robert oeckl and hendryk pfeiffer- the dual of pure non-abelian lattice gauge theory as a spin foam...

8/3/2019 Robert Oeckl and Hendryk Pfeiffer- The dual of pure non-Abelian lattice gauge theory as a spin foam model

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arXiv:hep-thl0008095 v2 15 Jan 2001

~ : ; : : : '" t J : I ~ " '" ;;: : : : ; : (l) (l)* CL tr: '-I l@P_, S ~ I e+~ G.: l > o S. '" ~ "0 e+ '" , . . , , . . , : : : :, ~ ' " I I- CJq '" (t)'" S 0 w e+ >IV '" e+ '" M-~ S 0+ S '" en V '" n C = ; ' ~ I= Do 0+ '" (YS '""" o '" :: : i= O (X) '< ~ en ,. , V e+ (1) (1) >---,""""d ' " CJq CJ q CJq > en ~. '" '< ~ tv r- ~ , . . , (t)"d n '" '" (l)e+ ,. ,

'"b 0n ~ 8 "

,. , : ; : : : (1) '" (1),. , ~. . . . . , C D ' I - = i n ~ M-S (l) ,. , (l) 0+ S (1) (1) e+ P"' e+ ~CJq 0+ ~ ~ C f C l "0 '" C f C l 0S , ., (l) c : ; . '" ~ ,. , S ,. , (1) o, >I l ? :Js ~ (l) f- S ~ ~ ~ "D0+ 0 i= O (1) 0 o, i= O M- 0~ 0+ ~ en '" i= O b-" ~ : : : :: ~ CL ,. , C f C l 5 : (1) i=O ~ v(l) s en (1) "0 "0 (t)i: p;- (l) i= O ~ f- e+ '" en en ~ "V (t) I= D t-10 CJ q e : . . . u- < '" e+ e+ e+ 0 a e+ ~ "V , . . . , IJ);- . . . . , '" e+ '< (1)tr: H-, : : : : : ; : '" ~ ~. P"' e+ '" P"' e+ >I l . . . . _ c-t-s 0+ 0 en (1) o '< e+ . . . . . .0+ ~ '-< e+ 0 "0 0 (1) '" ,. , M- (t) 0 I= D ~Jq ~ '" (l) C = ; ' ,. , (1) '" en e+ '" . . . . . . 0...l) I '< ,. , 0 "0 ,. , n (t) 0l) 0 > - ~ 0' ~e n "0 '" e+ '" '" f- >I l IJ)S tr: H-, e+ en ,. , P"' '" en u- < . . . . _ ~ n "D ~r: 0+ 0" 0 ,. , s ~. (1) en 0' : > ; - "0 '" ,. , 0+ ~ S S ~ 0' '--' Cf) >I l , _ _ . . . . . . . . . . IS 0 ~ e+ e+ en e+ C f C l ,. , > * ~ > -L (l) ~. (1) P"' (1) (1) "0 o '" ,. , S >I l n M-~ (l) '" n (1) ~ ,. , s i= O S 0- ~ . . . . . . b-" >I lCJ q -< '" P"' e+ '" C f C l '" > = (t) (t) ~ V''" :: s :: s '" e+ en b 0''" I - = i '" 8' (1) o , . . , . _ >I l ~ c,~ ,. , n n '" (1) s (1) e+ .. , n (1)0 i= O CL C = ; ' '" P"' e+ '" C f C l o '" ~ , . . . , (t) >I l ~ I= D >---'" i= O i= O en i';"' (1) (1) :: s n ,. , '" '< ~tJ : l . . . . . . .i= O ,. , 0 0' (") M- S" ~ s e : . . . n e+ en 0 e+ i= O , . . , . _ ~ . . . . . . (t) I= Di - - = J 0 :: s ,. , ,. , o n ,. , n ~0+ s : : : : : ; : P"' 0 '" '" i';"' "0 '" 0 C J t J : l ~~ ~ ~ 0 i= O '" ' " e n '" '" "0 0 >I l c, Sil CJ q ~ '" >I l , . . . ,(l) en en p_, ~ '" i= O f- ~ >---,. , '" (1) s 0' e+ en '" V ,. , b '< I= D+ ,. , tr: n 0 o, (1) (1) 0"" 0... : > ; - " 0., tr: (l) : > . ; "0 n C f C l ,. , P"' :: s L""t-' " '-< CJ q : ; : : : s 0 s '" (1) 0 '" '" , . . , '"d 0 . . . . L""t-~ ~ '" "0 . . . . . . ~' " tr: s 8 '" ,. , '" ,. , 0 ~ . . . . . . .(1) '" "0 ~ '" ct ' (1)tr: 0+ '" ~ C f C l e+ M n8 " (l) C f C l o n ,. , '" C D ' I >---(l) (l) s ~ (1) i= O > (t) (t) ~ (1)S (l) 0+ i= O '" (1) en en 0., 0 .0 : ; : : : ,. , ,. , en Q. , p_ , VS i= O '" (1) S en o ~ 0 . . : ~ 6 3 , . . , (Jq: ; : : : H-, 0 ~ P"' ~ (t) , . . . ,'" 0 '" (1) 0 M- -+' " '" 0 ,. , o H-, '" 0 (1) ~. I= D+ 0+ " '" "0 H-, en '" ~ . . . . . .lfJ '" o '"

~ (1) C f C l '" n : : : : (l) en P"' en a s ~ '" '" >I l' " "0 '" lfJ (1) ~ -< "0 en i=O C f C l a . . . . _ (Jq,. , ~lfJ "0 '" P"' (1) C f C l '" ~0 ~ s ,. , p_ , (1) . . . . , 0 n "0 (1) i= O "U (1): ; : : : 0" (l) (1) i= O '" < " ~ '" C f C l b-"' " 0: 0+ ~ en '" n ~ L""t-'" ~ en ~ s (1) P"' (1) ~fJ 8" "0 ,. , "0 ~ I p--" (l) 0 '" C f C l o ~ (1) ~CJq ~ 0+ '" s s '" ,. , s 0 P"' ~ . . . . . .(l) ~ ~ S 0 C f C l ~ 0 '" ,. , (1) ~ n (1) U' " (l) , ., 0' p_ , C = ; ' '" -< (1) '< 0 ~tJ : l 0 >+ 0 lfJ n ,. ,(l) ~ ,. , 0+ '" ~ 0 i= O '" ~ ~. '< t-1 ~., (l) (l) ~ S i= O ~ ~ 2 " '- . ; en I - = i ~ , . , S. (1) ~ P"' (1)t : : j . til en C = ; ' "d'" ~ s ~ C f C l(l) ,. , S 0 0 ~ (1) 0 (1) '" i= O ICL '" lfJ '" '" en C f C l I H-, en '" I tvS '-< "0 C0+ lfJ r o 0" C0 (l) 0+ C,. , (l) 0 IlfJ S 0 en' " I H-, "0 ~


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2 1 INTRODUCTION

wide class of Abelian lattice models (spin models, gauge theories and their higher rank tensorgeneralizations) in any Euclidean space-time dimension. For a review see e.g. [ 2 ] .

