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 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 theories 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.

In this paper we conduct a numerical study of the supersymmetric O(3) nonlinear sigma model. The lattice formulation we employ was derived in [1] and corresponds to a discretization of a twisted form of the continuum action. The twisting process exposes a nilpotent supercharge Q and allows the action to be rewritten in Q-exact form. These properties may be maintained on the lattice. We show how to deform the theory by the addition of potential terms which preserve the supersymmetry. A Wilson mass operator may be introduced in this way with a minimal breaking of supersymmetry. We additionally show how to rewrite the theory in the language of Kähler-Dirac fields and explain why this avenue does not provide a good route to discretization. Our numerical results provide strong evidence for a restoration of full supersymmetry in the continuum limit without fine tuning. We also observe a non-vanishing chiral condensate as expected from continuum instanton calculations.

A supersymmetric extension of the nonlinear O(3) sigma model in two spacetime dimensions is investigated by means of Monte Carlo simulations. We argue that it is impossible to construct a lattice action that implements both the O(3) symmetry as well as at least one su-persymmetry exactly at finite lattice spacing. It is shown by explicit calculations that previously proposed discretizations fail to reproduce the exact symmetries of the target manifold in the continuum limit. We provide an alternative lattice action with exact O(3) symmetry and compare two approaches based on different derivative operators. Using the nonlocal SLAC derivative for the quenched model on moderately sized lattices we extract the value σ(2, u 0) = 1.2604(13) for the step scaling function at u 0 = 1.0595, to be compared with the exact value 1.261210. For the supersymmetric model with SLAC derivative the discrete chiral symmetry is maintained but we encounter strong sign fluctuations, rendering large lattice simulations ineffective. By applying the Wilson prescription, supersymmetry and chiral symmetry are broken explicitly at finite lattice spacing, though there is clear evidence that both are restored in the continuum limit by fine tuning of a single mass parameter.