I am a total newb to PVs, but I do know a bit about batteries, so I hope I can help here. The charger being connected to the battery shouldn't make a significant difference in how long the battery stays charged. Most of my experience with LiON batteries was with the first gen ones in mobile phones, not the newer LiFePo4 ones that are in use in lots of places today. However, either is going to lose some charge over time. The older style would lose about 5% of it's charge every day or so, and the new ones are probably 5-10% a week or something. So, long story long, if you charge one up and unplug the charger from the wall, it should stay pretty much charged, at least for a reasonable amount of time.