In this paper, we introduce the overlap Dirac operator, which satisfies the Ginsparg-Wilson relation, to the matter sector of two-dimensional N=(2,2) lattice supersymmetric QCD (SQCD) with preserving one of the supercharges. It realizes the exact chiral flavor symmetry on the lattice, to make possible to define the lattice action for general number of the flavors of fundamental and anti-fundamental matter multiplets and for general twisted masses. Furthermore, superpotential terms can be introduced with exact holomorphic or anti-holomorphic structure on the lattice. We also consider the lattice formulation of matter multiplets charged only under the central U(1) (the overall U(1)) of the gauge group G=U(N), and then construct lattice models for gauged linear sigma models with exactly preserving one supercharge and their chiral flavor symmetry.
The Ginsparg-Wilson algebra is the algebra underlying the Ginsparg-Wilson solution of the fermion doubling problem in lattice gauge theory. The Dirac operator of the fuzzy sphere is not afflicted with this problem. Previously we have indicated that there is a Ginsparg-Wilson operator underlying it as well in the absence of gauge fields and instantons. Here we develop this observation systematically and establish a Dirac operator theory for the fuzzy sphere with or without gauge fields, and always with the Ginsparg-Wilson algebra. There is no fermion doubling in this theory. The association of …
The fermionic determinant of a lattice Dirac operator that obeys the Ginsparg-Wilson relation factorizes into two factors that are complex conjugate of each other. Each factor is naturally associated with a single chiral fermion and can be realized as a overlap of two many body vacua.
We discuss that the Ginsparg-Wilson relation, which has the key role in the recent development of constructing lattice chiral gauge theory, can play an important role to define chiral structures in finite matrix models and noncommutative geometries.
We argue that lattice QCD with Ginsparg-Wilson fermions satisfies the Leutwyler-Smilga sum rules for the eigenvalues of the chiral Dirac operator. The result is obtained in the one flavor case, by rephrasing Leutwyler and Smilga's original analysis for the finite volume partition function. This is a further evidence that Ginsparg-Wilson fermions, even if breaking explicitly the chirality on the lattice in accordance to the Nielsen-Ninomiya theorem, mimic the main features of the continuum theory related to chiral symmetry.
It is shown that it is impossible to construct a free theory of fermions on infinite hypercubic Euclidean lattice in four dimensions that is: (a) ultralocal, (b) respects symmetries of hypercubic lattice, (c) corresponding kernel satisfies D gamma5 + gamma5 D = D gamma5 D (Ginsparg-Wilson relation), (d) describes single species of massless Dirac fermions in the continuum limit.
Ginsparg-Wilson relation and admissibility condition have the key role to construct lattice chiral gauge theories. They are also useful to define the chiral structure in finite noncommutative geometries or matrix models. We discuss their usefulness briefly.
We evaluate chiral anomaly on the noncommutative torus with the overlap Dirac operator satisfying the Ginsparg-Wilson relation in arbitrary even dimensions. Utilizing a topological argument we show that the chiral anomaly is combined into a form of the Chern character with star products.
We propose a construction of a 2-dimensional lattice chiral gauge theory. The construction may be viewed as a particular limit of an infinite warped 3-dimensional theory. We also present a "single-site'' construction using Ginsparg-Wilson fermions which may avoid, in both 2 and 4 dimensions, the problems of waveguide-Yukawa models.
The Vafa-Witten arguments on the realization of parity and flavour symmetries in the QCD vacuum do not apply to two lattice regularizations of QCD which are able to reproduce the chiral anomaly: Wilson fermions and Ginsparg-Wilson fermions. We show here how using the last regularization one can get, from first principles, that the more standard order parameters for these symmetries take a vanishing vacuum expectation value for any number of flavours, if quarks have a non-vanishing mass.
It is shown that it is impossible to construct a free theory of fermions on infinite hypercubic Euclidean lattice in even number of dimensions that: (a) is ultralocal, (b) respects the symmetries of hypercubic lattice, (c) chirally nonsymmetric part of its propagator is local, and (d) describes single species of massless Dirac fermions in the continuum limit. This establishes non-ultralocality for arbitrary doubler-free Ginsparg-Wilson fermionic action with hypercubic symmetries ("strong non-ultralocality"), and complements the earlier general result on non-ultralocality of infinitesimal Ginsp…
A Yukawa-Higgs model with Ginsparg-Wilson (GW) fermions, proposed recently by Bhattacharya, Martin and Poppitz as a possible lattice formulation of chiral gauge theories, is studied. A simple argument shows that the gauge boson always acquires mass by the Stückelberg (or, in a broad sense, Higgs) mechanism, regardless of strength of interactions. The gauge symmetry is spontaneously broken. When the gauge coupling constant is small, the physical spectrum of the model consists of massless fermions, massive fermions and \emph{massive} vector bosons.
