Ginsparg-Wilson Operator Research Articles

Lattice formulation of 2D = (2,2) SQCD and twisted supercharges

We discuss on two kinds of lattice formulation of two-dimensional = (2, 2) supersymmetric QCD preserving one of its supercharges. The preserved supercharge is of the same form obtained by topological twist. One formulation is based on A model twist, and the other is based on B model twist. For the A model twist case, by employing the Ginsparg-Wilson operator, the lattice model can be constructed for general number of flavors of fundamental and anti-fundamental matter multiplets under the gauge group G = U(N) or SU(N). It preserves both of one supercharge (Q) and chiral flavor symmetry of matter multiplets. Furthermore, superpotential can be introduced with exact holomorphic or anti-holomorphic structure on lattice. For the B model twist case, the lattice action takes a simpler form compared with the A model twist case, because the admissibility condition is not required to the formulation. It can be applied to a restricted cases of G = U(N) and of fundamental matters or anti-fundamental matters only.

Classical limits of scalar and tensor gauge operators based on the overlap Dirac matrix

It was recently proposed by the second author to consider lattice formulations of QCD in which complete actions, including the gauge part, are built explicitly from a given Dirac operator D. In a simple example of such theory, the gauge action is proportional to the trace of Ginsparg-Wilson operator D chosen to define the quark dynamics. This construction relies on the proposition that the classical limit of lattice gauge operator trD(x,x) is proportional to trF{sup 2}(x) (up to an additive constant). Here we show this for the case of the overlap Dirac operator using both analytical and numerical methods. We carry out the same analysis also for the tensor component of D, which is similarly related to the field-strength tensor F, and obtain results identical to our previous derivation that used a different approach. The corresponding proportionality constants are computed to high precision for a wide range of the negative mass parameter values, and it is verified that they are the same in finite and infinite volumes.

Remnant index theorem and low-lying eigenmodes for twisted mass fermions

We analyze the low-lying spectrum and eigenmodes of lattice Dirac operators with a twisted mass term. The twist term expels the eigenvalues from a strip in the complex plane and all eigenmodes obtain a non-vanishing matrix element with γ5. For a twisted Ginsparg–Wilson operator the spectrum is located on two arcs in the complex plane. Modes due to non-trivial topological charge of the underlying gauge field have their eigenvalues at the edges of these arcs and obey a remnant index theorem. For configurations in the confined phase we find that the twist mainly affects the zero modes, while the bulk of the spectrum is essentially unchanged.

Fuzzy Ginsparg-Wilson algebra: A solution of the fermion doubling problem

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 Ginsparg-Wilson algebra with the fuzzy sphere is surprising as the latter is not designed with this algebra in mind. The theory reproduces the integrated U(1)_A anomaly and index theory correctly.

CP breaking in lattice chiral gauge theory

The CP symmetry is not manifestly implemented for the local and doubler-free Ginsparg-Wilson operator in lattice chiral gauge theory. We precisely identify where the effects of this CP breaking appear.

Supersymmetry on the lattice and the Leibniz rule

The major obstacle to a supersymmetric theory on the lattice is the failure of the Leibniz rule. We analyze this issue by using the Wess–Zumino model and a general Ginsparg–Wilson operator, which is local and free of species doublers. We point out that the Leibniz rule could be maintained on the lattice if the generic momentum k μ carried by any field variable satisfies | ak μ |< δ in the limit a→0 for arbitrarily small but finite δ. This condition is expected to be satisfied generally if the theory is finite perturbatively, provided that discretization does not induce further symmetry breaking. We thus first render the continuum Wess–Zumino model finite by applying the higher derivative regularization which preserves supersymmetry. We then put this theory on the lattice, which preserves supersymmetry except for a breaking in interaction terms by the failure of the Leibniz rule. By this way, we define a lattice Wess–Zumino model which maintains the basic properties such as U(1)× U(1) R symmetry and holomorphicity. We show that this model reproduces continuum theory in the limit a→0 up to any finite order in perturbation theory; in this sense all the supersymmetry breaking terms induced by the failure of the Leibniz rule are irrelevant. We then suggest that this discretization may work to define a low energy effective theory in a non-perturbative way.

Ginsparg–Wilson operators and a no-go theorem

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.

CP breaking in lattice chiral gauge theories

The CP symmetry is not manifestly implemented for the local and doubler-free Ginsparg-Wilson operator in lattice chiral gauge theory. We precisely identify where the effects of this CP breaking appear. We show that they appear in: (I) Overall constant phase of the fermion generating functional. (II) Overall constant coefficient of the fermion generating functional. (III) Fermion propagator appearing in external fermion lines and the propagator connected to Yukawa vertices. The first effect appears from the transformation of the path integral measure and it is absorbed into a suitable definition of the constant phase factor for each topological sector; in this sense there appears no ``CP anomaly''. The second constant arises from the explicit breaking in the action and it is absorbed by the suitable weights with which topological sectors are summed. The last one in the propagator is inherent to this formulation and cannot be avoided by a mere modification of the projection operator, for example, in the framework of the Ginsparg-Wilson operator. This breaking emerges as an (almost) contact term in the propagator when the Higgs field, which is treated perturbatively, has no vacuum expectation value. In the presence of the vacuum expectation value, however, a completely new situation arises and the breaking becomes intrinsically non-local, though this breaking may still be removed in a suitable continuum limit. This non-local CP breaking is expected to persist for a non-perturbative treatment of the Higgs coupling.

Lattice chiral symmetry, Yukawa couplings and the Majorana condition

It is shown that the conflict between lattice chiral symmetry and the Majorana condition in the presence of Yukawa couplings, which was noted in our previous Letter, is related in an essential way to the basic properties of Ginsparg–Wilson operators, namely, locality and species doubling.

Lattice chiral symmetry and the Wess–Zumino model

A lattice regularization of the supersymmetric Wess–Zumino model is studied by using Ginsparg–Wilson operators. We recognize a certain conflict between the lattice chiral symmetry and the Majorana condition for Yukawa couplings, or in Weyl representation a conflict between the lattice chiral symmetry and Yukawa couplings. This conflict is also related, though not directly, to the fact that the kinetic (Kähler) term and the superpotential term are clearly distinguished in the continuum Wess–Zumino model, whereas these two terms are mixed in the Ginsparg–Wilson operators. We illustrate a case where lattice chiral symmetry together with naive Bose–Fermi symmetry is imposed by preserving a SUSY-like symmetry in the free part of the Lagrangian; one-loop level non-renormalization of the superpotential is then maintained for finite lattice spacing, though the finite parts of wave function renormalization deviate from the supersymmetric value. All these properties hold for the general Ginsparg–Wilson algebra independently of the detailed construction of lattice Dirac operators.

Axial anomaly and Ginsparg–Wilson fermions in the lattice Dirac sea picture

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 $r>0.5$. The generalization to 3+1 dimensions is also discussed.