56107847
Fe b 20 18 YITP-1757 O ( a ) Improvement of 2 D N = ( 2 , 2 ) Lattice SYM Theory
We perform a tree-level O(a) improvement of two-dimensional N = (2, 2) supersymmetric Yang-Mills theory on the lattice, motivated by the fast convergence in numerical simulations. The improvement respects an exact supersymmetry Q which is needed for obtaining the correct continuum limit without a parameter fine tuning. The improved lattice action is given within a milder locality condition in which the interactions are decaying as the exponential of the distance on the lattice. We also prove that the path-integral measure is invariant under the improved Q-transformation. masanori.hanada@colorado.edu kadoh@keio.jp s.matsu@phys-h.keio.ac.jp fusugino@gmail.com
14361074
Electric - magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory
Abstract We study the vacuum structure and don spectrum of N =2 supersymmetric gauge theory in four dimensions, with gauge group SU(2). The theory turns out to have remarkably rich and physical properties which can nonetheless be described precisely; exact formulas can be obtained, for instance, for electron and dyon masses and the metric on the moduli space of vacua. The description involves a version of Olive-Montonen electric-magnetic duality. The “strongly coupled” vacuum turns out to be a weakly coupled theory of monopoles, and with a suitable perturbation confinement is described by monopole condensation.
18466754
Electric-magnetic duality in supersymmetric non-Abelian gauge theories
Abstract We demonstrate electric-magnetic duality in N = 1 supersymmetric non-Abelian gauge theories in four dimensions by presenting two different gauge theories (different gauge groups and quark representations) leading to the same non-trivial long distance physics. The quarks and gluons of one theory can be interpreted as solitons (non-Abelian magnetic monopoles) of the elementary fields of the other theory. The weak coupling region of one theory is mapped to a strong coupling region of the other. When one of the theories is Higgsed by an expectation value of a squark, the other theory is confined. Massless glueballs, baryons and Abelian magnetic monopoles in the confining description are the weakly coupled elementary quarks (i.e. solitons of the confined quarks) in the dual Higgs description.
5430278
Supersymmetry on a Spatial Lattice
We construct a variety of supersymmetric gauge theories on a spatial lattice, including N = 4 supersymmetric Yang-Mills theory in 3+1 dimensions. Exact lattice supersymmetry greatly reduces or eliminates the need for fine tuning to arrive at the desired continuum limit in these examples.
16803120
Lattice supersymmetry and topological field theory
It is known that certain theories with extended supersymmetry can be discretized in such a way as to preserve an exact fermionic symmetry. In the simplest model of this kind, we show that this residual supersymmetric invariance is actually a BRST symmetry associated with gauge fixing an underlying local shift symmetry. Furthermore, the original lattice action is then seen to be entirely a gauge fixing term. The corresponding continuum theory has a topological field theory interpretation. We look, in detail, at one example - supersymmetric quantum mechanics which possesses two such BRST symmetries. In this case, we show that the lattice theory can be obtained by blocking out of the continuum in a carefully chosen background metric. Such a procedure will not change the Ward identities corresponding to the BRST symmetries since they correspond to topological observables. Thus, at the quantum level, the continuum BRST symmetry is preserved in the lattice theory. Similar conclusions are reached for the two-dimensional complex Wess-Zumino model and imply that all the supersymmetric Ward identities are satisfied exactly on the lattice. Numerical results supporting these conclusions are presented.
16016563
Supersymmetry on a euclidean spacetime lattice 2. Target theories with eight supercharges
We formulate euclidean spacetime lattices whose continuum limits are supersymmetric Yang-Mills theories with eight supercharges in two and three dimensions. The lattice actions are themselves supersymmetric.
9616108
A lattice formulation of super Yang-Mills theories with exact supersymmetry
We construct SU( N ) super Yang-Mills theories with extended supersymmetry on hypercubic lattices of various dimensions keeping one or two supercharges exactly. It is based on topological field theory formulation for the super Yang-Mills theories. Gauge fields are represented by compact unitary link variables, and the exact supercharges on the lattice are nilpotent up to gauge transformations. This presentation is based on the works [JHEP 0401 (2004) 015, JHEP 0403 (2004) 067, F. Sugino, in preparation].
