Semi-steady non-commutative crepant resolutions via regular dimer models
  A consistent dimer model gives a non-commutative crepant resolution (= NCCR) of a $3$-dimensional Gorenstein toric singularity. In particular, it is known that a consistent dimer model gives a nice class of NCCRs called steady if and only if it is homotopy equivalent to a regular hexagonal dimer model. Inspired by this result, we introduce the notion of semi-steady NCCRs, and show a consistent dimer model gives a semi-steady NCCR if and only if it is homotopy equivalent to a regular dimer model. 
Yusuke Nakajima
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