Lattice Supersymmetry via Twisting

We describe how the usual supercharges of extended supersymmetry may be {\it twisted} to produce a BRST-like supercharge $Q$. The usual supersymmetry algebra is then replaced by a twisted algebra and the action of the twisted theory is shown to be generically $Q$-exact. In flat space the twisting procedure can be regarded as a change of variables carrying no physical significance. However, the twisted theories can often be transferred to the lattice while preserving the twisted supersymmetry. As an example we construct a lattice version of the two-dimensional supersymmetric sigma model.Comment: Contribution to Lattice2004(theory

A Note on the Action in d>4 Dynamical Triangulations

For dynamical triangulations in dimensions d<=4 the most general action has two couplings. We note that the most general action for d=5 has three couplings. We explore this larger coupling space using Monte Carlo simulations. Initial results indicate evidence for non-trivial phase structure.Comment: 3 page contribution to Lattice'97 proceeding

Phase diagram of four-dimensional dynamical triangulations with a boundary

We report on simulations of DT simplicial gravity for manifolds with the topology of the 4-disk. We find evidence for four phases in a two-dimensional parameter space. In two of these the boundary plays no dynamical role and the geometries are equivalent to those observed earlier for the sphere $S^4$. In another phase the boundary is maximal and the quantum geometry degenerates to a one dimensional branched polymer. In contrast we provide evidence that the fourth phase is effectively three-dimensional. We find discontinuous phase transitions at all the phase boundaries.Comment: 13 pages, late

Singular Vertices and the Triangulation Space of the D-sphere

By a sequence of numerical experiments we demonstrate that generic triangulations of the $D-$sphere for $D>3$ contain one {\it singular} $(D-3)-$simplex. The mean number of elementary $D-$simplices sharing this simplex increases with the volume of the triangulation according to a simple power law. The lower dimension subsimplices associated with this $(D-3)-$simplex also show a singular behaviour. Possible consequences for the DT model of four-dimensional quantum gravity are discussed.Comment: 15 pages, 9 figure

First results from simulations of supersymmetric lattices

We conduct the first numerical simulations of lattice theories with exact supersymmetry arising from the orbifold constructions of \cite{Cohen:2003xe,Cohen:2003qw,Kaplan:2005ta}. We consider the \cQ=4 theory in $D=0,2$ dimensions and the \cQ=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 ${\cal N}=4$ super Yang-Mills may be achievable in the near future.Comment: 25 pages, 14 figures, 9 tables. 3 references adde

Simulations of Dynamically Triangulated Gravity -- an Algorithm for Arbitrary Dimension

Recent models for discrete euclidean quantum gravity incorporate a sum over simplicial triangulations. We describe an algorithm for simulating such models in general dimensions. As illustration we show results from simulations in four dimensionsComment: 14 pages, 6 figures, CERN-TH.7286/9

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 ${\cal N}=4$ super Yang-Mills theory.Comment: Plenary talk at the symposium Quantum Theory and Symmetries, Lexington, Kentucky, July 2009. References adde

Dirac-K\"{a}hler fermions and exact lattice supersymmetry

We discuss a new approach to putting supersymmetric theories on the lattice. The basic idea is to start from a {\it twisted} formulation of the underlying supersymmetric theory in which the fermions are represented as grassmann valued antisymmetric tensor fields. The original supersymmetry algebra is replaced by a twisted algebra which contains a scalar nilpotent supercharge $Q$. Furthermore the action of the theory can then be written as the $Q$-variation of some function. The case of ${\cal N}=2$ super Yang-Mills theory in two dimensions is discussed in some detail. We then present our proposal for discretizing this theory and derive the resultant lattice action. The latter is local, free of spectrum doubling, gauge invariant and preserves the scalar supercharge invariance exactly. Some preliminary numerical results are then presented. The approach can be naturally generalized to yield a lattice action for ${\cal N}=4$ super Yang-Mills in four dimensions.Comment: 22 pages, 3 figures. Plenary talk given at Lattice 2005 Dublin July 25-30. 1 reference correcte

Twisted lattice supersymmetry and applications to AdS/CFT

I review recent approaches to constructing supersymmetric lattice theories focusing in particular on the concept of topological twisting. The latter technique is shown to expose a nilpotent, scalar supersymmetry which can be implemented exactly in the lattice theory. Using these ideas a lattice action for $\mathcal{N}=4$ super Yang-Mills in four dimensions can be written down which is gauge invariant, free of fermion doublers and respects one out of a total of 16 continuum supersymmetries. It is shown how these exact symmetries together with the large point group symmetry of the lattice strongly constrain the possible counterterms needed to renormalize the theory and hence determine how much residual fine tuning will be needed to restore all supersymmetries in the continuum limit. We report on progress to study these renormalization effects at one loop. We go on to give examples of applications of these supersymmetric lattice theories to explore the connections between gauge theories and gravity.Comment: 16 pages. Plenary talk at Lattice 201