A Dual Algorithm for Non-abelian Yang-Mills coupled to Dynamical Fermions
aa r X i v : . [ h e p - l a t ] F e b A DUAL ALGORITHM FOR NON-ABELIAN YANG-MILLS COUPLED TODYNAMICAL FERMIONSJ. WADE CHERRINGTON Department of Applied Mathemati s, University of Western Ontario, London, Ontario, Canadae-mail: j herrinuwo. aAbstra t. We extend the dual algorithm re ently des ribed for pure, non-abelian Yang-Mills on the latti eto the ase of latti e fermions oupled to Yang-Mills, by onstru ting an ergodi Metropolis algorithm fordynami fermions that is lo al, exa t, and built from gauge-invariant boson-fermion oupled on(cid:28)gurations.For on reteness, we present in detail the ase of three dimensions, for the group SU (2) and staggeredfermions, however the algorithm readily generalizes with regard to group and dimension. The treatment ofthe fermion determinant makes use of a polymer expansion; as with previous proposals making use of thepolymer expansion in higher than two dimensions, the riti al question for pra ti al appli ations is whetherthe presen e of negative amplitudes an be managed in the ontinuum limit.1. Ba kgroundDespite ontinued progress in algorithms and hardware, the in lusion of dynami al fermions in latti egauge al ulations ontinues to in ur signi(cid:28) ant omputational expense. To motivate our proposal for anovel fermion algorithm, we brie(cid:29)y review how dynami al fermions are urrently addressed. Re all thatdynami fermions oupled to a gauge (cid:28)eld on a D -dimensional hyper ubi latti e for D ≥ are governed byan a tion of the form(1) S [ g e , ψ v , ψ v ] = S G [ g e ] + S F [ g e , ψ v , ψ v ] , where the g e are valued in the gauge group G at the edges of the latti e and ψ v are the fermion (cid:28)elds de(cid:28)nedat the verti es of the latti e.Unlike gauge group variables, it is not pra ti al to dire tly simulate Grassmann variables on the om-puter. A ommon approa h to dynami al fermion simulation starts by integrating out the fermion variablesappearing in S F [ g e , ψ v , ψ v ] , to give a fun tion of the gauge variables known as the fermion determinant (itsspe i(cid:28) form is reviewed in Se tion 2). The fermion determinant an be ombined with the kineti partof the gauge boson amplitude e − S G [ g e ] to give an e(cid:27)e tive a tion for the gauge variables from whi h sim-ulations on a omputer an in prin iple pro eed. However, the fermion determinant renders this e(cid:27)e tivea tion non-lo al (cid:22) it ouples together gauge variables that are arbitrarily distant in the latti e. This posesa onsiderable problem for the simulation, sin e omputing the hange in the e(cid:27)e tive a tion due to a small hange in any variable be omes very expensive, growing prohibitively with in reasing latti e volume. Avariety of algorithms have been devised to work with the fermion determinant; a des ription of some of themethods ommonly employed an be found for example in [6℄.After reviewing the des ription of single omponent, staggered free fermions in terms of self-avoidingpolymers (as was done for example in [9℄), we review what happens when a similar pro edure is applied tomulti- omponent fermion (cid:28)elds minimally oupled to gauge (cid:28)elds. In this ase ea h polymer on(cid:28)guration orresponds to a Wilson loop fun tional; i.e. the tra e of a produ t of representation matri es around thepolymer. Be ause there is more than one omponent of the fermion (cid:28)elds in the non-abelian oupled ase,the stri t self-avoiding onstraint of the single omponent ase is weakened; that is, for an n - omponentfermion, up to n dire ted polymer lines an enter and leave a given vertex. The pi ture has long beenknown (cid:22) it is essentially that of a hopping parameter expansion of the fermion determinant, des ribed forexample in [14℄. Unlike many past appli ations, in the present ase no ut-o(cid:27) in the power of the hoppingparameter or otherwise is applied. Be ause we seek an exa tly dual model, all polymers are in luded in the on(cid:28)gurations onsidered.For ea h polymer diagram that arises in the free ase, upon applying the duality transformation for thegroup-valued (cid:28)eld the result is a sum of on(cid:28)gurations onsisting of all losed, bran hed, olored surfa es (spin1oams) with open one-dimensional boundaries de(cid:28)ned by the polymer diagram. The totality of spin foamsasso iated with all polymer diagrams (in luding the trivial empty polymer) de(cid:28)nes the joint on(cid:28)gurationspa e. Cru ially, lo al hanges to the dual on(cid:28)gurations (either polymer or surfa e stru ture) lead to lo al hanges in the dual amplitude.The two theoreti al inputs for this onstru tion, a polymer de omposition of the fermion determinant (i.e.hopping parameter expansion as des ribed in [14℄) and a dual non-abelian model (e.g. [13℄ and referen estherein), have been present in the literature for some time, and as we shall see the onstru tion of the jointdual model at the formal level is a rather straightforward synthesis of these onstituent models. However,unlike (the simplest implementations of) onventional latti e gauge simulations, (cid:28)nding any pra ti al algo-rithm for a dual model has proven somewhat non-trivial in the non-abelian ase for dimensions greater thantwo. The algorithm proposed here builds upon the dual non-abelian algorithm of [5℄ that has re ently beentested in the pure Yang-Mills se tor. In addition to pure spin foam moves, we onstru t a set of moves thata t on polymer stru ture and spe ify the type of vertex amplitudes that arise due to the harges arried bythe polymer. Currently, an implementation of this algorithm is being tested and will be reported on in aforth oming work.For ontext, it should be noted that a similar pi ture was present in the work of Aro a et al. [1℄ andFort [7℄, whi h dealt with the abelian ase of U (1) and proposed using a Hamiltonian that leads to a di(cid:27)erentLagrangian formulation, where the ensemble is built from a restri ted subset of the on(cid:28)gurations that arisein the Kogut-Susskind ase. In future work, we believe the non-abelian generalization of [1℄ may be a veryinteresting alternative to the Kogut-Susskind formulation