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Pseudoparticle methods for topological invariants greater than one
Volume 74B, number 1, 2
PHYSICS LETTERS
27 March 1978
PSEUDOPARTICLE METHODS FOR TOPOLOGICAL INVARIANTS GREATER THAN ONE Charles NASH
School of Mathematics, Trinity College Dublin, Ireland Received 5 December 1977
A method is presented for constructing gauge fields A ~(x) with topological invariant greater than one. These can give rise to singular field tensors F~v in certain cases.
l f A , is a pseudoparticle solution then on the sphere S 3 at infinity it may be written as
g - 1(x) 3~zg(x) ,
(2)
and
l trfdxeu~FuvF~:n/g 2
87r2
(3)
where n is an integer and is the so called topological invariant. The fact that the action is an extremum follows, in the example of ref. [1 ], from the fact that Au satisfies 1
Fur(x) = -~ euvaflFafl(x) = F u r ( x ) .
fax (Fur - 1 euva~Fa~)2 >1O,
i.e., S [A] ~> 2nTr2/g 2 ,
gtg= I.
(6)
This solution has n = 1. We would like to be able to write down fields A u with topological invariants n > 1. The key to doing this is provided b y the topological group which gives rise to the integer invariant n. We now need to draw on some simple topological arguments to describe the construction o f fields with n > 1. The form o f the field A u at infinity may be thought o f as giving rise to a map from S 3 to the gauge group SU(2). Under this map, which we call g, x goes to the SU(2) m a t r i x g ( x ) : g: S 3 ~ S U ( 2 ) ,
(5)
is saturated. Indeed we see from this that the extremum obtained when Fur = F u r is an absolute minimum. In ref. [1], the solution Au(x) takes the form (~ are the Pauli matrices):
x ~g(x).
(7)
But SU(2) may be parametrized on the surface of the sphere S 3. Thus we may think o f g as a map from $3 to $3: g: S 3 ~ S 3 ,
(4)
This means that the inequality tr
g(x) = (x4I + ix" a ) / V / ~ ,
(1)
where g(x) belong to the matrix representation o f the Yang-Mills gauge group G. (In this paper we take G to be SU(2).) The two important properties which are true for pseudoparticles are:
S [A] = ¼ t r f d x F ~ F ~ : 2mr2/g2,
Au(x) = [x2/(x 2 + ~k2)]i f (x) O#g(x) = A#(,g,),
x -~g(x).
(8)
It is well known that two such maps g l and g2 are not continuously deformable into one another: g may "cover" the sphere once, twice, etc. Maps that are deformable into one another are called homotopic; and equivalence classes with respect to h o m o t o p y can be made into a group - a so called h o m o t o p y group. In this case the group is written 7r3($3 ) and turns out to be simply the additive group o f integers. This accounts for the integer n, the topological invariant. Now we possess a n A u for which n = 1, i.e. a g ( x ) which represents the unit element of 7r3($3). But since 7r3($3) is simply the integers, to obtain an A u with n = 2 we 73
Volume 74B, number 1,2
PHYSICS LETTERS
simply have to "add" two unit elements 7T3($3). We have to "add" the map g to itself. All we need is the definition of what is meant by addition of two maps gl and g2" In general this definition is rather unwieldy. However in this situation we can make use o f an alternative definition which makes use of the fact that the second S 3 in eq. (8) is not only a sphere but that it also carries the group structure of SU(2). The definition of addition that we need is [2] :
27 March 1978
written explicitly because they are both equal and have the value unity. The remaining terms can, by simple properties of the trace be reduced to the three integrals: euva3 tr f d x g t F u v ( g ) g x ~ ( g ) ,
(14) euvc,ts tr
fdxF.v(g)×~#g),
euva~str
fdx×.~×.~.
