Hiroto So

Taming the Leibniz rule on the lattice

We study a product rule and a difference operator equipped with Leibniz rule in a general framework of lattice field theory. It is shown that the difference operator can be determined by the product rule and some initial data through the Leibniz rule. This observation leads to a no-go theorem that it is impossible to construct any difference operator and product rule on a lattice with the properties of (i) translation invariance, (ii) locality and (iii) Leibniz rule. We present a formalism to overcome the difficulty by an infinite flavor extension or a matrix expression of a lattice field theory.

Ginsparg–Wilson relation and lattice supersymmetry

Abstract The Ginsparg–Wilson (GW) relation is extended for supersymmetric free theories on a lattice. Exact lattice supersymmetry (SUSY) can be defined without any ambiguities in difference operators. The lattice action constructed by a block-spin transformation is invariant under the symmetry. U(1)R symmetry on the lattice is also realized as one of exact symmetries. For an application, the extended GW relation is given for a two-dimensional model with chiral-multiplets. It is argued that the relation may be generalized for interacting cases.

Toward a Super Yang-Mills Theory on the Lattice

We present an entirely new approach toward a realization of supersymmetric Yang-Mills theory on the lattice. The action consists of the staggered fermion and the plaquette variables distributed in the Euclidean space with a particular pattern. The system is shown to have fermionic symmetries relating the fermion and link variables.

Leibniz rule and exact supersymmetry on lattice: A Case of supersymmetrical quantum mechanics

We propose a new formulation of lattice theory. It is given by a matrix form and suitable for satisfying Leibniz rule on lattice. The theory may be interpreted as a multi-flavor system. By realizing the difference operator as a commutator, we may obtain exact supersymmetric theories on lattice explicitly. Some problems such as locality and single flavor reduction are also commented.

Towards the Super Yang-Mills Theory on the Lattice

We present an entirely new approach toward a realization of supersymmetric Yang-Mills theory on the lattice. The action consists of the staggered fermion and the plaquette variables distributed in the Euclidean space with a particular pattern. The system is shown to have fermionic symmetries relating the fermion and link variables.

Symmetry and Z2-Orbifolding Approach in Five-Dimensional Lattice Gauge Theory

In a lattice gauge-Higgs unification scenario using a Z_2-orbifolded extra-dimension, we find a new global symmetry in a case of SU(2) bulk gauge symmetry. It is a global symmetry on sites in a fixed point with respect to Z_2-orbifolding, independent of the bulk gauge symmetry. It is shown that the vacuum expectation value of a Z_2-projected Polyakov loop is a good order parameter of the new symmetry. The effective theory on lattice is also discussed.

BRS symmetry, the quantum master equation, and the Wilsonian renormalization group

Recently we made a proposal for realization of an effective BRS symmetry along the Wilsonian renormalization group flow. In this paper we show that the idea can be naturally extended to the most general gauge theories. Extensive use of the antifield formalism is made to reveal some remarkable structure of the effective BRS symmetry. The average action, defined with a continuum analog of the block spin transformation, obeys the quantum master equation (QME), provided that a UV action does so. We show that the RG flow described by the exact flow equations is generated by canonical transformations in the field-antifield space. Using the relation between the average action and the Legendre effective action, we establish the equivalence between the QME for the average action and the modified WardTakahashi identity for the Legendre action. The QME remains intact when the regularization is removed.

Exact symmetries realized on the renormalization group flow

Abstract We show that symmetries are preserved exactly along the (Wilsonian) renormalization group flow, though the IR cutoff deforms concrete forms of the transformations. For a gauge theory the cutoff dependent Ward–Takahashi identity is written as the master equation in the antifield formalism: one may read off the renormalized BRS transformation from the master equation. The Maxwell theory is studied explicitly to see how it works. The renormalized BRS transformation becomes non-local but keeps off-shell nilpotency. Our formalism is applicable for a generic global symmetry. The master equation considered for the chiral symmetry provides us with the continuum analog of the Ginsparg–Wilson relation and the Luschers symmetry.

Regularized quantum master equation in the wilsonian

Using the Pauli-Villars regularization, we make a perturbative analysis of the quantum master equation (QME), Σ = 0, for the Wilsonian effective action. It is found that the QME for the UV action determines whether exact gauge symmetry is realized along the renormalization group (RG) flow. The basic task of solving the QME can be reduced to compute the Troost-van Niuwenhuizen-Van Proeyen jacobian factor for the classical UV action. When the QME cannot be satisfied, the non-vanishing Σ is proportional to a BRS anomaly, which is shown to be preserved along the RG flow. To see how the UV action fulfills the QME in anomaly free theory, we calculate the jacobian factor for a pure Yang-Mills theory in four dimensions.

A criterion for lattice supersymmetry: cyclic Leibniz rule

A bstractIt is old folklore that the violation of Leibniz rule on a lattice is an obstruction for constructing a lattice supersymmetric model. While it is still true for full supersymmetry, we show that a slightly modified form of the Leibniz rule, which we call cyclic Leibniz rule (CLR), is actually a criterion for the existence of partial lattice supersymmetry. In one dimension, we find sets of lattice difference operator and field multiplication smeared over lattice which satisfy the CLR under some natural assumptions such as translational invariance and locality. Thereby we construct a model of supersymmetric lattice quantum mechanics without spoiling locality. The CLR relation is coincident with the condition that the vanishing of the so-called surface term in the construction by lattice Nicolai map. We can construct superfield formalism with arbitrary superpotential. This also enables us to apply safely a localization technique to our model, because the kinetic term and the interaction terms of our model are independently invariant under the supersymmetry transformation. A preliminary attempt in finding a solution for the higher dimensional case is also discussed.