used here, parti ularly if the imbalan e betweennegative and positive amplitudes and the redu tion of spe ies doubling des ribed in [1℄ an be arried overto the non-abelian ase.The outline of the paper is as follows. In Se tion 2, we review the origin of the fermion determinantand dis uss its expansion in terms of polymers. In Se tion 3, we brie(cid:29)y review the dual omputationalframework for non-abelian, pure Yang-Mills theory on the latti e. In Se tion 4, we show a natural way to to ombine these frameworks and formulate ergodi moves for the oupled fermion-boson system. In Se tion 5,we o(cid:27)er some on lusions and des ribe our program for ongoing numeri al work based on this algorithm andits extensions. Appendix A des ribes the vertex amplitudes that arise in the joint ase, while Appendix Bexpands on Se tion 2 to des ribe the tra e stru ture of polymers with multiply o upied verti es.2. Polymer des ription of fermions on the latti eIn this se tion, we start from a onventional latti e dis retization of free fermions, following [12℄ inessentials and notation. In the usual manner, the fermion determinant is arrived at by exa t integrationof the Grassmann variables. Following [9℄, the fermion determinant is then expanded into states, ea h ofwhi h is represented by a family of disjoint, losed oriented loops, in luding trivial and degenerate (cid:16)loops(cid:17),the monomers and dimers, respe tively. A typi al polymer on(cid:28)guration (in the D = 2 massive ase) isillustrated 1 We now review expli itly how the fermion determinant and polymer pi ture ome about. Notethis se tion is purely for pedagogi al purposes and to (cid:28)x notation to be used later; those familiar with thehopping parameter expansion of the fermion determinant an safely skip it.2.1. Kogut-Susskind staggered fermions (cid:22) free ase. To illustrate the on ept and introdu e termi-nology, we treat a single spe ies of fermions with no additional indi es.We start with the naieve latti e (cid:28)elda tion for free staggered fermions(2) S = X x ∈ V Ψ x ( γ µ ∂ µ + m ) Ψ x , where γ µ are the (Eu lidean) Dira matri es and V is the set of latti e verti es.Using the entral di(cid:27)eren e for the partial derivative, this be omes(3) S = X x ∈ V a D ( m (Ψ x Ψ x ) − a D X µ =1 (Ψ x γ µ Ψ x +ˆ µ − Ψ x +ˆ µ γ µ Ψ x ) ) , ycle DimerMonomer Figure 1. Part of a typi al on(cid:28)guration in the polymer expansion of the fermion deter-minant in two dimensions (massive ase).where D is the dimension2 of the hyper- ubi latti e, m is the mass, a is the latti e spa ing, and µ labelsone of the D dire tions of the latti e; ˆ µ is the unit latti e ve tor asso iated to the µ th dire tion. Following[12℄, we hange to a basis whi h diagonalizes the gamma matri es, rewriting the free a tion as(4) S = X x ∈ V ( M ( ψ x ψ x ) − K D X µ =1 α xµ (cid:0) ψ x ψ x +ˆ µ − ψ x +ˆ µ ψ x (cid:1)) , with α xµ ≡ ( − x + ··· + x µ − for µ ∈ { , , ..., D } where the x i are the omponents of the latti e site four-ve torand M K = ma . The edge dependent sign fa tor α xµ arises from the hosen diagonalization.To express the result of integration over the Grassmann variables it is onvenient to introdu e the quarkmatrix Q , de(cid:28)ned in terms of latti e regularized a tion as(5) S F [ g e , ψ v , ψ v ] = X x,y ∈ V ψ y Q [ g e ] yx ψ x . For later referen e, we in lude dependen e on the gauge degrees of freedom in our de(cid:28)nition of Q . We nextapply the well known result [12℄ that the Grassmann integral over the fermion (cid:28)elds at every vertex evaluatesto(6) Z Y v ∈ V dψ v dψ v ! e − S F [ g e ,ψ v ,ψ v ] = Z Y v ∈ V dψ v dψ v ! e − P x,y ∈ V ψ y Q [ g e ] yx ψ x = det Q [ g e ] , the determinant of the quark matrix.Next we re all the ontinuum form of the a tion for massive fermions oupled to a gauge (cid:28)eld. In themassive ase, the quark matrix an be written as(7) Q yx = M δ yx − K yx , M is the fermion mass and K xy is the hopping matrix that is non-zero for nearest neighbor pairs ( x, y ) . By inspe tion of (4), we write the quark matrix as(8) Q yx = M δ yx − K D X µ =1 α xµ ( δ y,x − ˆ µ − δ y − ˆ µ,x ) . We now apply a well known identity for the determinant of (any) matrix Q ,(9) det Q = X π sgn ( π ) Y x Q xπ ( x ) , where π ranges over the set Π of all permutations of the indi es of Q and sgn ( π ) is the sign of the permutation.In the ase of the quark matrix Q , the matrix indi es being permuted orrespond to verti es of the latti e.Thus, one is led to onsider the produ t Q x Q xπ ( x ) for every permutation of latti e verti es.The next step is to re ognize that every permutation π an be de omposed into a omposition of disjoint,non-trivial y li permutations π c , π = Q i π ci . For a matrix with all entries non-zero, these permutationsmay involve sets of verti es that are arbitrarily separated on the latti e. However, the quark matri es thatarise in pra ti e have a very spe i(cid:28) stru ture (originating in the latti e dis retization from nearest neighborapproximations of the derivative operator); the only non-zero matrix elements onsist of nearest-neighborpairs (and in the massive ase, on-diagonal). Thus, the non-zero ontributions an be analyzed as follows.A given permutation π a(cid:27)e ts any vertex trivially (the vertex is sent to itself) or as part of a non-trivial y li permutation. In the massive ase, verti es that are permuted trivially give monomer fa tors equalto the mass M . In the massless ase where diagonal entries vanish, any trivial permutation will lead to avanishing ontribution; thus every vertex must parti ipate in a y li permutation in the massless ase.For the non-trivial permutations, it is useful to distinguish two ases that a vertex may parti ipatein. Permutations that swap a pair of neighboring verti es are referred to as dimers; all other non-trivialpermutations onsist of non-trivial loops of edges on the latti e; we shall refer to these as y les. Given theseobservations on the stru ture of Q , we an now write an expression for det Q in more expli it detail as(10) det Q yx = X π ∈ Π sgn ( π ) M N m Y ( xy ) ∈ p K xy . Where N m are the number of monomers. The produ t is over all dire ted edges ( xy ) that are