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Supersymmetric Yang-Mills (SYM) theories in four dimensions exhibit many interesting nonperturbative phenomena that can be studied by means of Monte Carlo lattice simulations. However, the lattice regularization breaks supersymmetry explicitly, and in general a fine-tuning of a large number of parameters is required to correctly extrapolate the theory to the continuum limit. From this perspective, it is important to preserve on the lattice as many symmetries of the original continuum action as possible. Chiral symmetry for instance prevents an additive renormalization of the fermion mass. A (modified) version of chiral symmetry can be preserved exactly if the Dirac operator fulfills the Ginsparg-Wilson relation. In this contribution, we present an exploratory nonperturbative study of

Supersymmetric Yang-Mills (SYM) theories are promising extensions of the Standard Model to energies of the order and beyond the TeV scale. However, as the efforts for the experimental discovery of the predicted superparticles are still unsuccessful, the interest to supersymmetry (SUSY) has been recently motivated by the possibility of understanding in supersymmetric theories many nonperturbative features of strong interactions. Yang-Mills theories exhibit many emergent low-energy phenomena, like confinement and chiral symmetry breaking, that are difficult to understand analytically. The conjecture that there exist supersymmetric models that share the same physical properties if their coupling constant

The electromagnetic duality has been proposed originally for a certain class of theories in Refs.

The presence of many different fermion and scalar fields in the Lagrangian of

The tuning problem can be simplified drastically if a (modified) chiral symmetry is preserved on the lattice by using Ginsparg-Wilson fermions. For instance, the fermion mass would be protected by chiral symmetry, and Yukawa interactions would couple correctly left-handed and right-handed spinors to the corresponding scalar fields. The advantages of chiral fermions might be crucial for simulations of certain supersymmetric theories. Ginsparg-Wilson fermions allow one to simulate in principle

Models with enlarged SUSY can be constructed from simpler supersymmetric theories living in a space-time with extra dimensions. The pure

In this paper we explore overlap lattice simulations for the pure

The simplest Yang-Mills theory with a single conserved supercharge is

Intact supersymmetry implies an exact cancellation between fermion and boson loops in a perturbative asymptotic expansion, allowing an analytic calculation of many quantities and properties that would be out of the reach of perturbation theory in nonsupersymmetric theories. A remarkable example is the exact

The overlap formalism allows one to construct a fermion action with an exact (modified) chiral symmetry even at nonzero lattice spacing. The overlap operator is defined as

The massless overlap operator fulfills the modified chiral anticommutator

The sign function of the Hermitian Dirac-Wilson operator can be computed from an eigenvalue decomposition, but for practical numerical calculation it is better expressed as

Chiral symmetry prevents additive renormalization of the fermion mass; therefore, the renormalized mass is proportional to

The fermion propagator in the massless case

The implementation of the hybrid Monte Carlo (HMC) algorithm for overlap gluinos has to face several challenges compared to the corresponding QCD simulations. There are two main obstacles. The first is the general problem of topology and fermion zero modes, and the second is the specific problem of the Majorana nature of the gluino field. Several strategies can be applied to control these problems, like the domain-wall representation of the Ginsparg-Wilson relation or simulations at fixed topology, which all imply a deviation from the exact chiral symmetric action. We apply instead a polynomial approximation of the sign function as a simple strategy to control the simulations.

Supersymmetry requires that the gluino mass is exactly zero. Even in the continuum, the zero mass limit is quite peculiar and requires a careful consideration of the order of chiral and infinite volume limit. As a consequence of chiral symmetry, related problems arise with the overlap operator on the lattice. Each lattice configuration featuring a nontrivial topology has at least one zero mode of the overlap operator, such that its determinant and its Boltzmann probabilistic weight are both exactly zero. This simple consequence of preserving chiral symmetry exactly effectively prevents topological charge fluctuations and the generation of instantonlike configurations. However, zero modes and topological objects are also a fundamental feature of the mechanisms responsible for gluino condensation

The integration of Majorana fermions leads to the Pfaffian of the operator

Several possibilities to overcome the challenge of the HMC algorithm due to the nondifferentiability of the sign function have been discussed in the literature

Our approach is different from the preliminary studies with overlap gluinos presented in Ref.

As already explained the smoothing of the sign problem by a polynomial approximation reduces the problem of near zero eigenvalues, which can be considered as a controlled way to relax the Ginsparg-Wilson relation. A polynomial of order

Simulations with overlap fermions are significantly more expensive compared those with Wilson fermions. The calculation of the force and of the inverse of the overlap Dirac operator requires at each iteration the evaluation of the approximation of the sign function. Our main studies to check the validity of the proposed algorithm have been done for the gauge group SU(2) on an

Summary table of analyzed runs.

The gauge action is discretized using a combination of

Polynomial approximation of the sign function for

The rational approximation used for the computation of the force can be chosen to be in principle different from the one used for the Metropolis or heat bath steps. We approximate the fourth root of the Dirac operator by two rational approximations with four and 11 fractions, used in combination with multiple timescale integrators. The rational approximations used for the Metropolis step require instead a higher accuracy and are the sum of 37 fractions. The spectral intervals used to construct the rational approximations are chosen to fit all eigenvalues of the overlap operator, including the lowest modes.

