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Gauge fields on the continuum and lattice tori
Nuclear Physics B325 (1989) 138-160 North-Holland, Amsterdam
GAUGE FIELDS ON THE CONTINUUM AND LATTICE TORI D.R. LEBEDEV, M.I. POLIKARPOV and A.A. ROSLY Institute of Theoretical and Experimental Physics, Moscow 117259, USSR
Received 5 December 1988
We consider topological properties of gauge fields on the four-dimensional euclidean torus and, in particular, the relation between boundary conditions on the torus and topological classes of gauge fields. Using the methods of geometry of complex tori we obtain explicit expressions for self-dual solutions and twisted boundary conditions of the 't Hooft type for U(N) and SU(N) gauge theory on the euclidean torus. We consider also lattice toron fields and discuss possible computer experiments for lattices with nontrivial boundary conditions.
1. Introduction
The infrared regularization of the q u a n t u m theory can be done if we consider q u a n t u m fields enclosed in a finite volume. It is natural to consider the four-dimensional hypercube with the edges of length L~, g = 1 . . . . . 4, and to impose the periodic b o u n d a r y conditions. This means that the gauge-invariant quantities like tr F~2 must be periodic functions on the four-dimenisonal torus, T 4, i.e. the gauge fields on opposite faces of the hypercube are related by a gauge transformation. These b o u n d a r y conditions first proposed by t' H o o f t [1] correspond in general to a fractional topological charge in the S U ( N ) gauge theory. I n the euclidean field theory it is necessary to carry out the summation over all topological classes of the fields. For example the standard formula for the partition function is
z = E f A.Q e
s ,
@
(1)
where Q~
1 16rr 2
fT4F~rffl~vd4x
(2)
is the topological charge. We consider antihermitian gauge fields and for that reason the minus sign appears in the r.h.s, of eq. (2). For the evaluation of the sum (1), for example in the quasiclassical approximation, it is important to know the topologi0550-3213/89/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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cally nontrivial solutions of the classical equations of motion. An example of the self-dual solutions with the constant field strength tensor, satisfying the nontrivial (twisted) boundary conditions was given by t' Hooft [2]. These solutions called torons have fractional topological charges. It turned out that torons are responsible for the nonzero value of the gluino condensate in the N = 1 supersymmetric Yang-Mills theory [3]. In the S U ( N ) gluodynamics on the euclidean torus with twisted boundary conditions the theory of holomorphic vector bundles on complex tori is useful [4-7]. This theory was applied by van Baal [8] for calculation of quantum fluctuations around abelian torons. The next example is the classification of the minima of the twisted Eguchi-Kawai model [9,10]. As it was found in refs. [11,12] these minima are described by the unitary representation of the finite Heisenberg group. These representations are realized on the space of 0-functions on complex (abelian) tori with characteristics of special kind [4,7]. On the other hand the twisted Eguchi-Kawai model is related to the gauge theory with twisted boundary conditions. In the present paper we give another example of this relation. We use the holomorphic geometry of complex toil in order to construct the most general boundary conditions and corresponding solutions of the classical equations of motion with the constant field strength tensor (torons). In a particular case our solutions are the t' Hooft's torons. We also consider the topological properties of the gauge fields on the four-dimensional torus and discuss possible applications to lattice gauge theories. This paper is organized as follows. In sect. 2 we describe the topology of gauge fields on the torus, and discuss an additional topological charge corresponding to the first Chern class. We also stress the relation between the boundary conditions on the torus and topological classes of gauge fields. This relation allows one to rewrite eq. (1) in the form
z = E fDA e -s,
(3)
b.c.
where F.b.c. is the sum over the boundary conditions corresponding to different topological classes. In sect. 3 we consider the explicit expressions for the topological charge for U ( N ) and S U ( N ) gauge fields on the torus. Due to the existence of an additional topological charge, a h-term analogous to the usual 0-term can be added to the action. In sect. 4 we introduce complex coordinates in order to make use of the relation between the U(1) gauge theory on the four-dimenisonal torus and the theory of holomorphic vector bundles.
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In sect. 5 we construct U ( N ) torons from U(1) gauge fields considered in sect. 4. In sect. 6 we go back from complex to real coordinates and construct toron fields with integer topological charges. In sect. 7 we go from U ( N ) to S U ( N ) gauge group and obtain S U ( N ) torons with the topological charge Q = - p / N , p = 0 . . . . . N - 1. Our construction yields a generalization of the results of ref. [13]. In sect. 8 we reformulate our results for the lattice gauge theory: we obtain the lattice toron field which is the solution of the lattice equations of motion and lattice self-dual equations. We also discuss the lattice version of the Bianchi identities, and a special kind of lattice equations of motion which contain all the solutions of the (anti-)self-duality equations. In sect. 9 we give some conclusions and discuss possible computer experiments in lattices with nontrivial boundary conditions.
2. Boundary conditions and topological properties of gauge fields on a torus In this section we describe some general properties of gauge fields on the four-dimensional torus T 4. Topological characteristics of these fields o n T 4 differ slightly from those in the space R4; they are defined by the gauge group G and by the boundary conditions. We say that a field in a gauge theory is defined on a four-dimensional torus if (1) There exist four numbers L~ (bt = 1 . . . . . 4) called the periods of the torus such that any gauge invariant quantity O(x) satisfies the periodicity conditions
O(x) = O(x + l~,).
(4)
The components of the four-vectors l~ are defined as follows l~ = L, 8~. (2) Gauge covariant quantities may be gauge transformed when x ---, x + i~. For example, a coloured scalar field ~J(x), j = 1 . . . . . N, satisfies
(5) These matrices of gauge transformations, Vy(x), should satisfy 't Hooft's self-consistency equations [1, 2], which can be obtained as follows. Consider two paths from point x to point x + !~ + Iv: (x + i~) + l~ and (x + Iv) + i~, see fig. 1. The value of the field ~ ( x + 1~ + 1,) cannot depend on the path, and the necessary condition for that is: +
= v.(x +
(6)
The gauge transformation
• '(x) = v ( x )
(7)
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x÷lu
141
x+lv÷l~
x+l~_ Fig. 1. Two possible paths from point x to point x + I. + I v.
corresponds to
= v(x =
0).
(8)
Thus the torus, T 4, is characterized by four periods L~ and by four matrices V~(x) which we shall later call boundary conditions. The boundary conditions on the torus are closely related to topological properties of the gauge fields defined on the torus. It is well known from the theory of vector bundles that the gauge fields belong to the same topological class if and only if the corresponding boundary conditions are homotopically equivalent. Two boundary conditions V,(x) and V~(x) are called homotopically equivalent if there exists a continuous deformation V,-~ V~ which preserves the self-consistency equation (6). Moreover the boundary conditions V~(x) and V~(x) are topologically equivalent if and only if they are gauge equivalent, i.e. if there exists a smooth matrix function g(x) in the hypercube {0 ~
X(x.= 0).
(9)
The above discussion shows the equivalence of the two expressions (1) and (3) for the partition function. Now it is clear that in order to find the topological characteristics of the gauge field we should find out what prevents us from deforming the boundary conditions V~(x) to the trivial ones V~(x) - 1. It occurs that one element from the homotopical group ~r3(G) and six elements from Th(G) may prevent this deformation. The element from %(G) is due to the deformation on the border of the four-dimensional hypercube and elements from ¢rl(G) are due to deformations on the borders of the two-dimensional faces. Since 7r2(G) ---0 there are no obstacles for the deformation on the two-dimensional borders of the three-dimensional faces. We consider connected gauge groups G, i.e. %(G) = 0. The homotopy groups for U ( N ) and S U ( N ) gauge groups are listed in the table 1. This table is in accordance with the well-known fact that the four-dimensional U ( N ) gauge fields are characterized by the first and the second Chern classes c1~~
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TABLE 1 G
7h(G)
~r3(G)
U(1)
Z
0
U(N)
7
;7
SU(N)
0
Z
~'u
Z
SU(N)/ZN
(/~ > v) and c 2. For the gauge group G = S U ( N ) the topological classes are characterized by a single integer number which is the standard topological charge. For the other gauge groups there exist additional characteristics, i.e. six integer numbers for G = U ( N ) , and six integer numbers defined modulo N for G = S U ( N ) / / Z N . The last gauge group corresponds to a theory without fields in the fundamental representation (or, more precisely, only with fields which are not affected by the transformations from the center of the gauge group). Two well-known examples of such theories are pure gluodynamics and N = 1 supersymmetric Yang-Mills theory.
3. Topological charges of gauge fields on a torus We consider now explicit expressions for the topological charges. First consider the U ( N ) gauge group. As it was explained in sect. 2 besides the standard integer topological charge 1
q
16q.:2fT4Gl,,Gm, d4x,
G,,= ZB,-
+
(10)
8,],
(11)
there exists additional topological invariants on T 4. They can be defined as follows. Consider the U(1) gauge field
b~,(x) = t r B , ( x ) .
(12)
Then it can be proven [14] that six numbers 1
Ll'/~"
Cl~'~ = 2wi L1L2L3L 4
ff.
d4x
(~ > v)
f~,~ = O~,b~- O~b~, (13), (14)
are integers and correspond to the first Chern class. Thus for the U ( N ) gauge group the seven integer numbers q and cl, . define topological classes of the gauge fields B~,(x) on T 4.
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The SU(N) gauge field A,(x) can be obtained from B,(x) as follows: 1
A , ( x ) = B,(x) - ~ b , . 1N ,
(15)
where 1 N is the unit N x N matrix. The SU(N) field strength tensor is 1
(16) The standard definition of the topological charge Q for the SU(N) gauge fields in the case of such fields on the torus can in general give a non-integer number [1, 2]. (An agreement with mathematical terminology would be achieved in the case of gauge fields with non-integer Q if one called them the SU(N)//7/N gauge fields.) Indeed, in the case of the above fields A, we find 1
Q-
P
16~rZf T F ~ f i ~ d 4 x = q + ~ ,
(17)
where q is defined by eq. (10) and 1
v = 16rr 2 fT/~ f ~ d4x = Clf,~~1~,~.