part of a dimeror y le of the permutation.2.2. Coupled ase. To ouple fermions to the gauge (cid:28)elds, the ordinary derivative is repla ed by the ovariant one, thereby introdu ing the gauge variables g e whi h a t on the fermions through the matri esof the representation orresponding to the harge of the fermion. For spe i(cid:28) ity, we will onsider fermions harged in the representation of G labelled by c . One an show [12℄ that the latti e a tion for staggeredfermions oupled to the gauge (cid:28)eld be omes(11) S = X x M ( ψ x ψ x ) + K D X µ α xµ (cid:0) ψ x U † xµ ψ x +ˆ µ − ψ x +ˆ µ U xµ ψ x (cid:1)! . In ontrast to the (simpli(cid:28)ed one omponent) free ase, there is in general a ve tor possessing multipleGrassmann variable omponents at ea h vertex, and matri es U ( g e ) that a t non-trivially on this ve tor. Interms of our permutation expansion, the quark matrix now takes on omponent as well as vertex labels asfollows:(12) S F [ g e , ψ v , ψ v ] = X x,y ∈ V ψ jy Q [ g e ] jiyx ψ ix . Comparing (12) to (11) we identify the multi omponent quark matrix as(13) Q jiyx [ g e ] = M δ yx − K D X µ =1 α xµ (cid:0) ( U † xµ ) ij δ y,x − ˆ µ − ( U xµ ) ij δ y − ˆ µ,x (cid:1) . The g e dependen e is through the representation matri es U e , where e is labelled in one of the two onven-tions introdu ed above. The determinant formula an again be applied; upon doing so, the expansion into4 x x x L x i=2i=1 Figure 2. Graphi al representation of one (solid) of all possible permutations (dashed)asso iated with a y le of verti es of length L .permutations takes on the form(14) det Q [ g e ] = X π sgn ( π ) Y x,i Q iπ I ( x,i ) xπ V ( x,i ) . Observe that the permutations π now a t on both vertex and omponent indi es of Q ; the a tion of π on indi es and verti es an be separated into the maps π I and π V , respe tively. As in the free ase, thelo ality stru ture allows one to identify non-vanishing, polymer-like ontributions. For π where vertex indexmapping is trivial, the only non-vanishing entries are those for whi h π I ( x, i ) = i as the massive term is aninner produ t with no ross-terms. Thus, the monomers ontribute fa tors M n , where n are the number of omponents of the fermion ve tors.As in the single- omponent ase, for verti es that parti ipate in non-trivial permutations, one still hasthat only permutations whi h move x → π ( x ) where π ( x ) is a nearest neighbor are non-vanishing. However,due to the presen e of multiple omponents to the fermion (cid:28)eld, permutations an shift both omponents ata vertex simultaneously.Con(cid:28)gurations whi h involve non-trivial shifts of more than a single omponent (multiply o upied vertex ontributions) are dis ussed in the Appendix B. In the remainder of this se tion, we will restri t ourselvesto the ase where only a single omponent parti ipates in the shift, to illustrate in a simple setting how thetra e of a produ t of U e matri es omes about.By inspe tion of (11), we see for a given nearest neighbor vertex shift, all possible permutations of indi esare allowed, sin e in the general ase U ijxµ has all non-vanishing entries. To ontinue our analysis we fa torthe permutation into a part that a ts on verti es π V (these orrespond to the dimers and y les of the free ase) and a part that a ts on omponent indi es π I . We now write the fermion determinant as(15) det Q [ g e ] = X π V sgn ( π V ) X π I Y x Q iπ I ( x,i ) xπ V ( x ) + (multiply o upied vertex ontributions) . Fo using our attention on the (cid:28)rst term, we note that only one omponent per vertex is shifted in ea h of theprodu ts of the sum (multiple omponent shifts at a vertex are pre isely what is in luded in the se ond term,dis ussed in Appendix B). For a given π V , we an represent π I as an ordered sequen e of arrows through thedis rete n -point spa e living above every vertex a ted on by π V ; see Figure 2. Note that for a given dimeror y le π V , all possible index sequen es orrespond to permutations of the same order (thus sgn ( π V ) an befa tored out). The (cid:28)nal step in the analysis is to re ognize that the sum over all paths is simply the tra eof the matrix produ t of representation matri es around the y le.We de(cid:28)ne D as the restri tion of det Q [ g e ] to permutations involving singly o upied verti es; that is,the (cid:28)rst term of (15). D an be onstru ted out of loops with a single asso iated tra e as follows: D = X π V sgn ( π V ) X π I Y x Q iπ I ( i ) xπ V ( x ) (16) = X π V sgn ( π V )( nM ) N m K N e Y ( xy ) ∈ π V α ( xy ) U i i ( x x ) U i i ( x x ) U i i ( x x ) · · · U i L i ( x L x ) = X π V sgn ( π V )( nM ) N m K N e Y ( xy ) ∈ π V α ( xy ) Tr Y ( xy ) ∈ π V U ( xy ) , where N e is the total number of edges where verti es are shifted in the permutation. In the se ond line, theEinstein summation onvention for repeated indi es is used. Observe that ( xy ) denotes an oriented edge.5igure 3. Visualization of 2- omponent fermion (cid:28)eld on latti e. Polymers that appearto have open ends lose on the opposite side of latti e due to periodi boundary onditions.Depending on whether ( xy ) is along or opposing the anoni al orientation, one has either a produ t of U ( xy ) or U ( yx ) = U † ( xy ) . The visualization of a typi al polymer on(cid:28)guration on a D = 3 latti e (in luding thosewith multiply o upied verti es) appears in Figure 3 below.So far our dis ussion has been generi with regard to group, dimension, and fermion harge; the onlymajor hoi e has been staggered fermions rather than an alternative latti e dis retization. In the remainderof this work, we restri t our attention to D = 3 , G = SU (2) , and n - omponent massive fermion (cid:28)elds hargedwith half-integer spin c and minimally oupled to SU (2) .3. Spin Foam Des ription of Pure Gauge Theory on the Latti eHaving des ribed what we shall refer to as the free (no oupling to gauge (cid:28)elds) fermion partition fun tionin terms of losed latti e polymers, in this se tion we brie(cid:29)y review the dual formulation of pure (no ouplingto fermions) Yang-Mills, whi h leads to losed, olored, bran hed latti e surfa es. We shall see in the nextse tion that these pi tures naturally ombine to