9316110
Twisted superspace on a lattice
Abstract We propose a new formulation which realizes exact twisted supersymmetry for all the supercharges on a lattice by twisted superspace formalism. We show explicit examples of N = 2 twisted supersymmetry invariant BF and Wess–Zumino models in two dimensions. We introduce mild lattice non-commutativity to preserve Leibniz rule on the lattice. The formulation is based on the twisted superspace formalism for N = D = 2 supersymmetry which was proposed recently. From the consistency condition of the non-commutativity of superspace, we find an unexpected three-dimensional lattice structure which may reduce into two-dimensional lattice where the superspace describes semilocally scattered fermions and bosons within a double size square lattice.
15367828
Two-dimensional compact N = (2,2) lattice super-Yang-Mills theory with exact supersymmetry
Abstract We construct two-dimensional N = ( 2 , 2 ) lattice super-Yang–Mills theory, where the gauge and Higgs fields are all represented by U ( N ) compact variables, with keeping one exact supercharge along the line of the papers [F. Sugino, JHEP 0401 (2004) 015, hep-lat/0311021 , F. Sugino, JHEP 0403 (2004) 067, hep-lat/0401017 , F. Sugino, JHEP 0501 (2005) 016, hep-lat/0410035 ]. Interestingly, requirements of the exact supersymmetry as well as of the compact gauge and Higgs fields lead to the gauge group U ( N ) rather than SU ( N ) . As a result of the perturbative renormalization argument, the model is shown to flow to the target continuum theory without any fine-tuning. Different from the case of noncompact Higgs fields, the path integral along the flat directions is well defined in this model.
17259101
Lattice formulation of (2,2) supersymmetric gauge theories with matter fields
We construct lattice actions for a variety of (2,2) supersymmetric gauge theories in two dimensions with matter fields interacting via a superpotential.
42435766
From Twisted Supersymmetry to Orbifold Lattices
We show how to derive the supersymmetric orbifold lattices of Cohen et al. (1, 2) and Kaplan et al. (3) by direct discretization of an appropriate twisted supersymmetric Yang-Mills theory. We examine in detail the four supercharge two dimensional theory and the theory with sixteen supercharges in four dimensions. The continuum limit of the latter theory is the well known Marcus twist of N = 4 Yang-Mills. The lattice models are gauge invariant and possess one exact supersymmetry at non-zero lattice spacing.
14678892
Two-dimensional = (2, 2) supersymmetric lattice gauge theory with matter fields in the fundamental representation
In this paper, we construct a lattice formulation for two-dimensional = (2, 2) supersymmetric gauge theory with matter fields in the fundamental representation. We first construct it by the orbifolding procedure from Yang-Mills matrix theory with eight supercharges. We show that we can obtain the same lattice formulation by extending the geometrical discretization scheme. This suggests that the equivalence between the two schemes holds even for theories with matter fields.
16982932
Lattice formulation of two-dimensional N=(2,2) SQCD with exact supersymmetry
Abstract We construct a lattice model for two-dimensional N = ( 2 , 2 ) supersymmetric QCD (SQCD), with the matter multiplets belonging to the fundamental or anti-fundamental representation of the gauge group U ( N ) or SU ( N ) . The construction is based on the topological field theory (twisted supercharge) formulation and exactly preserves one supercharge along the line of the papers [F. Sugino, JHEP 0401 (2004) 015; F. Sugino, JHEP 0403 (2004) 067; F. Sugino, JHEP 0501 (2005) 016; F. Sugino, Phys. Lett. B 635 (2006) 218] for pure supersymmetric Yang–Mills theories. In order to avoid the species doublers of the matter multiplets, we introduce the Wilson terms and the model is defined for the case of the number of the fundamental matters ( n + ) equal to that of the anti-fundamental matters ( n − ). If some of the matter multiplets decouple from the theory by sending the corresponding anti-holomorphic twisted masses to the infinity, we can analyze the general n + ≠ n − case, although the lattice model is defined for n + = n − . By computing the anomaly of the U ( 1 ) A R-symmetry in the lattice perturbation, we see that the decoupling is achieved and the anomaly for n + ≠ n − is correctly obtained.