In this paper, we generalize the transformation to the case of lattice gauge theory with anon-Abelian gauge group G (our proofs are valid for compact Lie groups and finite groups).The resulting dual model generalizes the well-known results for the Abelian case and is de-scribed in terms of the finite-dimensional unitary representations of G, their representationmorphisms (intertwiners) and the character expansion of the Boltzmann weight. In particular,all group integrations are explicitly performed.

The method we use is the Peter- Weyl decomposition of the algebra Calg( G) of represen-tation functions of G, the 'algebraic functions' on G. This decomposition can be viewed asa generalization of Fourier transformation to functions on a compact non-Abelian Lie group.It is convenient to exploit the Hopf algebra structure of Calg( G) and to employ a purely al-gebraic description of the Haar measure. The duality transformation follows the lines of thewell-known Abelian case, but some attention and geometric intuition is necessary to makethe generalized gauge constraint appear in a local form in the dual model.

The duality transformation establishes the equality of the partition function and the ex-pectation value of the non-Abelian Wilson loop (the generic gauge invariant expression whichis given by a spin network) with their corresponding purely algebraic expressions in the dualmodel. The dual model is found to have a Boltzmann weight of such a form that the strongcoupling regime of non-Abelian lattice gauge theory is mapped to the weak coupling regimeof the dual model. In addition, the strong coupling expansion can be applied to this reformu-lation of non-Abelian gauge theory in a systematic way.

The duality relation is stated in Theorems 3.1 and 3.3 which form the main result of thispaper. The dual model is a system in statistical mechanics whose configurations are spinfoams on the lattice. These configurations are assigned Boltzmann weights and are subjectto certain constraints. Spin foams have been introduced in the study of quantum gravity, seee.g. [3-6] and the recent introductory article [7]. The configurations of our dual model areclosed spin foams on the lattice according to the definition given in [6]. The transformationthus provides an explicit example for the relation of lattice gauge theory with a particularspin foam model.

The dual model reduces to the known results for non-Abelian gauge theory in 2 dimensionswhere the partition function is particularly simple, as well as to the known results for Abelianlattice gauge theory in arbitrary dimension, i.e. in the cases G =U(1), 'll and 'lln.

Whereas the duality transformation in the Abelian case is known to work in a similar wayfor spin models, gauge theories and higher rank tensor models on the lattice [2], this is notthe case for the non-Abelian generalization. Even though it can be applied to spin models ina straight forward way, the resulting 'gauge' constraint cannot be cast in a local form.

The motivation for deriving a dual description of non-Abelian lattice gauge theory arisesfrom conceptual issues such as the 'dual superconductor' picture of confinement and from rig-


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3

orous studies in the framework of constructive quantum field theory as well as from technicaland numerical problems in lattice gauge theory.

At present only a few ways of studying gauge theories in their strong coupling regimeare known - in particular if there are no additional symmetries like supersymmetry. Inthe famous paper by Wilson [8], the lattice formulation of U ( I ) gauge theory was used inconjunction with the high temperature expansion which is known from statistical mechanicsand which plays the role of a strong coupling expansion.

The duality transformation for Abelian lattice gauge theories (see e.g. [2]) can be seen onthe one hand as a result of attempts to make this strong coupling expansion systematic. Onthe other hand there is the picture of 'dual superconductivity' as an explanation of confinementgoing back to ideas of t'Hooft and Mandelstam, see e.g. [9,10]. Whereas in a superconductorelectrically charged quasi-particles condense and force the magnetic flux into quantized tubes,the picture is that in a gauge theory with confinement, magnetic monopoles condense. Thisleads to the formation of electric flux tubes which are responsible for the linearity of the staticpotential between opposite external electric charges.

In this picture the magnetic monopoles appear as collective excitations of the gauge theory.They are found to be quantized topological defects [11,12]. The Abelian duality transfor-mation allows one to spot these topological degrees of freedom. All group integrations areperformed, and the partition function of Abelian lattice gauge theory is rewritten in newvariables such that the topological degrees of freedom have the form of ordinary expectationvalues of the dual fields.

In the Abelian case, the lattice approach to gauge theories in conjunction with the du-ality transformation and related techniques has lead to a number of remarkable results. Wemention the existence of a phase transition in U ( I ) lattice gauge theory in d = 4 [13,14],the existence of world-lines of magnetic monopoles in the same model which behave likeinfra-particles [11,12], the fact that these are responsible for the phase transition by conden-sation of magnetic monopoles, and finally the absence of a deconfinement phase transitionin d = 3 [15]. The picture of dual superconductivity leading to confinement in the Abeliancase is well established, see e.g. the study of the monopole degrees of freedom in Monte-Carlosimulations [16] and their properties at the deconfining phase transition [17] of U(1) latticegauge theory in d =4.

Although there are strong conjectures both from lattice studies of QCD and from resultsin supersymmetric Yang-Mills theory that confinement in non-Abelian gauge theories can bedescribed in a similar way by dual superconductivity, no analogous approach is available fornon-Abelian gauge theory. The exact duality transformation presented in this paper is meantto be a first step in this direction.

Finally, we comment on the relations of this work with other approaches. Firstly, in theAbelian case, the dual model is again a gauge theory if certain cohomologies of the space-timelattice are trivial. The case of general topology is studied in [18]. Furthermore, for SU (2 )


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4 2 PRELIMINARIES

lattice gauge theory, there are results by explicit computations which relate this theory tocertain simplex models of gravity [19,20].

Furthermore, there has been an interesting categorial approach to models which could bedual to non-Abelian gauge theory [21]. It uses the same colouring of links and plaquettesof the lattice with representations and intertwiners, respectively, as our dual model does onthe dual lattice in d = 3 and seems to enjoy many close similarities. However, the gaugedegrees of freedom proposed in [21] are not in a straight forward way a symmetry of thedually transformed model as derived in this paper. Nevertheless, the ideas developed theremight prove useful for a further study of the dual model.

We thank M.B. Halpern for bringing the paper [22] to our attention. In [22] a plaquetteformulation of lattice gauge theory is derived emphasizing the non-Abelian Bianchi identity.Using the mathematical methods that we describe in Sections 2.1 to 2.4, it can be shown thatthe approach of [22] can be extended to a full duality transformation similar to the treatmentin our Section 3. The result would be the same as given in Theorem 3.1. Finally, we thank areferee for drawing our attention to the paper [23] where a duality transformation of classicalYang-Mills theory in continuous space-time based on the loop approach is suggested.

This paper is organized as follows. In Section 2, we recall all definitions and structureswhich are needed to present the duality transformation. In particular these are the structureof the Hopf algebra of representation functions Calg( G) of G and the lattice formulation ofgauge theories.

In Section 3, we present the duality transformation in detail first for the partition function,and then for the non-Abelian generalizations of the Wilson loop. We specialize our result tothe known cases of non-Abelian lattice gauge theory in 2 dimensions and to Abelian latticegauge theory in arbitrary dimension in Section 4.

Finally, in Section 5 we indicate how the dual formulation can be used in conjunction withstrong coupling expansion techniques and discuss open questions and directions for furtherresearch.