It is shown that the Ginsparg-Wilson relation implies an exact symmetry of the fermion action, which may be regarded as a lattice form of an infinitesimal chiral rotation. Using this result it is straightforward to construct lattice Yukawa models with unbroken flavour and chiral symmetries and no doubling of the fermion spectrum. A contradiction with the Nielsen-Ninomiya theorem is avoided, because the chiral symmetry is realized in a different way than has been assumed when proving the theorem.
Important recent discoveries suggest that Ginsparg-Wilson-Luscher (GWL) symmetry has analogous dynamical consequences for the theory on the lattice as chiral symmetry does in the continuum. While it is well known that inherent property of lattice chiral symmetry is fermion doubling, we show here that inherent property of GWL symmetry is that the infinitesimal symmetry transformation couples fermionic degrees of freedom at arbitrarily large lattice distances (non-ultralocality). The consequences of this result for ultralocality of symmetric actions are discussed.
We show that, under certain general assumptions, any sensible lattice Dirac operator satisfies a generalized form of the Ginsparg-Wilson relation (GWR). Those assumptions, on the other hand, are mostly dictated by large momentum behaviour considerations. We also show that all the desirable properties often deduced from the standard GWR hold true of the general case as well; hence one has, in fact, more freedom to modify the form of the lattice Dirac operator, without spoiling its nice properties. Our construction, a generalized Ginsparg-Wilson relation (GGWR), is satisfied by some known propos…
We construct a topological charge of gauge field configurations on a fuzzy S^2xS^2 by using a Dirac operator satisfying the Ginsparg-Wilson relation. The topological charge defined on the fuzzy S^2xS^2 can be interpreted as a noncommutative (or matrix) generalization of the 2nd Chern character on S^2xS^2. We further calculate the number of chiral zero modes of the Dirac operator in topologically nontrivial gauge configurations. Generalizations of our formulation to fuzzy (S^2)^k are also discussed.
We present a chiral solution of the Ginsparg-Wilson equation. This work is motivated by our recent proposal for nonperturbatively regulating chiral gauge theories, where five-dimensional domain wall fermions couple to a four-dimensional gauge field that is extended into the extra dimension as the solution to a gradient flow equation. Mirror fermions at the far surface decouple from the gauge field as if they have form factors that become infinitely soft as the distance between the two surfaces is increased. In the limit of an infinite extra dimension we derive an effective four-dimensional chi…
We investigate several properties of Ginsparg-Wilson fermion on fuzzy 2-sphere. We first examine chiral anomaly up to the second order of the gauge field and show that it is indeed reduced to the correct form of the Chern character in the commutative limit. Next we study topologically non-trivial gauge configurations and their topological charges. We investigate 't Hooft-Polyakov monopole type configuration on fuzzy 2-sphere and show that it has the correct commutative limit. We also consider more general configurations in our formulation.
If one uses a general class of Ginsparg-Wilson operators, it is known that CP symmetry is spoiled in chiral gauge theory for a finite lattice spacing and the Majorana fermion is not defined in the presence of chiral symmetric Yukawa couplings. We summarize these properties in the form of a theorem for the general Ginsparg-Wilson relation.
We show that, if the formula for the topological charge density operator suggested by fermions obeying the Ginsparg-Wilson relation is employed, it is possible to give a precise and unambiguous definition of the topological susceptibility in full QCD, chi_tL, for finite quark masses on the lattice. The lattice expression of chi_tL looks like the formal continuum one, in the sense that no power divergent subtractions are needed for its proper definition. As a consequence, the small mass behaviour of chi_tL leads directly to a multiplicative renormalizable definition of the chiral condensate tha…
We expand the most general lattice Dirac operator D in a basis of simple operators. The Ginsparg-Wilson equation turns into a system of coupled quadratic equations for the expansion coefficients. Our expansion of D allows for a natural cutoff and the remaining quadratic equations can be solved numerically. The procedure allows to find Dirac operators which obey the Ginpsparg-Wilson equation with arbitrary precision.
I implement a set of tricks for constructing lattice fermion actions which approximately realize the Ginsparg-Wilson relation, with very promising results from simulations.
