arXiv subjects

Simon Catterall

Publications and source records attributed to Simon Catterall.

120 records · Page 7Linked to original sources

arXiv2005

Dirac-Kähler 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.

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arXiv2005

Lattice formulation of ${\cal N}=4$ 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.

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arXiv2004

A geometrical approach to N=2 super Yang-Mills theory on the two dimensional lattice

We propose a discretization of two dimensional Euclidean Yang-Mills theories with N=2 supersymmetry which preserves exactly both gauge invariance and an element of supersymmetry. The approach starts from the twisted form of the continuum super Yang Mills action which we show may be written in terms of two real Kahler-Dirac fields whose components transform into each other under the twisted supersymmetry. Once the theory is written in this geometrical language it is straightforward to discretize by mapping the component tensor fields to appropriate geometrical structures in the lattice and by replacing the continuum exterior derivative and its adjoint by appropriate lattice covariant difference operators. The lattice action is local and possesses a unique vacuum state while the use of Kahler-Dirac fermions ensures the model does not exhibit spectrum doubling.

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arXiv2004

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.

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arXiv2004

Lattice Sigma Models with Exact Supersymmetry

We show how to construct lattice sigma models in one, two and four dimensions which exhibit an exact fermionic symmetry. These models are discretized and {\it twisted} versions of conventional supersymmetric sigma models with N=2 supersymmetry. The fermionic symmetry corresponds to a scalar BRST charge built from the original supercharges. The lattice theories possess local actions and in many cases admit a Wilson term to suppress doubles. In the two and four dimensional theorie s we show that these lattice theories are invariant under additional discrete symmetries. We argue that the presence of these exact symmetries ensures that no fine tuning is required to achieve N=2 supersymmetry in the continuum limit. As a concrete example we show preliminary numerical results from a simulation of the O(3) supersymmetric sigma model in two dimensions.

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arXiv2003

Exact Lattice Supersymmetry from Topological Field Theory

We discuss the connection between supersymmetric field theories and topological field theories and show how this connection may be used to construct local lattice field theories which maintain an exact supersymmetry. It is shown how metric independence of the continuum topological field theory allows us to derive the lattice theory by blocking out of the continuum in a deformed geometry. This, in turn, allows us to prove the cut-off independence of certain supersymmetric Ward identities.

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arXiv2003

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.

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arXiv2003

Lattice Supersymmetry and Topological Field Theory

It is known that certain theories with extended supersymmetry can be discretized in such a way as to preserve an exact fermionic symmetry. In the simplest model of this kind, we show that this residual supersymmetric invariance is actually a BRST symmetry associated with gauge fixing an underlying local shift symmetry. Furthermore, the starting lattice action is then seen to be entirely a gauge fixing term. The corresponding continuum theory is known to be a topological field theory. We look, in detail, at one example - supersymmetric quantum mechanics which possesses two such BRST symmetries. In this case, we show that the lattice theory can be obtained by blocking out of the continuum in a carefully chosen background metric. Such a procedure will not change the Ward identities corresponding to the BRST symmetries since they correspond to topological observables. Thus, at the quantum level, the continuum BRST symmetry is preserved in the lattice theory. Similar conclusions are reached for the two-dimensional complex Wess-Zumino model and imply that all the supersymmetric Ward identities are satisfied {\it exactly} on the lattice. Numerical results supporting these conclusions are presented.

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arXiv2002

Exact Lattice Supersymmetry: the Two-Dimensional N=2 Wess-Zumino Model

We study the two-dimensional Wess-Zumino model with extended N=2 supersymmetry on the lattice. The lattice prescription we choose has the merit of preserving {\it exactly} a single supersymmetric invariance at finite lattice spacing $a$. Furthermore, we construct three other transformations of the lattice fields under which the variation of the lattice action vanishes to $O(ga^2)$ where $g$ is a typical interaction coupling. These four transformations correspond to the two Majorana supercharges of the continuum theory. We also derive lattice Ward identities corresponding to these exact and approximate symmetries. We use dynamical fermion simulations to check the equality of the massgaps in the boson and fermion sectors and to check the lattice Ward identities. At least for weak coupling we see no problems associated with a lack of reflection positivity in the lattice action and find good agreement with theory. At strong coupling we provide evidence that problems associated with a lack of reflection positivity are evaded for small enough lattice spacing.

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arXiv2000

Phase diagram of three-dimensional dynamical triangulations with a boundary

We use Monte Carlo simulation to study the phase diagram of three-dimensional dynamical triangulations with a boundary. Three phases are indentified and characterized. One of these phases is a new, boundary dominated phase; a simple argument is presented to explain its existence. First-order transitions are shown to occur along the critical lines separating phases.

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arXiv2000

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.

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arXiv2000

A lattice path integral for supersymmetric quantum mechanics

We report on a study of the supersymmetric anharmonic oscillator computed using a euclidean lattice path integral. Our numerical work utilizes a Fourier accelerated hybrid Monte Carlo scheme to sample the path integral. Using this we are able to measure massgaps and check Ward identities to a precision of better than one percent. We work with a non-standard lattice action which we show has an {\it exact} supersymmetry for arbitrary lattice spacing in the limit of zero interaction coupling. For the interacting model we show that supersymmetry is restored in the continuum limit without fine tuning. This is contrasted with the situation in which a `standard' lattice action is employed. In this case supersymmetry is not restored even in the limit of zero lattice spacing. Finally, we show how a minor modification of our action leads to an {\it exact}, local lattice supersymmetry even in the presence of interaction.

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