all MacBook Airs from 2010 onwards need the EXACT matched partnumber for the batteries.
make sure you get the battery labeled as A1405.
do NOT get the battery labeled as A1377 or anything that they say works with any other MacBook Air than the mid 2011 model. (some websites list the same batteries for the 2010 and 2011 models: this is not correct and they're incompatible).
I have had these issues multiple times, had a freakin headache because suppliers kept sending me mismatched parts; always the same result: the MacBook Air freezes up and is using huge amounts of cpu on kernel task because of the mismatched battery.
If you DID order the A1405, then you've probably got a DOA, and you need to send it back in for a swap.