give the full intera ting partition fun tion in terms of aspa e of oupled on(cid:28)gurations.It an be shown (see for example [3, 13℄, and [5℄ for detail on G = SU (2) in three dimensions) that startingfrom the latti e dis retized a tion for pure Yang-Mills(17) Z B = Z Y e ∈ E dg e e − P p ∈ P S ( g e ) , one an transform to a spin foam formulation expressing the partition fun tion in terms of dual variables asfollows:(18) Z B = X j X i Y v ∈ V j v ( i v , j v ) Y e ∈ E N e ( i e , j e ) − ! Y p ∈ P e − β j p ( j p +1) (2 j p + 1) . Here V , E , and P denotes the verti es, edges, and plaquettes of the latti e, respe tively. The summationsover i and j range over all possible edge and plaquette labellings, respe tively. A plaquette labelling j assigns an irredu ible representation of SU (2) to ea h element of P . These representations are labelled bynon-negative half-integers (we will denote this set by N ), also referred to as spins; a labelling j is thus amap j : P → N . In the SU (2) , D = 3 ase, edges are also labeled by half-integer representations. thus anedge labelling is a map i : E → N . 6igure 4. Visualization of pure Yang-Mills va uum on latti e.Following [5℄, we de(cid:28)ne a (va uum) spin foam on(cid:28)guration as one summand in (18), i.e. a labelling ofboth plaquettes and edges by spins and intertwiners, respe tively. The amplitude assigned to a spin foamfa tors into a lo al produ t of amplitudes. The vertex amplitude ( j symbol) depends on the 12 plaquettesand 6 edges in ident to a vertex; the N e fa tors depend on the 4 plaquettes and the intertwiner labelling ofan edge, and there is a produ t of lo al plaquette fa tors.The lo ality of the spin foam formulation was applied in [5℄ to perform omputations that were veri(cid:28)edagainst onventional methods. 4. Dual Fermion-Boson SimulationsIn this se tion we des ribe how the dual pi tures of latti e fermions and gauge bosons presented in theprevious two se tions an be ombined to form a joint dual partition fun tion, built up of gauge-invariant on(cid:28)gurations with dis rete o upan y and representation labels.4.1. The Joint Partition Fun tion. Using the a tion S [ g e , ψ v , ψ v ] for the full theory we write the partitionfun tion as follows: Z J = Z Y e ∈ E dg e ! Z Y v ∈ V dψ v dψ v ! e − S F [ g e ,ψ v ,ψ v ] e − S G [ g e ] (19) = Z Y e ∈ E dg e ! det Q [ g e ] e − S G [ g e ] , where we have integrated out the fermioni variables to get the fermion determinant. We next use thepolymer expansion for the determinant in the ase of gauge- oupled Q , as des ribed in Se tion 2.2: Z J = Z Y e ∈ E dg e ! det Q [ g e ] e − S G [ g e ] (20) = Z Y e ∈ E dg e ! X γ sgn ( γ )( nM ) N m K N K Y ( xy ) ∈ γ α ( xy ) Tr Y ( xy ) ∈ γ U ( xy ) e − S G [ g e ] . Here N K is the number of K fa tors (one per unit of edge o upan y) in the polymer on(cid:28)guration; thede(cid:28)nition of polymer on(cid:28)guration for D = 3 , G = SU (2) is given in Se tion 4.3.1. In the se ond line, we7ave substituted the polymer expansion for fermion determinant. Next, we re all the form of the hara terexpansion (see [5℄ and referen es therein) for the amplitude based on the heat kernel a tion at a plaquette p ,(21) e − S p ( g ) = 1 K ( I, γ ) X j (2 j + 1) e − γ j ( j +1) χ j ( g ) , j = 0 , , , . . . Substituting the hara ter expansion into the previous equation, we have Z J = Z Y e ∈ E dg e ! X γ sgn ( γ )( nM ) N m K N K Y ( xy ) ∈ γ α ( xy ) Tr Y ( xy ) ∈ γ U ( xy ) (22) × Y p ∈ P X j p (2 j p + 1) e − γ j p ( j p +1) χ j ( g ) , where an overall onstant fa tor of K ( I, γ ) per plaquette has been dis arded. We show in Appendix A thatthe group integrals over produ ts of tra es and hara ters in ea h term of the hara ter expansion an beevaluated exa tly in terms of harged 18j symbols, provided a sum over intertwiner labels is made at ea hedge. Using the vertex and edge amplitudes of Appendix A, we an exhibit the joint dual partition fun tionas Z J = X γ ∈P X j X i s ( γ ) Y v ∈ V j v ( i v , j v , γ ) Y e ∈ E N e ( i e , j e , γ ) − Y p ∈ P e − β j p ( j p +1) (2 j p + 1) , (23)where s ( γ ) ≡ sgn ( γ ) Q ( xy ) ∈ γ α ( xy ) ( nM ) N m K N K ombines the two sign fa tors and a produ t of M and K fa tors. As we shall see in the next se tion, joint on(cid:28)gurations asso iated to a polymer γ arry in generalthree rather than a single intertwiner label i e for ea h edge belonging to the polymer; i here ranges over allthe intertwiner labels.Although the overall dual amplitude is still a produ t of lo al amplitudes as in the pure Yang-Mills ase,the presen e of the Wilson loop fun tionals asso iated to non-trivial polymers requires the va uum vertexand edge amplitudes to be modi(cid:28)ed in a way that we de(cid:28)ne in Appendix A; the result is a produ t ofmodi(cid:28)ed j v symbols and edge amplitudes N e that are harged a ording to the polymer ontent γ of the on(cid:28)guration.The joint ensemble that results here an be viewed as a generalization of the usual de(cid:28)nition of spin foamsto in lude one-dimensional stru ture orresponding to the presen e of fermioni harge. For a given polymer,there is a sum over all spin foams satisfying admissibility, whi h is modi(cid:28)ed at the polymer edges. From theworldsheet point of view [4℄, a polymer loop a ts as the sour e or sink of c fundamental sheets, where c isthe half-integer harge of the fermion.4.2. The Joint Fermion-Boson Con(cid:28)gurations. In this se tion we de(cid:28)ne expli itly the set of on(cid:28)gu-rations that in lude all those that give non-zero ontributions3 to the joint dual partition fun tion.Spe i(cid:28) ally, for a given polymer γ , we introdu e de(cid:28)nitions that will allow us to hara terize the set ofspin foam on(cid:28)gurations (plaquette olorings) that are admissible in the presen e of γ .As in the pure Yang-Mills ase [5℄, we assume a splitting has been made for ea h edge with j and j onone side and j and j on the other. Be ause of the presen e of harges c and c , the intertwiner is generally6-valent and three splittings have to be made. For dis ussing edge admissibility, we assume the splitting issu h that c and c are in the middle of the hannel as shown in Figure 5. Less