ERJ-170 Captain reported receiving a wind shear warning on climb out from MEM that may have been related to wake turbulence from a preceding B777.

Narrative

The winds were 180/4; very light. Took off behind a heavy B777. We were aware of the potential of wake turbulence on the climb out; since I had also heard the B777 was cleared for the same RNAV departure as us. The spacing was good between takeoffs; but still thinking about the wake turbulence potential. The climb out was perfectly smooth; and at about 1;200 feet AGL we received the wind shear caution. That took me by surprise as our aircraft had barely any chop on the climb; and there were no indications of wind shear at the field. It took me a little bit to mentally get from the wake turbulence maneuvers to the wind shear maneuver. Before I was getting ready to call max thrust; the indication had gone away. The aircraft had no indication of performance loss in the form of airspeed or altitude or otherwise; this could've been the reason why the caution took me by surprise.At the end of the day; I should've executed the wind shear maneuver. Better to be safe with the extra power. Although; looking at the situation; I feel it was a bad caution message since the aircraft was acting like it would in smooth air; and I have encountered much worse wake and turbulence below 1;500 feet and not had any caution in any of those situations. We were cognizant of potential wake turbulence; the takeoff separation we felt was adequate; we tried to fly a bit upwind and above the B777's flight path.