(18)
The last equality can be proven on the basis of definition (13). The numbers c1,, are integer, so v is also integer and Q (17) is in general non-integer. Similarly to eq. (5) when x is shifted by a period of the torus the field A,(x) is gauge transformed as follows
A (x + t.) = a (x) A (x) a;l(x) + a.(x) O. a;l(x).
(19)
The matrices of the gauge transformation, f2~(x), are expressed through U(N) matrices V~(x) t2~(x) = V~(x)det(V~(x))
1/u.
(20)
Substituting V, from eq. (20) into eq. (6) we obtain the self-consistency equation for the SU(N) gauge group [1,2]: ~ . ( x + !~) fa~(x) = ~2~(x + !~,)/2.(x) Z~,~.
(21)
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The additional coefficient
Z,~=det(V~(x+l~) V.(x)) I/N det(V.(x+i,)V~(x))
1/N
must be an element of the center of SU(N), and therefore cannot depend on x. It follows from eq. (21) that Z.~ = Z*~.. Therefore Z,. = exp(27rin.JN
),
(22)
where n . . = - n . . is the so-called twist tensor. The six integers n ~ , bt > v, correspond to the first Chern class, and therefore define topological classes of the gauge fields In sect. 2 we referred to the gauge group corresponding to the boundary conditions with Z . . ~ 1 as the S U ( N ) / Z u gauge group. If a theory admits fields in the fundamental representation (e.g. quarks in QCD) then Z ~ = 1 [1, 2]. The proof is as follows. Consider a field q)(x) in the fundamental representation. Then going around the boundary of the torus in the/~v plane we obtain from the boundary condition ¢b(x + i.) = and eq. (21) that ~ ( x ) = Z.~ (b(x), so Z~. = 1. The topological charge Q can be expressed in terms of the boundary conditions [15,16]:
A.(x).
~?.(x)Cb(x)
1
Q-
24~r2
E~z fd3°.eu,~Btr(%,%~%B)
1
+87r---TEfd2S,.e,,~Btr[~%(x,=L,)%B(x.=O)],
(23)
pw
VOV 1; fd301=dx2dx3dx4; fd2Sa2=fdx3dx4
where ~0= etc. It can also be proven that c1~~ and n~,. are equal modulo N. Therefore in accordance with the results of sect. 2 the topological characteristics of the gauge fields are explicitly related to the boundary conditions. N o w we make two almost obvious remarks on the quantization of the gauge theory on a torus. To be definite we consider the gauge group. With evident modifications these remarks are valid for the U ( N ) gauge group. At first we note that in the functional integral one must consider all topological classes of the fields, so instead of eq. (1), the partition functional is
SU(N)/7/N
z= E fDAQ,",,e s. Q, n~v
The expression (2) is still valid.
(24)
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Secondly, in analogy with the 0-term we may consider a ?~-term, which corresponds to the additional topological characteristics n~. If we assume the Lorentzinvariance in the infinite volume limit, then this generalization is unambiguous, and in the functional integral (24) e - s must be replaced by
e sl = e(-S+ieQ+~iX~/N),
(25)
V = ~nl~f, lilt,,, -- ~ep,,~a,8 n~p rla,8
(26)
where
and the coefficients in eq. (25) are such that e -sl is unambiguous for ?~ defined modulo N.
4. 0-functions and the abelian gauge group
Here and in sects. 5 and 6 we propose a regular method of solving eqs. (6) and (21) for the U ( N ) and S U ( N ) / Z u gauge groups. For given boundary conditions (V~(x) for the U ( N ) gauge group, ~2~(x) and Z,~ for the S U ( N ) / Z u gauge group) we find some solutions of the classical equations of motion. These solutions are generalizations of t' Hooft's torons. It is convenient to deal with two complex coordinates z = (Zl, z2) instead of the four real ones x. This helps one to reduce the problem of solving self-duality equations to certain problems of holomorphic geometry [17]. The coordinates z and x are related by a linear transformation explicitly specified below (eqs. (34) and (72)). Our approach is to consider a general scalar field • on the torus which satisfies eq. (5). Then the sufficient self-consistency condition for the V, in eq. (5) is given by eq. (6). It occurs that if we construct • from the two-dimensional 0-functions we obtain the boundary conditions V~ which satisfy eq. (6) and lead to the most general linear gauge fields (torons). As a first example we consider a scalar field (27) Here the two-dimensional 0-function is defined as follows [4, 7]: 0m,,m,,("/',Z) =
~
exp[rri(p+m')'r(p+m')t+
2~ri(p+m')(z+m")t],
(28)
p~Z 2
where z = (zl, z2) ~ C2; m ' = ( m l , m2), m" = ( m 3 , m4) E R 2 are characteristics of the 0-function, ~- is a 2 x 2 symmetric complex matrix which satisfies the condition Im r > 0.
(29)
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P = (Pl, P2) is a pair of integer numbers, the symbol t is used for the transformation of rows into columns. Some useful and important properties of 0-functions are given in appendix A. For q~(z) defined by eq. (27) eq. (A.1) has the following form ~(z +X.) =,o(z) O(z).
(30)
! X~,= n~,r + nj,t!
(31)
where
t tl and n~, n~, may be arbitrary pairs of integer numbers. Since ~ in eq. (30) is the analogue of l, in eq. (5) one must choose X~ linearly independent. The simplest choice for n~¢ and n,t! is the following
(n;) k = 8..k,,
't n , )
=~,-2,
(32)
k.
Now the explicit expressions for ?t, are )kl = (q'll, q'12) ,
X3 = (1,0),
)k2 = ('I"21, T22)'
~k4= (0,1).
(33)
Due to eq. (29) these X~ are linearly independent. The relation between x and z is obvious 4
z=
(34)
Y'~ ( x ~ / L ~ ) X ~ .
If x ~ x + l~, then z ~ z + X~, and eq. (30) is equivalent to eq. (5). The explicit expression for %(z) in eq. (30) is (see eq. (A.1))
(35)
% ( z ) = exp[2~ri%(z)], where a(z)
= m'n"
=
- n'm"-
5t n , r n , -- n ' z
1
-m.+2-2~'~-z., m,_2,
/~=1,2, /~ = 3,4.
(36)
Since Im % 4= 0 then [%[ 4= 1 and the gauge group corresponding to eqs. (27) and (30) is GL(1), i.e. the group of general linear transformations. Now we show how to obtain from the fields 4~(z) the field X,,,m,(z) which corresponds to the gauge group U(1). Under the shift of z by the period of the complex torus the field X,,,m,,(z) must transform similarly to ~(z) (30):
Xm' "(Z +
=
O,(z)xm.,,..(Z).
(37)
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Here the unitarity restriction is
v~(z)=exp[2~riB~(z)],
lm/3~(z) = 0 .
(38)
In order to find X we use the following ansatz
Xm,m,,(z) = ~ ( z ) e x p [ - ~ r T ( z ) ] .
(39)
Due to eqs. (30), (35) and (37)-(39) the condition Im/3, = 0 can be rewritten as follows I m % ( z ) = ½ [ y ( z ) - 7(z + X,)].
(40)
Taking into account eq. (31) and the explicit form of a,(z) (36) we obtain the solution of eq. (40) y(z)=(Imz)(Im~-)
l(Imz)t.
(41)
The phase of the U(1) gauge transformation (38) is /3~(z) = Re c~(z).
(42)
Note that the boundary conditions or(z ) automatically satisfy the self-consistency equations
v.(z +
X,.) v.(z)
= v.(z +
X.)
v~.(z).
(43)
Now we show how to obtain the U(1) gauge field corresponding to the boundary conditions v~,(z). The scalar field ~ ( z ) is holomorphic, therefore O ~ ( z ) = 0 , k = 1, 2. Taking in this equation • expressed through X,,,m,, by means of eq. (39) we obtain
(O~,+b~,)X,,,m,,(z)=O,
k = 1,2,
(44)
where b~k = e-~Y(z)0~k e ~7(z) = ~r0z, 7 ( z ) .
(45)
It is obvious that b~k is a gauge field in the "Dirac equation" (44), i.e. when X,,,,,~,, is gauge transformed, b~k(z ) transforms as a gauge field, in particular
bz,(z + )k,) = bzk(z ) -- 2~ri clskfl#(z ) . From the explicit expression for b~,, eqs. (41) and (45) it is almost obvious that other components of the gauge field are
bz, = -~rOz~y(z ) .
(46)
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Of, Fig. 2. Division of the torus with the periods )'1, X2 into N tori with the periods fl, ./2.
Eqs. (45) and (46) define a U(1)-gauge field which corresponds to the b o u n d a r y conditions v ( z ) (38)*.
5. Torons in the case of the U ( N ) gauge group In this section we give an explicit construction of a toron field for the case of the U ( N ) gauge group. The mathematical motivation of the suggested algorithm is discussed in refs. [18,19]. Instead of vectors ) t we consider vectors
f~=?~i/3i,
fi+2 = ~,i+2,
i=1,2,
(47)
where 3 i are positive integer numbers, which satisfy the constraint N = 31 32 .
(48)
T h e definition (47) of vectors fk means that in the plane (1, 2) we divide the initial four-dimensional cube into N small cubes as it is shown in fig. 2. The N - c o m p o n e n t scalar field g,(z), which is an analog of Xm,m.(Z) for N = 1, is constructed as follows:
~ ( z ) = ( X , ~ , m . ( z ) , x , , , , , ~ . ( z + f l ) . . . . . Xm,m,,(z+g) .... ),
(49)
where
g = k ~ f l + k 2 f 2,
k , = 0 . . . . . 3i - 1 .
(50)
This means that each c o m p o n e n t of the field ~ ( z ) corresponds to one of the small cubes shown in fig. 2. To be definite we specify the following order of components * The field • is called the field in the holomorphic gauge. The gauge transformation (39) is called transition from holomorphic to unitary gauge. The above construction of the gauge field b., b_ is nothing but the standard method of obtaining a gauge field corresponding to an hermitian metric in a holomorphic vector bundle.
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in eq. (49). First we fix k 2 = 0 and make k 1 vary from 0 to 8 1 - 1 , then we fix k 2 --- 1 etc. Taking into account fig. 2 and eq. (37) it is simple to derive the following transformation property of the field ~b(z)
+(~ +/.)= w.(~)~(~),
(5~)
where 0 0
1 0
...
