^{*}

^{†}

^{‡}

^{3}.

The realization of center and chiral symmetries in

Supersymmetric Yang-Mills theories (SYM) have been a useful laboratory to extend our understanding of nonperturbative phenomena. For instance, the study of supersymmetry combined with gauge symmetry has led to the discovery of duality as a feature of supersymmetric gauge theories. Besides the well-known gauge-gravity duality between the conformal

Numerical lattice simulations are the natural tool to explore nonperturbatively the phase diagram of strongly interacting gauge theories, such as

The investigations of chiral symmetry breaking of

Recently the gradient flow has been proposed as a regularization-scheme independent smoothing technique that is able to simplify drastically the renormalization of lattice bare composite operators

In this work we present an extended study of the phase diagram of

On the lattice, spinors are site variables and Yang-Mills fields

A nonzero gaugino mass breaks supersymmetry softly and the bare parameter

At zero temperature, the YM vacuum is expected a confining medium for external static color-electric charges. It can be probed through the Polyakov loop, which is the path ordered product of the links in the fundamental representation along a line which wraps in the compact direction

In the case of QCD, there is no real confinement-deconfinement phase transition but a crossover. In the confined phase, the quark-antiquark potential grows linearly until it is screened by another quark-antiquark pair popping up from the vacuum. In contrast, the identification of the deconfinement phase transition is clear for

In this section

The exact value of the proportionality constant

There have been many attempts to generalize the calculations of the condensate towards other gauge theories. The computed value has been compared to the results of one-flavour QCD based on the orientifold planar equivalence. This comparison is based on several assumptions that are discussed in Sec.

In thermal quantum field theory, temperature corresponds to the inverse radius of a compactified direction of the path integral with thermal boundary conditions, i.e., periodic for bosons and antiperiodic for fermions. The phases of

Fortunately, both the Polyakov loop and the gaugino condensate are exact order parameters for the respective phase transitions in

In previous investigations

Motivated in the context of trivializing maps

The term

The flow equations have a smoothening effect on the fields, which are Gaussian-like smeared over an effective radius

Written explicitly, the flowed bare gaugino condensate is

The numerical integration of the flow is relatively straightforward, and the methods we employ to compute the flowed gluino condensate are described in Appendix

The gradient flow method allows for the computation of the gaugino condensate at zero temperature, without an additive renormalization. We consider four ensembles from our previous investigations of SU(2) SYM at

The gradient flow scale

As shown in Fig.

Extrapolation of the chiral condensate to vanishing renormalized gaugino mass. The gaugino condensate scales almost linearly with the lattice spacing

Our results are of the same significance as the ones obtained with domain-wall and overlap fermions in

The comparison of our numerical result with strong and weak coupling instanton calculations

The second main purpose of this study is to investigate the realization of chiral and center symmetries in SU(2) SYM at finite temperature. Some first results in this direction were obtained in

The results are summarized in Figs.

The disconnected part of the susceptibility is expected to represent the largest contribution to the phase transition’s peak

Temperature dependence of the bare gaugino condensate is shown for

Binder cumulant of the Polyakov loop for the same

At the top: Chiral susceptibility for

This temperature approximately coincides with the deconfinement phase transition, which was found in

Chiral extrapolation of the deconfinement phase transition. The deconfinement temperature extrapolated to the chiral limit is

Our results show that, up to numerical uncertainties, in

The agreement of the two transitions is obtained up to the systematic uncertainties of our numerical determination which includes possible supersymmetry breaking lattice artefacts. Based on our previous investigations of particle spectrum and Ward identities

An interesting final quantity determined by our study is the deconfinement temperature in the chiral/supersymmetric limit. We compare our result to the deconfinement critical temperature of SU(2) YM found in

The result that deconfinement and restoration of chiral symmetry occur at the same critical point is far from trivial and can only be confirmed by nonperturbative methods. Lattice simulations can compute an estimation of the order parameters, but they do not provide immediately a qualitative physical interpretation of the mechanisms responsible for such results. A deep understanding of the dynamics of non-Abelian gauge theories in general and QCD in particular is, indeed, still missing and concepts like supersymmetry and string theory arose in the attempt to achieve it. An interpretation developed in the context of string theory might indeed provide such a deeper understanding of the mechanism behind our results. We provide here some rough arguments based on t’Hooft anomaly matching and a more detailed picture derived from the string theory perspective.

The coincidence of the transitions is in agreement with the predictions of

In

The framework is actually M-theory. The brane model is not equivalent to SYM but is in the same universality class. SYM is obtained when taking the IIA, i.e., ten dimensional limit.

The effective theory on the world-volume of the four-brane is aThe gradient flow has enabled us to explore the confining and the chiral properties of the

We have also explored the phase diagram of the theory at nonzero temperature. The chiral condensate provides a signal compatible with a second order phase transition at smaller gaugino masses. In comparison to our previous investigations without gradient flow in

We are currently working towards new applications of the gradient flow. One immediate further step is to investigate thermal SYM for SU(3) gauge group. Another very interesting direction is to take profit of the method to study the vacuum’s structure of the theory at zero temperature. In addition we plan to investigate possible applications for the renormalization of the supercurrent in theories with extended supersymmetry

The configurations and the parameters of this study have been created in the long term effort of the DESY-Münster collaboration to study the properties of SYM. We thank in particular Gernot Münster and Istvan Montvay for helpful comments and discussions. The authors gratefully acknowledge the Gauss Centre for Supercomputing e. V. (

In this Appendix we describe the path for the numerical computation of the gradient flow

Further, the vector

The discrete version of the gluino condensate

Here

Condensate, chiral susceptibility and Binder cumulant of the Polyakov loop at

Adjoint pion masses for each