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Exact lattice supersymmetry

Simon Catterall, David B. Kaplan, Mithat Unsal

TL;DR

The paper surveys the main strategies for defining and simulating supersymmetric theories on the lattice, focusing on preserving a subset of SUSY at nonzero lattice spacing through twisting and orbifold/deconstruction. It shows that exact lattice-implemented scalar supercharges can steer the IR toward the target continuum theories, notably enabling controlled approaches to ${\cal N}=4$ SYM in 4D via either Marcus-type twisting or orbifold lattices with $A_4^*$ geometry. Key insights include the equivalence of twisting and orbifold approaches in gauge theories, the role of Dirac-Kähler fermions in avoiding doublers, and the use of Nicolai maps and topological perspectives to constrain quantum corrections. The review also discusses the renormalization challenges, moduli-space dynamics, and open problems surrounding higher-dimensional lattice SUSY, with an eye toward nonperturbative tests and potential links to string theory and quantum gravity.

Abstract

We provide an introduction to recent lattice formulations of supersymmetric theories which are invariant under one or more real supersymmetries at nonzero lattice spacing. These include the especially interesting case of ${\cal N}=4$ SYM in four dimensions. We discuss approaches based both on twisted supersymmetry and orbifold-deconstruction techniques and show their equivalence in the case of gauge theories. The presence of an exact supersymmetry reduces and in some cases eliminates the need for fine tuning to achieve a continuum limit invariant under the full supersymmetry of the target theory. We discuss open problems.

Exact lattice supersymmetry

TL;DR

The paper surveys the main strategies for defining and simulating supersymmetric theories on the lattice, focusing on preserving a subset of SUSY at nonzero lattice spacing through twisting and orbifold/deconstruction. It shows that exact lattice-implemented scalar supercharges can steer the IR toward the target continuum theories, notably enabling controlled approaches to SYM in 4D via either Marcus-type twisting or orbifold lattices with geometry. Key insights include the equivalence of twisting and orbifold approaches in gauge theories, the role of Dirac-Kähler fermions in avoiding doublers, and the use of Nicolai maps and topological perspectives to constrain quantum corrections. The review also discusses the renormalization challenges, moduli-space dynamics, and open problems surrounding higher-dimensional lattice SUSY, with an eye toward nonperturbative tests and potential links to string theory and quantum gravity.

Abstract

We provide an introduction to recent lattice formulations of supersymmetric theories which are invariant under one or more real supersymmetries at nonzero lattice spacing. These include the especially interesting case of SYM in four dimensions. We discuss approaches based both on twisted supersymmetry and orbifold-deconstruction techniques and show their equivalence in the case of gauge theories. The presence of an exact supersymmetry reduces and in some cases eliminates the need for fine tuning to achieve a continuum limit invariant under the full supersymmetry of the target theory. We discuss open problems.

Paper Structure

This paper contains 62 sections, 217 equations, 22 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Supersymmetry
  3. The supersymmetry algebra
  4. Counting supercharges
  5. $R$ symmetries
  6. Why study lattice supersymmetry?
  7. Accidental supersymmetry and twisted supercharges
  8. ${\cal N}=1$ supersymmetry in $d=4$ and accidental susy without scalars
  9. Accidental SUSY with scalars?
  10. Twisted supersymmetry
  11. Supersymmetric quantum mechanics on the lattice
  12. Algebra - two supersymmetries
  13. Naive discretization
  14. Feynman diagrams and power counting
  15. Exact lattice supersymmetry
  16. ...and 47 more sections

Figures (22)

  • Figure 1: The profile of the domain wall fermion mass in the fifth dimension, showing the chiral zero-modes (L,R) bound to the two domain walls where the fermion mass switches sign.
  • Figure 2: The lattice point group in supersymmetric lattices cannot be considered to be a subgroup of just the Lorentz group, but rather it is a subgroup of the product of the Lorentz group and and the $R$-symmetry group, $G_R$.
  • Figure 3: Boson and fermion masses vs lattice spacing for naive action
  • Figure 4: One loop Feynman graphs for SUSYQM
  • Figure 5: Counterterm corrected boson/fermion masses (From Giedt et al. Giedt_qm)
  • ...and 17 more figures