0
0
1
UI(Z )
0
0
1
0 0
Wl(z ) =
® 182 ,
O 0
w:(z)=%
i
1 0
... 1
GO 0 v~(z) 0
W3(z) = v3(z)ls, ® 182,
1
0 W4(z ) = v4(z)ls, ® 182.
(52)
The generalization of eqs. (45) and (46) consistent with (49) is Bz = -~rdiag( Oz,7(z ) ..... Oz.l( z + g ) . . . . ) ,
B~, = ~rdiag( O~,y( z ) . . . . , 05,y(z + g ) , . . . ).
(53)
The above construction provides the unitary matrices Bz, B~, and W,; moreover the toron fields (53) correspond to the boundary conditions W~ (52); which satisfy the self-consistency conditions
w~(~ +L) W.(z) = <(z +&) W~(z).
(54)
It occurs that there exists a gauge transformation Bz, "+ R ( z ) O z R - l ( z ) + R ( Z ) B z R - I ( z ) ,
B~,--,R(~) O~R-l(z) + R(z)B~R-I(~),
(551
which leads to toron fields proportional to the unit matrix 18~ ® 1,2. The explici
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expression for such R (z) is
R( z ) = diag(1 ..... exp[2~riBr.,( z)] .... ), flr.,(z) = R e ( - z . r -
(56)
½r.r- r) - r. m",
(57)
where r denotes the two-dimensional row (kl/8~, kJ82) and therefore g = r. z. The gauge fields are
B~ = -~rOz, y ( z ) . lst® l,2-- ~ri[(Imr)-l(Imz)t]i .18 ®182 , B~=~rcg~y(z).l~a®182=rri[(Imr)-t(Imz)t]i.la
®lsz.
(58)
The field strength tensor is
Fz,zj = F~& = 0,
F~,,, = ~r(Im r ) ~ ~. la, ® la~,
(59)
where by definition
F~,~=O=B~-OzBz+ [B~,, B~,]
(60)
etc. The fields (58) correspond to the gauge transformed boundary conditions
V~(z) --- R(z + f,)W~,(z)R -1
(61)
or explicitly 171(2) = Psx ® 18~exp[2rriB(s&,o)-,(z)],
V3(z) --- Qnl ® 182exp{2~rim1) '
Vz(z ) = 181 ® Ps~exp(ZcriB(o,8~),,(z) },
V4(z ) = 1 8 1 ® Q ~ exp{2~rim2).
(62)
Here 1
0
0
o..
1
0 0
~=
0
1
0
exp(2~ri/8)
08
0
0
0
1
0
0
1
...
0
...
exp[2~ri(8- 1)/8] (63)
are matrices which satisfy the commutation relations of the finite Heisenberg group [4, 7]:
P8 Q8 = Q8 Psexp(2~ri/8 ) .
(64)
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6. Going from complex to real coordinates Now we rewrite the formulae of sect. 5 in terms of the real coordinates x. The relation of x and z (34) is not the most general. In order to find the proper generalization of eq. (34) let us begin from eq. (59) for the field strength F written in complex coordinates. In that form F is proportional to the following (1,1)-form
o: = ½iY'. (Im r),.J dz i A d~j
(65)
i,j
which takes integer values on the vectors
~i=Siei,
~2+i=Y'~rijej,
i,j=1,2;
e l = ( 1 , O ),
e2=(O,l);
J ~o(~,,~j)=0,
o~(~,,~2+j)=8~6,,j,
~o(~2+i,~2+j)=0,
i,j=1,2.
(66)
That is
where /18 = diag(81, 82). 4 Under an SL(4. Z) change of the basis ~t, - Ewla~v~v, a . . ~ Z and det(a,.) = + 1. the matrix % . transforms as follows ,0j. = ~o(~,. ~'.) = S.~.¢a,~a.~¢. One has the following general assertion [7]. Any nondegenerate skew-symmetric integer matrix n~.. can be transformed to the canonical form (67) by an SL(4. Z) transformation: t
n=a where a ~ SL(4, 7/), 8g ~ Z and n,,, and 81 is a divisor of 82. It is easy to check that
81
(0
_A 8
a t,
(68)
is the greatest common divisor of the numbers
pf(n) = dTCd~n =
8~ "t,,"~B__ log~va~m . -- 8182
"
Any integer skew-symmetric matrix can thus be represented in the following form 2
n~'vN= E i=1
8/-1(a~,i
or,2+ i -- a,,i a~,,2+,) •
(69)
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Let us now choose the vectors f~ in eq. (47) as a real basis in 122: 4
2
E u.f~= Eziei, ~=i
zinC,
u~n.
i=1
One can verify that the field strength F in eq. (59) has in coordinates u. the following form F = ~rY'~(Im r)i. 1 dz i A d2jl~, ® 1~2 i,j
= -2¢ri~ 67' du i A du2+il.,
® 162.
(70)
i
Let us pass to the basis f, relation
= E4=,a~,~L with a ~ SL(4,77), and find x, from the
4
4
2
E (x./L.)f/,= E u.f.= E ziei/~=I
E
)
a~,,i+ Y'~ a~,,2+j6j-l"rij (x~,/L.).
~t=l
(71)
i=1
4(2
Then Zi=
/~=1
(72)
j=l
The transformation (71)-(72) makes four vectors f~ in 122 from four vectors l. in ~ 4 with coordinates (l.)~= L.82, that is to say, we obtain a euclidean torus T 4 from the torus which results from C 2 divided by the lattice generated by the vectors f~. Computing F in the coordinates x. with the help of eqs. (69) and (70) we find F=
~ri n~. ~ L - -A~ 1 8d1 ® x1 8 2 .d x .,
,
(73)
where n~JN are given by eq. (69). It follows that the self-duality condition for the fields (58) in terms of x~ is given by the constraint n l~v
1
nab
(74)
This is precisely the condition found by t' Hooft [2]. Now we obtain the expression for the fields (58) in the coordinates x. of eq. (72). Using the relation 2(
Ozi
05i )
D.R. Lebedevet al. / Gaugefields
153
eq. (58) for Bz and B~ and eqs. (71)-(72) we get
Here K , , is a symmetric tensor which can be computed explicitly and which does not depend on x,. Due to the symmetry of K ~ the second term in the r.h.s, of this formula can be removed by the gauge transformation B e ~ VB, V-1 + VO,V-1 with
V ( x ) = exp ~
E K..x~x~ l~l ~ 1~ e U ( N ) . /x, t,'
Thus, finally, rri
n~,~xv
(75)
In order to compute the matrices of gauge transformations corresponding to the boundary conditions let us define an N-component scalar field qo(z) connected with q~(z) by the gauge transformation (56): ~u(z) =R(z).~(z) or, explicitly, ~(Z)
= ( X m ,m ,,( Z ) . . . . . X m , + r , m , , ( z ) . . . . )t.
Then the boundary conditions for the gauge field (75) can be found from the equations cp(z + f d ) = Vd(z)ep(z),
i~= 1 . . . . . 4
and eq. (72). The result is
(76)
The above constructed gauge field has the topological charge q = - 1 as can be easily obtained from eqs. (10) and (73). A simple generalization of the algorithm allows to obtain the higher values of q. Let diag(A (i)) = diag( A O). . . . . A(k)),
(77)
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D.R. Lebedev et aL / Gauge fields
where A (i) are matrices. Then the U(N)-gauge field
{ ~ri
n(i)x
B~ = diag ~ - ~ ~
~"--" 1~ ) ® l~i)
}
(78)
has the topological charge q = - k , N = E~=IN (~) and satisfies the boundary condition
v~k(x)
= daagl P~t~¢ Qs!~] ® P~? Qn~]~exp_ - ~ - ~
As before N o ) = 8(~)~(~)1,,2,n~ ) is related to
8~i), 6~2~) by
~
-, .
(79)
eq. (69), and a ~o ~ SL(4, 7/).
7. Torons for S U ( N ) gauge group
In sect. 3 we have shown how to obtain an S U ( N ) gauge field from the given U ( N ) gauge field, see formulae (15), (16) and (20). But if we use this procedure for the toron field (75) we obtain an SU(N)-field with zero topological charge (or, equivalently, with boundary conditions V~ which do not depend on x). The following modification however leads to nontrivial S U ( N ) fields. Consider a new U ( N ) gauge field, onew=
0
V new =
Op '
(v)
0
0
lp
(80)
'
where the field B and the boundary conditions V' are defined by eqs. (75) and (76). The rank of the gauge group is now
N =p + 8162,
p = 0 ..... N - 1.
(81)
Using eq. (15) for B "ew we obtain the SU(N)-gauge field
A~(x)=diag
~rip N(N-p)~
n~,vx~
qri
L-~I~I®ln2'-N-~
n~,~x~ } L~
lp
'
(82)
which corresponds to the topological charge
Q=-I+
81 62 N
P N'
p = 0 .... , N - 1 .
(83)
D.R. Lebedev et aL / Gauge fields
155
The boundary condition matrix I2s is expressed by eq. (20) through V new, 12s(x ) = diag(
d;1Ps~Q~3®P~,~Q~ ×exp
N(N-p)~ ns~xJL~ ,exp
-~---~-,
lp
, (84)
where d s = det( P(,a Q~,3 ® p{;2 Qg;,,). The matrices I2s satisfy eqs. (21) and (22) with ns, given by eq. (69). If eq. (74) is valid then A s (82) is a self-dual field. In general even if eq. (74) is not valid A s (82) is a solution of the equation of motion
[A.,
(85)
= 0
An anti-self-dual field can be obtained by the change of the orientation of the coordinate system. This can be done for example by the substitution x 1 ---, x 2, X 2 ---> X l .
A more general S U ( N ) toron field can be obtained if we use U ( N ) fields defined by eqs. (78) and (79). In this case the topological charge of the S U ( N ) toron is
Q = -k+
Y'~..
N
,
N (/)=
l,J
~2 ~'~6~'~+p.