of two SU(2) matrices. The symbol ~ means is homotopic or continuously deformable into. Thus what we are saying is that i f g ( x ) i s as given in eq. (6) then, if we define Au(h ) by:
These integrals when computed turn out to be zero. Hence n = 2 as we wished to establish. The generalisation to larger values o f n can proceed similarly. The evaluation of the trace algebra in these integrals can be involved. We shall now describe a method for mechanising these traces. First we proceed by analogy with the group SL(2, c) to construct the general SU(2) matrix g. Let
A.(h)
X = [ x4+ix3
gl + g2 ~ glg2 • (9) The map glg2 on the r.h.s, o f e q . (9) is the map which sends x to gl(x)g2(x), where gl(x)g2(x) is the product
= [ x 2 / ( x 2 + ~2)1 (g2(x)) t 3 . g 2 ( x ) ,
(10)
Au(h) will have topological invariant n = 2. Note of course that hth = I as it should. We can now check this. Using the definition of Fur and the convenient identity 3ugt g = _g? Dug , we compute straightaway that (we have set the coupling constant equal to unity to avoid confusion with g(x))
l, ixl - x 2
ixl + x 2 ) x 4 ix 3
.
(15)
Then X has the property that d e t X = (xl) 2 +(x2) 2 +(x3) 2 + (x4) 2 = x 2 , (16)
X : xl~TIg ,
where r u are the matrices T1, ..., 74 given by
Fuv(h ) = OuAvv(h) - OvAu(h ) + [Au(h), Av(h)] = Fuv(g ) +gtFuv(g)g + Xuv(g ) ,
~=i~, (1 1)
r4=l.
(17)
A general element of SU(2) is:
where
g(x) = x/,¢~2 .
Fu~(g) = OuA.(g) - O~Au(g) + [A (g), Av(g)],
It clearly satisfies g#g = I and det g = 1. The general trace calculation that we have is then seen to be o f the form tr (glg2 ""gn)" However,
X . v ( g ) = [X2X2/(X2 + ~k2)4 ]
X ( [ a . g t, augt ]g2 + ( g t ) 2 [ B v g ,
Oug]) .
(12)
According to the definition (3) of the invariant integer n we have
n=
1 trfdx
8~r2
Gu v~(Fu v(g) + gt Fu v(g)g + X u v(g)}
X (Fa~(g) + gt Fat3(g)g + Xa#(g)), 1
= 8rr2
(13)
tr f dx euva~Fuv(g) Faa(g )
+ 1 tr 8rr 2
f dx euw3gt Fuv(g ) F~3(g)g + ....
The first two of the nine terms in eq. (13) we have 74
g(x)
=
(1s)
xxrX/~/~-,
(19)
so
tr(glg2'"gn) -xxt
Xkn tr(TXl...rXn). (x2)~/2
(20)
The problem is then reduced to calculation of T x l ' x v , where TXl...xn = tr (r M ... rXn).
(21)
The quantities Tkt...k n can be computed as needed by use of the commutation and anticommutation laws for the r x. Also it is convenient to use the r x as a basis for the space of 2 × 2 matrices via the relations
Volume 74B, number 1, 2
M=axrx,
ax-gtr(r
PHYSICS LETTERS M),
(22)
where M is any 2 X 2 matrix. Although this work is in euclidean 4-space use may be made of the Minkowski metric tensor guy to facilitate calculations involving gt rather than g. Ifgt` v is as usual, guy = diag ( 1 , - 1 , - 1 , - 1 ) ,
(23)
then we may represent gt by g] = X~ (r;~)]/V/~ = X j x / V ~ ,
(25)
Clearly r x = (rx)* and this representation is of some convenience when computing the traces derived above. It is obvious to ask if the action S [A(g2)] is finite and if it is whether the action is extremised. The answer is that the action is finite and non-perturbative in its origin but that the action is not extremised. The actual value of the action is given by evaluating the integrals of eq. (26) in accordance with the rules given above: SIA(g2)] _atr_l f d x ( G v ( g ) + gtG~(g)g + ×t`~(g))2
=½trfdxF2~(g)+¼tr f d x × 2 v ( g ) .
Dt`(g 2) = at` + At`(g2) .