We demonstrate that in the topologically trivial gauge sector the Ginsparg-Wilson relation for lattice Dirac operators admits an exactly gauge invariant path integral formulation of the Weyl fermions on a lattice.
Defining chiral lattice gauge theories in the Ginsparg-Wilson formalism is complicated by the so-called fermion measure problem. It has been proven for the abelian theories that smooth well-behaved fermion measure exists if and only if the anomaly-free condition is granted, and the same was shown to hold in perturbative theories for non-abelian gauge groups, but the non-perturbative proof is absent. In this paper, we consider a simpler problem in 2-d and present a proof for the existence of smooth and gauge invariant fermion measure on the gauge field configuration space with zero field streng…
It was shown recently by I. Horvath that lattice fermions obeying the standard form of the Ginsparg-Wilson relation cannot be ultralocal. However, there are more general forms of the Ginsparg-Wilson relation, which also guarantee the physical properties related to chirality, but which are not covered by Horvath's consideration. Here we present a proof which applies to all Ginsparg-Wilson fermions, demonstrating that they can only be local in the sense of an exponential decay of their couplings, but not ultralocal.
We show rigorously that for general Ginsparg-Wilson fermions the dimensions of the geometric eigenspace and of the algebraic one for zero modes agree so that the index theorem on the lattice is not spoiled by unwanted additional terms.
We formulate a Ginsparg-Wilson relation on a fuzzy 2-sphere for matter in the adjoint representation of the gauge group. Because of the Ginsparg-Wilson relation, an index theorem is satisfied. Our formulation is applicable to topologically nontrivial configurations as monopoles. It gives a solid basis for obtaining chiral fermions, which are an important ingredient of the standard model, from matrix model formulations of the superstring theory, such as the IIB matrix model, by considering topological configurations in the extra dimensions. We finally discuss whether this mechanism really works…
A specific algebraic realization of the Ginsparg-Wilson relation in the form $γ_{5}(γ_{5}D)+(γ_{5}D)γ_{5} = 2a^{2k+1}(γ_{5}D)^{2k+2}$ is discussed, where $k$ stands for a non-negative integer and $k=0$ corresponds to the commonly discussed Ginsparg-Wilson relation. From a view point of algebra, a characteristic property of our proposal is that we have a closed algebraic relation for one unknown operator $D$, although this relation itself is obtained from the original proposal of Ginsparg and Wilson, $γ_{5}D+Dγ_{5}=2aDγ_{5} αD$, by choosing $α$ as an operator containing $D$ (and thus Dirac matr…
The Fixed Point Dirac Operator and Chirally Improved Fermions both use large numbers of gauge paths and the full Dirac structure to approximate a solution of the Ginsparg-Wilson equation. After a brief review of the two approaches we present recent results for quenched QCD with pion masses down to 210 MeV. We discuss the limits and advantages of approximate parameterizations and outline future perspectives.
The improvement of fermionic operators for Ginsparg-Wilson fermions is investigated. We present explicit formulae for improved Green's functions, which apply both on-shell and off-shell.
We analyze general solutions of the Ginsparg-Wilson relation for lattice Dirac operators and formulate a necessary condition for such operators to have non-zero index in the topologically nontrivial background gauge fields.
The concept of lattice modified symmetry formulations is adapted to the parity symmetry of gauge fields and applied to the pure Abelian Chern-Simons action in three dimensions. We derive an analogue of the Ginsparg-Wilson relation for the parity anti-symmetry, which is motivated from the perfect lattice action, and which we denote as the Chern-Simons-Ginsparg-Wilson relation (CSGWR). In addition to the overlap type solutions, we construct explicitly simple and local polynomial solutions to the CSGWR. We show that these actions are exactly invariant under a lattice modified parity transformatio…
We discuss the improvement of bilinear fermionic operators for Ginsparg-Wilson fermions. We present explicit formulae for improved Green's functions, which apply both on-shell and off-shell.
A chiral random matrix model with locality is constructed and examined. The Nielsen-Ninomiya no-go theorem is circumvented by the use of a generally applicable modified Dirac operator which respects the Ginsparg-Wilson relation. We observe the expected universal behaviour of the eigenvalue density in the microscopic limit.
A new method for finding approximate solutions of the Ginsparg-Wilson equation is tested in 2-d. The Dirac operator is first constructed and then used in a dynamical simulation of the 2-flavor Schwinger model. We find a very small mass of the pi-particle implying almost chirally symmetric fermions. The generalization of our method to 4-d is straightforward.
