We analyse the pure Chern-Simons theory on an Euclidean infinite lattice. We
point out that, as a consequence of its symmetries, the Chern-Simons theory
does not have an integrable kernel. Due to the linearity of the action in the
derivatives, the situation is very similar to the one arising in the lattice
formulation of fermionic theories. Doubling of bosonic degrees of freedom is
removed by adding a Maxwell term with a mechanism similar to the one proposed
by Wilson for fermionic models.Comment: Lattice 2000, 4 pages, Late

Berruto, F.

Diamantini, M. C.

Sodano, P.

English

arXiv

We analyse the pure Chern-Simons theory on an Euclidean infinite lattice. We point out that, as a consequence of its symmetries, the Chern-Simons theory does not have an integrable kernel. Due to the linearity of the action in the derivatives, the situation is very similar to the one arising in the lattice formulation of fermionic theories. Doubling of bosonic degrees of freedom is removed by adding a Maxwell term with a mechanism similar to the one proposed by Wilson for fermionic models.We analyse the pure Chern-Simons theory on an Euclidean infinite lattice. We point out that, as a consequence of its symmetries, the Chern-Simons theory does not have an integrable kernel. Due to the linearity of the action in the derivatives, the situation is very similar to the one arising in the lattice formulation of fermionic theories. Doubling of bosonic degrees of freedom is removed by adding a Maxwell term with a mechanism similar to the one proposed by Wilson for fermionic models

Diamantini, M.C.