2 Preliminaries2.1 The Hopf algebra of representation functionsIn this section we collect definitions and basic statements related to the algebra of represen-tation functions Calg( G) of G. These and the results presented in the next section about thePeter- Weyl decomposition and the Peter- Weyl theorem are basically text book knowledge,see e.g. [24,25]. We recall the basic facts to fix our notation and present a purely algebraictreatment of the relevant results which we have not found elsewhere in this form.

Let G be a finite group or a compact Lie group. We denote finite-dimensional complexvector spaces on which G is represented by Vp and by p : G - - - + Aut Vp the corresponding group


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2.1 The Hopf algebra of representation functions 5

homomorphisms. Since each finite-dimensional complex representation of G is equivalent toa unitary representation, we select a set R containing one unitary representation of G foreach equivalence class of finite-dimensional representations. The tensor product, the directsum and taking the dual are supposed to be closed operations on this set. This amounts to aparticular choice of representation isomorphisms PI i 6 J P 2 + - - t P 3 etc., P j E R, which is implicitin our formulas. We furthermore denote by R ~R the subset of irreducible representations.

If PER , we write p " for the dual representation and denote the dual vector space of Vpby Vp*. The dual representation is given by p ": G f - - - 7 Aut Vp* , where

(2.1)1.e. (p*(g)1])(v) =1](p(g-l )v) for all v E Vp.

For the unitary representations Vp, PER, we have standard (sesquilinear) scalar products(-;.) and orthonormal bases (Vj) in such a way that the basis (Vj) of Vp is dual to the basis(1]j) of Vp* , i.e. 1] j (Vk) =Sjk. This means that duality is given by the scalar product,

1< j, k < dim v; (2.2)There exists a one-dimensional 'trivial' representation of G which we denote by V[I] ~ < C .

The functions

t~2: G - - - - t < C , g f - - - 7 1](p(g)v), (2.3)where PER , v E Vp and 1] E Vp* , are called the representation functions of G. They form acommutative and associative unital algebra over < C ,

(2.4)whose operations are given by

._ t(Pffw) ()1)+V,V+W g ,

._ t(prg ;w ) ()1)!j:)v,v!j:)w g ,

(2.5a)(2.5b)

where p, a E R and v E Vp, w E Vo-, 1] E Vp* , 1) E V ; and g E G. The zero element of Calg(G)is given by t ~ ~ 6g ) =0 and its unit element by t~I,L( g ) =1 where we have normalized 1]( v) =1.

The algebra Calg(G) is furthermore equipped with a Hopf algebra structure with thecoproduct 6.: Calg(G) - - - - t Calg(G) i 6 J Calg(G) ~ Calg(G X G), the co-unit s:Calg(G) - - - - t < C andthe antipode S : Calg(G) - - - - t Calg(G) which are defined by

6 .t~2 (g, h ) ._ t(p ) (g . h ) (2.6a)1 7 ,V ,ct(p) ._ t~2(1) , (2.6b)1),V

St~2(g) ._ t( p) ( g- I) (2.6c)1 7 ,V ,


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6 2 PRELIMINARIES

where pER and v E Vp, T f E V/ and g, hE G.Since G is a finite group or a compact Lie group, all finite-dimensional representations of

G are completely reducible. Moreover, all representations of G X G are tensor products ofrepresentations of G such that we have an isomorphism of algebras Calg(G X G) ~ Calg(G) i 6 JCalg(G) which is used in the definition of the coproduct. This tensor product is algebraic,and there is no need for a topology or a completion of the tensor product at this point.

In the standard orthonormal bases, the representation functions are given by the coeffi-cients of representation matrices,

(2.7)such that the coproduct corresponds to the matrix product,

dim V,6.t~~(g,h)= L t~}(g)t)~(h),j=l

(2.8)

while the antipode refers to the inverse matrix, St~~(g) = (p(g)- l )mn ' and the co-unitdescribes the coefficients of the unit matrix, ct~~ =6m n. Furthermore, the antipode relatesa representation with its dual,

which is just the conjugate representation because on the other hand(2.10)

Here the bar denotes complex conjugation.

2.2 Peter- Weyl decomposition and theoremThe structure of the algebra Calg(G) can be understood if Calg(G) is considered as a repre-sentation of G X G by combined left and right translation of the function argument,

(2.11)It can then be decomposed into its irreducible components as a representation of G X G:Theorem 2.1 (Peter- Weyl Decomposition). Let G be a finite group or a compact Liegroup.

1. There is an isomorphismCalg(G) ~GxG E9(Vp* i 6 J Vp),

pERof representations of G X G. Here the direct sum is over one unitary representative ofeach equivalence class of finite-dimensional irreducible representations of G. The direct

(2.12)

summands V/ i 6 J Vp are irreducible as representations of G X G.


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2.2 Peter- Weyl decomposition and theorem 7

2. The direct sum in (2.12) is orthogonal with respect to the L2 scalar product on Calg(G)which is formed using the Haar measure on G on the left hand side, and using thestandard scalar products on the right hand side, namely

(2.13)

where p , a E R are irreducible. The Haar measure is denoted by fa dg and normalizedsuch that fa dg =1.

Remark 2.2. 1. If G is finite, the Haar measure coincides with the normalized summationover all group elements.

2. The decomposition (2.12) directly corresponds to our notation of the representationfunctions t };~(g) if p ER is irreducible.

3. Each representation function f E Calg(G) has a decomposition according to (2.12),(2.14)

such that we find for the L2-norm

2 '""' 1 2I l f l l p = D dim V I l f p ll ,pER p

(2.15)

where fp E Vp* i 6 J Vp , PER, and all except finitely many fp are zero. Here I I f p l 1 2 is thetrace norm for the finite-dimensional space Vp* i 6 J V p ~ End Vp .

The analytical aspects of Calg(G) can now be stated.Theorem 2.3 (Peter-Weyl Theorem). Let G be a compact Lie group. Then Calg(G) isdense in L2 (G).

Remark 2.4. 1. We use the Peter- Weyl theorem to complete Calg(G) with respect to theL2 norm to L2(G). Functions f E L2 (G) then correspond to square summable seriesin (2.14). These series are thus invariant under a reordering of summands, and theirlimits commute with group integrations. We will make use of these invariances in theduality transformation.

2. If G is a finite group, Calg(G) is a finite-dimensional vector space such that the corre-sponding results hold trivially.


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8 2 PRELIMINARIES

2.3 Character decompositionIf G is a finite group or a compact Lie group, the characters X(p ) : G - - - + < C associated with thefinite-dimensional unitary representations pER of G are obtained from the representationfunctions by

dim V,X (p)= '" t(p). DJJ.

j=1(2.16)

Each class function f E Calg(g) has a character decompositionwhere ~ =dim V p J X(p ) (g )f(g ) dg .

