Simon Catterall
Recently, new theoretical ideas have allowed the construction of lattice actions which are explicitly invariant under one or more supersymmetries. These theories are local and free of doublers and in the case of Yang-Mills theories also possess exact gauge invariance. In this talk these ideas are reviewed with particular emphasis being placed on ${\cal N}=4$ super Yang-Mills theory.
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Recent development in numerical simulations of supersymmetric Yang-Mills (SYM) theories on the lattice is reviewed.
The N=2 supersymmetric Yang-Mills theory is formulated on the lattice. The feasibility of numerical simulations is discussed.
We construct lattice actions for a variety of (2,2) supersymmetric gauge theories in two dimensions with matter fields interacting via a superpotential.
The partition function (quantum transition amplitude) of the gauge system with gauge group $Z_2$ coupled with Majorana fermions is calculated on the regular 3D cubic lattice.
We comment on several issues concerning both the physics and authorship of article hep-lat/0305020.
We summarise the latest results of our collaboration concerning N=1 supersymmetric Yang-Mills theory in four dimensions on the lattice. We investigate the expected formation of supersymmetric multiplets of the lightest particles and the behaviour of the topological susceptibility approaching the supersymmetric limit of the theory.
Supersymmetric gauge theories in four dimensions have taught us many important physics lessons. These can both inform and be informed by future work on the lattice. I focus on three issues: the properties of supersymmetric Yang-Mills theory and its relation to the non-supersymmetric case; the properties of gauge theories with matter and their relation to real QCD; and, briefly, the recent discovery that gauge theories and string theories are more deeply connected than ever previously realized. Specific questions for lattice gauge theorists to consider are raised in the context of the first two…
The perspectives of numerical simulations in supersymmetric quantum field theories with vector-like gauge symmetries are discussed. A numerical simulation algorithm for SU(2) gauge theory with gluinos is studied and the first results on the glueball-gluinoball spectrum are presented.
We propose a numerical method of estimating various physical quantities in lattice (supersymmetric) quantum mechanics. The method consists only of deterministic processes such as computing a product of transfer matrix, and has no statistical uncertainties. We use the numerical quadrature to define the transfer matrix as a finite dimensional matrix, and find that it effectively works by rescaling variable for sufficiently small lattice spacings. For a lattice supersymmetric quantum mechanics, the correlators can be estimated without statistical errors, and the effective masses coincide with the…
SUSY Ward identities for the N=1 SU(2) SUSY Yang-Mills theory are studied on the lattice in a non-perturbative numerical approach. As a result a determination of the subtracted gluino mass is obtained.
We summarize our investigations of several aspects of $\mathcal{N}=1$ supersymmetric Yang-Mills (SYM) theory. We present our final results for SU(3) $\mathcal{N}=1$ SYM simulated with Wilson fermions. We also discuss the first test of the simulations of the theory with overlap gluinos. Finally, we present some recent progresses concerning the phase structure of the compactified theory on $R^3\times S^1$.
We investigate the lattice regularization of $\mathcal{N} = 4$ supersymmetric Yang-Mills theory, by stochastically computing the eigenvalue mode number of the fermion operator. This provides important insight into the non-perturbative renormalization group flow of the lattice theory, through the definition of a scale-dependent effective mass anomalous dimension. While this anomalous dimension is expected to vanish in the conformal continuum theory, the finite lattice volume and lattice spacing generically lead to non-zero values, which we use to study the approach to the continuum limit. Our n…
We present ongoing investigations of maximally supersymmetric Yang--Mills ($Q = 16$ SYM) theory in three space-time dimensions. At low temperatures and large $N$ this theory is related to black branes in higher-dimensional quantum gravity. Building on previous work that focused on the homogeneous `D2' phase of the theory, we are now exploring phase transitions between this D2 phase and the localized `D0' phase.