symmetri splittings arepossible, but we restri t our attention to this ase in the following.Let γ denote a polymer on(cid:28)guration of harge c . We de(cid:28)ne the set of γ -admissible plaquette on(cid:28)gurationsas those on(cid:28)gurations whose labellings satisfy c -edge admissibility at every edge in the latti e, where c -admissibility is de(cid:28)ned as follows:De(cid:28)nition 4.1 ( c -edge admissibility). The spins assigned to plaquettes in ident to an edge are said to be c -edge admissible if the parity and triangle inequality onditions are satis(cid:28)ed. The harge insertions c and3Due to ex eptional zeros there may be on(cid:28)gurations that are admissible by the onditions de(cid:28)ned in this se tion, but arenonetheless zero. As in the pure Yang-Mills ase [5℄, we assume ex eptional zeros are su(cid:30) iently isolated that ergodi ity onadmissibles is equivalent to ergodi ity on non-zero on(cid:28)gurations.8 j i ic i j j Figure 5. Symmetri splitting of a 6-valent SU (2) spin network vertex. c may be 0 or c , a ording to whether the edge is un harged, singly harged, or doubly harged. Writing j , j , j and j for the four spins in ident to a given edge, these onditions are(1) Parity: j + j + j + j + c + c is an integer.(2) Triangle Inequality: for ea h permutation x ≡ ( x , x , x , x , x , x ) of the harge and spin vari-ables ( c , c , j , j , j , j ) we have x + x + x + x + x ≥ x . These onditions are equivalent to the existen e of a non-zero invariant ve tor in the SU (2) representation c ⊗ c ⊗ j ⊗ j ⊗ j ⊗ j .The allowed range of intertwiner labels i , i c and i depend on the in ident spin labels, on ea h other andon c through the verti es where the harges c and c enter the diagram. We now state the ondition for andadmissible spin foam (plaqeutte and edge intertwiner labelling) in the presen e of an arbitrary polymer γ :De(cid:28)nition 4.2 ( γ -admissible spin foam). A spin foam is γ -admissible if and only if for every edge e ∈ E :(1) The plaquettes in ident to e are c -edge admissible in the sense of De(cid:28)nition 4.1, with c and c assigned depending on the o upan y of e by γ .(2) Ea h vertex of Figure 5 is admissible. Expli itly, the following onditions are simultaneously satis(cid:28)ed: i + j + j , i + i c + c , i + i c + c and i + j + j are integersand i ∈ [ | j − j | , j + j ] ∩ [ | i c − c | , | i c + c | ] ,i c ∈ [ | i − c | , i + c ] ∩ [ | i − c | , | i + c | ] ,i ∈ [ | j − j | , j + j ] ∩ [ | i c − c | , | i c + c | ] . For a given polymer γ , we denote the set of γ -admissible spin foams by F Aγ .4.3. The Joint Moves and Algorithm. In this se tion we de(cid:28)ne moves that transform from one joint on(cid:28)guration to another. Together, they are ergodi and obey detailed balan e and an thus be used in aMetropolis or other Markov hain Monte Carlo algorithms.De(cid:28)nition 4.3 (Pure spin foam move). A pure spin foam move onsists of a single appli ation of the ube,edge, or homology move. In terms of their e(cid:27)e t on plaquette spins, these moves are as de(cid:28)ned in [5℄.However, their e(cid:27)e t on intertwiner labels needs to be generalized to a ount for the extra intertwinerlabellings introdu ed by polymers.Be ause polymer moves require simultaneous hanges in spin foam stru ture, we use the term (cid:16)pure(cid:17) todistinguish spin moves that leave the polymer stru ture un hanged. We now des ribe the generalization ofea h pure spin foam move to a ount for extra intertwiner labels.De(cid:28)nition 4.4 (Generalized ube move). As des ribed in De(cid:28)nition 2.4 and Appendix A.3 of [5℄. Withreferen e to ompatible intertwiner moves of Type A, no hanges in intertwiner labels are ne essary. ForType B edges, all three labels are in reased or de reased by the same half-unit of spin.9e(cid:28)nition 4.5 (Generalized edge move). As des ribed in De(cid:28)nition 2.5 of [5℄. Rather than hanging thesingle intertwiner label, the three intertwiner labels i ( e ) , i ( e ) and i c ( e ) are ea h randomly hanged by − , , or (to preserve parity) units of spin.De(cid:28)nition 4.6 (Generalized homology move). As des ribed in De(cid:28)nition 2.6 of [5℄, but all three intertwinerlabels are in reased or de reased by one half-unit of spin.4.3.1. Polymer moves. In Se tion 2.2, we saw how sums over permutations in Π an be en oded into tra esof matrix produ ts, ordered a ording to the orientation of the permutation. Our example was restri ted tosingly o upied verti es, and is generalized in Appendix B. Combining these ases, we see that the sum over allpermutations in Π an be represented by tra es of produ ts of matri es, if we in lude diagrams orrespondingto all possible routings at multiply o upied verti es. This new set of obje ts, oriented diagrams with routingsat multiply o upied verti es, we refer to as polymers, and denote by P . It is important to distinguish thepolymers from their (cid:28)ner-grained onstituents, the permutations Π , as polymers an be oupled naturallyinto the spin foam partition fun tion, whereas individual permutations annot be using the methods here.In this se tion we present a set of moves that are ergodi on the spa e of polymers P on a 3-dimensionalhyper ubi latti e. The polymer states at an edge in the - omponent ase onsidered here are as follows.Assuming a global orientation has been sele ted for the edges, an edge an be uno upied, singly o upied,or doubly o upied with the o upied ases arrying both positive and negative orientation. The o upan ydata at ea h edge an thus be assigned from the set {− , − , , , } . We shall use the term (cid:16)line of (cid:29)ux(cid:17)inter hangeably with dire ted polymer line.De(cid:28)nition 4.7 (Plaquette move). A plaquette and plaquette orientation ( lo kwise or ounter lo kwise) israndomly sele ted. To ea h edge, a delta o upan y of +1 or − (with signs given a ording to plaquetteorientation) is assigned, and added to the present o upan y. If the resulting o upan y on any edgehas magnitude greater than 2, the move is immediately reje ted. At ea h multi-valent vertex, a hoi e ofrouting is made with equal probability. A proposed move that removes o upan y of edges in ident onmultiply o upied verti