Gf(g) =LX(p ) (g ) fp ,

pER(2.17)

The completion of Calg( G) to L2 (G ) is compatible with this decomposition.2.4 Projector description of the Haar measureFor the duality transformation, it is important to understand the Haar measure on G in thepicture of the Peter-Weyl decomposition (2.12). We describe the Haar measure in terms ofprojectors.Proposition 2.5. Let G be a finite group or a compact Lie group and pER be a finite-dimensional unitary representation of G with the orthogonal decomposition

kV p ~ E 9 VTJ '

j=1Tj E R, k E IN , (2.18)

into irreducible components Tj. Let r': V p - - - + VTJ ~ V p be the G-invariant orthogonalprojectors associated with the above decomposition. Assume that precisely the first E com-ponents Tl, ... ,Tg, 0 < e < k , are equivalent with the trivial representation. Then the Haarmeasure of a representation function t~t1 :::;m, n :::;im Vp, is given by

e e eJ t~~ (g ) dg =L(-. ; pU ) vn) =L(U ) Vm ; p U ) vn) =Lp SP p~ j) ,G J=1 J=1 J=1

(2.19)

where p SP =r;(p(j)vm ) denotes the matrix elements of the j-th projector. Here r; E Vp * is thenormalized linear form which is zero everywhere except on the one-dimensional sub-spacesVTJ < v; 1< j < t.Proof. The representation function is Peter- Weyl decomposed by inserting n = 2 : J = 1 p(j)twice into the right hand side of t~~(g) = (vm;p(g)vn). We use hermiticity p(j)t = pU) ,G-invariance [pU),p(g)] =0 and transversality p(i)p(j) =6ijP(j) to obtain

kJ t~~ (g ) dg =LJ (p(j)vm ; p (g )p (j)V n ) dg .G J=1 G

(2.20)


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2.5 The Lattice formulation of non-Abelian gauge theories 9

Since the Haar measure is bi-invariant, all terms vanish except those corresponding to the Tj,1 < j < twhich are equivalent to the trivial representation. D2.5 The Lattice formulation of non-Abelian gauge theoriesThe purpose of this section is to fix a notation in which we can write down the partitionfunction and expectation values of non-Abelian lattice gauge theory and which is suitable toformulate the duality transformation. For all other issues we refer the reader to the standardtext books on lattice gauge theory, e.g. [26,27], and references therein.

We consider a regular hyper-cubic lattice corresponding to an Euclidean space-time ofdimension d> 2. The lattice points (vertices) are denoted by tuples of integer numbers

(2.21)where the lattice is of size NI- ' in the IL-th dimension. We denote the unit vectors along thelattice axes by

Ii:= (0 , .. . ,O ,~, 0 , .. . ,0 ),I-'

1< IL < d. (2.22)Thus i+ Ii refers to the neighbour of the point iin the direction IL . We choose periodic (i.e.toroidal) boundary conditions and identify iNI-" Ii = = ifor all rzE {I, ... ,d}. It is crucial forthe existence of various integrals that we work on a finite lattice. Periodic boundary conditionsare the most convenient choice for our purpose. The non-trivial homologies introduced bythe periodic boundary conditions do not play any role for the duality transformation in theform presented below.

The set of all links (edges) is called AI, the set of all plaquettes (faces, squares) A2, andmore generally the set of all k-cells Ak. These are specified by

(2.23)In particular, the sets Ak , 0 :::;k :::; d , are all finite. The k-cells are considered unoriented, e.g.we do not want to distinguish the plaquette (i , IL , l I) from (i , u, IL). In our notation both arerepresented in the standard way (i , IL , l I) where IL < u.The configurations of lattice gauge theory are the maps

(2.24)which assign a group element gil-' E G to each link (i , IL ) E Al of the lattice. These groupelements correspond to the parallel transports of the gauge connection along the links.

The path integral measure depends on these configurations only via the plaquette product(see also Figure 1),

(2.25)


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10 2 PRELIMINARIES

Figure 1: The oriented product oflink variables around a given plaquette (i , /1 , z ; ) EA2 as given by (2.25).

which is the path ordered product of the link variables around a given plaquette (i , /1 , z ; ) E A2For arbitrary generating functions i .p: A0 - - - + G, any class function of G, evaluated on dgi/Lv,

is invariant under the gauge transformation(2.26)

Let G be a compact Lie group (or a finite group) and fa dg denote the Haar measure. Thepath integral integrates over all configurations, i.e. it consists of one integration over G foreach link. We denote this integration by

J V g = ( I T J dgi/L):= J dgi/L J dgi/L( i , /L )EAl a a aone fa for each link (2.27)The path integral measure of lattice gauge theory is this integration together with a Boltzmannweight exp( -S(dg)) . Here the (local) action S is given by a sum over all plaquettes,

S(dg):= L s(dgi/Lv),( i , /L ,v)EA2

(2.28)

where s: G - - - + IR is an L2 integrable class function on G which is bounded from below. Theaction is thus manifestly gauge invariant.

The full partition function of lattice gauge theory with gauge group G finally reads

z =J V g exp( -S(dg)) = ( I T J dgi/L) I T f(dgi/Lv),( i , /L )EAl a ( i , /L ,v)EA2

(2.29)

where f(g) = exp( -s(g)) is a positive real and L2 integrable class function on G. Thestandard example for the action is the Wilson action,

(2.30)


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2.6 Gauge invariant quantities 11

where x(p) : G - - - + < C is the character of a unitary matrix representation p of G, usually thefundamental representation. The inverse temperature f 3 encodes the coupling constant.

In (2.29) and in the following we are careful not to waste letters of the alphabet fordummy indices. The indices i, f1 of the first product sign are just there to indicate that groupintegration is performed for each link of the lattice. We adopt the convention that iand f1in this case do not have any meaning outside the enclosing brackets. So we can use the sameletters again after the closing bracket.

2.6 Gauge invariant quantitiesOf course, Wilson loops are gauge invariant expressions. However, in non-Abelian latticegauge theory, not all gauge invariant expressions are given by Wilson loops. The genericgauge invariant expressions are so-called spin networks which include branchings of the Wilsonlines. This is familiar, for example, from the expression which is used to determine the staticthree-quark potential.

Here we formalize this generalization and give the following slightly technical definition:Definition 2.6. Let T: Al - - - + R, ( j , K , ) f - - - 7 Tjr;, associate a finite-dimensional irreducible uni-tary representation of G with each link of the lattice. Choose furthermore for each latticepoint j E A an intertwiner

d dQ(j): Tj_/i,/L - - - + Tj/L'/L= /L= (2.31)

which maps from the tensor product of the representations of the d 'incoming' links to thetensor product of the d 'outgoing' links at the point j E AO .

The non-Abelian Wilson loop (or spin network) associated with T and QU ) is defined by(2.32)

x ( I T Q(j) .. ).(b_11 b_d d),(aJl ... aJd)oJ, J ,jEAThe indices (bj_i,l bj_d,d) of QU ) refer to the tensor factors of the domain of QU ) ('incom-ing') while the (ajl ajd) refer to the image ('outgoing'), cf. (2.31).

In (2.32) we have abbreviated the summation of the vector indices ajr; and bjr; for all linksby

(2.33)

one 2: for each linkThis notation is frequently used in the duality transformation in Section 3.


