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Equivalence of the Z4 and the Z2 × Z2 lattice gauge theories
Volume 98B, number 1,2
PHYSICS LETTERS
1 January 1981
EQUIVALENCE OF THE Z 4 AND THE Z 2 × Z 2 LATTICE GAUGE THEORIES H. GROSSE x
Institut fiir Theoretische Physik, Universitdt Wien, Vienna, Austria C.B. LANG 2 and H. NICOLAI
CERN, Geneva, Switzerland Received 21 October 1980
We prove the equivalence of the Z4 and Z2 × Z2 lattice gauge theories in any dimension and discuss some of the consequences of this result.
The lattice formulation of gauge theories [1 ] has provided an elegant way to introduce a cut-off and to discuss confinement [1,2]. Abelian ZN gauge theories are of interest for two reasons. On one hand, although they have a twoor three-phase structure, they are simple enough to be used to develop techniques that may be useful for the further investigation of U(1) or SU(N) gauge theories [3 ]. On the other hand, there is a dose connection between the gauge theory of SU(N) and of its centre ZN [4] and a better understanding of the abelian theory may be important for further progress in the non-abelian SU(N) theory. It is known that in two-dimensional spin systems the Z 4 model decouples into two independent Z 2 models [5]. The corresponding gauge models may be explicitly solved in two dimensions where they exhibit the same relationship. There are indications that this equivalence carries over to four dimensions. Both theories are self-dual (Korthals Altes, Yoneya [4], Drouffe [6]) with the critical coupling of the Z 4 theory being twice that of the Z 2 theory. Monte Carlo results [7] suggest that both theories have first order phase transitions; duality arguments confirm this suggestion for the Z 2 theory but leave the Z 4 transition type undecided [6]. Recent strong coupling expansions for the string tension [8] show an equality up to 12th order for d = 3 and 4. It is the aim of this paper to prove that the Z 4 theory with coupling constant/~ is fully equivalent to a Z 2 × Z 2 theory with coupling constant/3/2 in any dimension. All Z 4 quantities (free energy, string tension, etc.) may be expressed through the corresponding Z 2 quantities and the singularity structures of both theories are simply related to each other. The partition function of a ZN gauge theory with coupling/3 on a finite lattice with V points is [2] Z(ZN'/3' V) = f ~ l
dUl I1 exp[[3 Re x~N)(Up)] ,
(1)
P where the product is over all links and plaquettes. The characters of ZN are given by
X(pN)(e2rrin/N) = e 2~ripn/N
(2)
and the Haar measure is defined by
f ZN
F(U) dU =-1 ~
F(~).
(3)
~v ~-EZN
I Work supported in part by "Fonds zur F6rderung der wissenschaflichen Forschung in Osterreich", project number 3659. 2 On leave of absence from Institut fiir Theoretische Physik, Universit//t Graz, Austria. 0 03"1-9163/81/0000-0000/$ 02.50 © North-Holland Publishing Company
69
Volume 98B, number 1,2
PHYSICS LETTERS
1 January 1981
The plaquette variable Up is the oriented product of its boundary link variables; reversing the orientation of a link is the same as taking its complex conjugate. Specializing to Z 4, we make use of the character expansion [2] to rewrite the exponential of the Z 4 action as follows exp[/3 Re X1 (Up)] = cosh2 ½/3{1 + tanh
1/3[xI(Up) + x3(Up) ] + tanh2 ~13x2(Up)} .
(4)
For later convenience, we also define
XA(U)-xI(U)
+ x3(U) = U+ U + ,
XB(U)=--x2(U)= V 2 = U +2
,
(5a,b)
so the Z 4 partition function becomes .1 ~2:(Z4,/3, V) = (cosh
1~) vd(d-x)
f zv
(dUl} 11
[1 + tXA(Up)
+ t2XB(Up)] .
