Nucl. Phys. B 257 [FSI4] (1985) 531. [6] P. Di Vecchia ct al., Nucl. Phys. B 190 [FS3] ( 1981 ) 719: G. Martinelli, R. Petronzio and M.A Virasoro. Nucl. Phys. B205 (1982) 255. [7] B. Berg and M. Liischer, Nucl. Phys. B 190 [FS3] ( 1981 ) 412: B. Berg, Phys. Left. B 104 ( 1981 ) 475; M. Liischer. Nucl. Phys. B 200 [FS4] (1982) 61. [ 8 ] D. Percher and M. Liischer, Nucl. Phys. B 225 [ FS9 ] ( 1983 ) 53. [ 9 ] A. Phillips and D. Stone, Commun. Math. Phys. 103 ( 1986 ) 599. [ 10J M. Kremer et al., Nucl. Phys. B 305 (1988) 109. [ l l ] J . Hock, Rutherford Laboratory preprint RAI,-88-098 (1988).