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Is it possible to put a supersymmetric theory on a lattice?

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Several lattice models have recently been shown to display emergent supersymmetry at length scales long enough that the lattice can be coarse-grained into a continuum (e.g. see here, here, and here). Could a lattice model display exact supersymmetry even at the lattice length scales? Clearly the answer is no, because the lattice breaks the Poincare subgroup of the supersymmetry group down to $S \times \mathbb{R}$, where $S$ is the lattice's space group and the $\mathbb{R}$ corresponds to time translational invariance.

According to this answer, the supersymmetry Lie supergroup $G$ corresponding to 3+1D SUSY with $N$ fermionic generators and no additional internal symmetries is Inonu-Wigner contracted $OSP(4/N)$. Does there exist a Hamiltonian, defined on a lattice with space group $S$, with a symmetry Lie supergroup $H < G$ such that the bosonic part of $H$ is $S \times \mathbb{R}$ and the fermionic part of $H$ is nontrivial? (In other words, I want to reduce the full supersymmetry supergroup $G$ down to a sub-supergroup $H$, such that the bosonic part of $G$ reduces from the Poincare group down to a lattice space group (times time translation), but without reducing the fermionic part all the way down to the identity, which would eliminate the supersymmetry entirely.) This seems to me like the natural way to restrict supersymmetry to a lattice.


This post imported from StackExchange Physics at 2017-01-11 10:27 (UTC), posted by SE-user tparker

asked Dec 24, 2016 in Theoretical Physics by tparker (305 points) [ revision history ]
edited Jan 11 by Dilaton

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