(86)
i=1
8. T o r o n s on the lattice
The dynamical variables of the lattice S U ( N ) gauge theory are SU(N)-matrices
exp[
x+~
attached to the links which connect the sites x and x +/2 of the four-dimensional lattice. Here we use the standard notations, /2 is the vector in the direction /~, [/2[ - a, a is the lattice spacing. In the lattice formulae x is the four-vector which defines the coordinates of a site of the lattice. Substituting in eq. (87) the toron field (82) we obtain
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D.R. Lebedev et al. / Gaugefields
This "lattice toron" has the following three important properties. First, the boundary conditions which correspond to the field (88) are nontrivial
Ux+ l~,lt = ~?v( x ) Ox.g ~?; ( x + ~ )
(89)
with the matrices ~2. defined by eq. (84). Second, toron field (88) is an exact solution of the equations of motion:
~ . , ( P x , , ~ - Ux+-~,~Px ~ , ~ , . U x - z , . ) - c . c .
=0,
(90)
P
which correspond to the lattice theory with Wilson's action: S=
~ 1-~trP~..~ x,.,.
.
(91)
The plaquette matrix Px,.~ entering eqs. (90) and (91) is:
Px,~,. = Ux., Ux+~,. Ux++~,~,U£+,~•
(92)
Substituting eq. (88) into (92) we obtain that
Px ",~ = exp diag
N(N-p)
~L~
l pL ~ ln' ® la2, - - -N ~ - ~- Lt,
(93)
commutes with Ux ~,, and therefore (88) is a solution of eqs. (90). Thirdly, it follows from eq. (93) that if and only if the equality (74) is valid the toron field (88) satisfies the lattice analogue of the self-duality equations
Px,..=Px,~.
(94)
Here the operation - is defined by the relations /~x,12 = ex.34,
]~x,13 = ex,42,
/~x.14 = Px.z3,
Z Px,~,. = Px, t,."
(95)
If in eq. (88) we change the orientation of the system of coordinates (e.g. x~ ~ x 2, x 2 ~ Xl) then the lattice anti-self-duality equations are satisfied: P~... =/Sx. ,,.
(96)
Thus as in the continuum limit the lattice (anti)toron fields satisfy (anti)self-duality equations and equations of motion. For the general lattice fields if (anti-)self-duality equations are satisfied the equations of motion (90) may not be valid.
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D.R. Lebedev et al. / Gauge fields
X
X
----
Fig. 3. Graphical proof of the lattice Bianchi identity (97).
Below we suggest the lattice equations of motion which are valid for all solutions of eqs. (94) and (96) at any value of lattice spacing a. Consider the lattice analogue of the Bianchi identity: W~,x~~ Wx,,X. Wx,,, x = 1,
(97)
where there is no summation over repeated indices and (98) The graphical proof of identity (97) is given in fig. 3. In the limit a ~ 0 this identity becomes the usual Bianchi identity. If we replace P~, ~, in eqs. (97) and (98) by the dual quantities (95), i.e. substitute in eq. (97) (99) then the resulting equations admit by construction all the solutions of the self-duality equations (94) and in the continuum limit they become the Yang-Mills equations. Any solution of anti-self-duality equations (96) is also a solution of eqs. (97) and (99). This can be proven by the hermitian conjugation of eqs. (97) and (99).
9. Conclusion and possible applications We have found linear solutions (torons) of the equations of motion of the U ( N ) and SU(N) Yang-Mills theory on the torus with given boundary conditions. It occurs that the torons are closely related to the two-dimensional 0-functions. Due to the linear dependence of torons on the coordinates it is possible to find the lattice analogues of the continuum formulae.
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D.R. Lebedev et al. / Gauge fields
We hope that our remarks may be useful for numerical experiments in lattice gauge theories. In a perfect case numerical experiments have to be performed on the lattice of such a size L x L X L x L, that L is much larger than the correlation length, ~, so that in each domain of size R ( L >> R >> ~) the summation over the topological charges could be performed with the weight defined by the dynamics. For example, in the model of the instanton vacuum [20] the distribution of the instanton charge is gaussian. This distribution was also obtained by numerical simulation of the SU(2) lattice gauge theory [21]. The relation between the boundary conditions and the topological charge discussed in sects. 2 and 3 may be important for numerical experiments at high temperature, T - 1/~. In the continuum the integration of the lagrangian over the time is performed from 0 to/3 = 1/T. The functional integral is calculated under the condition that the physical quantities are periodic in time, for example t r F Z ( x , 0 ) = tr F ~2 ( x , fl).
(100)
This statement follows from the standard expression for the statistical sum, Z = tr e-#n, where the trace is taken over the physical states. In ref. [22] the "periodic instanton" (a periodic in the euclidean time solution of the equation of motion carrying the unit topological charge) has been found. The "periodic instanton" has a singularity in the center which is responsible for the topological charge. So the coexistence of the trivial boundary conditions and the unit topological charge is due to the singular gauge. If we consider nonsingular fields, then the summation over topological classes is equivalent to the summation over the classes of boundary conditions. Therefore at a finite temperature the boundary conditions for topologically nontrivial nonsingular fields are A.(x,O)
=
(101)
Here Aft = 12A~12-1+ 123~I2-1 and the boundary condition (100) is obviously satisfied. In the computer simulations of lattice gauge theories instead of eq. (101) the trivial boundary condition A~(x,0)=Aj,(x, fl) is usually used. Due to the discretization some traces of singular fields may be left on the lattice. One can employ certain numerical experiments in order to understand whether the summation over the classes of boundary conditions is important for lattice gauge theories. All the numerical calculations are performed (in most cases) with trivial boundary conditions which correspond to Q - - 0 in the continuum limit*. So a nonzero value of ( Q 2 ) is a consequence of lattice discretization. By the freezing of the quantum fluctuations it is possible to obtain the one-instanton configuration on the lattice * The Monte-Carlo simulationsof the SU(3) lattice gauge theory were performedby the authors of ref. [231.
D.R. Lebedet; et al. / Gauge fields
159
[21, 24], but the continuous field which corresponds to that configuration consists of one instanton with the radius of order of the correlation length and of one anti-instanton with the radius of order of the lattice spacing. Therefore the total topological charge is equal to zero due to the existence of a dislocation. Finally, we discuss the method of summation over the boundary conditions in the S U ( N ) lattice gauge theory. The enumeration of different n,~ is trivial ( n ~ is equal to n~, if for all #, p n ~ = n',, modulo N). The topological charge is expressed through pf(n), see eq. (83). If eq. (74) is valid then the lattice toron field (88) is self-dual and corresponds (at least in the continuum limit) to the minimum of the action. For the summation over the integer topological charges which correspond to multi-instanton configurations we must find the matrices ~2~(x) which satisfy the selfconsistency equation (21) with the trivial twist (Zu~ = 1). For the SU(2) gauge group such matrices corresponding to an integer charge Q are
~'~p.= g( xp. = t~) g- l( xls.= O) , g=
xI Z x0~
Q,
% = (1, iff),
(102)
(103)
where o k are the Pauli matrices. The variation of x 0 in eqs. (102) and (103) is equivalent to the gauge transformation (8) of the boundary condition (102), (103). A gauge field which corresponds to this boundary condition can be chosen in the following form:
A~,(x) = f ( x ) ( O ~ , g ( x ) ) g ( x ) - 1 ,
(104)
where f ( x ) is a function such that f ( x ) = 1 on the boundary and f ( x ) ~ 0 when x--* x 0 in such a way that makes A t nonsingular at x o. Analogously the lattice gauge field has the form
Vx,t~ = f ( x ) g ( x ) g - l ( x
+ ~).
(105)
We are planning to perform numerical experiments in order to find how the nontrivial boundary conditions influence the quantum lattice fields. Some interesting results for the quantum gauge theory on the torus were obtained recently in ref. [25]. The authors are grateful to A.Yu. Morozov, O.V. Ogievetsky and Yu.A. Simonov for useful remarks and valuable discussions, and to van Baal for sending the reprints of articles [11] and [25]. It is a pleasure to thank A. Di Giacomo and G. Paffuti for helpful conversations. One of us (M.I.P.) is grateful to the theory group of Physics department of Pisa University for the kind hospitality.
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D.R. Lebedev et al. / Gauge fields
Appendix The following properties of the O-functions are important in our considerations. (1) For any n', n" ~ 7/× Z
Om, = exp(-~rin'rn'-
2~rin'z)exp[2~ri(m'n"- n'm")] On,,.,,,(~, z).
(A.1)
(2) For any u', u" ~ R2
=exp{Z~ri[-½u"ru'-u'(z + u"
+m")]}
Om,+u,m,,+u,,('r,Z ).
(A.2)
(3) For any X'E ~=IZX/, X " ~ ez=xzX2+i, 0m,+x,,,~,,+x,,(% z) = exp(2~rim'?~") 0,,, ,~,,(~-, z).
