Non-positive fermion determinants in lattice supersymmetry

We find that fermion determinants are not generally positive in a recent class of constructions with explicit lattice supersymmetry. These involve an orbifold of supersymmetric matrix models, and have as their target (continuum) theory (2,2) 2-dimensional super-Yang-Mills. The fermion determinant is shown to be identically zero for all boson configurations due to the existence of a zeromode fermion inherited from the “mother theory.” Once this eigenvalue is factored out, the fermion determinant generically has arbitrary complex phase. We discuss the implications of this result for simulation of the models. [email protected] Introductory remarks. Models with explicit lattice supersymmetry have been discussed in the literature by a few groups. For example, latticizations of superYang-Mills [1, 2], supersymmetric quantum mechanics [3, 4], the 2d Wess-Zumino model [5, 4], and direct constructions in the spirit of the Ginsparg-Wilson relation [6] have all been considered. In this letter we will be interested in the super-Yang-Mills constructions that lead to a Euclidean lattice theory [1]. The method of building such models is based on deconstruction of extra dimensions [7, 8]. The corresponding interpretation in terms of the world-volume theory of D-branes has led to the latticizations of 2d, 3d and 4d supersymmetric gauge theories. These lattice constructions are all arrived at by orbifold projections of supersymmetric matrix models; i.e., in each case we quotient a matrix model by some discrete symmetry group of the theory. Degrees of freedom that are not invariant with respect to the combined action of the orbifold generators are projected out. Thus, throughout this letter we will have occasion to speak of “orbifolded” matrix models and “nonorbifolded” matrix models. A major motivation for efforts to latticize supersymmetric models is that some nonperturbative aspects of supersymmetric field theories are not accessible by the usual techniques, such as holomorphy. One hope of a lattice supersymmetry program of research is that it would lead to, e.g., simulations that would provide further data on supersymmetric field theories, especially those that include super-Yang-Mills. With explicit lattice supersymmetry the target (continuum) theory may be obtained in a more controlled fashion. Indeed, in some cases it may be obtained without the need for fine-tuning [1, 2]. In this enterprise, it is of great practical importance that the fermion determinant, obtained integrating over the fermion degrees of freedom in the partition function, be positive. For let φ be the lattice bosons in the theory, and ψ, ψ̄ the lattice fermions. For a detailed discussion, we refer the reader to [1]. For a recent review of existing work on this broad topic, and a complete list of relevant references, see [9].