es must make further random hoi e of routing that mat hes the routing presentor be reje ted. This is ne essary to preserve detailed balan e. If the move is not reje ted, the spin of thesele ted plaquette is randomly de reased or in reased by c to satisfy parity. Be ause a hange in the polymero upan y for es a hange in both the plaquette spin and the harge stru ture at an edge, the a(cid:27)e tedintertwiner labels (at ea h edge of the a(cid:27)e ted plaquette) must hange in a way that is ompatible; if notthe result will be immediately c -edge inadmissible for the new harging.This is the most fundamental polymer- hanging move, and onne ts a very large region of the spa eof polymers ontributing to the fermion determinant. One an see trivially that the plaquette move an reate fundamental loops of either orientation when applied to (cid:16)empty(cid:17) spa e (plaquettes of zero o upan yedges). Figure 6 illustrates how the plaquette move an deform an existing y le. The same plaquette moveof opposite orientation would lead to one of multiple routings with a doubly o upied edge, as shown byFigure 10 in Appendix B.Be ause an equally weighted routing hoi e is made amongst several alternatives when the plaquette moveof (for example) Figure 10 is made, the move that reverses a parti ular routing to get the initial loop ba kshould o ur with proportional probability, in order to satisfy detailed balan e. This will unfortunately leadto a lowered a eptan e rate, parti ularly in regimes with high o upan y.To identify intertwiner moves ompatible with a plaquette move, one must give the hoi e of splitting(grouping of j , . . . , j into pairs) around the edges of a plaquette, in the same way that intertwiner moves ompatible with a ube move depends on the splitting (see A.3 of [5℄). For on iseness and generality, wehave made the present de(cid:28)nition of the algorithm splitting independent. Forth oming work on numeri alsimulations with this algorithm will evaluate alternative splittings and des ribe the appropriate ompatiblemoves.De(cid:28)nition 4.8 (Global ir le move). For integer valued harge c , a global ir le moves adds a single line of harge and a minimal ylinder of c harged plaquette spins spanning the latti e. The origin and orientationof the ylinder is randomly sele ted to lie on a plaquette of one of three orthogonal latti e planes throughthe origin. 10igure 6. Stret hing of a loop by a plaquette moveFor half-integer c , two lines of harge spanning the latti e are added and a minimal surfa e onsisting ofa line of c - harged plaquettes is added between the harge lines. The position and orientation of the sheetis randomly sele ted to lie on an edge of one of the three orthogonal latti e planes through the origin.As in the pure gauge theory ase, the non-trivial global topology of latti es with periodi boundary onditions leads one to moves that reate and destroy stru ture on a global s ale, in this ase lines of harge.In the half-unit harge ase, a se ond line needs to be added to absorb the (cid:29)ux introdu ed by the (cid:28)rst;the (cid:16)smallest(cid:17) possible global move pla es a se ond line of half-unit harge immediately beside the (cid:28)rst.In the ase of integer valued harge, the lowest energy (and hen e smallest hange in amplitude) stru turesatisfying admissibility is formed by wrapping the smallest possible ylinder supported by the latti e; be ausethere are n fundamental irreps, n an be in ident on one side normal to the line, wrap into a ylinder, andenter the other to satisfy c -edge admissibility.De(cid:28)nition 4.9 (Dimer move). An edge is randomly sele ted and a single unit of o upan y is added orremoved. If the result is not onsistent with the presen e of dimers and y le edges (i.e. a y le that ranthrough the edge is broken by the removal of o upan y), the move is reje ted.Singly and doubly harged dimers annot be onstru ted by plaquette moves. Note a singly harged dimer ontributes Tr ( U e U † e ) = n with weight K while a doubly harged dimer ontributes Tr ( U e U † e ) Tr ( U e U † e ) − Tr ( U e U † e U e U † e ) = n − n with weight K (unlike general polymers, we ombine the two routings into one on(cid:28)guration). Both types of dimers evaluate to onstants with respe t to the gauge variables, and thusdon't ouple to the gauge bosons.De(cid:28)nition 4.10 (Jun tion move). A vertex is randomly sele ted, and if multiply o upied, the routing of(cid:29)ux is hanged.A multiply o upied vertex in the ase of a 2- omponent fermion (cid:28)eld has two (cid:29)ux paths, whi h an berouted in two di(cid:27)erent ways. When a multiply o upied vertex (cid:28)rst appears as a result of a polymer move,one routing is randomly sele ted (similarly in the inverse ase, with weightings to preserve detailed balan eas dis ussed above). Thus, the jun tion moves are stri ly speaking unne essary for ergodi ity, but may beused to improve performan e of Metropolis algorithm.4.3.2. The Algorithm. Combining the polymer and spin foam moves, given, we give a statement for a Me-tropolis algorithm ergodi on the joint ensemble.Algorithm 4.11. (Joint Fermion-Boson Algorithm). An iteration of the joint algorithm onsists of hoosingone of the seven previously de(cid:28)ned moves, whi h an be organized as as follows:11 oint Move Homology moveGlobal circle moveCube moveEdge move Generalized PureSpin Foam Move Plaquette movePolymer move Dimer moveJunction move The algorithm an be tuned to improve a eptan e rate by adjusting the relative frequen y of the attemptedmove types.The algorithm also tra ks the sign of the on(cid:28)guration hanges with the reation and destru tion offermion loops, whi h an o ur with any of the polymer moves. The produ t of monomer fa tors M andhopping fa tors K are also updated for polymer moves. It is important to emphasize that hanges in boththe sign and other fa tors require only lo al onsideration of the polymer moves.With regard to lo ality, one sees that the (pure spin foam) homology move and global ir le moves willlead to updates osting on the order of L and L respe tively, where L is a hara teristi side length ofthe latti e. In the pure Yang-Mills ase analyzed numeri ally in [5℄, the