(A.3)
References [1] G. 't [21 [31 [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
[25]
Hooft, Nucl, Phys. B153 (1979) 141; G. 't Hooft, Acta Phys. Austriaca Suppl. XXII (1980) 531 G. 't Hooft, Comm. Math, Phys. 81 (1981) 267 E. Cohen and C. Gomez, Phys. Rev. Lett. 52 (1984) 237 D. Mumford, Tata Lectures on Thetas vols. I, II, Progress in Math. vol. 28 (Birkhauser, Berlin, 1983) D. Mumford, Abelian Varieties (Bombay, 1968) P. Griffiths and J. Harris, Principles of algebraic geometry, vol. 1 (New York, 1978) J. Igusa, Theta Functions (Springer, Berlin, 1972) P. van Baal, Comm. Math. Phys. 94 (1984) 397 T. Eguchi and H. Kawai, Phys. Rev. Lett. 48 (1982) 1063 A. Gonzalez-Arroyo and M. Okawa, Phys. Rev. D27 (1983) 2397 P. van Baal and B. van Geemen, Proc. K. Ned. Akad. Wet. B89 (1986) 39; J. Math. Phys. 27(2) (1986) 455 D.R. Lebedev and M.I. Polikarpov, Nucl. Phys. B269 (1986) 285 H. Schenk, Comm. Math. Phys. 116 (1988) 177; P. Braam and P. van Baal, Nahm's Transformation for Instantons, CERN-TH.5108/88 T. Eguchi, P.B. Gelkey and A.J. Hanson, Phys. Rep. 66 (1980) 213 M. Lucher, Comm. Math. Phys. 85 (1982) 39 P. van Baal, Comm. Math. Phys. 85 (1982) 529 M. Atyah, Geometry of Yang-Mills fields: Lezioni Fermiani, Pisa, 1979 H. Moricawa, Nagoya Math. J. 41 (1971) 101 D.R. Lebedev, M.I. Polikarpov and A. Rosly, Yad. Phys. 48 (1989) (in Russian) D.I. Dyakonov and V.Yu. Petrov, Nucl. Phys. B245 (1984) 259; B272 (1986) 457; Phys. Lett. B147 (1984) 351 M.I. Polikarpov and A.I. Veselov, Nucl. Phys. B297 (1988) 34 B. Harrington and H. Shepard, Phys. Rev. D17 (1978) 2212 B. S~Sderberger and L. LSnnblad, Twist and finite size effects for the source method, preprint LU TP 87-16 (1987) M. Teper, Phys. Lett. B162 (1985) 357; B171 (1986) 81, 86; E.M. Ilgenfritz, M.L. Laursen, M. Mi~ller-Preugker, G. Schierholz and H. Schiller, Nucl. Phys. B268 (1986) 693 P. van Baal and J. Koller, Annals of Phys. 174 (2) (1987) 299; Nucl. Phys. B302 (1988) 1
GAUGE FIELDS ON THE CONTINUUM AND LATTICE TORI D.R. LEBEDEV, M.I. POLIKARPOV and A.A. ROSLY Institute of Theoretical and Experimental Physics, Moscow 117259, USSR
Received 5 December 1988
We consider topological properties of gauge fields on the four-dimensional euclidean torus and, in particular, the relation between boundary conditions on the torus and topological classes of gauge fields. Using the methods of geometry of complex tori we obtain explicit expressions for self-dual solutions and twisted boundary conditions of the 't Hooft type for U(N) and SU(N) gauge theory on the euclidean torus. We consider also lattice toron fields and discuss possible computer experiments for lattices with nontrivial boundary conditions.
1. Introduction
The infrared regularization of the q u a n t u m theory can be done if we consider q u a n t u m fields enclosed in a finite volume. It is natural to consider the four-dimensional hypercube with the edges of length L~, g = 1 . . . . . 4, and to impose the periodic b o u n d a r y conditions. This means that the gauge-invariant quantities like tr F~2 must be periodic functions on the four-dimenisonal torus, T 4, i.e. the gauge fields on opposite faces of the hypercube are related by a gauge transformation. These b o u n d a r y conditions first proposed by t' H o o f t [1] correspond in general to a fractional topological charge in the S U ( N ) gauge theory. I n the euclidean field theory it is necessary to carry out the summation over all topological classes of the fields. For example the standard formula for the partition function is
z = E f A.Q e
s ,
@
(1)
where Q~
1 16rr 2
fT4F~rffl~vd4x
(2)
is the topological charge. We consider antihermitian gauge fields and for that reason the minus sign appears in the r.h.s, of eq. (2). For the evaluation of the sum (1), for example in the quasiclassical approximation, it is important to know the topologi0550-3213/89/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
D.R. Lebedev et al. / Gauge fields
139
cally nontrivial solutions of the classical equations of motion. An example of the self-dual solutions with the constant field strength tensor, satisfying the nontrivial (twisted) boundary conditions was given by t' Hooft [2]. These solutions called torons have fractional topological charges. It turned out that torons are responsible for the nonzero value of the gluino condensate in the N = 1 supersymmetric Yang-Mills theory [3]. In the S U ( N ) gluodynamics on the euclidean torus with twisted boundary conditions the theory of holomorphic vector bundles on complex tori is useful [4-7]. This theory was applied by van Baal [8] for calculation of quantum fluctuations around abelian torons. The next example is the classification of the minima of the twisted Eguchi-Kawai model [9,10]. As it was found in refs. [11,12] these minima are described by the unitary representation of the finite Heisenberg group. These representations are realized on the space of 0-functions on complex (abelian) tori with characteristics of special kind [4,7]. On the other hand the twisted Eguchi-Kawai model is related to the gauge theory with twisted boundary conditions. In the present paper we give another example of this relation. We use the holomorphic geometry of complex toil in order to construct the most general boundary conditions and corresponding solutions of the classical equations of motion with the constant field strength tensor (torons). In a particular case our solutions are the t' Hooft's torons. We also consider the topological properties of the gauge fields on the four-dimensional torus and discuss possible applications to lattice gauge theories. This paper is organized as follows. In sect. 2 we describe the topology of gauge fields on the torus, and discuss an additional topological charge corresponding to the first Chern class. We also stress the relation between the boundary conditions on the torus and topological classes of gauge fields. This relation allows one to rewrite eq. (1) in the form
z = E fDA e -s,
(3)
b.c.
where F.b.c. is the sum over the boundary conditions corresponding to different topological classes. In sect. 3 we consider the explicit expressions for the topological charge for U ( N ) and S U ( N ) gauge fields on the torus. Due to the existence of an additional topological charge, a h-term analogous to the usual 0-term can be added to the action. In sect. 4 we introduce complex coordinates in order to make use of the relation between the U(1) gauge theory on the four-dimenisonal torus and the theory of holomorphic vector bundles.
D.R. Lebedev et aL / Gauge fields
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In sect. 5 we construct U ( N ) torons from U(1) gauge fields considered in sect. 4. In sect. 6 we go back from complex to real coordinates and construct toron fields with integer topological charges. In sect. 7 we go from U ( N ) to S U ( N ) gauge group and obtain S U ( N ) torons with the topological charge Q = - p / N , p = 0 . . . . . N - 1. Our construction yields a generalization of the results of ref. [13]. In sect. 8 we reformulate our results for the lattice gauge theory: we obtain the lattice toron field which is the solution of the lattice equations of motion and lattice self-dual equations. We also discuss the lattice version of the Bianchi identities, and a special kind of lattice equations of motion which contain all the solutions of the (anti-)self-duality equations. In sect. 9 we give some conclusions and discuss possible computer experiments in lattices with nontrivial boundary conditions.
2. Boundary conditions and topological properties of gauge fields on a torus In this section we describe some general properties of gauge fields on the four-dimensional torus T 4. Topological characteristics of these fields o n T 4 differ slightly from those in the space R4; they are defined by the gauge group G and by the boundary conditions. We say that a field in a gauge theory is defined on a four-dimensional torus if (1) There exist four numbers L~ (bt = 1 . . . . . 4) called the periods of the torus such that any gauge invariant quantity O(x) satisfies the periodicity conditions
O(x) = O(x + l~,).
(4)
The components of the four-vectors l~ are defined as follows l~ = L, 8~. (2) Gauge covariant quantities may be gauge transformed when x ---, x + i~. For example, a coloured scalar field ~J(x), j = 1 . . . . . N, satisfies
(5) These matrices of gauge transformations, Vy(x), should satisfy 't Hooft's self-consistency equations [1, 2], which can be obtained as follows. Consider two paths from point x to point x + !~ + Iv: (x + i~) + l~ and (x + Iv) + i~, see fig. 1. The value of the field ~ ( x + 1~ + 1,) cannot depend on the path, and the necessary condition for that is: +
= v.(x +
(6)
The gauge transformation
• '(x) = v ( x )
(7)
D.R. Lebedev et al. / Gauge fields
x÷lu
141
x+lv÷l~
x+l~_ Fig. 1. Two possible paths from point x to point x + I. + I v.
corresponds to
= v(x =
0).
(8)
Thus the torus, T 4, is characterized by four periods L~ and by four matrices V~(x) which we shall later call boundary conditions. The boundary conditions on the torus are closely related to topological properties of the gauge fields defined on the torus. It is well known from the theory of vector bundles that the gauge fields belong to the same topological class if and only if the corresponding boundary conditions are homotopically equivalent. Two boundary conditions V,(x) and V~(x) are called homotopically equivalent if there exists a continuous deformation V,-~ V~ which preserves the self-consistency equation (6). Moreover the boundary conditions V~(x) and V~(x) are topologically equivalent if and only if they are gauge equivalent, i.e. if there exists a smooth matrix function g(x) in the hypercube {0 ~
X(x.= 0).
(9)
The above discussion shows the equivalence of the two expressions (1) and (3) for the partition function. Now it is clear that in order to find the topological characteristics of the gauge field we should find out what prevents us from deforming the boundary conditions V~(x) to the trivial ones V~(x) - 1. It occurs that one element from the homotopical group ~r3(G) and six elements from Th(G) may prevent this deformation. The element from %(G) is due to the deformation on the border of the four-dimensional hypercube and elements from ¢rl(G) are due to deformations on the borders of the two-dimensional faces. Since 7r2(G) ---0 there are no obstacles for the deformation on the two-dimensional borders of the three-dimensional faces. We consider connected gauge groups G, i.e. %(G) = 0. The homotopy groups for U ( N ) and S U ( N ) gauge groups are listed in the table 1. This table is in accordance with the well-known fact that the four-dimensional U ( N ) gauge fields are characterized by the first and the second Chern classes c1~~
D.R. Lebedev et al. / Gaugefields
142
TABLE 1 G
7h(G)
~r3(G)
U(1)
Z
0
U(N)
7
;7
SU(N)
0
Z
~'u
Z
SU(N)/ZN
(/~ > v) and c 2. For the gauge group G = S U ( N ) the topological classes are characterized by a single integer number which is the standard topological charge. For the other gauge groups there exist additional characteristics, i.e. six integer numbers for G = U ( N ) , and six integer numbers defined modulo N for G = S U ( N ) / / Z N . The last gauge group corresponds to a theory without fields in the fundamental representation (or, more precisely, only with fields which are not affected by the transformations from the center of the gauge group). Two well-known examples of such theories are pure gluodynamics and N = 1 supersymmetric Yang-Mills theory.
3. Topological charges of gauge fields on a torus We consider now explicit expressions for the topological charges. First consider the U ( N ) gauge group. As it was explained in sect. 2 besides the standard integer topological charge 1
q
16q.:2fT4Gl,,Gm, d4x,
G,,= ZB,-
+
(10)
8,],
(11)
there exists additional topological invariants on T 4. They can be defined as follows. Consider the U(1) gauge field
b~,(x) = t r B , ( x ) .
