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

+ 4 like - 0 dislike
269 views

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
edited Jan 11, 2017

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:$\varnothing\hbar$ysicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.