(26)
(29)
Eq. (29) reduces to i2 Dt`(g 2 )e~t`voFvo(g 2 ) -- O, i.e. Dt` (g2) ½ eXt`vo(Fvxo(°) + g? Fvo(g)g + Xvo(g))
(30)
= Dt`(g 2) (Fxt` (g) + g'~Fxt`(g)g + ½ ext`voXvo(g)) = O. On the other hand Dt`(g 2) Ft`.(g 2) is just: Dt` (g2) (Ft` v(g) + g? Ft` v(g)g + Xt`v(g))"
(24)
where r x is defined by using gt`v:
rx = g a s " "
27 March 1978
(31 )
Therefore if we compare eqs. (30), (31) we find that ifDt`(g2)Ft`v(g2 ) is zero then 1
Dt`(g 2) (Xt`v(g) - : et`vxoXvo(g)) = 0 .
(32)
By explicit calculation it can be verified that eq. (32) does not hold and hence we do not have an extremum of the action. It is also of interest to consider modifications to At`(g 2) in an attempt to minimise the action S[A ]. If we replace At`(g 2) by At`(h) given by
At`(h ) = Jk (x) g t Ot`g + fx,g ~ (g? Ot`g)g , with fx = X2/( X2 + X2), ¢~,' = X2/( X2 + X'2), we find that
Ft`v(h ) = Ft`v(g, X) + gt Ft`v(g, h' )g + )-(t`v(g) .
(34)
The notation in eq. (34) is as follows: Ft`v(g , X) and Ft`v(g, X') are the field tensors that result when At` is given by fag ? Ot`g and fx,g ? 3t`g respectively; Xt`v(g) is given by
The cross product terms in eq. (26) have dropped out by the use of Ft`~(g) = ,gt`~(g) and the fact that the integrals in eq. (14) are zero. We obtain S [A(g2)] = 47r2/g 2 + lOrr2/9g 2 ,
(27)
where g is now the coupling constant. This value for the action means that S [A(g2)] has not attained the absolute mininmm possible from eq. (5). To see whether eq. (27)represents an extremum of S[A(g2)] we must check the field equations for A(g2). That is to say evaluate
Dt`F.~ = ot`G~ + [A., G , ] •
(28)
The vanishing of eq. (28) indicates the presence of an extremum. This is most simply done in the following way. The Jacobi identity allows us to write 1
-5 ex•voDu(.g with
2
) [Dr(g2), Do(g2)] = 0 ,
(35) Xt`~ = (fx'A - A = ) { [ O v g t , OAt lg 2 + (gt)2 lavg ' Ot,g ] }. It is again quite straightforward to compute the action S [A(,~)] for this field configuration At`(h). The result may be written S[A(h)] = 4rr 2 + 4
fdx tr Xt`vXt` .
(36)
(The coupling constant has again been set equal to unity.) However it is simple to compute the integral over Xt`vXt`v and verify that it vanishes in the limit where X --*0. Thus in the limit we have: lim S [A(h)l = 4rr 2 . x--*0
(37)
It would appear then that S[A] is minimised. Unfortunately this limiting procedure leads to a singular field tensor Ft`v(-h ). For if we compute Ft`v(g , X), the first term in eq. (34), we obtain the expression 75
Volume 74B, number 1, 2 F~v(g, ~.) = [~k2/(X2 + ~k2)2 ]
PHYSICS LETTERS
"(rtaZ;--T T~f) /.'
,tt
'
(38)
Eq. (38) has a singular limit when X ~ 0. It tends to the product rr26 (x/t~) ~ ( x / ~ ) . In summary then, if we are willing to avoid this singular behaviour o f Fuv(-h) by not taking the limit X -+ 0 at this stage, then we can minimise S[A ] by taking X ~ 0 in the formula (36) after the X integrations have been performed. Thus it is possible to miniraise S [A(h)] at the expense of introducing a singular field tensor Fuv(h ). Further work along these lines to
76
27 March 1978
construct more well-behaved solutions by the methods of this paper is clearly o f some interest and importance.
References [1] A. Belavin, A. Polyakov, A. Schwartz and Y. Tyupkin, Phys. Lett. 59B (1975) 85. [2] B. Eckmann, Commun. Math. Helv. 14 (1941) 234; N. Steenrod, The topology of fibre bundles (Princeton Univ. Press, 1951).
