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.Comment: 21 pages, 8 figures. v2: References added, two figures were modified for clarity. v3: Normalisation of lattice coupling corrected by factor of two resulting in change of estimate for c_cri

Lattice formulation of N=4{\cal N}=4 super Yang-Mills theory

We construct a lattice action for ${\cal N}=4$ super Yang-Mills theory in four dimensions which is local, gauge invariant, free of spectrum doubling and possesses a single exact supersymmetry. Our construction starts from the observation that the fermions of the continuum theory can be mapped into the component fields of a single real anticommuting Kahler-Dirac field. The original supersymmetry algebra then implies the existence of a nilpotent scalar supercharge $Q$ and a corresponding set of bosonic superpartners. Using this field content we write down a $Q$-exact action and show that, with an appropriate change of variables, it reduces to a well-known twist of ${\cal N}=4$ super Yang-Mills theory due to Marcus. Using the discretization prescription developed in an earlier paper on the ${\cal N}=2$ theory in two dimensions we are able to translate this geometrical action to the lattice.Comment: 15 pages. 1 reference correcte

Relations among Supersymmetric Lattice Gauge Theories via Orbifolding

We show how to derive Catterall's supersymmetric lattice gauge theories directly from the general principle of orbifolding followed by a variant of the usual deconstruction. These theories are forced to be complexified due to a clash between charge assignments under U(1)-symmetries and lattice assignments in terms of scalar, vector and tensor components for the fermions. Other prescriptions for how to discretize the theory follow automatically by orbifolding and deconstruction. We find that Catterall's complexified model for the two-dimensional N=(2,2) theory has two independent preserved supersymmetries. We comment on consistent truncations to lattice theories without this complexification and with the correct continuum limit. The construction of lattice theories this way is general, and can be used to derive new supersymmetric lattice theories through the orbifolding procedure. As an example, we apply the prescription to topologically twisted four-dimensional N=2 supersymmetric Yang-Mills theory. We show that a consistent truncation is closely related to the lattice formulation previously given by Sugino.Comment: 20 pages, LaTeX2e, no figur

Matrix formulation of superspace on 1D lattice with two supercharges

Following the approach developed by some of the authors in recent papers and using a matrix representation for the superfields, we formulate an exact supersymmetric theory with two supercharges on a one dimensional lattice. In the superfield formalism supersymmetry transformations are uniquely defined and do not suffer of the ambiguities recently pointed out by some authors. The action can be written in a unique way and it is invariant under all supercharges. A modified Leibniz rule applies when supercharges act on a superfield product and the corresponding Ward identities take a modified form but hold exactly at least at the tree level, while their validity in presence of radiative corrections is still an open problem and is not considered here.Comment: 25 page

Towards lattice simulation of the gauge theory duals to black holes and hot strings

A generalization of the AdS/CFT conjecture postulates a duality between IIA string theory and 16 supercharge Yang-Mills quantum mechanics in the large N 't Hooft limit. At low temperatures string theory describes black holes, whose thermodynamics may hence be studied using the dual quantum mechanics. This quantum mechanics is strongly coupled which motivates the use of lattice techniques. We argue that, contrary to expectation, the theory when discretized naively will nevertheless recover continuum supersymmetry as the lattice spacing is sent to zero. We test these ideas by studying the 4 supercharge version of this Yang-Mills quantum mechanics in the 't Hooft limit. We use both a naive lattice action and a manifestly supersymmetric action. Using Monte Carlo methods we simulate the Euclidean theories, and study the lattice continuum limit, for both thermal and non-thermal periodic boundary conditions, confirming continuum supersymmetry is recovered for the naive action when appropriate. We obtain results for the thermal system with N up to 12. These favor the existence of a single deconfined phase for all non-zero temperatures. These results are an encouraging indication that the 16 supercharge theory is within reach using similar methods and resources.Comment: 49 pages, 14 figure