(6)
p
Expanding the product and exploiting the orthogonality relations between ×A and ×B, one obtains a sum of terms, each of which may be associated with a two dimensional polyhedral complex [3]. Each such complex may be divided into regular components, i.e., two dimensional surfaces whose boundary is homeomorphic to a circle. Conversely, any complex may be constructed by joining regular components along common parts of their boundaries. In d dimensions, the number of surfaces meeting at any common piece of their boundaries cannot exceed 2 ( d - 1). In the case of the Z 4 theory, each regular component carries either the index A or B. Surfaces of type A are orientable, whereas those of type B may not be due to the self-adjointness [cf. eq. (5b); e.g., XA(ABC) ¢ XA(AB+C) whereas XB(ABC) = XB(AB+C)]. It is easy to verify that only those complexes where an even number of type A surfaces are joined to an arbitrary number of type B surfaces at the intersection of their boundaries can possibly contribute to the Z 4 partition function (6). Each type A surface, consisting o f P plaquettes, is accompanied by a factor tP; for type B, the factor is t 2P. Due to the local orientability of type A surfaces, two possibilities may arise. In the first case, the polyhedral complex may be built by joining together closed orientable type A surfaces which are homeomorphic to spheres with any number of handles cut along branch lines; an arbitrary number of type B surfaces may be added. Then, all regular components of type A may be oriented so that at any boundary piece half of them are directed in one direction and the other half in the opposite direction. One way to see this is that progressively more complicated complexes may be constructed by breaking up "old" branching lines and attaching "new" ones; in this process, the required orientation is always preserved (see fig. 1). In the second case, the complex is built up from some orientable and some non-orientable surfaces of type A (e.g., Klein bottles) closed up to an integral along common branching lines. Then the non-orientable surfaces contribute along their boundary line factors like XA(UU) which may be replaced by XB(U). Stated otherwise, the nonorientable type A contribution may be treated like a type B surface with the same boundary. These two observations are of crucial importance for our equivalence proof. The general expression for an integral over a boundary piece U bordering on a type A surfaces and ~ type B surfaces, which is encountered in the course of evaluating (6) reads #1 t ~ tanh/3/2 throughout this paper.
t _
Fig. 1. 70
Volume 98B, number 1,2
PHYSICS LETTERS
1 January 1981
f/~1"= XA(URi)/[-]I'=XB(US/)dU,
(7a)
Z4 where, by R i and S/, we denote the pieces of boundary connected to U. For Z4, the result of the integration is *:" (7a) = 0 ,
if a is odd,
a/2 I xB(si) lq r k []1"=
=
permutations k=l
+
l
+n) (mod 2)] ,
otherwise.
(7b)
The sum includes all 2 a/2 permutations where Yk is either XA(R2k_IR2k) or XA(R2k_IR2k); + n is the number of times a "twisted character" of type XA(RR +) occurs in the product. The Z 2 gauge theory has only one non-trivial representation, which is self-adjoint. An element U of Z 2 X Z 2 is represented as (u, 5). For the Z 2 X Z 2 theory, the character expansion of the exponentiated action with coupling ~/3 is 1
•
exp[½/~(u + 5)] = cosh2 ½fl[1 + t(u + 5) + t2ua].
(8)
In complete analogy with (3) and (4) we define ~A(5r) = u + ti,
~B(U) ~- uti
(9a,b)
and the partition function is Z ( Z 2 X Z 2, ½e, V) = (cosh ½13)Vd(d-1)
f (Z2 x Z2)v
{dO,}
N
[1 + t~A(Up) + t2,yB(Up) ]
(10)
P
with obvious notation. In particular, ~A(~rp) = )~A(U1U2U3U4) = Up + ~tp = UmU2U3U4 + ~1~2~3~ 4 .