Numerical simulations of supersymmetric theories on the lattice are intricate and challenging with respect to their theoretical foundations and algorithmic realisation. Nevertheless, the simulations of a four-dimensional supersymmetric gauge theory have made considerable improvements over the recent years. In this contribution we summarise the results of our collaboration concerning the mass spectrum of this theory. The investigation of systematic errors allows now a more precise estimate concerning the expected formation of supersymmetric multiplets of the lightest particles. These multiplets…
Species doubling is a problem that infects most numerical methods that use a spatial lattice. An understanding of species doubling can be found in the Nielsen-Ninomiya theorem which gives a set of conditions that require species doubling. The transverse lattice approach to solving field theories, which has at least one spatial lattice, fails one of the conditions of the Nielsen-Ninomiya theorem nevertheless one still finds species doubling for the standard Lagrangian formulation of the transverse lattice. We will show that the Supersymmetric Discrete Light Cone Quantization (SDLCQ) formulation…
We present preliminary numerical results from a lattice study of the two-dimensional O(3) non-linear sigma model. In the continuum this model possesses N=2 supersymmetry. The lattice formulation we use retains an exact (twisted) supersymmetry except for a soft breaking associated with a Wilson mass term needed to remove the doubles. Our numerical results show that the partition function is independent of coupling as predicted by supersymmetry.
The one loop corrections to the supersymmetric Ward identities (WIs) in the discretized N=1 SU(2) supersymmetric Yang-Mills theory can be investigated by means of lattice perturbation theory. The supersymmetry (SUSY) is explicitly broken by the lattice discretization as well as by the introduction of Wilson fermions. However, the renormalization of the supercurrent can be carried out in a scheme that restores the nominal continuum WIs. We present our work in progress which is concerned with the 1-loop renormalization of the local supercurrent, i.e. with the perturbative computation of the corr…
In this work we perform a perturbative study of the Noether supercurrent operator in the context of Supersymmetric ${\cal N}{=}1$ Yang-Mills (SYM) theory on the lattice. The supercurrent mixes with several other operators, some of which are not gauge invariant, having the same quantum numbers. We determine, to one loop order, the renormalization and all corresponding mixing coefficients by computing relevant Green's functions of each one of the mixing operators with external elementary fields. Our calculations are performed both in dimensional and lattice regularization. From the first regular…
We discuss N=2 supersymmetric quantum mechanics on the lattice using the fermion loop formulation. In this approach the system naturally decomposes into a bosonic and fermionic sector. This allows us to deal with the sign problem arising in the context of broken supersymmetry due to the vanishing of the Witten index. Employing transfer matrix techniques we obtain exact results at finite lattice spacing and are hence able to study how the continuum limit is approached. In particular, we determine how supersymmetry is restored and how, in the case of broken supersymmetry, the goldstino mode emer…
Simulations of supersymmetric models on the lattice with (spontaneously) broken supersymmetry suffer from a fermion sign problem related to the vanishing of the Witten index. We propose a novel approach which solves this problem in low dimensions by formulating the path integral on the lattice in terms of fermion loops. The formulation is based on the exact hopping expansion of the fermionic action and allows the explicit decomposition of the partition function into bosonic and fermionic contributions. We devise a simulation algorithm which separately samples the fermionic and bosonic sectors,…
This talk is an overview of our recent investigations of supersymmetric and near conformal gauge theories. We have studied extensively $\mathcal{N}=1$ super Yang-Mills theory, most recently with the gauge group SU(3). In addition we have investigated theories that show indications for a conformal behavior at an infrared fixed point. We have included a mixed fundamental and adjoint fermion action setup in our studies. I will explain how this is related to the investigation of supersymmetric QCD on the lattice and present some first studies of the main obstacles that need to be addressed in the …
We study the lattice model for the supersymmetric Yang-Mills theory in two dimensions proposed by Cohen, Kaplan, Katz, and Unsal. We re-examine the formal proof for the absence of susy breaking counter terms as well as the stability of the vacuum by an explicit perturbative calculation for the case of U(2) gauge group. Introducing fermion masses and treating the bosonic zero momentum mode nonperturbatively, we avoid the infra-red divergences in the perturbative calculation. As a result, we find that there appear mass counter terms for finite volume which vanish in the infinite volume limit so …
Owing to confinement, the fundamental particles of N=1 Supersymmetric Yang-Mills (SYM) theory, gluons and gluinos, appear only in colourless bound states at zero temperature. Compactifying the Euclidean time dimension with periodic boundary conditions for fermions preserves supersymmetry, and confinement is predicted to persist independently of the length of the compactified dimension. This scenario can be tested non-perturbatively with Monte-Carlo simulations on a lattice. SUSY is, however, broken on the lattice and can be recovered only in the continuum limit. The partition function of compa…
We provide details of the lattice construction of five-dimensional maximally supersymmetric Yang-Mills theory. The lattice theory is supersymmetric, gauge invariant and free from spectrum doublers. Such a supersymmetric lattice formulation is interesting as it can be used for non-perturbative explorations of the five-dimensional theory, which has a known gravitational dual.