(12)
Then it can be proven [14] that six numbers 1
Ll'/~"
Cl~'~ = 2wi L1L2L3L 4
ff.
d4x
(~ > v)
f~,~ = O~,b~- O~b~, (13), (14)
are integers and correspond to the first Chern class. Thus for the U ( N ) gauge group the seven integer numbers q and cl, . define topological classes of the gauge fields B~,(x) on T 4.
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The SU(N) gauge field A,(x) can be obtained from B,(x) as follows: 1
A , ( x ) = B,(x) - ~ b , . 1N ,
(15)
where 1 N is the unit N x N matrix. The SU(N) field strength tensor is 1
(16) The standard definition of the topological charge Q for the SU(N) gauge fields in the case of such fields on the torus can in general give a non-integer number [1, 2]. (An agreement with mathematical terminology would be achieved in the case of gauge fields with non-integer Q if one called them the SU(N)//7/N gauge fields.) Indeed, in the case of the above fields A, we find 1
Q-
P
16~rZf T F ~ f i ~ d 4 x = q + ~ ,
(17)
where q is defined by eq. (10) and 1
v = 16rr 2 fT/~ f ~ d4x = Clf,~~1~,~.
(18)
The last equality can be proven on the basis of definition (13). The numbers c1,, are integer, so v is also integer and Q (17) is in general non-integer. Similarly to eq. (5) when x is shifted by a period of the torus the field A,(x) is gauge transformed as follows
A (x + t.) = a (x) A (x) a;l(x) + a.(x) O. a;l(x).
(19)
The matrices of the gauge transformation, f2~(x), are expressed through U(N) matrices V~(x) t2~(x) = V~(x)det(V~(x))
1/u.
(20)
Substituting V, from eq. (20) into eq. (6) we obtain the self-consistency equation for the SU(N) gauge group [1,2]: ~ . ( x + !~) fa~(x) = ~2~(x + !~,)/2.(x) Z~,~.
(21)
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The additional coefficient
Z,~=det(V~(x+l~) V.(x)) I/N det(V.(x+i,)V~(x))
1/N
must be an element of the center of SU(N), and therefore cannot depend on x. It follows from eq. (21) that Z.~ = Z*~.. Therefore Z,. = exp(27rin.JN
),
(22)
where n . . = - n . . is the so-called twist tensor. The six integers n ~ , bt > v, correspond to the first Chern class, and therefore define topological classes of the gauge fields In sect. 2 we referred to the gauge group corresponding to the boundary conditions with Z . . ~ 1 as the S U ( N ) / Z u gauge group. If a theory admits fields in the fundamental representation (e.g. quarks in QCD) then Z ~ = 1 [1, 2]. The proof is as follows. Consider a field q)(x) in the fundamental representation. Then going around the boundary of the torus in the/~v plane we obtain from the boundary condition ¢b(x + i.) = and eq. (21) that ~ ( x ) = Z.~ (b(x), so Z~. = 1. The topological charge Q can be expressed in terms of the boundary conditions [15,16]:
A.(x).
~?.(x)Cb(x)
1
Q-
24~r2
E~z fd3°.eu,~Btr(%,%~%B)
1
+87r---TEfd2S,.e,,~Btr[~%(x,=L,)%B(x.=O)],
(23)
pw
VOV 1; fd301=dx2dx3dx4; fd2Sa2=fdx3dx4
where ~0= etc. It can also be proven that c1~~ and n~,. are equal modulo N. Therefore in accordance with the results of sect. 2 the topological characteristics of the gauge fields are explicitly related to the boundary conditions. N o w we make two almost obvious remarks on the quantization of the gauge theory on a torus. To be definite we consider the gauge group. With evident modifications these remarks are valid for the U ( N ) gauge group. At first we note that in the functional integral one must consider all topological classes of the fields, so instead of eq. (1), the partition functional is
SU(N)/7/N
z= E fDAQ,",,e s. Q, n~v
The expression (2) is still valid.
(24)
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145
Secondly, in analogy with the 0-term we may consider a ?~-term, which corresponds to the additional topological characteristics n~. If we assume the Lorentzinvariance in the infinite volume limit, then this generalization is unambiguous, and in the functional integral (24) e - s must be replaced by
e sl = e(-S+ieQ+~iX~/N),
(25)
V = ~nl~f, lilt,,, -- ~ep,,~a,8 n~p rla,8
(26)
where
and the coefficients in eq. (25) are such that e -sl is unambiguous for ?~ defined modulo N.
4. 0-functions and the abelian gauge group
Here and in sects. 5 and 6 we propose a regular method of solving eqs. (6) and (21) for the U ( N ) and S U ( N ) / Z u gauge groups. For given boundary conditions (V~(x) for the U ( N ) gauge group, ~2~(x) and Z,~ for the S U ( N ) / Z u gauge group) we find some solutions of the classical equations of motion. These solutions are generalizations of t' Hooft's torons. It is convenient to deal with two complex coordinates z = (Zl, z2) instead of the four real ones x. This helps one to reduce the problem of solving self-duality equations to certain problems of holomorphic geometry [17]. The coordinates z and x are related by a linear transformation explicitly specified below (eqs. (34) and (72)). Our approach is to consider a general scalar field • on the torus which satisfies eq. (5). Then the sufficient self-consistency condition for the V, in eq. (5) is given by eq. (6). It occurs that if we construct • from the two-dimensional 0-functions we obtain the boundary conditions V~ which satisfy eq. (6) and lead to the most general linear gauge fields (torons). As a first example we consider a scalar field (27) Here the two-dimensional 0-function is defined as follows [4, 7]: 0m,,m,,("/',Z) =
~
exp[rri(p+m')'r(p+m')t+
2~ri(p+m')(z+m")t],
(28)
p~Z 2
where z = (zl, z2) ~ C2; m ' = ( m l , m2), m" = ( m 3 , m4) E R 2 are characteristics of the 0-function, ~- is a 2 x 2 symmetric complex matrix which satisfies the condition Im r > 0.
(29)
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P = (Pl, P2) is a pair of integer numbers, the symbol t is used for the transformation of rows into columns. Some useful and important properties of 0-functions are given in appendix A. For q~(z) defined by eq. (27) eq. (A.1) has the following form ~(z +X.) =,o(z) O(z).
(30)
! X~,= n~,r + nj,t!
(31)
where
t tl and n~, n~, may be arbitrary pairs of integer numbers. Since ~ in eq. (30) is the analogue of l, in eq. (5) one must choose X~ linearly independent. The simplest choice for n~¢ and n,t! is the following
(n;) k = 8..k,,
't n , )
=~,-2,
(32)
k.
Now the explicit expressions for ?t, are )kl = (q'll, q'12) ,
X3 = (1,0),
)k2 = ('I"21, T22)'
~k4= (0,1).
(33)
Due to eq. (29) these X~ are linearly independent. The relation between x and z is obvious 4
z=
(34)
Y'~ ( x ~ / L ~ ) X ~ .
If x ~ x + l~, then z ~ z + X~, and eq. (30) is equivalent to eq. (5). The explicit expression for %(z) in eq. (30) is (see eq. (A.1))
(35)
% ( z ) = exp[2~ri%(z)], where a(z)
= m'n"
=
- n'm"-
5t n , r n , -- n ' z
1
-m.+2-2~'~-z., m,_2,
/~=1,2, /~ = 3,4.
(36)
Since Im % 4= 0 then [%[ 4= 1 and the gauge group corresponding to eqs. (27) and (30) is GL(1), i.e. the group of general linear transformations. Now we show how to obtain from the fields 4~(z) the field X,,,m,(z) which corresponds to the gauge group U(1). Under the shift of z by the period of the complex torus the field X,,,m,,(z) must transform similarly to ~(z) (30):
Xm' "(Z +
=
O,(z)xm.,,..(Z).
(37)
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147
Here the unitarity restriction is
v~(z)=exp[2~riB~(z)],
lm/3~(z) = 0 .
(38)
In order to find X we use the following ansatz
Xm,m,,(z) = ~ ( z ) e x p [ - ~ r T ( z ) ] .
(39)
Due to eqs. (30), (35) and (37)-(39) the condition Im/3, = 0 can be rewritten as follows I m % ( z ) = ½ [ y ( z ) - 7(z + X,)].
(40)
Taking into account eq. (31) and the explicit form of a,(z) (36) we obtain the solution of eq. (40) y(z)=(Imz)(Im~-)
l(Imz)t.
(41)
The phase of the U(1) gauge transformation (38) is /3~(z) = Re c~(z).
(42)
Note that the boundary conditions or(z ) automatically satisfy the self-consistency equations
v.(z +
X,.) v.(z)
= v.(z +
X.)
v~.(z).
(43)
Now we show how to obtain the U(1) gauge field corresponding to the boundary conditions v~,(z). The scalar field ~ ( z ) is holomorphic, therefore O ~ ( z ) = 0 , k = 1, 2. Taking in this equation • expressed through X,,,m,, by means of eq. (39) we obtain
(O~,+b~,)X,,,m,,(z)=O,
k = 1,2,
(44)
where b~k = e-~Y(z)0~k e ~7(z) = ~r0z, 7 ( z ) .
(45)
It is obvious that b~k is a gauge field in the "Dirac equation" (44), i.e. when X,,,,,~,, is gauge transformed, b~k(z ) transforms as a gauge field, in particular
bz,(z + )k,) = bzk(z ) -- 2~ri clskfl#(z ) . From the explicit expression for b~,, eqs. (41) and (45) it is almost obvious that other components of the gauge field are
bz, = -~rOz~y(z ) .
(46)
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Of, Fig. 2. Division of the torus with the periods )'1, X2 into N tori with the periods fl, ./2.
Eqs. (45) and (46) define a U(1)-gauge field which corresponds to the b o u n d a r y conditions v ( z ) (38)*.
5. Torons in the case of the U ( N ) gauge group In this section we give an explicit construction of a toron field for the case of the U ( N ) gauge group. The mathematical motivation of the suggested algorithm is discussed in refs. [18,19]. Instead of vectors ) t we consider vectors
f~=?~i/3i,
fi+2 = ~,i+2,
i=1,2,
(47)
where 3 i are positive integer numbers, which satisfy the constraint N = 31 32 .