homology moves have negligiblein(cid:29)uen e beyond very small latti e sizes. It remains to be seen how this is modi(cid:28)ed in the joint ase, andhow large an in(cid:29)uen e the global ir le moves have.Within the s ope of the urrent work, the expe tation values of observables depending on dual degrees offreedom are omputed in the usual manner, by averaging the observable over the Markov hain generatedby the Metropolis algorithm. For Wilson loop type observables ommonly studied, the expe tation value isa tually a ratio of dual harged and dual va uum partition fun tions, with stati harge orresponding tothe Wilson loop observable present in the harged partition fun tion and a va uum partition fun tion givenby Z J of equation (23). The omputation of Wilson loop observables of pure Yang-mills and dynami alfermions will be reported on in forth oming work by the author.5. Outlook and Con lusionsWe present here a lo al, exa t algorithm for Metropolis simulation of the fermion-boson va uum. Thedetails have been provided for the ase of D = 3 staggered Kogut-Susskind fermions oupled to a Yang-Mills SU (2) (cid:28)eld; however the algorithm has a straightforward generalization to other dimensions and gaugegroups.A limitation of the algorithm as urrently given is the spe ies doubling inherent in the (unrooted) Kogut-Susskind formulation (e.g. in four dimensions there will be four spe ies). An alternative to Kogut-Susskindfermions whi h addresses spe ies doubling was developed by Aro a et al. [1℄ and Fort [7℄. We are urrentlyinvestigating this modi(cid:28)ed fermion a tion in the non-abelian, higher dimensional ontext. Another approa hwould be to go through a similar pro edure using Wilson fermions, in whi h the unwanted doublers be omevery heavy in the ontinuum limit.The ru ial question for any new fermion algorithm is its performan e relative to the highly evolveddynami fermion methods that exist within the onventional latti e gauge ommunity today. In three-dimensions, slow-down at weaker oupling has been observed in re ent work on the pure Yang-Mills ase [5℄.12he situation in D = 4 is not well understood and is urrently the subje t of numeri al work by the author,as are improvements in the original D = 3 ase.With regard to the ontinuum limit, we expe t the most riti al question for the algorithm proposed hereis the seriousness of the sign problem. A hard sign problem has been dis ussed as a general feature of thepolymer expansion in the ontinuum (small mass) limit [11℄. While os illating signs an be over ome forlatti e fermions in ertain two dimensional theories [11, 15℄, the author is not aware of methods that havesu essfully addressed the sign problem for D > . As both the fermion and dual Yang-Mills (spin foam)amplitudes an arry negative signs, an important question is how the signs intera t; i.e. the problem ofsigns may be harder or easier than for either the free fermion polymer expansion or dual Yang-Mills alone,depending on how the signs orrelate.Although numeri al developments are required to begin evaluating this proposal, we believe the approa hmay be of onsiderable interest. We (cid:28)nd it remarkable that the fermion expansion into polymers and thegauge (cid:28)eld dualization into spin foams (both of whi h have been extensively explored on there own), ombinetogether in a way that is very ompelling geometri ally, and allows a lo al, exa t Metropolis simulation usinggauge-invariant on(cid:28)gurations arrying entirely dis rete labels.A knowledgement. The author would like to thank Dan Christensen, Florian Conrady, and Igor Khavkinefor valuable dis ussions. The author was supported by NSERC.Appendix A. Charged nJ SymbolsA.1. The dual model with harges. In this appendix, we deal spe i(cid:28) ally with D = 3 , G = SU (2) .Following the dis ussion in the appendix of [5℄, we re all that the dual partition fun tion (in the absen e of harge) has the form(24) Z = X { j p } Z Y e ∈ E dg e Y p ∈ P c j p χ j p ( g p ) , where summation over j p is over unitary irredu ible representations of SU (2) . At this point it is onvenientto spe ialize to a D = 3 ubi latti e with periodi boundary onditions; orientation hoi e is as given inthe appendix of [5℄. With this hoi e of orientation, the holonomy around a plaquette p is g p = g g g − g − ,where g , g , g and g are the group elements asso iated to the edges of the plaquette p , starting with anappropriate edge and going y li ally. Re all that the inverse g − i is used if the orientation of edge i doesnot agree with that of p . Thus(25) χ j p ( g p ) = U j p ( g ) ba U j p ( g ) cb U j p ( g − ) dc U j p ( g − ) ad , where U j ( g ) ba denotes a matrix element with respe t to a basis of the j representation. If we insert (25)into (24) and olle t together fa tors depending on the group element g e , we get a produ t of independentintegrals over the group, ea h of the form(26) Z dg e U j ( g e ) b a U j ( g e ) b a U j ( g − e ) b a U j ( g − e ) b a = Z dg e j j j j g e g e −1 g e −1 g e . Here and below we use a graphi al notation for tensor ontra tions, as in [5℄.Equation (26) de(cid:28)nes a proje tion operator on the spa e of linear maps j ⊗ j → j ⊗ j , so it an beresolved into a sum over a basis of intertwiners I i : j ⊗ j → j ⊗ j (27) Z dg e j j j j g e g e −1 g e −1 g e = X i I i I ∗ i h I ∗ i , I i i = X i j j j j I i j j j j I i i i j j j j , I ∗ i : j ⊗ j → j ⊗ j are hosen su h that the tra e h I ∗ i ′ , I i i of the omposite I ∗ i ′ I i iszero whenever i ′ = i and non-zero if i ′ = i . The proje tion property is readily veri(cid:28)ed.We next de(cid:28)ne Z γ , the partition fun tion harged a ording to the polymer γ , as follows(28) Z γ = X { j p } Z Y e ∈ E dg e Tr Y e ∈ γ U ec ( g e ) i + e i − e ! Y p ∈ P c j p χ j p ( g p ) . Colle ting matrix fa tors by dependen e on edge variable g e , we (cid:28)nd in addition to the matri es from thefour in ident plaquettes, a matrix from the edge e with harge c belonging to the polymer γ :4(29) Z dg e U j ( g e ) b a U j ( g e ) b a U j ( g − e ) b a U j ( g − e ) b a U ec ( g e ) i + e i − e = Z dg e j j j cj g e −1 g e g e g e g e −1 . As in the pure ase, the group integral an be resolved into