We construct lattice action for five-dimensional maximally supersymmetric Yang-Mills theory. This supersymmetric lattice formulation can be used to explore the non-perturbative regime of the continuum target theory, which has a known gravitational dual.
In numerical investigations of supersymmetric Yang-Mills theory on a lattice, the supersymmetric Ward identities are valuable for finding the critical value of the hopping parameter and for examining the size of supersymmetry breaking by the lattice discretisation. In this article we present an improved method for the numerical analysis of supersymmetric Ward identities, which takes into account the correlations between the various observables involved. We present the first complete analysis of supersymmetric Ward identities in $\mathcal{N}=1$ supersymmetric Yang-Mills theory with gauge group …
We study and simulate N=2 supersymmetric Wess-Zumino models in one and two dimensions. For any choice of the lattice derivative, the theories can be made manifestly supersymmetric by adding appropriate improvement terms corresponding to discretizations of surface integrals. In particular, we check that fermionic and bosonic masses coincide and the unbroken Ward identities are fulfilled to high accuracy. Equally good results for the effective masses can be obtained in a model with the SLAC derivative (even without improvement terms). In two dimensions we introduce a non-standard Wilson term in …
Certain classes of supersymmetric gauge theories, including the well known N=4 supersymmetric Yang-Mills theory, that takes part in the AdS/CFT correspondence, can be formulated on a Euclidean spacetime lattice using the techniques of exact lattice supersymmetry. Great ideas such as topological field theories, Dirac-Kaehler fermions, geometric discretization all come together to create supersymmetric lattice theories that are gauge-invariant, doubler free, local and exact supersymmetric. We discuss the recent lattice constructions of supersymmetric Yang-Mills theories in two and three dimensio…
N=1 Supersymmetric QCD on the lattice using overlap fermions
本文以 N=1 SQCD 为原型，利用重叠费米子与 Ginsparg-Wilson 关系构建保持格点手征对称性的公式，通过引入辅助场减少抵消项，为超对称规范理论的非微扰格点研究奠定基础。
In $\mathcal{N}=1$ supersymmetric Yang-Mills theory, regularised on a space-time lattice, in addition to the breaking by the gluino mass term, supersymmetry is broken explicitly by the lattice regulator. In addition to the parameter tuning in the theory, the supersymmetric Ward identities can be used as a tool to investigate lattice artefacts as well as to check whether supersymmetry can be recovered in the chiral and continuum limits. In this paper we present the numerical results of an analysis of the supersymmetric Ward identities for our available gauge ensembles at different values of the…
We show that a large class of Euclidean extended supersymmetric lattice gauge theories constructed in [hep-lat/0302017 - hep-lat/0503039] can be regarded as compact formulations by using the polar decomposition of the complex link fields. In particular, the gauge part of the supersymmetric lattice action is the standard Wilson action. This formulation facilitates the construction of gauge invariant operators.
