Supersymmetric Lattice Research Articles

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.

Observation of supersymmetric scattering in photonic lattices.

Supersymmetric (SUSY) optical structures display a number of intriguing properties that can lead to a variety of potential applications, ranging from perfect global phase matching to highly efficient mode conversion and novel multiplexing schemes. Here, we experimentally investigate the scattering characteristics of SUSY photonic lattices. We directly observe the light dynamics in such systems and compare the reflection/transmission properties of SUSY partner structures. In doing so, we demonstrate that discrete settings constitute a promising testbed for studying the different facets of optical supersymmetry.

Lattice formulation for 2d $ \mathcal{N} $ = (2, 2), (4, 4) super Yang-Mills theories without admissibility conditions

We present a lattice formulation for two-dimensional 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, N=(8,8) supersymmetric Yang-Mills theory and N=(2,2) supersymmetric QCD.

Supersymmetric lattices

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 definition of the continuum supersymmetric field theory. In this talk these ideas are reviewed with particular emphasis being placed on = 4 super Yang-Mills theory.

Supersymmetric optical structures.

We show that supersymmetry can provide a versatile platform in synthesizing a new class of optical structures with desired properties and functionalities. By exploiting the intimate relationship between superpatners, one can systematically construct index potentials capable of exhibiting the same scattering and guided wave characteristics. In particular, in the Helmholtz regime, we demonstrate that one-dimensional supersymmetric pairs display identical reflectivities and transmittivities for any angle of incidence. Optical supersymmetry is then extended to two-dimensional systems where a link between specific azimuthal mode subsets is established. Finally, we explore supersymmetric photonic lattices where discreteness can be utilized to design lossless integrated mode filtering arrangements.

A criterion for lattice supersymmetry: cyclic Leibniz rule

It is old folklore that the violation of Leibniz rule on a lattice is an obstruction for constructing a lattice supersymmetric model. While it is still true for full supersymmetry, we show that a slightly modified form of the Leibniz rule, which we call cyclic Leibniz rule (CLR), is actually a criterion for the existence of partial lattice supersymmetry. In one dimension, we find sets of lattice difference operator and field multiplication smeared over lattice which satisfy the CLR under some natural assumptions such as translational invariance and locality. Thereby we construct a model of supersymmetric lattice quantum mechanics without spoiling locality. The CLR relation is coincident with the condition that the vanishing of the so-called surface term in the construction by lattice Nicolai map. We can construct superfield formalism with arbitrary superpotential. This also enables us to apply safely a localization technique to our model, because the kinetic term and the interaction terms of our model are independently invariant under the supersymmetry transformation. A preliminary attempt in finding a solution for the higher dimensional case is also discussed.

Area law violations in a supersymmetric model

We study the structure of entanglement in a supersymmetric lattice model of fermions on certain types of decorated graphs with quenched disorder. In particular, we construct models with controllable ground state degeneracy protected by supersymmetry and the choice of Hilbert space. We show that in certain special limits these degenerate ground states are associated with local impurities and that there exists a basis of the ground state manifold in which every basis element satisfies a boundary law for entanglement entropy. On the other hand, by considering incoherent mixtures or coherent superpositions of these localized ground states, we can find regions that violate the boundary law for entanglement entropy over a wide range of length scales. More generally, we discuss various desiderata for constructing violations of the boundary law for entanglement entropy and discuss possible relations of our work to recent holographic studies.

Supersymmetric lattice fermions on the triangular lattice: superfrustration and criticality

We study a model for itinerant, strongly interacting fermions where a judicious tuning of the interactions leads to a supersymmetric Hamiltonian. On the triangular lattice this model is known to exhibit a property called superfrustration, which is characterized by an extensive ground state entropy. Using a combination of numerical and analytical methods we study various ladder geometries obtained by imposing doubly periodic boundary conditions on the triangular lattice. We compare our results to various bounds on the ground state degeneracy obtained in the literature. For all systems we find that the number of ground states grows exponentially with system size. For two of the models that we study we obtain the exact number of ground states by solving the cohomology problem. For one of these, we find that via a sequence of mappings the entire spectrum can be understood. It exhibits a gapped phase at 1/4 filling and a gapless phase at 1/6 filling and phase separation at intermediate fillings. The gapless phase separates into an exponential number of sectors, where the continuum limit of each sector is described by a superconformal field theory.

