Supersymmetric lattice theories on curved space

TL;DR

This paper constructs Hamiltonian lattice theories with exact supersymmetry on arbitrary triangulations, enabling a curved-space discretization via Kähler-Dirac fields. The bosonic sector uses a novel noncanonical commutator while the fermionic sector employs a nilpotent supercharge Q, with H = Q^2 linking to a reduced staggered fermion in flat space and to Kähler-Dirac dynamics in curved space. On flat lattices, the framework yields 2^d flavors and rich symmetry structure, including parity, time reversal, and non-invertible shift symmetries; on random triangulations, parity is lost but time reversal and supersymmetry persist, governed by the Euler characteristic χ through a cohomology-based zero-mode structure. Coupling to background gauge fields doubles the supersymmetry and provides a probe of anomaly structure, while the triangulation generalization connects to cohomological topological quantum field theories and sets the stage for future dynamical gauge field implementations.

Abstract

We show how to construct Hamiltonian lattice theories with one exact supersymmetry on arbitrary triangulations of curved space in any number of dimensions. Both bosons and fermions satisfy discrete Kähler-Dirac equations. The quantization of the fermions proceeds by imposing conventional anti-commutation relations while the bosons require a modification of the usual canonical commutator. On regular lattices we construct parity, time reversal and translation-by-one (shift) symmetries. We argue that the latter are generically non-invertible symmetries. We also show how to couple these degrees of freedom to background gauge fields which leads to a theory with enhanced supersymmetry.

Figures (3)

• Figure 1: Periodic staggered boson in 1D
• Figure 2: Relativistic staggered boson algebra in 2D
• Figure 3: Four coloring of sites for center