121013058
Ginsparg‐Wilson Formulation of 2D N = (2,2) SQCD with Exact Lattice Supersymmetry
We discuss on a lattice formulation of 2D N = (2,2) SQCD preserving one of its supercharges. In particular, the overlap Dirac operator, which satisfies the Ginsparg‐Wilson relation, is introduced to the matter sector of the theory. It exactly realizes chiral flavor symmetry on lattice, to make possible to define the lattice action for general number of the flavors of fundamental and anti‐fundamental matter multiplets. Furthermore, superpotential terms can be introduced with exact holomorphic or anti‐holomorphic structure on lattice. It is applicable to 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 to lattice models for gauged linear sigma models with exactly preserving one supercharge and their chiral flavor symmetry.
117051651
Lattice formulation of two-dimensional $$ \mathcal{N} = \left( {{2},{2}} \right) $$ super Yang-Mills with SU(N ) gauge group
A bstractWe propose a lattice model for two-dimensional SU(N ) $$ \mathcal{N} = \left( {{2},{2}} \right) $$ super YangMills model. We start from the CKKU model for this system, which is valid only for U(N ) gauge group. We give a reduction of U(1) part keeping a part of supersymmetry. In order to suppress artifact vacua, we use an admissibility condition.
7958478
Restoration of supersymmetry on the lattice: Two-dimensional N = (2,2) supersymmetric Yang-Mills theory
Abstract By numerically investigating the conservation law of the supercurrent, we confirm the restoration of supersymmetry in Sugino's lattice formulation of the two-dimensional N = ( 2 , 2 ) supersymmetric SU ( 2 ) Yang–Mills theory with a scalar mass term. Subtlety in the case without the scalar mass term, that appears to ruin perturbative power counting, is also pointed out.
14739225
Some physics of the two-dimensional N=(2,2) supersymmetric Yang–Mills theory: Lattice Monte Carlo study
Abstract We illustrate some physical application of a lattice formulation of the two-dimensional N = ( 2 , 2 ) supersymmetric SU ( 2 ) Yang–Mills theory with a (small) supersymmetry breaking scalar mass. Two aspects, power-like behavior of certain correlation functions (which implies the absence of the mass gap) and the static potential V ( R ) between probe charges in the fundamental representation, are considered. For the latter, for R ≲ 1 / g , we observe a linear confining potential with a finite string tension. This confining behavior appears distinct from a theoretical conjecture that a probe charge in the fundamental representation is screened in two-dimensional gauge theory with an adjoint massless fermion, although the static potential for R ≳ 1 / g has to be systematically explored to conclude real asymptotic behavior in large distance.
3080563
Lattice study of two-dimensional N=(2,2) super Yang-Mills theory at large N
We study two-dimensional $\mathcal{N}=(2,2)$ $SU(N)$ super Yang-Mills theory on Euclidean two-torus using Sugino's lattice regularization. We perform the Monte Carlo simulation for $N=2,3,4,5$ and then extrapolate the result to $N=\ensuremath{\infty}$. With the periodic boundary conditions for the fermions along both circles, we establish the existence of a bound state in which scalar fields clump around the origin, in spite of the existence of a classical flat direction. In this phase the global $({\mathbb{Z}}_{N}{)}^{2}$ symmetry turns out to be broken. We provide a simple explanation for this fact and discuss its physical implications.
118404128
Absence of sign problem in two-dimensional $ \mathcal{N} = \left( {2,2} \right) $ super Yang-Mills on lattice
We show that $ \mathcal{N} = \left( {2,2} \right) $ SU(N) super Yang-Mills theory on lattice does not have sign problem in the continuum limit, that is, under the phase-quenched simulation phase of the determinant localizes to 1 and hence the phase-quench approximation becomes exact. Among several formulations, we study models by Cohen-Kaplan-Katz-Unsal (CKKU) and by Sugino. We confirm that the sign problem is absent in both models and that they converge to the identical continuum limit without fine tuning. We provide a simple explanation why previous works by other authors, which claim an existence of the sign problem, do not capture the continuum physics.
119246836
On the sign problem in 2D lattice super Yang-Mills
A bstractIn recent years a new class of supersymmetric lattice theories have been proposed which retain one or more exact supersymmetries for non-zero lattice spacing. Recently there has been some controversy in the literature concerning whether these theories suffer from a sign problem. In this paper we address this issue by conducting simulations of the $ \mathcal{N} $ = (2, 2) and $ \mathcal{N} $ = (8, 8) supersymmetric Yang-Mills theories in two dimensions for the U(N ) theories with N = 2, 3, 4, using the new twisted lattice formulations. Our results provide evidence that these theories do not suffer from a sign problem in the continuum limit. These results thus boost confidence that the new lattice formulations can be used successfully to explore non-perturbative aspects of four-dimensional $ \mathcal{N} $ = 4 supersymmetric Yang-Mills theory.