On the sign problem in 2D lattice super Yang-Mills

In 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 N=(2, 2) and 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 N=4 supersymmetric Yang--Mills theory.

Detailed analysis of the continuum limit of a supersymmetric lattice model in1D

We present a full identification of lattice model properties with their field theoreticalcounterparts in the continuum limit for a supersymmetric model for itinerant spinlessfermions on a one-dimensional chain. The continuum limit of this model is described by an superconformal field theory (SCFT) with central chargec = 1. We identify states and operators in the lattice model with fields in the SCFT and werelate boundary conditions on the lattice to sectors in the field theory. We use thedictionary we develop in this paper to give a pedagogical explanation of a powerful tool tostudy supersymmetric models based on spectral flow (Huijse 2008 Phys. Rev. Lett. 101146406). Finally, we employ the developed machinery to explain numerically observedproperties of the particle density on the open chain presented in Beccaria and De Angelis(2005 Phys. Rev. Lett. 94 100401).

Topological gravity on the lattice

In this paper we show that a particular twist of $ \mathcal{N} = 4 $ super Yang-Mills in three dimensions with gauge group SU(2) possesses a set of classical vacua corresponding to the space of flat connections of the complexified gauge group $ {\text{SL}}\left( {2,\mathbb{C}} \right) $ . The theory also contains a set of topological observables corresponding to Wilson loops wrapping nontrivial cycles of the base manifold. This moduli space and set of topological observables is shared with the Chern Simons formulation of three dimensional gravity and we hence conjecture that the Yang-Mills theory gives an equivalent description of the gravitational theory. Unlike the Chern Simons formulation the twisted Yang-Mills theory possesses a supersymmetric and gauge invariant lattice construction which then provides a possible non-perturbative definition of three dimensional gravity.

Deformed matrix models, supersymmetric lattice twists and 𝒩 = ¼ 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.

First results from simulations of supersymmetric lattices

We conduct the first numerical simulations of lattice theories with exact super- symmetry arising from the orbifold constructions of (1, 2, 3). We consider the Q = 4 theory in D = 0,2 dimensions and the Q = 16 theory in D = 0,2,4 dimensions. We show that the U(N) theories do not possess vacua which are stable non-perturbatively, but that this problem can be circumvented after truncation to SU(N). We measure the distribution of scalar field eigenvalues, the spectrum of the fermion operator and the phase of the Pfaffian arising after integration over the fermions. We monitor supersymmetry breaking effects by measuring a simple Ward identity. Our results indicate that simulations of N = 4 super Yang-Mills may be achievable in the near future.

Upper bounds on the Witten index for supersymmetric lattice models by discrete Morse theory

The Witten index for certain supersymmetric lattice models treated by de Boer, van Eerten, Fendley, and Schoutens, can be formulated as a topological invariant of simplicial complexes, arising as independence complexes of graphs. We prove a general theorem on independence complexes, using discrete Morse theory: if G is a graph and D a subset of its vertex set such that G ∖ D is a forest, then ∑ i dim H ̃ i ( Ind ( G ) ; Q ) ≤ | Ind ( G [ D ] ) | . We use the theorem to calculate upper bounds on the Witten index for several classes of lattices. These bounds confirm some of the computer calculations by van Eerten on small lattices. The cohomological method and the 3-rule of Fendley et al. is a special case of when G ∖ D lacks edges. We prove a generalized 3-rule and introduce lattices in arbitrary dimensions satisfying it.

Two-dimensional 𝒩 = (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.

Geometry of orbifolded supersymmetric lattice gauge theories

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.

From twisted supersymmetry to orbifold lattices

We show how to derive the supersymmetric orbifold lattices of Cohen et al. \cite{Cohen:2003xe,Cohen:2003qw} and Kaplan et al. \cite{Kaplan:2005ta} 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 ${\cal N}=4$ Yang-Mills. The lattice models are gauge invariant and possess one exact supersymmetry at non-zero lattice spacing.

Lattice supersymmetry: equivalence between the link approach and orbifolding

We examine the relation between supersymmetric lattice gauge theories constructed by the link approach and by orbifolding and show that they are equivalent. We discuss the number of preserved supersymmetries.

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 = (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 = 2 supersymmetric Yang-Mills theory. We show that a consistent truncation is closely related to the lattice formulation previously given by Sugino.

Classification of supersymmetric lattice gauge theories by orbifolding

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 distinct groups, five of which have two preserved supercharges while the others have only one.