Supersymmetry on the lattice is explicitly broken by the gluino mass and lattice artifacts. However, it can be restored in the continuum limit by fine tuning the parameters based on the renormalized Ward identities. On the renormalization step not only the mass but also the renormalization of the supercurrent needs to be addressed. Here we present a lattice investigation to obtain the renormalization factors of the supercurrent for $\mathcal{N}$=1 Super-Yang Mills theory in a gauge invariant renormalization scheme. We also provide the conversion factors which are necessary in order to translat…
We formulate supersymmetric Euclidean spacetime Ad* lattices whose classical continuum limits are U(N) supersymmetric Yang-Mills theories with sixteen supercharges in d=1,2,3 and 4 dimensions. This family includes the especially interesting N=4 supersymmetry in four dimensions, as well as a Euclidean path integral formulation of Matrix Theory on a one dimensional lattice.
Lattice simulation of supersymmetric gauge theories is not straightforward. In some cases the lack of manifest supersymmetry just necessitates cumbersome fine-tuning, but in the worse cases the chiral and/or Majorana nature of fermions makes it difficult to even formulate an appropriate lattice theory. We propose to circumvent all these problems inherent in the lattice approach by adopting a non-lattice approach in the case of one-dimensional supersymmetric gauge theories, which are important in the string/M theory context.
A lattice-type regularization of the supersymmetric field theories on a supersphere is constructed by approximating the ring of scalar superfields by an integer-valued sequence of finite dimensional rings of supermatrices and by using the differencial calculus of non-commutative geometry. The regulated theory involves only finite number of degrees of freedom and is manifestly supersymmetric.
The spectrum of particles in supersymmetric Yang-Mills theory is expected to contain a spin 1/2 bound state of gluons and gluinos, the gluino-glue particle. We study the mass of this particle in softly broken supersymmetric Yang-Mills theory on a lattice by means of numerical simulations. The main focus is the estimation of finite size effects. We extrapolate the mass first to the infinite volume and then to the limit of a vanishing gluino mass. The results indicate that finite size effects are tolerable on lattices of moderate size, and that remaining deviations from supersymmetry are probabl…
We perform a pilot study of the perturbative renormalization of a Supersymmetric gauge theory with matter fields on the lattice. As a specific example, we consider Supersymmetric ${\cal N}{=}1$ QCD (SQCD). We study the self-energies of all particles which appear in this theory, as well as the renormalization of the coupling constant. To this end we compute, perturbatively to one-loop, the relevant two-point and three-point Green's functions using both dimensional and lattice regularizations. Our lattice formulation involves the Wilson discretization for the gluino and quark fields; for gluons …
Fermion boundary conditions play a relevant role in revealing the confinement mechanism of N=1 supersymmetric Yang-Mills theory with one compactified space-time dimension. A deconfinement phase transition occurs for a sufficiently small compactification radius, equivalent to a high temperature in the thermal theory where antiperiodic fermion boundary conditions are applied. Periodic fermion boundary conditions, on the other hand, are related to the Witten index and confinement is expected to persist independently of the length of the compactified dimension. We study this aspect with lattice Mo…
We perform a pilot study of the perturbative renormalization of a Supersymmetric gauge theory with matter fields on the lattice. As a specific example, we consider Supersymmetric ${\cal N}{=}1$ QCD (SQCD). We study the self-energies of all particles which appear in this theory, as well as the renormalization of the coupling constant. To this end we compute, perturbatively to one-loop, the relevant two-point and three-point Green's functions using both dimensional and lattice regularizations. Our lattice formulation involves the Wilson discretization for the gluino and quark fields; for gluons …
We prove that the prescription for construction of supersymmetric lattice gauge theories by orbifolding and deconstruction directly leads to Catterall's geometrical discretization scheme in general. These two prescriptions always give the same lattice discretizations when applied to theories of p-form fields. We also show that the geometrical discretization scheme can be applied to more general theories.