The axial anomaly of Ginsparg-Wilson fermion operator $D$ is discussed in general for the operator $R$ which enters the chiral symmetry breaking part in the Ginsparg-Wilson relation. The axial anomaly and the index of $D$ as well as the exact realization of the Atiyah-Singer index theorem on the lattice are determined solely by the topological characteristics of the chirally symmetric operator $D_c$ in the chiral limit $ R \to 0 $.
A generalized anti-hermitian staggered Dirac operator is formulated. Its relation with noncommutative geometry is briefly reviewed. Once this anti-hermitian operator is modified to be ``$γ^5$-hermitian'', it will provide a new solution to Ginsparg-Wilson relation, basing on an abstract algebraic analysis of Neuberger's overlap construction and a redefinition of chirality.
The Ginsparg-Wilson relation is extended to interacting field theories with general linear symmetries. Our relation encodes the remnant of the original symmetry in terms of the blocked fields and guides the construction of invariant lattice actions. We apply this approach in the case of lattice supersymmetry. An additional constraint has to be satisfied because of the appearance of a derivative operator in the symmetry transformations. The solution of this constraint leads to non-local SLAC-type derivatives. We investigate the corresponding kinetic operators on the lattice within an exact solu…
Lattice fermions obeying the Ginsparg-Wilson relation do correctly represent the physical properties related to chirality. This can be achieved by local fermions, which involve an infinite number of couplings, however. For practical purposes, it is useful to first construct approximate Ginsparg-Wilson fermions within a short range. We report on a successful construction in QCD at β=6. The good quality of the approximation is observed from the spectrum, which is situated close to a Ginsparg-Wilson circle. These fermions also provide an excellent approximation to rotational symmetry and they are…
A novel feature of a Ginsparg-Wilson lattice Dirac operator is discussed. Unlike the Dirac operator for massless fermions in the continuum, this lattice Dirac operator does not possess topological zero modes for any topologically-nontrivial background gauge fields, even though it is exponentially-local, doublers-free, and reproduces correct axial anomaly for topologically-trivial gauge configurations.
We construct a 4-d lattice Dirac operator D using a systematical expansion in terms of simple operators on the lattice. The Ginsparg-Wilson equation turns into a system of coupled equations for the expansion coefficients of D. We solve these equations for a finite parametrization of D and find an approximate solution of the Ginsparg-Wilson equation. We analyze the spectral properties of our D for various ensembles of quenched SU(3) configurations. Improving the gauge field action considerably improves the spectral properties of our D.
We consider partially quenched, mixed chiral perturbation theory with staggered sea and Ginsparg-Wilson valence quarks in order to extract a chiral-continuum extrapolation expression for the vector meson mass up to order O(a^2), at one-loop level. Based on general principles, we accomplish the task without explicitly constructing a sophisticated, heavy vector meson chiral Lagrangian.
We derive Ginsparg-Wilson relation for a lattice chiral symmetry in theories with self-interacting fermions. Auxiliary scalar and pseudo-scalar fields are introduced on a coarse lattice to give an effective description of the fermionic interactions. We obtain particular solutions to the Ginsparg-Wilson relation and other Ward-Takahashi identities in a closed form. These non-perturbative solutions can be used to construct a chiral invariant action and an invariant path-integral measure on the coarse lattice. The resulting partition function exhibits the exact chiral symmetry in the fermionic sy…
We discuss a number of lattice fermion actions solving the Ginsparg-Wilson relation. We also consider short ranged approximate solutions. In particular, we are interested in reducing the lattice artifacts, while avoiding (or suppressing) additive mass renormalization. In this context, we also arrive at a formulation of improved domain wall fermions.
We determine the free field hypercubic Dirac operator which is optimally close to satisfying the Ginsparg-Wilson relation. Inserting this operator into the overlap formula, we show that the analytic locality bound on the resulting overlap Dirac operator is substantially stronger than in the standard case. This improvement generally persists in gauge backgrounds when the plaquette variables are all close to unity.
We present a new staggered discretization of the Dirac operator. In comparison with standard staggered fermions, real and imaginary parts are located in different nodes. Doubling gives only a doublet of Dirac fermions which we propose to interpret as a physical (lepton or quark) doublet. Contrary to usual staggered fermions, we have no exact chiral symmetry but obtain a generalization of the Ginsparg-Wilson relation.