CERN Document Server

arXiv:hep-lat/0011052v1  10 Nov 20001On the Doubling Phenomenon in Lattice Chern-Simons Theories ∗DFUPG-53-00F. Berruto(a), M.C. Diamantini†(b) and P. Sodano(a)a)Dipartimento di Fisica and Sezione I.N.F.N.,Universita´ di Perugia, Via Pascoli I-06123 Perugia, Italyb)CERN, Theory Division, CH-1211 Geneva, SwitzerlandWe analyse the pure Chern-Simons theory on an Euclidean infinite lattice. We point out that, as a consequenceof its symmetries, the Chern-Simons theory does not have an integrable kernel. Due to the linearity of the actionin the derivatives, the situation is very similar to the one arising in the lattice formulation of fermionic theories.Doubling of bosonic degrees of freedom is removed by adding a Maxwell term with a mechanism similar to theone proposed by Wilson for the fermionic theories.1. IntroductionIn odd space-time dimensions, there is the pos-sibility of adding a gauge invariant, topologicalChern-Simons (CS) term to the gauge field ac-tion. The CS term breaks both the parity andtime-reversal symmetries and, when coupled witha Maxwell or Yang-Mills term, leads to massivegauge excitations [1]. For an Abelian model inthree space-time dimensions, the pure CS La-grangian is defined asLCS =k2Aµǫµαν∂αAν , (1)where k is a dimensionless coupling constant.The pure CS theory is a topological field the-ory [2]. Being dominant at large distances, theCS action may be used as a low energy effectivefield theory for condensed matter systems[3].While in the continuum the pure CS theoryis exactly solvable, things are quite different onthe lattice: in fact the kernel defining the CS ac-tion exhibits a set of zeroes which are not dueto gauge invariance [4] and the theory is not in-tegrable even after gauge fixing. The action (1)is of first order in the derivatives, and the ap-pearence of extra zeros in its lattice formulationis reminescent of the “doubling” of fermions onthe lattice [5]. Doubling phenomena for bosonic∗Presented by P. Sodano at Lattice ’00, Bangalore, India.†Supported by a Swiss National Science Foundationfellowship.degrees of freedom have been already evidencedin Euclidean lattice gravity [6] and in the anal-ysis of the coupling of gravity to matter on thelattice [7].Previous studies of pure CS theory on the lat-tice have been carried out using the Hamiltonianformalism in [8], by introducing a mixed CS ac-tion with two gauge fields with opposite parity orby means of two gauge fields living on the links oftwo dual lattices (thereby obtaining in both casesa parity even action) [9].Here we shall evidence that as a result of thetheorem proved in [10] the non-integrability ofthe CS kernel is a general feature of any gauge-invariant, local, parity odd and cubic symmetricgauge theory on an infinite Euclidean lattice pro-vided that under parity Aµ(~x) −→ Aµ(−~x − µˆ)and Aµ(~p) −→ −Aµ(−~p). Since the addition of aMaxwell term regularizes the CS action, the pres-ence of the extra zeros in the CS action did notcause any problem in previous investigations ofthe Maxwell-CS action on the lattice [4,11]. Al-though in physical applications one is naturallylead to study the dynamics of gauge models de-scribed by a Maxwell-CS action, the proper def-inition of a pure CS action on the lattice is aninteresting problem in its own right since it couldopen the way to a lattice evaluation of topologicalinvariants [2].The question naturally arises if it is possible todefine a parity on the lattice so that the Chern-2Simons term is parity odd but the kernel is inte-grable.2. Euclidean Chern-Simons theory on thelatticeWe consider an infinite Euclidean cubic lat-tice with lattice spacing a, which we set to unity(a = 1). We shall denote a lattice site by the vec-tor ~x and a link between ~x and ~x+ µˆ (µ = 0, 1, 2)by (~x, µˆ). Forward and backward difference op-erators are given by dµf(~x) = f(~x+ µˆ)− f(~x) =(Sµ−1)f(~x) and dˆµf(~x) = f(~x)−f(~x− µˆ) = (1−S−1µ )f(~x), where Sµf(~x) = f(~x + µˆ), S−1µ f(~x) =f(~x − µˆ) are the forward and backward shift op-erators respectively. Summation by parts on thelattice interchanges the forward and backwardderivatives:∑~x f(~x)dµg(~x) = −∑~x dˆµf(~x)g(~x).The lattice Fourier transformation of the gaugefield Aµ is given byAµ(~x) =∫Bd3p(2π)3e−i~p·~xe−ipµ/2Aµ(~p) . (2)Due to the phase factor e−ipµ/2, Aµ(~p) is an-tiperiodic if pµ −→ pµ + 2π(2n + 1), withn integer. The integration over momenta ineq.(2) is restricted to the Brillouin zone B ={pµ| − π ≤ pµ ≤ π , µ = 0, 1, 2}. Under parity,which on an Euclidean cubic lattice correspondsto the simultaneous inversion of all three direc-tions,Aµ(~x) −→ Aµ(−~x− µˆ) (3)andAµ(~p) −→ −Aµ(−~p) . (4)The CS action on an Euclidean lattice derived byFro¨lich and Marchetti [4] is:S =∑~xAµ(~x)K˜µν(~x− ~y)Aν(~y) , (5)where K˜µν = Kµν + Kˆµν , andKµν(~x− ~y) = S~yµ ǫµαν d~yα δ~x,~y , (6)Kˆµν(~x− ~y) = S−1,~yν ǫµαν dˆ~yα δ~x,~y . (7)K and Kˆ are exchanged by summation by parts.Although both K and Kˆ define a gauge invari-ant and parity odd kernel, Bose symmetry [12]requires that only the linear combination K˜ =K + Kˆ respects this symmetry and thus providesan acceptable definition of the lattice CS action.The operator K˜(p) = K(p) + Kˆ(p) has, apartfrom the zero mode associated with gauge invari-ance, eigenvalues given byλ˜(p) = ±2√√√√1 + cos2∑µ=0pµ√√√√3−2∑µ=0cos pµ (8)and thus λ˜ = 0 whenever cos∑2µ=0 pµ = −1, i.e.when∑2µ=0 pµ = (2n+1)π. The CS action (5) istherefore not integrable.The properties of K and Kˆ parallel the onesof the forward and backward derivatives, whichin momentum space read dµ → eipµ/2pˆµ anddˆµ → e−ipµ/2pˆµ with pˆµ = 2 sin pµ/2: they do nothave extra zeroes inside the Brillouin zone, buttheir linear combination d + dˆ → 2 cos(pµ/2)pˆµhas zeros at the border of the Brillouin zonepµ = ±π.The appearance of the extra zeroes is not dueto the specific form of the kernel in (5). In fact,as proven in [10], if the actionS =∑~x,~yAµ(~x)Gµν(~x− ~y)Aν(~y) (9)i) is local on the lattice;ii) is gauge invariant: dˆxµGµν(~x−~y) = dˆyνGµν(~x−~y) = 0;iii) is odd under parity;(9) is not integrable.The proof of the theorem is easier in mo-memtum space. Bose symmetry and par-ity oddness imply that the kernel G˜µν(~p) =eipµ/2Gµν(~p)e−ipν/2 (no sum over µ and ν) is an-tisymmetric:G˜µν(~p) = −G˜νµ(~p) . (10)The Poincare´ lemma [14] then enable us torewrite G˜µν(~p) in term of a scalar function f(~p):G˜µν(~p) = ǫρµν pˆρf(~p). Making use of the fact thatGµν(~p) must be a periodic function of the mo-menta, it is easy to see that for p0 = p1 = p2 =3±π one getsf(±π,±π,±π) = −f(±π,±π,±π) = 0 . (11)Since the spectrum of Gµν(~p) is given by G(~p) =±|f(~p)|√∑2µ=0 pˆ2µ, eq.