PHYSICS LETTERS
27 March 1978
PSEUDOPARTICLE METHODS FOR TOPOLOGICAL INVARIANTS GREATER THAN ONE Charles NASH
School of Mathematics, Trinity College Dublin, Ireland Received 5 December 1977
A method is presented for constructing gauge fields A ~(x) with topological invariant greater than one. These can give rise to singular field tensors F~v in certain cases.
l f A , is a pseudoparticle solution then on the sphere S 3 at infinity it may be written as
g - 1(x) 3~zg(x) ,
(2)
and
l trfdxeu~FuvF~:n/g 2
87r2
(3)
where n is an integer and is the so called topological invariant. The fact that the action is an extremum follows, in the example of ref. [1 ], from the fact that Au satisfies 1
Fur(x) = -~ euvaflFafl(x) = F u r ( x ) .
fax (Fur - 1 euva~Fa~)2 >1O,
i.e., S [A] ~> 2nTr2/g 2 ,
gtg= I.
(6)
This solution has n = 1. We would like to be able to write down fields A u with topological invariants n > 1. The key to doing this is provided b y the topological group which gives rise to the integer invariant n. We now need to draw on some simple topological arguments to describe the construction o f fields with n > 1. The form o f the field A u at infinity may be thought o f as giving rise to a map from S 3 to the gauge group SU(2). Under this map, which we call g, x goes to the SU(2) m a t r i x g ( x ) : g: S 3 ~ S U ( 2 ) ,
(5)
is saturated. Indeed we see from this that the extremum obtained when Fur = F u r is an absolute minimum. In ref. [1], the solution Au(x) takes the form (~ are the Pauli matrices):
x ~g(x).
(7)
But SU(2) may be parametrized on the surface of the sphere S 3. Thus we may think o f g as a map from $3 to $3: g: S 3 ~ S 3 ,
(4)
This means that the inequality tr
g(x) = (x4I + ix" a ) / V / ~ ,
(1)
where g(x) belong to the matrix representation o f the Yang-Mills gauge group G. (In this paper we take G to be SU(2).) The two important properties which are true for pseudoparticles are:
S [A] = ¼ t r f d x F ~ F ~ : 2mr2/g2,
Au(x) = [x2/(x 2 + ~k2)]i f (x) O#g(x) = A#(,g,),
x -~g(x).
(8)
It is well known that two such maps g l and g2 are not continuously deformable into one another: g may "cover" the sphere once, twice, etc. Maps that are deformable into one another are called homotopic; and equivalence classes with respect to h o m o t o p y can be made into a group - a so called h o m o t o p y group. In this case the group is written 7r3($3 ) and turns out to be simply the additive group o f integers. This accounts for the integer n, the topological invariant. Now we possess a n A u for which n = 1, i.e. a g ( x ) which represents the unit element of 7r3($3). But since 7r3($3) is simply the integers, to obtain an A u with n = 2 we 73
Volume 74B, number 1,2
PHYSICS LETTERS
simply have to "add" two unit elements 7T3($3). We have to "add" the map g to itself. All we need is the definition of what is meant by addition of two maps gl and g2" In general this definition is rather unwieldy. However in this situation we can make use o f an alternative definition which makes use of the fact that the second S 3 in eq. (8) is not only a sphere but that it also carries the group structure of SU(2). The definition of addition that we need is [2] :
27 March 1978
written explicitly because they are both equal and have the value unity. The remaining terms can, by simple properties of the trace be reduced to the three integrals: euva3 tr f d x g t F u v ( g ) g x ~ ( g ) ,
(14) euvc,ts tr
fdxF.v(g)×~#g),
euva~str
fdx×.~×.~.