(48)
T h e definition (47) of vectors fk means that in the plane (1, 2) we divide the initial four-dimensional cube into N small cubes as it is shown in fig. 2. The N - c o m p o n e n t scalar field g,(z), which is an analog of Xm,m.(Z) for N = 1, is constructed as follows:
~ ( z ) = ( X , ~ , m . ( z ) , x , , , , , ~ . ( z + f l ) . . . . . Xm,m,,(z+g) .... ),
(49)
where
g = k ~ f l + k 2 f 2,
k , = 0 . . . . . 3i - 1 .
(50)
This means that each c o m p o n e n t of the field ~ ( z ) corresponds to one of the small cubes shown in fig. 2. To be definite we specify the following order of components * The field • is called the field in the holomorphic gauge. The gauge transformation (39) is called transition from holomorphic to unitary gauge. The above construction of the gauge field b., b_ is nothing but the standard method of obtaining a gauge field corresponding to an hermitian metric in a holomorphic vector bundle.
D.R. Lebedevet al. / Gaugefields
149
in eq. (49). First we fix k 2 = 0 and make k 1 vary from 0 to 8 1 - 1 , then we fix k 2 --- 1 etc. Taking into account fig. 2 and eq. (37) it is simple to derive the following transformation property of the field ~b(z)
+(~ +/.)= w.(~)~(~),
(5~)
where 0 0
1 0
...
0
0
1
UI(Z )
0
0
1
0 0
Wl(z ) =
® 182 ,
O 0
w:(z)=%
i
1 0
... 1
GO 0 v~(z) 0
W3(z) = v3(z)ls, ® 182,
1
0 W4(z ) = v4(z)ls, ® 182.
(52)
The generalization of eqs. (45) and (46) consistent with (49) is Bz = -~rdiag( Oz,7(z ) ..... Oz.l( z + g ) . . . . ) ,
B~, = ~rdiag( O~,y( z ) . . . . , 05,y(z + g ) , . . . ).
(53)
The above construction provides the unitary matrices Bz, B~, and W,; moreover the toron fields (53) correspond to the boundary conditions W~ (52); which satisfy the self-consistency conditions
w~(~ +L) W.(z) = <(z +&) W~(z).
(54)
It occurs that there exists a gauge transformation Bz, "+ R ( z ) O z R - l ( z ) + R ( Z ) B z R - I ( z ) ,
B~,--,R(~) O~R-l(z) + R(z)B~R-I(~),
(551
which leads to toron fields proportional to the unit matrix 18~ ® 1,2. The explici
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D.R. Lebedev et al. / Gauge fields
expression for such R (z) is
R( z ) = diag(1 ..... exp[2~riBr.,( z)] .... ), flr.,(z) = R e ( - z . r -
(56)
½r.r- r) - r. m",
(57)
where r denotes the two-dimensional row (kl/8~, kJ82) and therefore g = r. z. The gauge fields are
B~ = -~rOz, y ( z ) . lst® l,2-- ~ri[(Imr)-l(Imz)t]i .18 ®182 , B~=~rcg~y(z).l~a®182=rri[(Imr)-t(Imz)t]i.la
®lsz.
(58)
The field strength tensor is
Fz,zj = F~& = 0,
F~,,, = ~r(Im r ) ~ ~. la, ® la~,
(59)
where by definition
F~,~=O=B~-OzBz+ [B~,, B~,]
(60)
etc. The fields (58) correspond to the gauge transformed boundary conditions
V~(z) --- R(z + f,)W~,(z)R -1
(61)
or explicitly 171(2) = Psx ® 18~exp[2rriB(s&,o)-,(z)],
V3(z) --- Qnl ® 182exp{2~rim1) '
Vz(z ) = 181 ® Ps~exp(ZcriB(o,8~),,(z) },
V4(z ) = 1 8 1 ® Q ~ exp{2~rim2).
(62)
Here 1
0
0
o..
1
0 0
~=
0
1
0
exp(2~ri/8)
08
0
0
0
1
0
0
1
...
0
...
exp[2~ri(8- 1)/8] (63)
are matrices which satisfy the commutation relations of the finite Heisenberg group [4, 7]:
P8 Q8 = Q8 Psexp(2~ri/8 ) .
(64)
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151
6. Going from complex to real coordinates Now we rewrite the formulae of sect. 5 in terms of the real coordinates x. The relation of x and z (34) is not the most general. In order to find the proper generalization of eq. (34) let us begin from eq. (59) for the field strength F written in complex coordinates. In that form F is proportional to the following (1,1)-form
o: = ½iY'. (Im r),.J dz i A d~j
(65)
i,j
which takes integer values on the vectors
~i=Siei,
~2+i=Y'~rijej,
i,j=1,2;
e l = ( 1 , O ),
e2=(O,l);
J ~o(~,,~j)=0,
o~(~,,~2+j)=8~6,,j,
~o(~2+i,~2+j)=0,
i,j=1,2.
(66)
That is
where /18 = diag(81, 82). 4 Under an SL(4. Z) change of the basis ~t, - Ewla~v~v, a . . ~ Z and det(a,.) = + 1. the matrix % . transforms as follows ,0j. = ~o(~,. ~'.) = S.~.¢a,~a.~¢. One has the following general assertion [7]. Any nondegenerate skew-symmetric integer matrix n~.. can be transformed to the canonical form (67) by an SL(4. Z) transformation: t
n=a where a ~ SL(4, 7/), 8g ~ Z and n,,, and 81 is a divisor of 82. It is easy to check that
81
(0
_A 8
a t,
(68)
is the greatest common divisor of the numbers
pf(n) = dTCd~n =
8~ "t,,"~B__ log~va~m . -- 8182
"
Any integer skew-symmetric matrix can thus be represented in the following form 2
n~'vN= E i=1
8/-1(a~,i
or,2+ i -- a,,i a~,,2+,) •
(69)
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152
Let us now choose the vectors f~ in eq. (47) as a real basis in 122: 4
2
E u.f~= Eziei, ~=i
zinC,
u~n.
i=1
One can verify that the field strength F in eq. (59) has in coordinates u. the following form F = ~rY'~(Im r)i. 1 dz i A d2jl~, ® 1~2 i,j
= -2¢ri~ 67' du i A du2+il.,
® 162.
(70)
i
Let us pass to the basis f, relation
= E4=,a~,~L with a ~ SL(4,77), and find x, from the
4
4
2
E (x./L.)f/,= E u.f.= E ziei/~=I
E
)
a~,,i+ Y'~ a~,,2+j6j-l"rij (x~,/L.).
~t=l
(71)
i=1
4(2
Then Zi=
/~=1
(72)
j=l
The transformation (71)-(72) makes four vectors f~ in 122 from four vectors l. in ~ 4 with coordinates (l.)~= L.82, that is to say, we obtain a euclidean torus T 4 from the torus which results from C 2 divided by the lattice generated by the vectors f~. Computing F in the coordinates x. with the help of eqs. (69) and (70) we find F=
~ri n~. ~ L - -A~ 1 8d1 ® x1 8 2 .d x .,
,
(73)
where n~JN are given by eq. (69). It follows that the self-duality condition for the fields (58) in terms of x~ is given by the constraint n l~v
1
nab
(74)
This is precisely the condition found by t' Hooft [2]. Now we obtain the expression for the fields (58) in the coordinates x. of eq. (72). Using the relation 2(
Ozi
05i )
D.R. Lebedevet al. / Gaugefields
153
eq. (58) for Bz and B~ and eqs. (71)-(72) we get
Here K , , is a symmetric tensor which can be computed explicitly and which does not depend on x,. Due to the symmetry of K ~ the second term in the r.h.s, of this formula can be removed by the gauge transformation B e ~ VB, V-1 + VO,V-1 with
V ( x ) = exp ~
E K..x~x~ l~l ~ 1~ e U ( N ) . /x, t,'
Thus, finally, rri
n~,~xv
(75)
In order to compute the matrices of gauge transformations corresponding to the boundary conditions let us define an N-component scalar field qo(z) connected with q~(z) by the gauge transformation (56): ~u(z) =R(z).~(z) or, explicitly, ~(Z)
= ( X m ,m ,,( Z ) . . . . . X m , + r , m , , ( z ) . . . . )t.
Then the boundary conditions for the gauge field (75) can be found from the equations cp(z + f d ) = Vd(z)ep(z),
i~= 1 . . . . . 4
and eq. (72). The result is
(76)
The above constructed gauge field has the topological charge q = - 1 as can be easily obtained from eqs. (10) and (73). A simple generalization of the algorithm allows to obtain the higher values of q. Let diag(A (i)) = diag( A O). . . . . A(k)),
(77)
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D.R. Lebedev et aL / Gauge fields
where A (i) are matrices. Then the U(N)-gauge field
{ ~ri
n(i)x
B~ = diag ~ - ~ ~
~"--" 1~ ) ® l~i)
}
(78)
has the topological charge q = - k , N = E~=IN (~) and satisfies the boundary condition
v~k(x)
= daagl P~t~¢ Qs!~] ® P~? Qn~]~exp_ - ~ - ~
As before N o ) = 8(~)~(~)1,,2,n~ ) is related to
8~i), 6~2~) by
~
-, .
(79)
eq. (69), and a ~o ~ SL(4, 7/).
7. Torons for S U ( N ) gauge group
In sect. 3 we have shown how to obtain an S U ( N ) gauge field from the given U ( N ) gauge field, see formulae (15), (16) and (20). But if we use this procedure for the toron field (75) we obtain an SU(N)-field with zero topological charge (or, equivalently, with boundary conditions V~ which do not depend on x). The following modification however leads to nontrivial S U ( N ) fields. Consider a new U ( N ) gauge field, onew=
0
V new =
Op '
(v)
0
0
lp
(80)
'
where the field B and the boundary conditions V' are defined by eqs. (75) and (76). The rank of the gauge group is now
N =p + 8162,
p = 0 ..... N - 1.