(11) implies that the ker-nel G(~p) exhibits extra zeroes at the edges of theBrillouin zone and is thus not integrable.Relaxing the assumption iii) one may studythe general form of a gauge invariant local ac-tion in three dimensions. With the help of thePoincare´ lemma [14] it is easy to show that thekernel G˜µν(~p) can be divided into the sum of par-ity even and parity odd terms. Since, due to lo-cality, G˜µν(~p) is an analytic function of ~p, it maybe expanded in Taylor series: all the terms hav-ing even power of the momenta are parity even,while the terms with odd power of the momentaare parity odd. The terms with the lowest num-ber of derivatives in this expansion are the CSterm defined in [4] and the Maxwell term, whosekernel on the lattice is:Mµν = −✷δµν + dµdˆν = KµρKˆρν , (12)where ✷ =∑2µ=0 dµdˆµ is the Laplacean in threedimensions. Since all the parity odd terms full-fill the assumptions of the theorem they generateextra zeroes in the spectrum. The only gauge in-variant way to regularize the CS action is thenthe addition of a parity even term such as theMaxwell term.For the Maxwell-CS theory on the lattice thekernel Γµν may be written asΓµν =14e2Mµν + ikGµν . (13)In (13) k is dimensionless and e2 has the dimen-sion of a mass; the Maxwell term is an irrele-vant operator and the CS action dominates in theinfrared region. The Fourier transform of Γµν ,apart from a zero mode due to gauge invariance,has eigenvalues given byλMCS(~p) =12e22∑µ=0(1− cos pµ) + ikG(~p) , (14)and, as it stands, it is free from extra zeroes in theBrillouin zone since the first term in (14), whichis the Fourier transform of the Maxwell kernel, iszero only at zero momentum, and at the cornersof the Brillouin zone, pµ = ±π, µ = 0, 1, 2, takesthe value λMCS(~p) = 3/e2.Since the CS action is purely immaginary, theaddition of the Maxwell term is used also in thecontinuum theory to provide a proper definitionof the functional integral in the partition functionof the pure CS theory. The CS limit is reachedalso there by taking the limit e2 −→ ∞ afterGaussian integration.The regularization of the extra zeros in the CSaction by adding a Maxwell term and therebyopening a gap in the fermion spectrum is similarto the mechanism of the Wilson fermion wherea gap is opened and the energy does not havesecondary minima at the non-zero corners of theBrillouin zone. As in the case of the Wilsonfermions [13], the regularization is done by meansof an irrelevant operator and the continuum limita −→ 0 is not changed by this addition. More-over, as the Wilson action explicitly breaks chiralsymmetry, the action obtained after the additionof the Maxwell term is not anymore defined underparity.3. Concluding remarksIn [10] we pointed out a no-go theorem in thelattice regularization of the pure CS theory, ifone requires locality, gauge invariance and par-ity oddness on the lattice. As we already pointedout in the introduction, a doubling phenomenonhas been already advocated for bosonic theorieson the lattice [6] [7]. In particular the authorsof [6] and [7] found a doubling phenomenon for aclass of lattice formulations of gravity and in thepresence of a gravitational background all matterfields exhibit the same degeneracy as the gravi-ton and the chiral fermion. This doubling phe-nomenon makes ill defined all matter fields in agravitational background and it can be removedjust in the flat space limit in the case of scalar andgauge bosonic fields. However, in [10] we found adoubling phenomenon for gauge fields completelyindependent from the metric of the manifold onwhich the fields live, since the pure CS theory is atopological theory. Our analysis evidences explic-4itly that doubling phenomena for fields defined ona discrete space-time manifold are a very generalphenomena due more to the linearity of the fieldaction in the derivatives than to the curvature ofthe manifold.The no-go theorem demonstrated in [10]strongly relies on the definition of the paritytransformation for the gauge field Aµ(~x) −→Aµ(−~x − µˆ). If one could define a new parityon the lattice so that the CS term is still oddbut the kernel is integrable, then the new defini-tion of parity should play a role analogous to theGinsparg-Wilson relation [15] for lattice fermionictheories. For these theories the Ginsparg-Wilsonrelation hints to a new definition of chiral sym-metry on the lattice, a generalization of the usualcontinuum chiral symmetry that is exactly recov-ered in the naive continuum limit.REFERENCES1. R. Jackiw and S. Templeton, Phys. Rev. D23(1981) 2291; J. Schonfeld, Nucl. Phys. B185(1981) 157; S. Deser, R. Jackiw and S. Tem-pleton, Phys. 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On the Doubling Phenomenon in Lattice Chern-Simons Theories

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On the Doubling Phenomenon in Lattice Chern-Simons Theories

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