Based upon the lattice Dirac operator satisfying the Ginsparg-Wilson relation, we investigate canonical formulation of massless fermion on the spatial lattice. For free fermion system exact chiral symmetry can be implemented without species doubling. In the presence of gauge couplings the chiral symmetry is violated. We show that the divergence of the axial vector current is related to the chiral anomaly in the classical continuum limit.
We study aspects concerning numerical simulations of Lattice QCD with two flavors of dynamical Ginsparg-Wilson quarks with degenerate masses. A Hybrid Monte Carlo algorithm is described and the formula for the fermionic force is derived for two specific implementations. The implementation with optimal rational approximation method is favored both in CPU time and memory consumption.
We derive the effective action of the light fermion field of the domain-wall fermion, which is referred as q(x) and \bar q(x) by Furman and Shamir. The inverse of the effective Dirac operator turns out to be identical to the inverse of the truncated overlap Dirac operator except a local contact term, which would give the chiral symmetry breaking in the Ginsparg-Wilson relation. We argue that there are direct relations between the low energy observables of the domain-wall QCD and observables of the Ginsparg-Wilson fermion described by the (truncated) overlap Dirac operator.
We analyse the structure of solutions of the Ginsparg-Wilson relation for lattice Dirac operator in topologically trivial gauge sector. We show that the properties of such solutions relating to the perturbative stability of the pole of the fermion propagator as well as to the structure of the Yukawa models based on these solutions are solely determined by the non-local chirally invariant part of these Dirac operators. Depending on the structure of this part, the pole in the fermion propagator may or may not be stable under radiative corrections. We illustrate this by explicit examples.
The Ginsparg-Wilson(G-W) relation for chiral symmetry is extended for a supersymmetrical(SUSY) case on a lattice. It is possible to define exact lattice supersymmetry which are devided into two different cases according to using difference operators. $U(1)_R$ symmetry on the lattice is also realized as one of exact symmetries. For an application, the extended G-W relation is given for a two-dimensional model with chiral multiplets.
We pursue Ginsparg and Wilsons' block spin approach in the derivation of the Ginsparg-Wilson relation and study the correspondence of the eigenmodes of the Dirac operators in the continuum and lattice theories. After introducing a suitable cut-off in the continuum theory, we identify unphysical modes of the lattice Dirac operator which do not correspond to any physical modes of the regulated continuum Dirac operator. We also consider zero modes in the continuum and lattice theories. Our studies give a physical interpretation of the expression of the index defined on a lattice and a formal argu…
We derive the effective action of the light fermion field of the domain-wall fermion, which is referred as $q(x)$ by Furman and Shamir. The inverse of the effective Dirac operator turns out to be identical to the inverse of the truncated overlap Dirac operator, except a local contact term which would give the chiral symmetry breaking in the Ginsparg-Wilson relation. This result allows us to relate the light fermion field and the fermion field described by the truncated overlap Dirac operator and to understand the chiral property of the light fermion through the exact chiral symmetry based on t…
We show how it is possible to give a precise and unambiguous implementation of the Witten--Veneziano formula for the eta' mass on the lattice, which looks like the formal continuum one, if the expression of the topological charge density operator, suggested by fermions obeying the Ginsparg--Wilson relation, is employed. By using recent numerical results from simulations with overlap fermions in 2 (abelian Schwinger model) and 4 (QCD) dimensions, one obtains values for the mass of the lightest pseudo-scalar flavour-singlet state that agree within errors with theoretical expectations and experim…
The axial anomaly equation in 1+1 dimensional QED is obtained on the lattice for fermions obeying the Ginsparg-Wilson relation. We make use of the properties of the Lattice Dirac sea to investigate the connection between the anomaly and the Ginsparg-Wilson operator in the Hamiltonian picture. The correct anomaly is reproduced for gauge fields whose characteristic time is much larger than the lattice spacing, which is the regime where the adiabatic approximation applies. A non-zero Wilson $r$ parameter is necessary to get the correct anomaly. The anomaly is shown to be independent of $r$ for $r…
The Baxter 8-vertex model is equivalent to a particular lattice formulation of a self-interacting, massive Dirac fermion theory. In the time-continuum limit, the lattice Hamiltonian (XYZ spin chain) can be explicitly transformed to a lattice Dirac Hamiltonian. We show that the kernel describing the quadratic part of this Hamiltonian satisfies a one-dimensional version of the Ginsparg-Wilson relation. The corresponding conserved charge is derived and compared with the conserved arrow number of the 8-vertex model.
