Kazuhiro Nagata

Exact Extended Supersymmetry on a Lattice: Twisted N=4 Super Yang-Mills in Three Dimensions

Abstract We propose a lattice action for two-dimensional super-Yang–Mills theory with a twisted N = 2 supersymmetry. The extended supersymmetry is fully and exactly realized on the lattice. The method employed is quite general and its extension to the N = 4 supersymmetry in four dimensions is briefly presented. The lattice has a new type of “fermionic” links, where odd Grassmann variables, supercharges and fermionic connections sit. The Leibniz rule is preserved on the lattice, although in a modified “shifted” form that takes into account the link nature of both derivatives and supercharges. Superfields are semi-local objects and superfield expansion is naturally embedded in the lattice structure. The Dirac–Kahler twist generates the extended twisted supersymmetry, turning the multiplicity of species doublers into the multiplicity due to the extended supersymmetry. In this way the balance between bosonic and fermionic degrees of freedom is preserved.

Twisted superspace on a lattice

Abstract We propose a new formulation which realizes exact twisted supersymmetry for all the supercharges on a lattice by twisted superspace formalism. We show explicit examples of N = 2 twisted supersymmetry invariant BF and Wess–Zumino models in two dimensions. We introduce mild lattice non-commutativity to preserve Leibniz rule on the lattice. The formulation is based on the twisted superspace formalism for N = D = 2 supersymmetry which was proposed recently. From the consistency condition of the non-commutativity of superspace, we find an unexpected three-dimensional lattice structure which may reduce into two-dimensional lattice where the superspace describes semilocally scattered fermions and bosons within a double size square lattice.

Twisted supersymmetric invariant formulation of Chern-Simons gauge theory on a lattice

We propose a twisted supersymmetric (SUSY) invariant formulation of the Chern-Simons theory on a Euclidean three-dimensional lattice. The SUSY algebra to be realized on the lattice is the N=4 D=3 twisted algebra that was recently proposed by DAdda et al. In order to keep the manifest anti-Hermiticity of the action, we introduce oppositely oriented supercharges. Accordingly, the naive continuum limit of the action formally corresponds to the Landau-gauge fixed version of the Chern-Simons theory with complex gauge group which was originally proposed by Witten. We also show that the resulting action consists of parity even and odd parts with different coefficients.

Twisted N=2 exact SUSY on the lattice for BF and Wess-Zumino⁎†

We formulate exact supersymmetric models on a lattice. We introduce noncommutativity to ensure the Leibniz rule. With the help of superspace formalism, we give supertransformations which keep the N=2 twisted SUSY algebra exactly. The action is given as a product of (anti)chiral superfields on the lattice. We present BF and Wess-Zumino models as explicit examples of our formulation. Both models have exact N=2 twisted SUSY in 2 dimensional space at a finite lattice spacing. In component fields, the action has supercharge exact form.

On the continuum and lattice formulations of N = 4 D = 3 twisted super Yang-Mills

Employing a twisted superspace with eight supercharges, we describe an off-shell formulation of N = 4?D = 3 twisted super Yang-Mills in the continuum spacetime which underlies the recent proposal of N = 4?D = 3 twisted super Yang-Mills on a lattice cite{DKKN3}. By a dimensional reduction from the N = 2?D = 4, we explore the two possible topological twists of N = 4?D = 3 and then show that the lattice formulation given in cite{DKKN3} is essentially categorized as the B-type. We also show that, amongst the two inequivalent twists of N = 4?D = 3, only the B-type SYM can be realized on the lattice consistently with the Leibniz rule and the gauge covariance on the lattice.

Exact lattice supersymmetry at large N

Employing a novel type of non-commutative product in the Dirac-Kahler twisted superspace on a lattice, we formulate a field theoretically rigid framework of extended supersymmetry on a lattice. This corresponds to a certain reformulation of the non-commutative superspace on a lattice which was essentially developed in ref. [1]. 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, while the non-planar contributions generally spoil the exactness of the supersymmetry at finite N on the lattice. These properties imply an exact lattice supersymmetry realization w.r.t. all the supercharges in the t Hooft large-N limit.

Lattice Formulation of the N=4 D=3 Twisted Super Yang-Mills

A lattice formulation of a three dimensional super Yang-Mills model with a twisted N=4 supersymmetry is proposed. The extended supersymmetry algebra of all eight supercharges is fully and exactly realized on the lattice with a modified Leibniz rule. The formulation we employ here is a three dimensional extension of the manifestly gauge covariant method which was developed in our previous proposal of Dirac-Kahler twisted N=2 super Yang-Mills on a two dimensional lattice. The twisted N=4 supersymmetry algebra is geometrically realized on a three dimensional lattice with link supercharges and the use of shifted (anti-)commutators. A possible solution to the recent critiques on the link formulation will be discussed.

Diquark representations for single heavy baryons with light staggered quarks

In the staggered fermion formulation of lattice QCD, we construct and classify the diquark operators to be embedded in the singly heavy baryons, qqQ. Along the same manner as in the staggered meson classifications given in [M.F.L. Golterman, Nucl. Phys. B273, 663 (1986).], we establish the group theoretical connections between continuum and lattice staggered diquark representations.