(11)
In two dimensions, ~ is already given by the factor in front of the integral and Z 4 and Z 2 X Z 2 are easily seen to concide. In d > 2, we again expand the product over the plaquettes in (10). As in the case of Z4, the contributions correspond to closed two dimensional polyhedral complexes with orientable and nonofientable type A and B surfaces; to each complex in Z 4 there corresponds exactly one in Z 2 X Z 2 . The operation which corresponds to complex conjugation in Z 4 is the interchange of u and fi and we accordingly define 0 + = (~, u) if ~r= (u, 5). The "twisted Z 2 X Z 2 characters" are XA(~r~"+) = UV + fro = X A ( U + F ) ,
XB(UV +) = uffg'o = ~B(Sr~').
(12)
Observe that ~'A(b'Lr+) = 2~B(~') and ~A(U5r) = 2 whereas exactly the opposite relations hold in the Z 4 theory! This implies that a non-orientable closed surface of type A, cut along branching lines with other surfaces, now contributes a factor like ~A(5@ +) which in turn equals ~B(~r) in Z 2 X Z 2 . The expression which is analogous to (7a) acquires the form a /3 Z2XZ 2 i---1
if ct is odd,
=0,
a/2
=f
S
/3
[-I fk/~=l~B(S/)8[(fJ+la+n')(mod2)],
permutations k = 1
otherwise.
(13)
*2 IIO=l Yk =- 1; 5(0) = 1 and 6(1) = 0.
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Volume 98B, number 1,2
PHYSICS LETTERS
1 January 1981
Again, the sum runs over all 2a/2 permutations where Yk is either XA(/~2k_L/~2k) or XA(/~2k_l/~k) but, in contradistinction to Z4, n* is the number of times an "untwisted character" ~(RR) occurs in the product. For the proof of our main result, we need the following identities (~/> 4) a/2 Z4: f
kI~--1[XA(UR2k-1)XA(U+R2k)] /~1XB(US])dU a/2
=f
~ x~(U1)k~=I [XA(UkR2k-1) rk dUkl]IqqxB(S])5[(½a+n)(m°d2)] , permutations = = a/2
Z2XZ2: f
(14a)
3
k~I=l [XA(fffR2k-1)~(A(ff-fR2k)]]~=l~(B(fffSf)dfff a/2
=;--permutationsE x~(U1) k__ISI 1=
3
[~(A(~fkR2k_l)Yk dUkljI-I ~(a(S])8[(½a+n*)(mod2)l ,
(14b)
+
where Yk equals either XA(UkR2k) or XA(U~R2k) (similarly for Yk); otherwise, the notation is the same as before. Note that the orientation of the Z 4 complex has been chosen in accordance with our previous observation. The relations (14) increase the number of integration variables by (a/2) - 1 at each junction. By repeated application of (14a,b) we may now reformulate all integrals of the form (Ta) and (13) so that each variable occurs twice, at most, in a type A character. The "old" variables appear only in the "twisted" combination UU+ in Z4 and in the "untwisted" combination UU in Z 2 × Z 2 while all "new" variables occur in both ways. From this formulation, one immediately recognizes the one-to-one correspondence between Z 4 and Z 2 × Z2: for each expression in Z 4 with k I "twisted" and k 2 "untwisted" combinations there exists one and only one expression in Z 2 × Z 2 with k 1 "untwisted" and k 2 "twisted" combinations of the corresponding boundary variables. The generic form of the sub-integrals is then f
XA(UX) ×A(U+Y) ×~(U) dU= XA(XY),
Z4 f
13even,
= XA(XY+), XA(U2) XA(~'Y) x~(U) dU = XA(XY) ,
Z2×Z2
(15a)
13odd ,
= XA(-~Y+),
13even, 3odd.
(15b)
The equivalence remains unaffected by the integrations since the relative twists within XY (XY) undergo the same changes in both theories. Therefore, both theories agree term by term when expanded on a finite lattice and we arrive at our main result 1
1
~(Z4, 13, V) = ~ (Z 2 X Z2, ~13, V) = ~2(Z 2, ~13, V) .
(16)
Being true for any finite V, this result remains valid in the thermodynamic limit V-+ oo. For the free energy
F(ZN, 13): vlim l l o g
Z(ZN,13, V),
(17)
we obtain from (16) F(Z4,/3) = 2F(Z2,½13).