We examine the relation between twisted versions of the extended supersymmetric gauge theories and supersymmetric orbifold lattices. In particular, for the $\mathcal{N}=4$ SYM in $d=4$, we show that the continuum limit of orbifold lattice reproduces the twist introduced by Marcus, and the examples at lower dimensions are usually Blau-Thompson type. The orbifold lattice point group symmetry is a subgroup of the twisted Lorentz group, and the exact supersymmetry of the lattice is indeed the nilpotent scalar supersymmetry of the twisted versions. We also introduce twisting in terms of spin groups…
The spectrum of supersymmetric Yang-Mills theory presented so far shows an unexpected gap between the bosonic and fermionic masses. This finding was in contradiction with the basic requirements of supersymmetry. In this work we will present new results indicating that the mass gap is reduced at a smaller lattice spacing. Hence lattice artifacts are the most likely explanation for it. These new results have been obtained at a larger beta value and on a larger lattice.
A lattice formulation of N=2 supersymmetric SYK model
本文提出在晶格上构建N=2超对称SYK模型，利用循环李导规则实现超对称性。
We study 4-dimensional SQCD with gauge group SU(Nc) and Nf flavors of chiral supermultiplets on the lattice. We perform extensive calculations of matrix elements and renormalization factors of composite operators in Perturbation Theory. In particular, we compute the renormalization factors of quark and squark bilinears, as well as their mixing at the quantum level with gluino and gluon bilinear operators. From these results we construct correctly renormalized composite operators, which are free of mixing effects and may be employed in non-perturbative studies of Supersymmetry. All our calculat…
In this work we study the renormalization of the SUSY Noether current in Supersymmetric $\mathcal{N}$ = 1 Yang-Mills (SYM) theory on the lattice. In particular, we study the mixing of the current with all other compatible operators of dimension 7/2 and 5/2, leading from the lattice-regularized to the $\overline{\rm MS}$-renormalized operator basis. We perform our task in two ways: (a) We compute, in dimensional regularization, the conversion factors relating the $\overline{\rm MS}$ scheme to an intermediate gauge-invariant coordinate-space scheme. In this second scheme, renormalization can be …
We investigate the pure gauge sector of Super-QCD, i.e. Super-Yang-Mills (SYM) theory, with focus on the bound states. To improve chiral symmetry as well as supersymmetry at finite lattice spacing, we use a deformed SYM lattice action. It contains a twist term, similar to the lattice formulation of twisted mass QCD. We present the status of our theoretical and numerical investigation.
In this paper, we construct a lattice formulation for two-dimensional N=(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.
In this paper we detail the lattice constructions of several classes of supersymmetric quiver gauge theories in two and three Euclidean spacetime dimensions possessing exact supersymmetry at finite lattice spacing. Such constructions are obtained through the methods of topological twisting and geometric discretization of Euclidean Yang--Mills theories with eight and sixteen supercharges in two and three dimensions. We detail the lattice constructions of two-dimensional quiver gauge theories possessing four and eight supercharges and three-dimensional quiver gauge theories possessing eight supe…
Recently a class of supersymmetric gauge theories have been successfully implemented on the lattice. However, there has been an ongoing debate on whether lattice versions of some of these theories suffer from a sign problem, with independent simulations for the ${\cal N} = (2, 2)$ supersymmetric Yang-Mills theories in two dimensions yielding seemingly contradictory results. Here, we address this issue from an interesting theoretical point of view. We conjecture that the sign problem observed in some of the simulations is related to the so called Neuberger 0/0 problem, which arises in ordinary …
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 supe…
We provide a general classification of supersymmetric lattice gauge theories that can be obtained from orbifolding of theories with four and eight supercharges. We impose at least one preserved supercharge on the lattice and Lorentz invariance in the naive continuum limit. Starting with four supercharges, we obtain one two-dimensional lattice gauge theory, identical to the one already given in the literature. Starting with eight supercharges, we obtain a unique three-dimensional lattice gauge theory and infinitely many two-dimensional lattice theories. They can be classified according to seven…