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12 3 THE DUALITY TRANSFORMATION

Proposition 2.7. The expression W ({Tj,,} , {Q U)} ) of the non-Abelian Wilson loop in (2.32)is gauge invariant under the transformation (2.26).Proof. Consider an arbitrary lattice point j E A 0. The gauge transformation (2.26) multipliesall 'incoming' links by


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3.1 The dual of the partition function 1 3

sition, see (2.17):d imVp

f (g) =L~x(p ) (g) =L~L t};~(g). ( 3 . 1 )p ER pER n= lThe partition function thus reads

d imVPi l 'VZ = ( I T J dgi/L) I T (L fp i l 'v L t~ ;::~ iI'V ( gi/L gi+ /i,vg i+ 1v ,/L g i))) ,

(i,/L)EAl G (i,/L,v)EA2 p il'vER n il'v= l(3.2)

where each plaquette (i , /1 , v) E A2 is coloured with an irreducible representation Pi /Lv E R,and the indices ni /Lv which originate from the traces, are summed once for each plaquette.

The next step is to employ the coproduct and the antipode, see (2.6a) and (2.6c), in orderto remove all group products and inverses from the arguments of the representation functions.Furthermore, we reorganize the summations:

( 3 . 3 )

x ( " " t(P il'v ) . ( . )t(p ,l'v ) . ( . - )S t(P il'v ) ( . - )S t(P il'v) ( . ))]D n 'I'V m ,I'V g '/L m ,I'V P ,I'V g '+ /L ,V p 'l'vq,l'v g ,+ v ,/L q 'I'V n ,I'V g ,v .nif1.v,mif1.v,Pif1.v,Qif1.V

In all places where we have applied the coproduct, written schematically as

m,p,q

new vector indices have entered which are associated with the plaquette (i , /1 , v) E A2 anddenoted by m i/L v , P i/L v and qi/Lv' They are summed over the range 1 ... dim VPi l ' v '

In order to perform the group integrations, we have to reorganize the product in (3.3)such that all representation functions whose argument refers to the same link are groupedtogether. Therefore we have to find all plaquettes which contain a given link (i , /1 ) E Al intheir boundary, i.e. all plaquettes which cobound the link. Figure 2 illustrates this situationfor d =3. The reorganized product reads

z = ( I T L) ( I T ~i I 'V) ( I T L)(i,/L,v) EA2 Pil 'V ER (i,/L,v) EA2 (i,/L,v) EA2 n il 'V ,m i l'V ,P iI 'V ,q il 'v

(3.4)

x

The expression (3.4) means that each plaquette is coloured with an irreducible represen-tation. There is furthermore a (dual Boltzmann) weight factor ~ per plaquette. Since we


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( i ,p,v),/,/

qm

-, -,\(i, A, p)

n

-, -, -,\( i -v ,p,v)1- v

Figure 2: All plaquettes whose boundary contains a given link (i , fL ) E AI. Thisfigure shows the situation in d = 3. In generic dimension d, there are 2(d - 1)such plaquettes. The coordinate axes are chosen such that 1 < A < u < v < d.We illustrate the numbering of the lattice points i, i-A, etc., which are neededto describe the relevant plaquettes (i - : X , A, f L ) , (i , fL , v), etc. with their correctorientations. Furthermore the letters n, m, p, q indicate to which lattice point thevector index summations ni/w, mi/w, Pi!-'v and qi!-'v of a given plaquette (i , fL , v)belong.

have reorganized the product of the representation functions t i j ( g ) such that those whoseargument refers to the same link (i , f L ) E Al are placed next to each other, the group integra-tions in (3.4) are performed for each link separately. In the integrand, the two products ILand I l u enumerate all plaquettes cobounding the link (i , f L ) in arbitrary dimension d and aresuch that 1 < A < u < v < d always. In dimension d, there are 2 (d - 1) factors for each linkdirection fl.

Next we eliminate the antipodes using (2.9). The group integrals in (3.4) thus read

(3 .5)

Now the group integrations can be performed using the expression (2.19) in terms of projec-


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3.1 The dual of the partition function 15

tors. We obtain

(3.6)AE{l, ... ,!,-l} AE{l, ... ,!,-l}

Here the sum is over a complete set Pif.L of inequivalent orthogonal projectors onto the trivialone-dimensional components in the decomposition of

(3.7)AE{1, ... ,f.L-1} vE{f.L+1, ... ,d }

into irreducible components. The dots "... " indicate that there are pairs p i 6 J p " of tensorfactors for all A E {I, ... , f L - I} and pairs p " i 6 J p for all 1/ E { f L + 1, ... , d} . This gives thecorrect result in arbitrary dimension and takes into account the orientation of the link (i , f L )in the boundary of the given plaquette. Opposite orientations of the link correspond to dualrepresentations. Similarly, the dots "... " in (3.6) indicate that there is one pair of indices foreach pair p i 6 J p* resp. p* i 6 J p which appears in (3.7).

With this step we have evaluated all group integrations over the links. As new degrees offreedom for the dual path integral, the colourings of all plaquettes with irreducible represen-tations of G have emerged:

z = ( I T L )( I T jPi!'v(i,f.L,v)EA2..__._,dual Boltzmann weight

) ( I T L )( i , / - L , v ) EA 2 ni,J- l. ,v ,mi f1.v ,Pi f1.v ,qi f1.vpart of thedual path integral

I T ( L p( ...)(...). p( ...)(...)(i,f.L)EAl PEPi!,

vector index summations

x (3 .8)one constraint for each link

The last sum has the form as in (3.6) for each link (i , f L ) appearing in the product.There are still the summations over the vector indices nif.Lv, ... ,qif.Lv for all plaquettes.

We call the factors arising from them and from the projectors gauge constraints because theygeneralize the conditions which ensure in the Abelian case that the integer 2-form appearingin the dual model, is co-closed. They seem to form a number of complicated non-localconstraints. However, the summations can again be reordered in a suitable way so thatthe constraints appear only locally in a certain sense. In order to see this, some geometricintuition is necessary.

The projectors which have appeared in the group integration fa dgif.L for the link (i , f L ) EA1 and their indices are of the form

(3 .9)


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p

p

i-/i-D

Figure 3: All plaquettes containing a given lattice point iE A0 in their boundaries.This figure shows the situation in d =3. In the generic case, there are 2d( d - 1)such plaquettes. This figure illustrates the numbering of points i-Ii - V, i-v,i-Ii and i which are used to specify the relevant plaquettes (i - Ii - V, /1 , v),(i - V, /1 , v), (i - Ii, /1 , v) and (i , /1 , v) in the (/1, v)-plane and similarly for the otherdirections, cf. (3.11). The letters n, m, p , q indicate which lattice points the vectorindex summations are associated with. The labelling for a given plaquette (i , /1 , v)starts with n at iand proceeds counter-clockwise with m, p, q.

i.e. the indices correspond to the plaquettes located at i - . x , i, ... , i - t/, i, ... for the firstprojector and similarly for the second projector.