(18)
From (18), we rederive the well-known relation between the critical couplings (at the self-dual point) 13cr(Z4) = 213or(Z2) = log(a + x/~). 72
(19)
Volume 98B, number 1,2
PHYSICS LETTERS
1 January 1981
Furthermore, the order o f the phase transition is the same in both theories, which settles one o f the questions cited in the introduction. Finally, one may, with little effort, derive from (16) the following relation for the Wilson-loop expectation values [1 ] Wc(Z4,/3) = Wc(Z2, ~/3),
(20)
where
Wc(ZN, I3)=--(Re XI(l~c Ul))ZN,~ .
(21)
This explains the result of ref. [8]. There is strong evidence for a roughening transition in the string tension [8,9]; again, the critical couplings in Z 4 and Z 2 must obey a relation like (19). We are grateful to W. Nahm for pointing out to us a gap in our original p r o o f and to P. Hasenfratz, B. Lautrup and M. Nauenberg for numerous discussions. H.G. thanks the CERN Theory Division for its hospitality.
References [1] K.G. Wilson, Phys. Rev. D10 (1974) 2445. [2] K. Osterwalder and E. Seiler, Ann. Phys. 110 (1978) 440. [3] R. Balian, J.M. Drouffe and C. Itzykson, Phys. Rev. D10 (1974) 3376; D11 (1975) 2098, 2104; D19 (1979) 2514 Erratum; J.M. Drouffe and C. Itzykson, Phys. Rep. 38C (1978) 133; J.B. Kogut, Rev. Mod. Phys. 51 (1979) 659. [4] G. 't Hooft, Nucl. Phys. B138 (1978) 1; C.P. Korthals Altes, Nucl. Phys. B143 (1978) 315; T. Yoneya, Nucl. Phys. B144 (1978) 195; G. Mack, Commun. Math. Phys. 65 (1979) 91; G. Mack and V.B. Petcova, Ann. Phys. 125 (1980) 117, J. FrtShlich, Phys. Lett. 83B (1979) 195. [5] M. Suzuki, Prog. Theor. Phys. 37 (1967) 770. [61 J.M. Drouffe, Nucl. Phys. B170 (FS1) (1980) 91. [7] M. Creutz, L. Jacobs and C. Rebbi, Phys. Rev. D20 (1979) 1915; G. Bhanot and M. Creutz, Phys. Rev. D21 (1980) 2892. [8] G. Mtinster and P. Weisz, DESY preprint 80/74 (June 1980). [9] A. Hasenfratz, E. Hasenfratz and P. Hasenfratz, CERN preprint TH. 2890 (June 1980); C. Itzykson, M. Peskin and J.B. Zuber, Phys. Lett. 95B (1980) 259; M. Liischer, G. MOnster and P. Weisz, DESY preprint 80/63 (July 1980).
73
PHYSICS LETTERS
1 January 1981
EQUIVALENCE OF THE Z 4 AND THE Z 2 × Z 2 LATTICE GAUGE THEORIES H. GROSSE x
Institut fiir Theoretische Physik, Universitdt Wien, Vienna, Austria C.B. LANG 2 and H. NICOLAI
CERN, Geneva, Switzerland Received 21 October 1980
We prove the equivalence of the Z4 and Z2 × Z2 lattice gauge theories in any dimension and discuss some of the consequences of this result.
