A geometrical approach toN=2 super Yang-Mills theory on the two dimensional lattice

Abstract

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 Kähler-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 Kähler-Dirac fermions ensures the model does not exhibit spectrum doubling.

Perturbative renormalization of lattice $ \mathcal{N} = 4 $ super Yang-Mills theory

“…This similarity between the divergence structure of the lattice theory and the continuum theory is strongly suggestive that the beta function of the lattice theory will also 8 The calculation we have done is somewhat formal as it ignores possible instabilities associated with the flat directions and specifically the U (1) trace mode of the scalars. It is possible that regulating these directions by eg introducing a mass term for the U (1) mode might modify our conclusions since supersymmetry is broken by such terms.…”

“…Indeed with one exception the only relevant counterterms correspond to renormalizations of existing terms in the action. We furthermore show by a computation of the effective action that the one new operator which cannot be excluded in the general analysis actually makes no appearance to all orders in perturbation theory 8 The renormalized action can then be written in terms of 4 coupling constants α i which take the value unity in the classical lattice action. We evaluate the renormalization of these couplings at one loop using lattice perturbation theory.…”

“…However, recent formulations of supersymmetric lattice theories, which retain exact supersymmetry at non zero lattice spacing offer the hope of at least partially evading these fine tuning problems -the existence of an exact lattice supersymmetry protecting the theory from many of these dangerous counterterms [4,5,6,7,8,9,10,11,12,13,14]. See also the recent reviews [15,16] for further references.…”

Vafa-Witten theory onN= 2 andN= 4 twisted superspace in four dimensions

“…Other models by using the Dirac-Kähler fermion without modified Leibniz rule possess the partial twisted supersymmetries. These models was investigated in [36][37][38][39][40][41][42][43][44][45][46]. In the recent development of lattice SUSY, there are matrix formulations on which we impose Z N orbifold conditions [47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63].…”

Lattice formulation of (2,2) supersymmetric gauge theories with matter fields

“…These tensors can then be given a geometric meaning, with p-index tensors being mapped to p-cells on a lattice. A lattice action is then constructed from the target theory which preserves the scalar supercharge even at finite lattice spacing [7][8][9][10][11][12][13][14][15][16]. This work was foreshadowed by an early proposal to use Dirac-Kähler fermions in the construction of a supersymmetric lattice Hamiltonian in one spatial dimension [17].…”