of two SU(2) matrices. The symbol ~ means is homotopic or continuously deformable into. Thus what we are saying is that i f g ( x ) i s as given in eq. (6) then, if we define Au(h ) by:
These integrals when computed turn out to be zero. Hence n = 2 as we wished to establish. The generalisation to larger values o f n can proceed similarly. The evaluation of the trace algebra in these integrals can be involved. We shall now describe a method for mechanising these traces. First we proceed by analogy with the group SL(2, c) to construct the general SU(2) matrix g. Let
A.(h)
X = [ x4+ix3
gl + g2 ~ glg2 • (9) The map glg2 on the r.h.s, o f e q . (9) is the map which sends x to gl(x)g2(x), where gl(x)g2(x) is the product
= [ x 2 / ( x 2 + ~2)1 (g2(x)) t 3 . g 2 ( x ) ,
(10)
Au(h) will have topological invariant n = 2. Note of course that hth = I as it should. We can now check this. Using the definition of Fur and the convenient identity 3ugt g = _g? Dug , we compute straightaway that (we have set the coupling constant equal to unity to avoid confusion with g(x))
l, ixl - x 2
ixl + x 2 ) x 4 ix 3
.
(15)
Then X has the property that d e t X = (xl) 2 +(x2) 2 +(x3) 2 + (x4) 2 = x 2 , (16)
X : xl~TIg ,
where r u are the matrices T1, ..., 74 given by
Fuv(h ) = OuAvv(h) - OvAu(h ) + [Au(h), Av(h)] = Fuv(g ) +gtFuv(g)g + Xuv(g ) ,
~=i~, (1 1)
r4=l.
(17)
A general element of SU(2) is:
where
g(x) = x/,¢~2 .
Fu~(g) = OuA.(g) - O~Au(g) + [A (g), Av(g)],
It clearly satisfies g#g = I and det g = 1. The general trace calculation that we have is then seen to be o f the form tr (glg2 ""gn)" However,
X . v ( g ) = [X2X2/(X2 + ~k2)4 ]
X ( [ a . g t, augt ]g2 + ( g t ) 2 [ B v g ,
Oug]) .
(12)
According to the definition (3) of the invariant integer n we have
n=
1 trfdx
8~r2
Gu v~(Fu v(g) + gt Fu v(g)g + X u v(g)}
X (Fa~(g) + gt Fat3(g)g + Xa#(g)), 1
= 8rr2
(13)
tr f dx euva~Fuv(g) Faa(g )
+ 1 tr 8rr 2
f dx euw3gt Fuv(g ) F~3(g)g + ....
The first two of the nine terms in eq. (13) we have 74
g(x)
=
(1s)
xxrX/~/~-,
(19)
so
tr(glg2'"gn) -xxt
Xkn tr(TXl...rXn). (x2)~/2
(20)
The problem is then reduced to calculation of T x l ' x v , where TXl...xn = tr (r M ... rXn).
(21)
The quantities Tkt...k n can be computed as needed by use of the commutation and anticommutation laws for the r x. Also it is convenient to use the r x as a basis for the space of 2 × 2 matrices via the relations
Volume 74B, number 1, 2
M=axrx,
ax-gtr(r
PHYSICS LETTERS M),
(22)
where M is any 2 X 2 matrix. Although this work is in euclidean 4-space use may be made of the Minkowski metric tensor guy to facilitate calculations involving gt rather than g. Ifgt` v is as usual, guy = diag ( 1 , - 1 , - 1 , - 1 ) ,
(23)
then we may represent gt by g] = X~ (r;~)]/V/~ = X j x / V ~ ,
(25)
Clearly r x = (rx)* and this representation is of some convenience when computing the traces derived above. It is obvious to ask if the action S [A(g2)] is finite and if it is whether the action is extremised. The answer is that the action is finite and non-perturbative in its origin but that the action is not extremised. The actual value of the action is given by evaluating the integrals of eq. (26) in accordance with the rules given above: SIA(g2)] _atr_l f d x ( G v ( g ) + gtG~(g)g + ×t`~(g))2
=½trfdxF2~(g)+¼tr f d x × 2 v ( g ) .
Dt`(g 2) = at` + At`(g2) .