The lattice formulation of gauge theories [1 ] has provided an elegant way to introduce a cut-off and to discuss confinement [1,2]. Abelian ZN gauge theories are of interest for two reasons. On one hand, although they have a twoor three-phase structure, they are simple enough to be used to develop techniques that may be useful for the further investigation of U(1) or SU(N) gauge theories [3 ]. On the other hand, there is a dose connection between the gauge theory of SU(N) and of its centre ZN [4] and a better understanding of the abelian theory may be important for further progress in the non-abelian SU(N) theory. It is known that in two-dimensional spin systems the Z 4 model decouples into two independent Z 2 models [5]. The corresponding gauge models may be explicitly solved in two dimensions where they exhibit the same relationship. There are indications that this equivalence carries over to four dimensions. Both theories are self-dual (Korthals Altes, Yoneya [4], Drouffe [6]) with the critical coupling of the Z 4 theory being twice that of the Z 2 theory. Monte Carlo results [7] suggest that both theories have first order phase transitions; duality arguments confirm this suggestion for the Z 2 theory but leave the Z 4 transition type undecided [6]. Recent strong coupling expansions for the string tension [8] show an equality up to 12th order for d = 3 and 4. It is the aim of this paper to prove that the Z 4 theory with coupling constant/~ is fully equivalent to a Z 2 × Z 2 theory with coupling constant/3/2 in any dimension. All Z 4 quantities (free energy, string tension, etc.) may be expressed through the corresponding Z 2 quantities and the singularity structures of both theories are simply related to each other. The partition function of a ZN gauge theory with coupling/3 on a finite lattice with V points is [2] Z(ZN'/3' V) = f ~ l
dUl I1 exp[[3 Re x~N)(Up)] ,
(1)
P where the product is over all links and plaquettes. The characters of ZN are given by
X(pN)(e2rrin/N) = e 2~ripn/N
(2)
and the Haar measure is defined by
f ZN
F(U) dU =-1 ~
F(~).
(3)
~v ~-EZN
I Work supported in part by "Fonds zur F6rderung der wissenschaflichen Forschung in Osterreich", project number 3659. 2 On leave of absence from Institut fiir Theoretische Physik, Universit//t Graz, Austria. 0 03"1-9163/81/0000-0000/$ 02.50 © North-Holland Publishing Company
69
Volume 98B, number 1,2
PHYSICS LETTERS
1 January 1981
The plaquette variable Up is the oriented product of its boundary link variables; reversing the orientation of a link is the same as taking its complex conjugate. Specializing to Z 4, we make use of the character expansion [2] to rewrite the exponential of the Z 4 action as follows exp[/3 Re X1 (Up)] = cosh2 ½/3{1 + tanh
1/3[xI(Up) + x3(Up) ] + tanh2 ~13x2(Up)} .
(4)
For later convenience, we also define
XA(U)-xI(U)
+ x3(U) = U+ U + ,
XB(U)=--x2(U)= V 2 = U +2
,
(5a,b)
so the Z 4 partition function becomes .1 ~2:(Z4,/3, V) = (cosh
1~) vd(d-x)
f zv
(dUl} 11
[1 + tXA(Up)
+ t2XB(Up)] .
(6)
p
Expanding the product and exploiting the orthogonality relations between ×A and ×B, one obtains a sum of terms, each of which may be associated with a two dimensional polyhedral complex [3]. Each such complex may be divided into regular components, i.e., two dimensional surfaces whose boundary is homeomorphic to a circle. Conversely, any complex may be constructed by joining regular components along common parts of their boundaries. In d dimensions, the number of surfaces meeting at any common piece of their boundaries cannot exceed 2 ( d - 1). In the case of the Z 4 theory, each regular component carries either the index A or B. Surfaces of type A are orientable, whereas those of type B may not be due to the self-adjointness [cf. eq. (5b); e.g., XA(ABC) ¢ XA(AB+C) whereas XB(ABC) = XB(AB+C)]. It is easy to verify that only those complexes where an even number of type A surfaces are joined to an arbitrary number of type B surfaces at the intersection of their boundaries can possibly contribute to the Z 4 partition function (6). Each type A surface, consisting o f P plaquettes, is accompanied by a factor tP; for type B, the factor is t 2P. Due to the local orientability of type A surfaces, two possibilities may arise. In the first case, the polyhedral complex may be built by joining together closed orientable type A surfaces which are homeomorphic to spheres with any number of handles cut along branch lines; an arbitrary number of type B surfaces may be added. Then, all regular components of type A may be oriented so that at any boundary piece half of them are directed in one direction and the other half in the opposite direction. One way to see this is that progressively more complicated complexes may be constructed by breaking up "old" branching lines and attaching "new" ones; in this process, the required orientation is always preserved (see fig. 1). In the second case, the complex is built up from some orientable and some non-orientable surfaces of type A (e.g., Klein bottles) closed up to an integral along common branching lines. Then the non-orientable surfaces contribute along their boundary line factors like XA(UU) which may be replaced by XB(U). Stated otherwise, the nonorientable type A contribution may be treated like a type B surface with the same boundary. These two observations are of crucial importance for our equivalence proof. The general expression for an integral over a boundary piece U bordering on a type A surfaces and ~ type B surfaces, which is encountered in the course of evaluating (6) reads #1 t ~ tanh/3/2 throughout this paper.