Discretization of supersymmetric theories is an old problem in lattice field theory. It has resisted solution until quite recently when new ideas drawn from orbifold constructions and topological field theory have been brought to bear on the question. The result has been the creation of a new class of lattice gauge theory in which the lattice action is invariant under one or more supersymmetries. The resultant theories are local and free of doublers and in the case of Yang-Mills theories also possess exact gauge invariance. In principle they form the basis for a truly non-perturbative definiti…
Supersymmetric Yang Mills theory is directly accessible to lattice simulations using current methodology, and can provide a non-trivial check of recent exact results in SQCD. In order to tune the lattice simulation to the supersymmetric point it is neccessary to understand the behavior of the theory with a small supersymmetry breaking gaugino mass. We introduce a soft breaking gaugino mass in a controlled fashion using a spurion analysis. We compute the gluino condensate, vacuum energy and bound-state masses as a function of the gaugino mass, providing more readily accessible predictions which…
We present some of the latest results from our numerical investigations of N=4 supersymmetric Yang--Mills theory formulated on a space-time lattice. Based on a construction that exactly preserves a single supersymmetry at non-zero lattice spacing, we recently developed an improved lattice action that is now being employed in large-scale calculations. Here we update our studies of the static potential using this new action, also applying tree-level lattice perturbation theory to improve the analysis of the potential itself. Considering relatively weak couplings, we obtain results for the Coulom…
In this white paper we summarise the construction and applications of lattice theories possessing exact supersymmetry focusing, in particular, on N=4 Yang-Mills theory. Lattice formulations of this theory allow for numerical simulation of the theory at strong coupling and hence give a window on non-perturbative physics away from the planar limit. This has important applications to our understanding of holographic approaches to quantum gravity and conformal field theories.
By numerically investigating the conservation law of the supercurrent, we confirm the restoration of supersymmetry in Sugino's lattice formulation of the two-dimensional $\mathcal{N}=(2,2)$ 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.
We analyze the Euclidean version of supersymmetric quantum mechanics on the lattice by means of a numerical path integral. We consider two different lattice derivatives and improve the actions containing them with respect to supersymmetry by systematically adding interaction terms with non-zero extent. To quantize this improvement, we measure boson and fermion masses and Ward identities for the naive as well as the improved models. The masses are degenerate in all models, but the magnitude of the Ward identities decreases significantly for both derivative operators using the improved actions. …
N=4 supersymmetric quantum mechanical model is formulated on the lattice. Two supercharges, among four, are exactly conserved with the help of the cyclic Leibniz rule without spoiling the locality. In use of the cohomological argument, any possible local terms of the effective action are classified into two categories which we call type-I and type-II, analogous to the D- and F-terms in the supersymmetric field theories. We prove non-renormalization theorem on the type-II terms which include mass and interaction terms with keeping a lattice constant finite, while type-I terms such as the kineti…
The introduction of a space-time lattice as a regulator of field theories breaks symmetries associated with continuous space-time, i.e.\ Poincar{é} invariance and supersymmetry. A non-zero gluino mass in the supersymmetric Yang-Mills theory causes an additional soft breaking of supersymmetry. We employ the lattice form of SUSY Ward identities, imposing that their continuum form would be recovered when removing the lattice regulator, to obtain the critical hopping parameter where broken symmetries can be recovered.
Non-perturbative investigations of $\mathcal N = 4$ supersymmetric Yang--Mills theory formulated on a space-time lattice have advanced rapidly in recent years. Large-scale numerical calculations are currently being carried out based on a construction that exactly preserves a single supersymmetry at non-zero lattice spacing. A recent development is the creation of an improved lattice action through a new procedure to regulate flat directions in a manner compatible with this supersymmetry, by modifying the moduli equations. In this proceedings I briefly summarize this new procedure and discuss t…
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