The crucial geometrical observation (Figure 2) is that all vector indices m_", ,niAIL' ... '1, A,A, /-Lqi-v,IL,v, niILv, ... which appear at the first projector correspond to the lattice point iwhereasall vector indices Pi:'; ,qiAIL' ... ' Pi-v" u :miILv, ... of the second projector correspond to the1 , / \ ,A, / - L )r")lattice point i+ Ii. Recall that the enumeration of the indices n, m, p , q for a given plaquette(i , /1 , v) E A2 was done starting with n at the point i, then proceeding counter-clockwise inthe (/1, v)-plane. It is thus possible to associate the summations over the n, m,p, qwith thelattice points i E Ao . For each link (i, /1 ) E AI,one of the two projectors in (3.6) then belongsto i, the other to i+ Ii. However, the projectors can be separated only if the summation overthe projectors PilL which is associated with a link rather than a point, can be removed from


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3.1 The dual of the partition function 17

these expressions.In the partition function, there is in total one such summation over projectors for each link

of the lattice. It is thus natural to consider these summations as part of the dual path integral.If this is done, we can reorganize the expressions such that all vector index summations andprojectors are associated with the lattice points iE A and such that they involve only datafrom the neighbouring links and plaquettes. This is what is meant by 'locality'. In Figure 3we show the lattice point iE A together with the 2d links and 2d( d - 1) plaquettes whichcontain i. The partition function finally reads

z=

dual path integral

I T jPil"v( i , /L ,v)EA2..__._,

dual Boltzmann weight

) I T C(i) .iEAO

(3.10)

where C(i) encompasses the vector index summations and projectors associated with thelattice point iE A0:

C(i)dim VPi-l1 -v,M,v dim VPi-v,f-I,V dim VPi-I1 ,M,v dim VPif-IvL L L L)Pi-!1-v,J-l.,v==.l Qi-v,f-I,v==l mi-MJf-I,v==l nif-Iv==l

(3.11)

AE{l, ... ,I"-l}

The first product parameterizes all possible planes by /1 < 1 / . The four sums are then asso-ciated with the four plaquettes (i - i i - V, /1 , 1 / ) , (i - V, /1 , 1 / ) , (i - i i , /1 , 1 / ) and (i , /1 , 1 / ) in the(/1, I/)-plane which contain the point i(Figure 3). The last product enumerates the 'outgoing'and 'incoming' links and contains the projectors associated with this link and with the pointi. Note that the projectors in (3.11) for a given lattice point i E AO involve only vector indiceswhose summation is part of the same C(i) .

The partition function (3.10) consists now of a sum over irreducible representations forall plaquettes and a sum over the projectors onto the trivial components in the tensor prod-uct (3.7) for all links. This tensor product involves the representations of all plaquetteswhich cobound the given link. In particular, if the tensor product does not contain a trivialcomponent, this sum over projectors is empty and does not contribute to the path integral.

These results are summarized in the following theorem:

Theorem 3.1 (Dual partition function). Let G be a compact Lie group or a finite group.The partition function (2.29) of lattice gauge theory with the gauge group G on a d-dimension-


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al finite lattice with periodic boundary conditions is equal to the expression

z= (IT 2:)(IT 2:)((i ,IL,V)EA2 pil-'vER (i ,IL)EAl P(il-'lEPil-'dual path integral

IT jPil-'v(i ,IL,V)EA2..__._,dual Boltzmann weight

) IT C(i) .iEAO (3.12)Here R denotes a set containing one unitary representation for each equivalence class of finite-dimensional irreducible representations of G. PilL denotes the set of all projectors onto thedifferent one-dimensional trivial components in the decomposition of the tensor product (3.7)into its irreducible components. C(i) describes a gauge constraint factor for each latticepoint iE A0 which is given by (3.11). The coefficients ~il-'V are defined by the characterdecomposition of the original Boltzmann weight exp( -s(g)),

~il-'V =dim VPil-'v J X(Pil-'v) ( g ) exp (- s ( g )) d g .G

(3.13)

Remark 3.2. 1. The dual partition function can be described in words as follows: Colourall plaquettes with finite-dimensional irreducible representations of G in all possibleways. Colour all links with projectors onto the trivial components in the tensor prod-ucts (3.7) (if there are any). The partition function contains a (local) dual Boltzmannweight factor which is the coefficient of the character expansion of the original Boltz-mann weight for each plaquette. The partition function contains furthermore a (local)gauge constraint factor C(i) , see (3.11), for each lattice point.

2. The two main differences to the Abelian case (see e.g. [2]) are the following: Firstly, inthe Abelian case only objects on a single level, namely the plaquettes, are coloured withinteger numbers (which characterize the finite-dimensional irreducible unitary represen-tations). In the non-Abelian case we have to colour the plaquettes with representationsand the links with intertwiners. The configurations of the dual model are thus spinfoams. Note that the choice of projectors PilL in (3.6) and (3.7) agrees up to canonicalisomorphisms with the choice of intertwiners in the definition of a closed spin foam asgiven in [6]. Here the assignment of 'incoming' and 'outgoing' faces has to be madeaccording to our standard orientations of plaquettes and links.Secondly, the integrand is not just a Boltzmann weight, but contains in addition thefactor C (i ) for each lattice point. In the Abelian case, this factor together with thesum over projectors enforces co-closed ness of the integer 2-form. The dual of Abeliangauge theory is again a gauge theory because if this 2-form is also co-exact, it can beintegrated and gauge degrees of freedom appear. In the non-Abelian case there is noobvious integration which would introduce gauge degrees of freedom.


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3.2 The dual of the non-Abelian Wilson loop 19

3.2 The dual of the non-Abelian Wilson loopThe duality transformation of the expectation value of the non-Abelian Wilson loop (2.34)proceeds along the same lines. However, the expressions become slightly more complicated dueto the presence of the additional integrand. In the following description of the transformation,we often refer to the calculations for the partition function in Section 3.1.

The expectation value of the non-Abelian Wilson loop is given by

(W({Tj , , } , {QU)})) ~ ( I T J dg iIL ) ( I T ! (dgi ILv)) ( I T L)(i,IL )EA l G (i,IL ,V )EA2 (j,,,)EA l

X ( I T t t J , , " t J J g J " ) ) ( I T Q g ) _11 ...b - d d ) , ( aJ l . . . a J d )) (3.14)(j,,,)EA l JEAo J, J ,

We insert the character decomposition (3.1), employ the coproduct and the antipode andreorganize the factors just as in the calculation for the partition function. The result gener-alizes (3.4) in which (3.5) has been inserted:

~ ( I T L) ( I T ~ i V V ) ( I T L) (3.15)( i , IL,V)EA2 pivvER ( i,IL ,V )EA2 (j,,,)EA l

( I T L )( I T Q g ~ - l ' l . . . b J - d d ), ( a Jl . . . a J d ))(i ,/-L,v)EA2 nip,v,mip,v,Pip,v,qip,v jEAO '

X

The features which are new compared with (3.4) and (3.5) are the summations over aj"and b j" for each link, the product over the intertwiners QU) for each lattice point and theadditional factor Thvb) (gi lL) in the integrand for each link (i , f 1 ) .a Z f 1 . Z f 1 .Now we can apply the projector expression for the Haar measure (2.19). Comparedwith (3.6), the additional factor in the integrand produces additional indices aiIL and bilL ofthe projectors,