118576717
Thermal phases of D1-branes on a circle from lattice super Yang-Mills
We report on the results of numerical simulations of 1 + 1 dimensional SU(N) Yang-Mills theory with maximal supersymmetry at finite temperature and compactified on a circle. For large N this system is thought to provide a dual description of the decoupling limit of N coincident D1-branes on a circle. It has been proposed that at large N there is a phase transition at strong coupling related to the Gregory-Laflamme (GL) phase transition in the holographic gravity dual. In a high temperature limit there was argued to be a deconfinement transition associated to the spatial Polyakov loop, and it has been proposed that this is the continuation of the strong coupling GL transition. Investigating the theory on the lattice for SU(3) and SU(4) and studying the time and space Polyakov loops we find evidence supporting this. In particular at strong coupling we see the transition has the parametric dependence on coupling predicted by gravity. We estimate the GL phase transition temperature from the lattice data which, interestingly, is not yet known directly in the gravity dual. Fine tuning in the lattice theory is avoided by the use of a lattice action with exact supersymmetry.
73699207
Restoration of supersymmetry in two-dimensional SYM with sixteen supercharges on the lattice
A bstractWe perform lattice simulations of two-dimensional supersymmetric Yang-Mills theory with sixteen supercharges with a lattice action which has two exact supercharges (Sugino lattice action). According to the gauge/gravity duality, the theory at finite temperature is expected to be well described by the corresponding black 1-branes, at low temperature in the large N limit. We aim to confirm the duality conjecture by comparing the lattice results with the theoretical predictions obtained in the gravity side. In this article, at the beginning of this study, we examine the supersymmetric Ward-Takahashi identity to test whether the lattice action reproduces the correct continuum theory. Numerical results of the SUSY WTI strongly suggest us that any cut-off effects, which break supersymmetry, disappear in the continuum limit. In addition, we study the issue of degenerate vacua and find that the admissiblilty condition or any other constraints of the link fields which guarantee the unique vacuum are not always needed.
118949487
Precision test of the gauge/gravity duality in two-dimensional N=(8,8) SYM
The $\epsilon - p$ is calculated from lattice simulations of two dimensional ${\cal N}=(8,8)$ $SU(N)$ SYM to test the gauge gravity duality. We employ the Sugino action with keeping two of sixteen supercharges exactly on the lattice. The thermodynamics of this gauge theory is described by the black 1-branes at low temperature. The internal energy density $\epsilon$ and the pressure $p$ can be analytically estimated in the gravitational theory. The lattice results in $N=12$ show that $\epsilon - p$ of the thermal SYM reproduces the prediction of the gravitational theory for the dimensionless temperature $T_{\rm eff}<0.4$. This clearly indicates that the duality is valid in this system.
56421224
Testing holography using lattice super-Yang-Mills theory on a 2-torus
We consider maximally supersymmetric SUðNÞ Yang-Mills theory in Euclidean signature compactified on a flat two-dimensional torus with antiperiodic (“thermal”) fermion boundary conditions imposed on one cycle. At large N, holography predicts that this theory describes certain black hole solutions in type IIA and IIB supergravity, and we use lattice gauge theory to test this. Unlike the one-dimensional quantum mechanics case where there is only the dimensionless temperature to vary, here we emphasize there are two more parameters which determine the shape of the flat torus. While a rectangular Euclidean torus yields a thermal interpretation, allowing for skewed tori modifies the holographic dual black hole predictions and results in another direction to test holography. Our lattice calculations are based on a supersymmetric formulation naturally adapted to a particular skewing. Using this we perform simulations up to N 1⁄4 16 with several lattice spacings for both skewed and rectangular tori. We observe the two expected black hole phases with their predicted behavior, with a transition between them that is consistent with the gravity prediction based on the Gregory-Laflamme transition.
