A Lattice Formulation of Two Dimensional Topological Field Theory

We propose a non-perturbative criterion to investigate whether supersymmetric lattice gauge theories preserving partial SUSY can have the desired continuum limit or not. Since the target continuum theories of the lattice models are extended supersymmetric gauge theories including the topological field theory (TFT) as a special subsector, the continuum limits of them should reproduce the properties of the TFT. Therefore, whether the property of the TFT can be recovered at the continuum limit becomes a non-perturbative criterion. Then we accept it as a criterion. In this paper, among the topological properties, we investigate the BRST cohomology on the two dimensional ${\mathcal N}=(4,4)$ CKKU lattice model without moduli fixing mass term. We show that the BRST cohomology in the target continuum theory cannot be realized from the BRST cohomology on the lattice. From this result, we obtain the possible implication that the ${\mathcal N} = (4,4)$ CKKU model cannot recover the target continuum theory if the non-perturbative effects are taken into account.