Exact lattice supersymmetry at largeN

Abstract

Employing a novel type of non-commutative product in the Dirac-Kähler twisted superspace on a lattice, we formulate a field theoretically rigid framework of extended supersymmetry on a lattice. As a first example of this treatment, we calculate one-loop (in some cases any loops) quantum corrections for a twisted Wess-Zumino model with N × N hermitian matrix superfields on a two dimensional lattice. The calculations are entirely given in a lattice superfield framework. We report that the mass and the coupling constant are exactly protected from the radiative corrections at non-zero lattice spacing as far as the planar diagrams are concerned, which implies the realization of exact lattice supersymmetry w.r.t. all the supercharges in the large-N limit.

Formulation of supersymmetry on a lattice as a representation of a deformed superalgebra

“…It was shown that the twisted supersymmetry can be derived by the Dirac-Kähler twisting procedure [10] in any dimensions: N = 2 in two dimensions, N = 4 in three dimensions, and N = 4 in four dimensions which coincides with the twisting derived by Marcus [11]. In these formulations the fermionic internal degrees of freedom can be defined semilocally on a lattice to be compatible with differential form nature of Dirac-Kähler fermion [6,7,8,9,12,13]. This type of the correspondence has been anticipated by the old works [14].…”

“…In the link approach, the central ansatz is that the supercharge Q lat A obeys the Leibniz rule of the form 3 11) where x + a A is to be interpreted as denoting an extended lattice site which goes to x in the naïve continuum limit. Introducing a translation or shift operator T a A in a "fundamental" representation such that 12) the Leibniz rule can be written as 13) showing that the operator T −1 a A Q lat A obeys a slightly different modified Leibniz rule. We may also write it in a symmetric form as…”

“…An invertible element χ ∈ H ⊗ H is called a (2-)cocycle when it satisfies that (χ ⊗ 1l)(∆ ⊗ id)χ = (1l ⊗ χ)(id ⊗∆)χ ((2-)cocycle condition). For a quasitriangular Hopf algebra (H, R) and a counital 2-cocycle χ, there exists a new Hopf algebra (H χ , R χ ) which has 13 The notion of braiding would be most suitably defined in terms of category theory. Here instead we just give a simple and intuitive description.…”

“…In the third approach [6,7,8,12,13,34,40], one defines lattice version of superalgebra where the momentum operators in the continuum superalgebra are replaced by finite difference operators on the lattice. This seems the most natural and naïve deformation of continuum algebra.…”

Two-dimensional 𝒩 = (2, 2) supersymmetric lattice gauge theory with matter fields in the fundamental representation

“…2 For a discussion on consistency in the deformation of supersymmetry on a lattice, see [24][25] [26] [22]. See also [27] for a further discussion on the consistency connecting with large-N limit. 3 See also [41]- [44] for numerical study of black hole thermodynamics and gauge/gravity duality.…”