118573147
Two-Dimensional Lattice for Four-Dimensional N=4 Supersymmetric Yang-Mills
We construct a lattice formulation of a mass-deformed two-dimensional N = (8, 8) super Yang-Mills theory with preserving two supercharges exactly. Gauge fields are represented by compact unitary link variables, and the exact supercharges on the lattice are nilpotent up to gauge transformations and SU(2)R rotations. Due to the mass deformation, the lattice model is free from the vacuum degeneracy problem, which was encountered in earlier approaches, and flat directions of scalar fields are stabilized giving discrete minima representing fuzzy S. Around the trivial minimum, quantum continuum theory is obtained with no tuning, which serves a nonperturbative construction of the IIA matrix string theory. Moreover, around the minimum of k-coincident fuzzy spheres, four-dimensional N = 4 U(k) super Yang-Mills theory with two commutative and two noncommutative directions emerges. In this theory, sixteen supersymmetries are broken by the mass deformation to two. Assuming the breaking is soft, we give a scenario leading to undeformed N = 4 super Yang-Mills on R without any fine tuning. As an evidence for the validity of the assumption, some computation of 1-loop radiative corrections is presented.
118737562
A proposal of a fine tuning free formulation of 4d $ \mathcal{N} = 4 $ super Yang-Mills
Recently, a nonperturbative formulation of 4d $ \mathcal{N} = 4 $ super Yang-Mills theory which does not require fine tuning at least to all order in perturbation theory has been proposed by combining two-dimensional lattice and matrix model techniques. In this paper we provide an analogous model by utilizing deconstruction approach of Kaplan et al. Two-dimensional lattice with a plane wave deformation is deconstructed from a matrix model and two additional dimensions emerge through the Myers effect. In other words we construct a D1-brane theory from which a D3-brane theory comes out. The action is much simpler than the previous formulation and hence numerical study, which enables us to test the AdS5/CFT4 duality at fully nonperturbative level, becomes much easier.
119311021
Topologically twisted N = (2, 2) supersymmetric Yang–Mills theory on an arbitrary discretized Riemann surface
We define supersymmetric Yang-Mills theory on an arbitrary two-dimensional lattice (polygon decomposition) with preserving one supercharge. When a smooth Riemann surface Σg with genus g emerges as an appropriate continuum limit of the generic lattice, the discretized theory becomes topologically twisted N = (2, 2) supersymmetric Yang-Mills theory on Σg. If we adopt the usual square lattice as a special case of the discretization, our formulation is identical with Sugino’s lattice model. Although the tuning of parameters is generally required while taking the continuum limit, the number of the necessary parameters is at most two because of the gauge symmetry and the supersymmetry. In particular, we do not need any fine-tuning if we arrange the theory so as to possess an extra global U(1) symmetry (U(1)R symmetry) which rotates the scalar fields.
119184438
Exact Results in Discretized Gauge Theories
We apply the localization technique to topologically twisted N=(2,2) supersymmetric gauge theory on a discretized Riemann surface (the generalized Sugino model). We exactly evaluate the partition function and the vacuum expectation value (vev) of a specific Q-closed operator. We show that both the partition function and the vev of the operator depend only on the Euler characteristic and the area of the discretized Riemann surface and are independent of the detail of the discretization. This localization technique may not only simplify numerical analysis of the supersymmetric lattice models but also connect the well-defined equivariant localization to the empirical supersymmetric localization.
118851659
Recent progress in lattice supersymmetry: from lattice gauge theory to black holes
Supersymmetry (SUSY) is a fascinating topic in theoretical physics, because of its unique and counterintuitive properties. It is expected to emerge as new physics beyond the standard model, and it is also a building block for supergravity and superstring theory. A number of exact results obtained via SUSY theories provide insights into field theory. However, the dynamics of many SUSY theories are not yet fully understood, and numerical study of SUSY theories through lattice simulations is promising as regards furthering this understanding. In this paper, I overview the current status of lattice SUSY by discussing its development in chronological order, and by reviewing some simple models. In addition, I discuss the numerical verification of gauge/gravity duality, which is one of the recent significant developments in this field.