t _
Fig. 1. 70
Volume 98B, number 1,2
PHYSICS LETTERS
1 January 1981
f/~1"= XA(URi)/[-]I'=XB(US/)dU,
(7a)
Z4 where, by R i and S/, we denote the pieces of boundary connected to U. For Z4, the result of the integration is *:" (7a) = 0 ,
if a is odd,
a/2 I xB(si) lq r k []1"=
=
permutations k=l
+
l
+n) (mod 2)] ,
otherwise.
(7b)
The sum includes all 2 a/2 permutations where Yk is either XA(R2k_IR2k) or XA(R2k_IR2k); + n is the number of times a "twisted character" of type XA(RR +) occurs in the product. The Z 2 gauge theory has only one non-trivial representation, which is self-adjoint. An element U of Z 2 X Z 2 is represented as (u, 5). For the Z 2 X Z 2 theory, the character expansion of the exponentiated action with coupling ~/3 is 1
•
exp[½/~(u + 5)] = cosh2 ½fl[1 + t(u + 5) + t2ua].
(8)
In complete analogy with (3) and (4) we define ~A(5r) = u + ti,
~B(U) ~- uti
(9a,b)
and the partition function is Z ( Z 2 X Z 2, ½e, V) = (cosh ½13)Vd(d-1)
f (Z2 x Z2)v
{dO,}
N
[1 + t~A(Up) + t2,yB(Up) ]
(10)
P
with obvious notation. In particular, ~A(~rp) = )~A(U1U2U3U4) = Up + ~tp = UmU2U3U4 + ~1~2~3~ 4 .
(11)
In two dimensions, ~ is already given by the factor in front of the integral and Z 4 and Z 2 X Z 2 are easily seen to concide. In d > 2, we again expand the product over the plaquettes in (10). As in the case of Z4, the contributions correspond to closed two dimensional polyhedral complexes with orientable and nonofientable type A and B surfaces; to each complex in Z 4 there corresponds exactly one in Z 2 X Z 2 . The operation which corresponds to complex conjugation in Z 4 is the interchange of u and fi and we accordingly define 0 + = (~, u) if ~r= (u, 5). The "twisted Z 2 X Z 2 characters" are XA(~r~"+) = UV + fro = X A ( U + F ) ,
XB(UV +) = uffg'o = ~B(Sr~').
(12)
Observe that ~'A(b'Lr+) = 2~B(~') and ~A(U5r) = 2 whereas exactly the opposite relations hold in the Z 4 theory! This implies that a non-orientable closed surface of type A, cut along branching lines with other surfaces, now contributes a factor like ~A(5@ +) which in turn equals ~B(~r) in Z 2 X Z 2 . The expression which is analogous to (7a) acquires the form a /3 Z2XZ 2 i---1
if ct is odd,
=0,
a/2
=f
S
/3
[-I fk/~=l~B(S/)8[(fJ+la+n')(mod2)],
permutations k = 1
otherwise.
(13)
*2 IIO=l Yk =- 1; 5(0) = 1 and 6(1) = 0.