(26)
(29)
Eq. (29) reduces to i2 Dt`(g 2 )e~t`voFvo(g 2 ) -- O, i.e. Dt` (g2) ½ eXt`vo(Fvxo(°) + g? Fvo(g)g + Xvo(g))
(30)
= Dt`(g 2) (Fxt` (g) + g'~Fxt`(g)g + ½ ext`voXvo(g)) = O. On the other hand Dt`(g 2) Ft`.(g 2) is just: Dt` (g2) (Ft` v(g) + g? Ft` v(g)g + Xt`v(g))"
(24)
where r x is defined by using gt`v:
rx = g a s " "
27 March 1978
(31 )
Therefore if we compare eqs. (30), (31) we find that ifDt`(g2)Ft`v(g2 ) is zero then 1
Dt`(g 2) (Xt`v(g) - : et`vxoXvo(g)) = 0 .
(32)
By explicit calculation it can be verified that eq. (32) does not hold and hence we do not have an extremum of the action. It is also of interest to consider modifications to At`(g 2) in an attempt to minimise the action S[A ]. If we replace At`(g 2) by At`(h) given by
At`(h ) = Jk (x) g t Ot`g + fx,g ~ (g? Ot`g)g , with fx = X2/( X2 + X2), ¢~,' = X2/( X2 + X'2), we find that
Ft`v(h ) = Ft`v(g, X) + gt Ft`v(g, h' )g + )-(t`v(g) .
(34)
The notation in eq. (34) is as follows: Ft`v(g , X) and Ft`v(g, X') are the field tensors that result when At` is given by fag ? Ot`g and fx,g ? 3t`g respectively; Xt`v(g) is given by
The cross product terms in eq. (26) have dropped out by the use of Ft`~(g) = ,gt`~(g) and the fact that the integrals in eq. (14) are zero. We obtain S [A(g2)] = 47r2/g 2 + lOrr2/9g 2 ,
(27)
where g is now the coupling constant. This value for the action means that S [A(g2)] has not attained the absolute mininmm possible from eq. (5). To see whether eq. (27)represents an extremum of S[A(g2)] we must check the field equations for A(g2). That is to say evaluate
Dt`F.~ = ot`G~ + [A., G , ] •
(28)
The vanishing of eq. (28) indicates the presence of an extremum. This is most simply done in the following way. The Jacobi identity allows us to write 1
-5 ex•voDu(.g with
2
) [Dr(g2), Do(g2)] = 0 ,
(35) Xt`~ = (fx'A - A = ) { [ O v g t , OAt lg 2 + (gt)2 lavg ' Ot,g ] }. It is again quite straightforward to compute the action S [A(,~)] for this field configuration At`(h). The result may be written S[A(h)] = 4rr 2 + 4
fdx tr Xt`vXt` .
(36)
(The coupling constant has again been set equal to unity.) However it is simple to compute the integral over Xt`vXt`v and verify that it vanishes in the limit where X --*0. Thus in the limit we have: lim S [A(h)l = 4rr 2 . x--*0
(37)
It would appear then that S[A] is minimised. Unfortunately this limiting procedure leads to a singular field tensor Ft`v(-h ). For if we compute Ft`v(g , X), the first term in eq. (34), we obtain the expression 75
Volume 74B, number 1, 2 F~v(g, ~.) = [~k2/(X2 + ~k2)2 ]
PHYSICS LETTERS
"(rtaZ;--T T~f) /.'
,tt
'
(38)
Eq. (38) has a singular limit when X ~ 0. It tends to the product rr26 (x/t~) ~ ( x / ~ ) . In summary then, if we are willing to avoid this singular behaviour o f Fuv(-h) by not taking the limit X -+ 0 at this stage, then we can minimise S[A ] by taking X ~ 0 in the formula (36) after the X integrations have been performed. Thus it is possible to miniraise S [A(h)] at the expense of introducing a singular field tensor Fuv(h ). Further work along these lines to
76
27 March 1978
construct more well-behaved solutions by the methods of this paper is clearly o f some interest and importance.
References [1] A. Belavin, A. Polyakov, A. Schwartz and Y. Tyupkin, Phys. Lett. 59B (1975) 85. [2] B. Eckmann, Commun. Math. Helv. 14 (1941) 234; N. Steenrod, The topology of fibre bundles (Princeton Univ. Press, 1951).