199427830
Numerical analysis of discretized N = (2,2) SYM on polyhedra
We perform a numerical simulation of the two-dimensional N = (2,2) supersymmetric YangMills (SYM) theory on the discretized curved space. The U(1)A anomaly of the continuum theory is maintained also in the discretized theory as an unbalance of the number of the fermions. In the process, we propose a new phase-quenched approximation, which we call the “anomaly-phasequenched (APQ) method", to make the partition function and observables well-defined by U(1)A phase cancellation. By adopting APQ method, we estimate the Ward-Takahashi identity for exact SUSY on lattice and clarify contribution of the pseudo zero-modes to the pfaffian phase.
115622501
Renormalization and lattice artifacts
Lectures given at the Summer School on "Modern perspectives in lattice QCD", Les Houches, August 3-28, 2009
3166786
Exact chiral symmetry, topological charge and related topics
Abstract It has been shown recently that Dirac operators satisfying the Ginsparg-Wilson relation provide a solution of the chirality problem in QCD at finite lattice spacing. We discuss different ways to construct these operators and their properties. The possibility to define lattice chiral gauge theories is briefly discussed as well.
14726029
Locality properties of Neuberger's lattice Dirac operator
Abstract The gauge covariant lattice Dirac operator D which has recently been proposed by Neuberger satisfies the Ginsparg-Wilson relation and thus preserves chiral symmetry. The operator also avoids a doubling of fermion species, but its locality properties are not obvious. We now prove that D is local (with exponentially decaying tails) if the gauge field is sufficiently smooth at the scale of the cutoff. Further analytic and numerical studies moreover suggest that the locality of the operator is in fact guaranteed under far more general conditions.
43230714
Topological quantum field theory
A twisted version of four dimensional supersymmetric gauge theory is formulated. The model, which refines a nonrelativistic treatment by Atiyah, appears to underlie many recent developments in topology of low dimensional manifolds; the Donaldson polynomial invariants of four manifolds and the Floer groups of three manifolds appear naturally. The model may also be interesting from a physical viewpoint; it is in a sense a generally covariant quantum field theory, albeit one in which general covariance is unbroken, there are no gravitons, and the only excitations are topological.
119312632
Lattice formulation for 2d $ \mathcal{N} $ = (2, 2), (4, 4) super Yang-Mills theories without admissibility conditions
A bstractWe present a lattice formulation for two-dimensional $ \mathcal{N} $ = (2, 2) and (4, 4) supersymmetric Yang-Mills theories that resolves vacuum degeneracy for gauge fields without imposing admissibility conditions. Cases of U(N) and SU(N) gauge groups are considered, gauge fields are expressed by unitary link variables, and one or two supercharges are preserved on the two-dimensional square lattice. There does not appear fermion doubler, and no fine-tuning is required to obtain the desired continuum theories in a perturbative argument. This formulation is expected to serve as a more convenient basis for numerical simulations. The same approach will also be useful to other two-dimensional supersymmetric lattice gauge theories with unitary link variables constructed so far — for example, $ \mathcal{N} $ = (8,8) supersymmetricYang-Mills theory and $ \mathcal{N} $ = (2,2) supersymmetricQCD.
22400252
The light bound states of supersymmetric SU(2) Yang-Mills theory
A bstractSupersymmetry provides a well-established theoretical framework for extensions of the standard model of particle physics and the general understanding of quantum field theories. We summarise here our investigations of N=1$$ \mathcal{N}=1 $$ supersymmetric Yang-Mills theory with SU(2) gauge symmetry using the non-perturbative first-principles method of numerical lattice simulations. The strong interactions of gluons and their superpartners, the gluinos, lead to confinement, and a spectrum of bound states including glueballs, mesons, and gluino-glueballs emerges at low energies. For unbroken supersymmetry these particles have to be arranged in supermultiplets of equal masses. In lattice simulations supersymmetry can only be recovered in the continuum limit since it is explicitly broken by the discretisation. We present the first continuum extrapolation of the mass spectrum of supersymmetric Yang-Mills theory. The results are consistent with the formation of super-multiplets and the absence of non-perturbative sources of supersymmetry breaking. Our investigations also indicate that numerical lattice simulations can be applied to non-trivial supersymmetric theories.
119372020
Exactly massless quarks on the lattice
Abstract It is suggested that the fermion determinant for a vector-like gauge theory with strictly massless quarks can be represented on the lattice as det 1+V 2 , where V=X(X † X) −1/2 and X is the Wilson-Dirac lattice operator with a negative mass term. There is no undesired doubling and no need for any fine tuning. Several other appealing features of the formula are pointed out.