71
Volume 98B, number 1,2
PHYSICS LETTERS
1 January 1981
Again, the sum runs over all 2a/2 permutations where Yk is either XA(/~2k_L/~2k) or XA(/~2k_l/~k) but, in contradistinction to Z4, n* is the number of times an "untwisted character" ~(RR) occurs in the product. For the proof of our main result, we need the following identities (~/> 4) a/2 Z4: f
kI~--1[XA(UR2k-1)XA(U+R2k)] /~1XB(US])dU a/2
=f
~ x~(U1)k~=I [XA(UkR2k-1) rk dUkl]IqqxB(S])5[(½a+n)(m°d2)] , permutations = = a/2
Z2XZ2: f
(14a)
3
k~I=l [XA(fffR2k-1)~(A(ff-fR2k)]]~=l~(B(fffSf)dfff a/2
=;--permutationsE x~(U1) k__ISI 1=
3
[~(A(~fkR2k_l)Yk dUkljI-I ~(a(S])8[(½a+n*)(mod2)l ,
(14b)
+
where Yk equals either XA(UkR2k) or XA(U~R2k) (similarly for Yk); otherwise, the notation is the same as before. Note that the orientation of the Z 4 complex has been chosen in accordance with our previous observation. The relations (14) increase the number of integration variables by (a/2) - 1 at each junction. By repeated application of (14a,b) we may now reformulate all integrals of the form (Ta) and (13) so that each variable occurs twice, at most, in a type A character. The "old" variables appear only in the "twisted" combination UU+ in Z4 and in the "untwisted" combination UU in Z 2 × Z 2 while all "new" variables occur in both ways. From this formulation, one immediately recognizes the one-to-one correspondence between Z 4 and Z 2 × Z2: for each expression in Z 4 with k I "twisted" and k 2 "untwisted" combinations there exists one and only one expression in Z 2 × Z 2 with k 1 "untwisted" and k 2 "twisted" combinations of the corresponding boundary variables. The generic form of the sub-integrals is then f
XA(UX) ×A(U+Y) ×~(U) dU= XA(XY),
Z4 f
13even,
= XA(XY+), XA(U2) XA(~'Y) x~(U) dU = XA(XY) ,
Z2×Z2
(15a)
13odd ,
= XA(-~Y+),
13even, 3odd.
(15b)
The equivalence remains unaffected by the integrations since the relative twists within XY (XY) undergo the same changes in both theories. Therefore, both theories agree term by term when expanded on a finite lattice and we arrive at our main result 1
1
~(Z4, 13, V) = ~ (Z 2 X Z2, ~13, V) = ~2(Z 2, ~13, V) .
(16)
Being true for any finite V, this result remains valid in the thermodynamic limit V-+ oo. For the free energy
F(ZN, 13): vlim l l o g
Z(ZN,13, V),
(17)
we obtain from (16) F(Z4,/3) = 2F(Z2,½13).
(18)
From (18), we rederive the well-known relation between the critical couplings (at the self-dual point) 13cr(Z4) = 213or(Z2) = log(a + x/~). 72
(19)
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PHYSICS LETTERS
1 January 1981
Furthermore, the order o f the phase transition is the same in both theories, which settles one o f the questions cited in the introduction. Finally, one may, with little effort, derive from (16) the following relation for the Wilson-loop expectation values [1 ] Wc(Z4,/3) = Wc(Z2, ~/3),
(20)
where
Wc(ZN, I3)=--(Re XI(l~c Ul))ZN,~ .
(21)
This explains the result of ref. [8]. There is strong evidence for a roughening transition in the string tension [8,9]; again, the critical couplings in Z 4 and Z 2 must obey a relation like (19). We are grateful to W. Nahm for pointing out to us a gap in our original p r o o f and to P. Hasenfratz, B. Lautrup and M. Nauenberg for numerous discussions. H.G. thanks the CERN Theory Division for its hospitality.
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