invariant intertwiners(30) Z dg e j j j cj g e −1 g e g e g e g e −1 = X i I i I ∗ i h I ∗ i , I i i = X i I i j j j j c * I i j j j j c c I *i I i j j j j . If, for ea h edge of the latti e, we (cid:28)x a term i in the above summation, the intertwiners I i and I ∗ i anbe ontra ted with those oming from the other edges, leading to a sum over intertwiner labellings at everyedge. Observe that at edges o upied by polymers, there is more than a single intertwiner spin label due tothe additional splittings (see Figure 5) introdu ed by the harge lines. At ea h vertex of the latti e, there willbe six intertwiners I i (some arrying multiple labels), and their ontra tion an be graphi ally representedas an o tahedral network plus additional lines depending on how the polymer passes through the vertex. Aswell, at ea h edge there will be a normalization fa tor orresponding to the denominator of equation (30).We onsider (cid:28)rst the va uum ase. In this ase, ea h edge arries only a single intertwiner label. Theresult is the j symbol entral to pure Yang-Mills spin foams,(31) +z−y −x+y+x −z . The verti es are labelled by the dire tions of the asso iated latti e edges emanating from the given latti evertex, namely ± x , ± y , and ± z . The value of the j symbol depends on the hoi e of basis elements I i and I ∗ i ′ in (30), the six summation indi es i labelling the edges, and the 12 in ident plaquette labels j .We now turn to the ase where there is a (single) polymer along one or more of the edges in ident to avertex. Ea h harged normalization fa tor N ≡ h I ∗ i , I i i in the denominator of (27) depends on the harge c of the fermion at that edge, the intertwiner labels i on that edge, and the labels of the four plaquettesin ident on that edge. In the presen e of external harges, the va uum j is modi(cid:28)ed depending on whether4In the ase where an edge is doubly o upied, the integral involves a sixth matrix (and the resolving intertwiners anadditional input and output arrow). It is straightforward to generalize the present analysis to this ase; the resulting doubly harged j symbols are shown in Appendix B. 14he line of harge pro eeds dire tly through the vertex or turns, leaving in a dire tion perpendi ular to entrydire tion. We all these ases harged j symbols and denote them by an overline, j ( j v , i e , γ ) where theadditional dependen e on gamma re(cid:29)e ts how the polymer harge is routed through the o tahedral network.Typi al harged j symbols are shown in Figure 7. Cases where the dire tion of the arrow on the harged +z−y −x+y+x −z +z−y −x+y+x −z Figure 7. Charged j symbols with (cid:29)ux lines passing through at right angles (left) andstraight through (right).lines is (cid:29)ipped will also o ur, depending on the polymer orientation. Additionally, as dis ussed in the nextse tion, more than a single line of (cid:29)ux an pass through a vertex, leading to harged j symbols of the formshown in Figures 8 and 11.In implementing numeri al ode for this algorithm, a hoi e of splitting (grouping of the four plaquettesinto ( j , j ) and ( j , j ) pairs on opposite sides of the splitting) is made and ea h vertex resolved into a 3-valentsub-network with up to three non-trivial intertwiner labels, as shown in Figure 5. At this point, re ouplingmoves (see A.2 of [5℄, and referen es therein) an be used to redu e the spin network to sums and produ tsof know spin networks su h as the j and theta networks, for whi h e(cid:30) ient algorithms are available. Itshould be noted, however, that di(cid:27)erent splittings lead to di(cid:27)ering e(cid:30) ien y in implementation, so some areand experimentation should be applied to (cid:28)nding an e(cid:30) ient splitting. Spe i(cid:28) splitting s hemes and theirperforman e evaluation will be reported on in forth oming numeri al work by the author and ollaborators. +z−y −x+y+x −z+z−y −x+y+x −z Figure 8. Charged j symbols with two pairs of (cid:29)ux. Cases with (cid:29)ux not in the sameplane are also possible.Appendix B. Polymers with multiply o upied verti esIn order to ouple to the spin foam representation of the gauge theory, we seek to olle t the permutation ontributions to the fermion determinant into tra es of produ ts of U e matri es around losed, oriented loopsof edges. The ase of permutations where a single omponent is shifted was dis ussed above in Se tion 2.2.For polymers where more than one omponent is shifted at a vertex, re overing a tra e formula is somewhatmore subtle.In Figure 9, we illustrate a ase where a vertex is multiply o upied; the orientation is su h that thereare two possible routings that resolve the ambiguity at that vertex. Neither diagram by itself orrespondsto the desired sum of permutation ontributions. In the matrix multipli ations and tra es (viewed as a sum15igure 9. Two routings asso iated with a polymer that self interse ts on e at a point.Figure 10. A move introdu ing a doubly o upied edge, for whi h there are two distin t routings.over all paths around a loop), there are terms in ea h orresponding to paths that are not permutations.However, the same undesired terms o ur with opposite sign in the two diagrams (as one involves paths thatform a single loop, the other paths that lie in two disjoint loops) so the sum of both aptures the sum ofpermutations asso iated with the polymer.A similar an ellation o urs when two loops share a single edge, i.e. the edge is multiply o upied.Be ause there are two multiply o uppied verti es, there are = 4 routings possible, however only two aretopologi ally distin t; two representatives appear in Figure 10. The an ellation of unphysi al paths betweenthe tra es over di(cid:27)erently routed polymers is well known from the hopping parameter expansion (HPE) ofthe fermion determinant as dis ussed for example in [14℄. As shown in Figure 11, the presen e of a doubly harged edge leads to a harged j spin network with a 6-valent node. As well, the harged normalizationfa tor N on a doubly harged edge is as given in the denominator of equation (27), but with an additional c - harged line parallel to the original c - harged line. 16 z−y −x+y+x −z (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)