The boundary conditions imposed on fermion fields are relevant when dealing with supersymmetry simulated on a small box. As the difference in thermal statistics between bosons and fermions breaks supersymmetry at nonzero temperature, we impose periodic boundary conditions to all fields in all directions.

The Lagrangian of

(a) Scale

The chirally extrapolated value of

Simulations at finer lattice spacing with supersymmetry preserving periodic boundary conditions might result into topological freezing. In our current setup, the topological charge is fluctuating between different topological sectors; see Fig.

Simulations with overlap gluinos open new perspectives beyond the solution to the problem of the tuning of supersymmetric Yang-Mills theories to the continuum limit. In particular, the vacuum structure of

In the overlap formalism the gluino condensate

The bare gluino condensate is shown for four different polynomial approximations in Fig.

(a) The extrapolation to the chiral limit of the gluino condensate

The striking feature of the chiral condensate is its smooth dependence on the order of the polynomial used to approximate the sign function, meaning that our approach has been able to capture the chiral properties of

Monte Carlo history of the gluino condensate on a lattice of volume

The gluino condensate is strictly positive in all our configurations. In the chiral limit, the second phase where the gluino condensate is negative can be reached simply after performing a chiral rotation, in the same way in which all vacuum expectation values of the Polyakov loop in the deconfined phase can be reached by a center symmetry transformation of the temporal links in a given time slice. The interface tension is however an interesting thermodynamic quantity that is difficult to determine in the current setup, as tunneling during a Monte Carlo simulation of the chiral condensate from positive to negative value is impossible, even on very small volumes. It would be interesting to study the dynamics of the domain-wall interpolating between such two phases, a possible solution to achieve this goal could be given by reweighting combined with a multicanonical approach

The dominant contribution to the gluino condensate comes from the eigenvalues of the overlap operator near the origin, and it is crucial to confirm our results by studying the lowest part of the spectrum of

Lowest part of the spectrum of the overlap operator as obtained from an ensemble of configurations on a lattice of volume

A full comparison between different determinations of the gluino condensate requires one to set a common scale and a common scheme to determine the multiplicative renormalization constants

In Ref.

We finally discuss in more detail the comparison of our polynomial approximation and the results from domain-wall fermions in Ref.

Periodic boundary conditions applied to all fields imply that the partition function being simulated is the Witten index, corresponding in the Hamiltonian formalism to the supertrace

The idea is to use the Witten index as a probe for supersymmetry restoration on the lattice. Any deviation from a constant behavior of

Fortunately, the ultraviolet divergence coming from the mixing with the identity operator does not depend on

The partial derivative with respect to the box size

Note that

Subtracted derivative of the Witten index

Along the same ideas as the Seiberg-Witten electromagnetic duality, several recent theoretical studies opened the possibility for an understanding of confinement in four-dimensional

The compactification of more than one space-time dimension has been studied in the context of the Eguchi-Kawai volume reduction

(a) Distribution of the Polyakov loop for periodic (red) and antiperiodic (blue) on a volume

The bound states of

The gluino-glue mass is measured from the exponential decay at large time distance of the zero momentum projected correlator of the operator

(a) Correlator of the gluino glue on the lattice volume

The signal in the

Correlator of the

The adjoint pion is not a physical particle of the theory, but it is defined in the context of partially quenched chiral perturbation theory. Its mass can be extracted from the connected part of the correlator of the

Pion mass as a function of

Based on this first exploratory studies, Ginsparg-Wilson fermions provide a starting point for several interesting further investigations of supersymmetric theories. In the following, we list a few directions which we are considering for our next studies.

The index theorem relates the difference of the zero eigenvalues with positive and negative chirality of the massless overlap operator to the topological charge

If two dimensions are compactified, and if their length is reduced to zero, the lower-dimensional effective theory has an enlarged supersymmetry. The two-dimensional model would be an ideal benchmark to study the renormalization properties of theories with extended supersymmetry constructed from

The improvements guaranteed by overlap fermions in supersymmetric Yang-Mills theory with more than one conserved supercharge or coupled to matter chiral superfields have not been fully addressed so far. Several difficulties appeared even in the simpler case of the Wess-Zumino model, in particular for the consistency between the Majorana condition and chiral symmetry in the Yukawa couplings

The main advantage of preserving exact chiral symmetry is that

The infrared observables we have studied in the current paper are either independent on the order

The Witten index, and in particular its derivative, allows one to check and confirm the expected boson degeneracy required by unbroken supersymmetry. Our numerical results are compatible with a constant Witten index up to lattice boxes as small as

We thank G. Münster and I. Montvay for helpful comments and discussions. The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. for funding this project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre. Further computing time has been provided on the compute cluster PALMA of the University of Münster. G. B. and C. L. acknowledge support from the Deutsche Forschungsgemeinschaft (DFG) Grant No. BE 5942/2-1.