Lattice Supersymmetry and Topological Field Theory

TL;DR

This paper demonstrates that certain lattice discretizations of supersymmetric theories preserve BRST symmetries exactly, ensuring the validity of supersymmetric Ward identities at the quantum level, supported by numerical evidence.

Contribution

It shows that lattice formulations of extended supersymmetric theories preserve BRST symmetries exactly, linking lattice actions to gauge fixing of topological field theories.

Findings

Lattice supersymmetry corresponds to BRST symmetry of gauge fixing.

Continuum BRST symmetry is preserved on the lattice.

Numerical results confirm exact supersymmetric Ward identities on the lattice.

Abstract

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.