Simon Catterall

Minimal walking on the lattice

We provide the first evidence of a walking dynamics for two color lattice Yang-Mills theory with two Dirac flavors in the symmetric representation of the gauge group.

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.

Phase diagram of SU(2) with 2 flavors of dynamical adjoint quarks

We report on numerical simulations of SU(2) lattice gauge theory with two flavors of light dynamical quarks in the adjoint of the gauge group. The dynamics of this theory is thought to be very different from QCD?the theory exhibiting conformal or near conformal behavior in the infrared. We make a high resolution survey of the phase diagram of this model in the plane of the bare coupling and quark mass on lattices of size 83 ? 16. Our simulations reveal a line of first order phase transitions extending from ? = 0 to ? = ?c ~ 2.0. For ??>??c the phase boundary is no longer first order but continues as the locus of minimum meson mass. For ??>??c we observe the pion and rho masses along the phase boundary to be light, independent of bare coupling and approximately degenerate. We discuss possible interpretations of these observations and corresponding continuum limits.

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 original lattice action is then seen to be entirely a gauge fixing term. The corresponding continuum theory has a topological field theory interpretation. 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 exactly on the lattice. Numerical results supporting these conclusions are presented.

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

We study the two-dimensional Wess-Zumino model with extended

Stripes from (non-commutative) stars

N=2

Lattice formulation of N=4 super Yang-Mills theory

supersymmetry on the lattice. The lattice prescription we choose has the merit of preserving 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

From Twisted Supersymmetry to Orbifold Lattices

{O(ga}^{2})

Extracting black hole physics from the lattice

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 mass gaps 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.

Lattice sigma models with exact supersymmetry

We show that lattice regularization of non-commutative field theories can be used to study non-perturbative vacuum phases. Specifically we provide evidence for the existence of a striped phase in two-dimensional non-commutative scalar field theory.