(3.16)


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These indices correspond to the additional tensor factor in the following decomposition: Theorthogonal projectors P E PI/L project onto the distinct one-dimensional trivial componentsin the decomposition of

(Pi-)_,A, /L i 6 J p iA /L ) i 6 J . . . i 6 J (pi-v, /L,v i 6 J Pi/Lv) i 6 J . . . i 6 J ~new

(3.17)AE{1, ... ,/L-1} v E{ /L +1 , ... ,d }

into its irreducible components, cf. (3.7).Similar to the calculation for the partition function, the vector index summations over the

n i/Lv , .. . , q i/Lv as well as over the ajr; and bjr;, the projectors p( ...)( ...) and the intertwiners QU)can be reorganized to form local expressions. This construction is entirely analogous to thederivation of (3.10) and (3.11). We obtain the following result which generalizes Theorem 3.1:Theorem 3.3 (Dual non-Abelian Wilson loop). Let G be a compact Lie group or afinite group and consider lattice gauge theory with gauge group G on a d-dimensional finitelattice with periodic boundary conditions. The normalized expectation value of the non-Abelian Wilson loop (2.34) is equal to the expression

I T )(i,/L,v)EA2..__._,

dual path integral dual Boltzmann weight

I T [ ( I T d di~il' dim~!:i'I') Q(i) .. C ( i ) ] . (3.18). D D ( bi_ l,l .. bi_ d,d) ,(a zl ... a ,d )'EAO /L= 1 a'l'= l bi_!:i,I'=l

x

Here P L denotes the set of all projectors onto the different trivial com~onents in the decom-position of the tensor product (3.17) into its irreducible components. C (i ) describes a gaugeconstraint factor for each lattice point iE A0 which is given by

C(i)dim VPi-l1 -v,M,v dim VPi-v,f-I,V dim VPi-I1 ,M,v dim VPif-IvL L L L)Pi-!1-v,J-l.,v==.l Qi-v,f-I,v==l mi-MJf-I,v==l nif-Iv==l

(3.19)

AE{l, ... ,I'-l}

Remark 3.4. The dual of the non-Abelian Wilson loop can be described in words as follows:Just as in the Abelian case, it is not an expectation value under the dual partition function,but looks rather like a modified partition function. In addition to the dual partition function,


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3.3 The constraints C (i ) on the duallattice 21

.> ;.?jI~ I I1 01 11 1 /1L _[.,./

Figure 4: The relation between objects on the original lattice (solid) and on thedual lattice (dashed) in d = 3. It is convenient to draw the points of the duallattice shifted by half a lattice constant in all positive directions and to draw itsaxes with reversed directions. Then the cube dual to a point is centered aroundthis point, the plaquette dual to a link is punctured by it etc ..

there are summations over the vector indices ajr; and bjr; which are necessary to multiplythe representation matrices which form the non-Abelian Wilson loop. The loop enters in twoplaces. First the intertwiners Q(i) appear for each lattice point iE A o . Furthermore, therepresentations Tjr; on the links which form the non-Abelian Wilson loop, enter the tensorproduct (3.17), and the corresponding indices ajr; and bjr; thus appear in (3.19). The factthat the presence of the Wilson loop changes the constraint factors C (i ) is familiar from theAbelian case. There it occurs in the expressions before the co-closed 2-form is integrated.

3.3 The constraints C( i) on the dual latticeIn the expression of the dual partition function (3.12) and (3.11), the factors C(i) look verycomplicated. They can be understood most easily on the dual lattice. We explain this ideafor the case d =3 where the relevant pictures can be drawn. Analogous constructions can bemade for arbitrary d ~ 2.

We construct the dual lattice in the standard way which is illustrated in Figure 4 for thecase d =3. To each k-cell (i , 111, ,11k ), 1 :::;111 < < 11 k :: :;d, of the original lattice, therecorresponds a (d - k)-cell (i , vI, ,Vd-k ), 1 :::;V I < < vd-k :::;d , of the dual lattice suchthat

{Ill, ... ,11k} U {V I, ... ,Vd-d ={I, .. . ,d} . (3.20)In the dual partition function on the original lattice, the plaquettes are coloured with

irreducible representations. The links are assigned projectors in a certain tensor productwhose factors are given by the representations belonging to the plaquettes that coboundthe link. Conversely, on the dual lattice in d = 3, the plaquettes cobounding a given linkcorrespond to the links in the boundary of a plaquette (see Figure 5). Thus we have to colourthe links of the dual lattice with irreducible representations. The plaquettes of the dual lattice


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22 4 SPECIAL CASES

u

Figure 5: In d =3, the dual objects of the plaquettes with cobound a given link(solid lines) are the links in the boundary of a plaq uette (dashed lines).

are then assigned projectors onto the trivial components of some tensor product. This tensorproduct is the product of the representations belonging to the links in the boundary of theplaquette.

Instead of the projectors onto trivial components, schematically(3.21)

we now write intertwiners(3.22)

using the isomorphisms of G-modules Hom a (Vp* i 6 J VT, < C ) ~a Homa(VT, Vp). The intertwinersF thus map from two links of a given plaquette to the other two links. Note that the F inherita normalization from the P coming from the implicit inclusion < C ~ PI i 6 J P2 i 6 J P~ i 6 J P:

The factors C(i) of (3.11) are associated with the cubes of the dual lattice. The expres-sion (3.11), interpreted on the dual lattice in d =3, contains one intertwiner per face of thecube as indicated in Figure 6. In this figure, the intertwiners F are represented by doublearrows leading from two links of each plaquette to the other two links. The arrows illus-trate how the intertwiners have to be composed to account for the contraction of the indicesin (3.11). The remaining indices are then summed over.

A similar visualization is straight forward for the factors C(i) in (3.19).

4 Special cases4.1 Abelian gauge theory in arbitrary dimensionIn this section we show how Theorem 3.1 reduces to the well-known results for G = U( l ) .Similar calculations are available for 'll or 'lln (Note that the transformation is applicable to


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4.1 Abelian gauge theory in arbitrary dimension 23

back of cube-,,,..... ,: '-: .

\" .,'.,-, .,,,., ,: "",, I,," .J

\\ \

\ \\ \

\ \\ \\ v!,\ ,,~

front of cube

Figure 6: The intertwiners F: P I i 6 J P 2 - - + P 3 i 6 J P 4 between the representations P jassociated with the links of the dual lattice in d = 3. This figure indicates howthe indices of the intertwiners F are contracted in the factor C (i).

'Il although 'Il is neither compact nor finite. This is because 'Il gauge theory is dual to U(l)gauge theory).

We start with the dual partition function (3.12). The unitary finite-dimensional irreduciblerepresentations of U(1) are all one-dimensional. They are given by homomorphisms g r - - + gk forg E U(l) and are characterized by integer numbers k E 'Il, i.e. R ~ 'Il. The dual representationis then given by g r - - + ".

