text: The purpose of this work is to investigate the role of the lattice in the optical Kubo sum rule in 
  the cuprates. We compute conductivities, optical integrals W , and ∆W between superconducting 
  and normal states for 2-D systems with lattice dispersion typical of the cuprates for four diﬀerent 
  models – a dirty BCS model, a single Einstein boson model, a marginal Fermi liquid model, and a 
  collective boson model with a feedback from superconductivity on a collective boson. The goal of 
  the paper is two-fold. First, we analyze the dependence of W on the upper cut-oﬀ (ωc) placed on 
  the optical integral because in experiments W is measured up to frequencies of order bandwidth. 
  For a BCS model, the Kubo sum rule is almost fully reproduced at ωc equal to the bandwidth. But 
  for other models only 70%-80% of Kubo sum rule is obtained up to this scale and even less so for 
  ∆W , implying that the Kubo sum rule has to be applied with caution. Second, we analyze the sign 
  of ∆W . In all models we studied ∆W is positive at small ωc, then crosses zero and approaches a 
  negative value at large ωc, i.e. the optical integral in a superconductor is smaller than in a normal 
  state. The point of zero crossing, however, increases with the interaction strength and in a collective 
  boson model becomes comparable to the bandwidth at strong coupling. We argue that this model 
  exhibits the behavior consistent with that in the cuprates. 