(81)
Using eq. (15) for B "ew we obtain the SU(N)-gauge field
A~(x)=diag
~rip N(N-p)~
n~,vx~
qri
L-~I~I®ln2'-N-~
n~,~x~ } L~
lp
'
(82)
which corresponds to the topological charge
Q=-I+
81 62 N
P N'
p = 0 .... , N - 1 .
(83)
D.R. Lebedev et aL / Gauge fields
155
The boundary condition matrix I2s is expressed by eq. (20) through V new, 12s(x ) = diag(
d;1Ps~Q~3®P~,~Q~ ×exp
N(N-p)~ ns~xJL~ ,exp
-~---~-,
lp
, (84)
where d s = det( P(,a Q~,3 ® p{;2 Qg;,,). The matrices I2s satisfy eqs. (21) and (22) with ns, given by eq. (69). If eq. (74) is valid then A s (82) is a self-dual field. In general even if eq. (74) is not valid A s (82) is a solution of the equation of motion
[A.,
(85)
= 0
An anti-self-dual field can be obtained by the change of the orientation of the coordinate system. This can be done for example by the substitution x 1 ---, x 2, X 2 ---> X l .
A more general S U ( N ) toron field can be obtained if we use U ( N ) fields defined by eqs. (78) and (79). In this case the topological charge of the S U ( N ) toron is
Q = -k+
Y'~..
N
,
N (/)=
l,J
~2 ~'~6~'~+p.
(86)
i=1
8. T o r o n s on the lattice
The dynamical variables of the lattice S U ( N ) gauge theory are SU(N)-matrices
exp[
x+~
attached to the links which connect the sites x and x +/2 of the four-dimensional lattice. Here we use the standard notations, /2 is the vector in the direction /~, [/2[ - a, a is the lattice spacing. In the lattice formulae x is the four-vector which defines the coordinates of a site of the lattice. Substituting in eq. (87) the toron field (82) we obtain
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D.R. Lebedev et al. / Gaugefields
This "lattice toron" has the following three important properties. First, the boundary conditions which correspond to the field (88) are nontrivial
Ux+ l~,lt = ~?v( x ) Ox.g ~?; ( x + ~ )
(89)
with the matrices ~2. defined by eq. (84). Second, toron field (88) is an exact solution of the equations of motion:
~ . , ( P x , , ~ - Ux+-~,~Px ~ , ~ , . U x - z , . ) - c . c .
=0,
(90)
P
which correspond to the lattice theory with Wilson's action: S=
~ 1-~trP~..~ x,.,.
.
(91)
The plaquette matrix Px,.~ entering eqs. (90) and (91) is:
Px,~,. = Ux., Ux+~,. Ux++~,~,U£+,~•
(92)
Substituting eq. (88) into (92) we obtain that
Px ",~ = exp diag
N(N-p)
~L~
l pL ~ ln' ® la2, - - -N ~ - ~- Lt,
(93)
commutes with Ux ~,, and therefore (88) is a solution of eqs. (90). Thirdly, it follows from eq. (93) that if and only if the equality (74) is valid the toron field (88) satisfies the lattice analogue of the self-duality equations
Px,..=Px,~.
(94)
Here the operation - is defined by the relations /~x,12 = ex.34,
]~x,13 = ex,42,
/~x.14 = Px.z3,
Z Px,~,. = Px, t,."
(95)
If in eq. (88) we change the orientation of the system of coordinates (e.g. x~ ~ x 2, x 2 ~ Xl) then the lattice anti-self-duality equations are satisfied: P~... =/Sx. ,,.
(96)
Thus as in the continuum limit the lattice (anti)toron fields satisfy (anti)self-duality equations and equations of motion. For the general lattice fields if (anti-)self-duality equations are satisfied the equations of motion (90) may not be valid.
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D.R. Lebedev et al. / Gauge fields
X
X
----
Fig. 3. Graphical proof of the lattice Bianchi identity (97).
Below we suggest the lattice equations of motion which are valid for all solutions of eqs. (94) and (96) at any value of lattice spacing a. Consider the lattice analogue of the Bianchi identity: W~,x~~ Wx,,X. Wx,,, x = 1,
(97)
where there is no summation over repeated indices and (98) The graphical proof of identity (97) is given in fig. 3. In the limit a ~ 0 this identity becomes the usual Bianchi identity. If we replace P~, ~, in eqs. (97) and (98) by the dual quantities (95), i.e. substitute in eq. (97) (99) then the resulting equations admit by construction all the solutions of the self-duality equations (94) and in the continuum limit they become the Yang-Mills equations. Any solution of anti-self-duality equations (96) is also a solution of eqs. (97) and (99). This can be proven by the hermitian conjugation of eqs. (97) and (99).
9. Conclusion and possible applications We have found linear solutions (torons) of the equations of motion of the U ( N ) and SU(N) Yang-Mills theory on the torus with given boundary conditions. It occurs that the torons are closely related to the two-dimensional 0-functions. Due to the linear dependence of torons on the coordinates it is possible to find the lattice analogues of the continuum formulae.
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D.R. Lebedev et al. / Gauge fields
We hope that our remarks may be useful for numerical experiments in lattice gauge theories. In a perfect case numerical experiments have to be performed on the lattice of such a size L x L X L x L, that L is much larger than the correlation length, ~, so that in each domain of size R ( L >> R >> ~) the summation over the topological charges could be performed with the weight defined by the dynamics. For example, in the model of the instanton vacuum [20] the distribution of the instanton charge is gaussian. This distribution was also obtained by numerical simulation of the SU(2) lattice gauge theory [21]. The relation between the boundary conditions and the topological charge discussed in sects. 2 and 3 may be important for numerical experiments at high temperature, T - 1/~. In the continuum the integration of the lagrangian over the time is performed from 0 to/3 = 1/T. The functional integral is calculated under the condition that the physical quantities are periodic in time, for example t r F Z ( x , 0 ) = tr F ~2 ( x , fl).
(100)
This statement follows from the standard expression for the statistical sum, Z = tr e-#n, where the trace is taken over the physical states. In ref. [22] the "periodic instanton" (a periodic in the euclidean time solution of the equation of motion carrying the unit topological charge) has been found. The "periodic instanton" has a singularity in the center which is responsible for the topological charge. So the coexistence of the trivial boundary conditions and the unit topological charge is due to the singular gauge. If we consider nonsingular fields, then the summation over topological classes is equivalent to the summation over the classes of boundary conditions. Therefore at a finite temperature the boundary conditions for topologically nontrivial nonsingular fields are A.(x,O)
=
(101)
Here Aft = 12A~12-1+ 123~I2-1 and the boundary condition (100) is obviously satisfied. In the computer simulations of lattice gauge theories instead of eq. (101) the trivial boundary condition A~(x,0)=Aj,(x, fl) is usually used. Due to the discretization some traces of singular fields may be left on the lattice. One can employ certain numerical experiments in order to understand whether the summation over the classes of boundary conditions is important for lattice gauge theories. All the numerical calculations are performed (in most cases) with trivial boundary conditions which correspond to Q - - 0 in the continuum limit*. So a nonzero value of ( Q 2 ) is a consequence of lattice discretization. By the freezing of the quantum fluctuations it is possible to obtain the one-instanton configuration on the lattice * The Monte-Carlo simulationsof the SU(3) lattice gauge theory were performedby the authors of ref. [231.
D.R. Lebedet; et al. / Gauge fields
159
[21, 24], but the continuous field which corresponds to that configuration consists of one instanton with the radius of order of the correlation length and of one anti-instanton with the radius of order of the lattice spacing. Therefore the total topological charge is equal to zero due to the existence of a dislocation. Finally, we discuss the method of summation over the boundary conditions in the S U ( N ) lattice gauge theory. The enumeration of different n,~ is trivial ( n ~ is equal to n~, if for all #, p n ~ = n',, modulo N). The topological charge is expressed through pf(n), see eq. (83). If eq. (74) is valid then the lattice toron field (88) is self-dual and corresponds (at least in the continuum limit) to the minimum of the action. For the summation over the integer topological charges which correspond to multi-instanton configurations we must find the matrices ~2~(x) which satisfy the selfconsistency equation (21) with the trivial twist (Zu~ = 1). For the SU(2) gauge group such matrices corresponding to an integer charge Q are
~'~p.= g( xp. = t~) g- l( xls.= O) , g=
xI Z x0~
Q,
% = (1, iff),
(102)
(103)
where o k are the Pauli matrices. The variation of x 0 in eqs. (102) and (103) is equivalent to the gauge transformation (8) of the boundary condition (102), (103). A gauge field which corresponds to this boundary condition can be chosen in the following form:
A~,(x) = f ( x ) ( O ~ , g ( x ) ) g ( x ) - 1 ,
(104)
where f ( x ) is a function such that f ( x ) = 1 on the boundary and f ( x ) ~ 0 when x--* x 0 in such a way that makes A t nonsingular at x o. Analogously the lattice gauge field has the form
Vx,t~ = f ( x ) g ( x ) g - l ( x
+ ~).
(105)
We are planning to perform numerical experiments in order to find how the nontrivial boundary conditions influence the quantum lattice fields. Some interesting results for the quantum gauge theory on the torus were obtained recently in ref. [25]. The authors are grateful to A.Yu. Morozov, O.V. Ogievetsky and Yu.A. Simonov for useful remarks and valuable discussions, and to van Baal for sending the reprints of articles [11] and [25]. It is a pleasure to thank A. Di Giacomo and G. Paffuti for helpful conversations. One of us (M.I.P.) is grateful to the theory group of Physics department of Pisa University for the kind hospitality.
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Appendix The following properties of the O-functions are important in our considerations. (1) For any n', n" ~ 7/× Z
Om, = exp(-~rin'rn'-
2~rin'z)exp[2~ri(m'n"- n'm")] On,,.,,,(~, z).
(A.1)
(2) For any u', u" ~ R2
=exp{Z~ri[-½u"ru'-u'(z + u"
+m")]}
Om,+u,m,,+u,,('r,Z ).
(A.2)
(3) For any X'E ~=IZX/, X " ~ ez=xzX2+i, 0m,+x,,,~,,+x,,(% z) = exp(2~rim'?~") 0,,, ,~,,(~-, z).
(A.3)
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[25]
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