Consider the tensor product (3.7) and specify the representations by integer numberski /LV E 'Il. Since all irreducible representations are one-dimensional, so are their tensor prod-ucts. The question is therefore just whether or not the tensor product (3.7) is equivalent tothe trivial representation. This is the case if and only if

M-l d2.__)ki-3 :,A ,M - k iAM ) + L (-k i - lJ,M,V + k iMV ) =O .A=1 V=M+l

(4.1)

In this case, there is exactly one projector onto a trivial component of the tensor productwhich is the identity map. If (4.1) does not hold, there is no such projector.

Furthermore, since all irreducible representations are one-dimensional, the summations inthe constraints C(i), see (3.11), disappear. Moreover, the projectors p( iM) do not have indicesand are all equal to 1 if they exist. Thus C (i ) =1 if (4.1) holds.


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24 4 SPECIAL CASES

The partition function (3.12) therefore reads for G =U(1),z = ( I T( i , /L ,v)EA2 (4.2)/L-l d

X ( I T S(l:_)ki-3:,A ,/L - kiA /L) + L (-ki-v,/L,V + ki/Lv))).( i , /L )EAl A=1 v=/L+l

Here we have used the notation S ( n ) =S O , n for n E ~.The dual path integral thus reduces to the summation over the integer numbers for each

plaquette while the dual Boltzmann weight is again given by the character decomposition ofthe original Boltzmann weight,

2 1 1 "hi/"V = 2~ J e -ik i/"v 'Pexp (_s(ei'P )) dip .o (4.3)

The S-constraint ensures that the integer 2-form ki/Lv is co-closed. This condition provides thedual model with the properties of a gauge theory. As is well-known, the partition function (4.2)describes the dual of U ( I ) lattice gauge theory, the so-called ~ gauge theory [2,16] and ishere presented on the original rather than on the dual lattice.

4.2 Non-Abelian gauge theory in two dimensionsIn this section we demonstrate how Theorem 3.1 reduces to the familiar result for non-Abelianlattice gauge theory in d =2. In this case the partition function is particularly simple.

We start again with the dual partition function (3.12). In d = 2, there are only twoplaquettes which cobound a given link (i , f1 ) E AI. Imagine the situation of Figure 2 in d =2.The tensor product (3.7) therefore consists of only two factors. It reads for links (i , 1) E Alin the I-direction,

(4.4)and for links (i , 2) in the 2 direction,

(4.5)Here we have suppressed the last two indices of Pi/Lv which are always f1 = 1 and 1/ =2. Inboth cases (4.4) and (4.5), there are trivial components in the tensor product if and only if

resp. (4.6)Since this holds for all iE A 0, the only contributions to the partition function are given byconfigurations which assign the same representation to all plaquettes. We thus have

z =L(];J)IA21 I T C(i) .pER iEAo

(4.7)


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25

Observe further that the projectors onto the trivial component in the tensor products (4.4)and (4.5) are both given by the trace, i .e . P(biIL ) = d O IV 6 ab. The constraint C(i ) can be easilya im pcalculated:

C(i )dim Vp dim Vp dim Vp dim VpL L L L p Jii~ ~n i p ~:~ L~ mi_l . p Ji;:~ ln i . P ~:~ L~ )q i-2

1 (4.8)3(dim Vp)Therefore the partition function reads

Z =L ( J ; , ) I A 21 (dim Vp)-3I A O I .pER

(4.9)

This is the well-known result for lattice gauge theory in two dimensions, see e.g. [ 2 8 ] .

5 DiscussionThe duality transformation given in Theorems 3.1 and 3.3 is a strong-weak duality. Forexample, the character decomposition of the Boltzmann weight (3.13) reads for the Wilsonaction of G =U( l ) ,

h e =h(p), k E~, (5.1)and for the Wilson action of G =SU(2 ) using the fundamental representation,

2 j E lN o . (5.2)Here the representations are parameterized by integers k resp. non-negative half-integers j,and In ( x ) denote the modified Bessel functions. The coefficients h e resp. J j are positive andcan thus be written h e =exp( -s*(k)) resp. J j =exp( -s*(j)). The p-dependence of the dualaction s" is such that high and low temperature regimes are exchanged or, in the language ofgauge theory, strong and weak coupling (see e.g. [2,28]). However, the coupling constant doesnot occur as a prefactor of the interaction terms of the dual model because its interactionsdo not arise from the dual Boltzmann weight but rather from the selection of projectors Pi lLand from the factors C(i), cf . (3.12). For details about the character decompositions of thevarious common actions in lattice gauge theory, see e.g. [27,28] and references therein.

Of course, it is also possible to define the action of non-Abelian lattice gauge theory interms of the character decomposition of its Boltzmann weight. For example, the hea t k e rn e laction (or generalized Villain action) is given by the choice

(5.3)


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26 REFERENCES

which makes the strong-weak duality manifest. Here Cp denotes the eigenvalue of the quadrat-ic Casimir operator (in a certain normalization) on the irreducible representation p of G. SinceCp is essentially quadratic in the highest weight of the representation p , it is apparent thathigher representations are exponentially suppressed in the dual path integral. The smaller f 3is chosen, the more pronounced is the suppression. The dual expressions (3.12) and (3.18) cantherefore serve as generating functions for the strong coupling expansion. For details aboutstrong coupling expansion techniques, see e.g. [ 2 8 ] .

Since the duality transformation for non-Abelian lattice gauge theory constructed in thispaper generalizes the Abelian case in the form written with an explicit gauge constraintrather than in the form which is integrated and exhibits gauge degrees of freedom, there isno immediate answer to the question whether the dual model has any gauge invariance andhow these degrees of freedom could be parametrized.

A non-Abelian generalization of the integration of a closed (and exact) k-cocycle to thecoboundary of a ( k - l)-cocycle up to a gauge freedom is not easy to find. If it exists, thispaper might help to assemble information on how a non-Abelian generalization of cohomologymight look like. Interesting in this context are the ideas developed in [21].

Degrees of freedom in the dual model which are always present, are related to the choiceof unitary representatives pER of each class of equivalent irreducible representations of G.The Clebsh-Gordan coefficients which enter the analysis extensively as the coefficients of thevarious projectors, depend on these choices.

In any case, the dual model can be expected to be the appropriate starting point to searchfor the non-Abelian magnetic degrees of freedom which generalize the magnetic monopolesof U(1) lattice gauge theory, and to provide a framework for a rigorous treatment of theirproperties.

As far as the strong coupling expansion is concerned, the crucial question is to what extentthe limit of this expansion is compatible with the continuum limit, i.e. whether propertiesderived via the strong coupling expansion hold (at least qualitatively) for all couplings. Recallthat the continuum limit of lattice QCD consists of a combination of sending the inversetemperature f 3 - - - + 00 and the lattice spacing a - - - + o .

AcknowledgementsBoth authors are grateful to DAAD for their scholarships. R.O. furthermore acknowledgessupport by EPSRC. We would like to thank in particular I.Drummond, J. Jersak, A. J. Mac-farlane, S. Majid, T. Neuhaus and K.-G. Schlesinger for valuable discussions, comments onthe manuscript and for drawing our attention to relevant literature.

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