Deformed matrix models, supersymmetric lattice twists and N=1/4 supersymmetry

A manifestly supersymmetric nonperturbative matrix regularization for a twisted version of N=(8,8) theory on a curved background (a two-sphere) is constructed. Both continuum and the matrix regularization respect four exact scalar supersymmetries under a twisted version of the supersymmetry algebra. We then discuss a succinct Q=1 deformed matrix model regularization of N=4 SYM in d=4, which is equivalent to a non-commutative $A_4^*$ orbifold lattice formulation. Motivated by recent progress in supersymmetric lattices, we also propose a N=1/4 supersymmetry preserving deformation of N=4 SYM theory on $\R^4$. In this class of N=1/4 theories, both the regularized and continuum theory respect the same set of (scalar) supersymmetry. By using the equivalence of the deformed matrix models with the lattice formulations, we give a very simple physical argument on why the exact lattice supersymmetry must be a subset of scalar subalgebra. This argument disagrees with the recent claims of the link approach, for which we give a new interpretation.Comment: 47 pages, 3 figure

A lattice study of the two-dimensional Wess Zumino model

We present results from a numerical simulation of the two-dimensional Euclidean Wess-Zumino model. In the continuum the theory possesses N=1 supersymmetry. The lattice model we employ was analyzed by Golterman and Petcher in \cite{susy} where a perturbative proof was given that the continuum supersymmetric Ward identities are recovered without finite tuning in the limit of vanishing lattice spacing. Our simulations demonstrate the existence of important non-perturbative effects in finite volumes which modify these conclusions. It appears that in certain regions of parameter space the vacuum state can contain solitons corresponding to field configurations which interpolate between different classical vacua. In the background of these solitons supersymmetry is partially broken and a light fermion mode is observed. At fixed coupling the critical mass separating phases of broken and unbroken supersymmetry appears to be volume dependent. We discuss the implications of our results for continuum supersymmetry breaking.Comment: 32 pages, 12 figure

Exact Vacuum Energy of Orbifold Lattice Theories

We investigate the orbifold lattice theories constructed from supersymmetric Yang-Mills matrix theories (mother theories) with four and eight supercharges. We show that the vacuum energy of these theories does not receive any quantum correction perturbatively.Comment: 14 pages, no figure, LaTeX2e, typos corrected, errors in references corrected, comments adde

Two-dimensional N=(2,2) super Yang-Mills theory on computer

We carry out preliminary numerical study of Sugino's lattice formulation \cite{Sugino:2004qd,Sugino:2004qdf} of the two-dimensional $\mathcal{N}=(2,2)$ super Yang-Mills theory (2d $\mathcal{N}=(2,2)$ SYM) with the gauge group \SU(2). The effect of dynamical fermions is included by re-weighting a quenched ensemble by the pfaffian factor. It appears that the complex phase of the pfaffian due to lattice artifacts and flat directions of the classical potential are not problematic in Monte Carlo simulation. Various one-point supersymmetric Ward-Takahashi (WT) identities are examined for lattice spacings up to $a=0.5/g$ with the fixed physical lattice size $L=4.0/g$, where $g$ denotes the gauge coupling constant in two dimensions. WT identities implied by an exact fermionic symmetry of the formulation are confirmed in fair accuracy and, for most of these identities, the quantum effect of dynamical fermions is clearly observed. For WT identities expected only in the continuum limit, the results seem to be consistent with the behavior expected from supersymmetry, although we do not see clear distintion from the quenched simulation. We measure also the expectation values of renormalized gauge-invariant bi-linear operators of scalar fields.Comment: 24 pages, 10 figures, the distribution of the complex phase of the pffafian is also measured, the final version to appear in JHE

Lattice formulation of two-dimensional N=(2,2) super Yang-Mills with SU(N) gauge group

We propose a lattice model for two-dimensional SU(N) N=(2,2) super Yang-Mills 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.Comment: 16 pages, 3 figures; v2: typo crrected; v3: 18 pages, a version to appear in JHE