Wookie is correct, Brown from the Ignition Key Switch. The Orange, sometimes Yellow, goes thru the main wiring harness to a connector block to the LH handlebar switch. That's where the L and R is determined. You'll see a Green and Grey? You need to open up the LH switchblock and have a lookie-see.
I don't think you need to open the switch at this point given that the turn signals illuminate as expected (just don't flash) I would open a cold beer and wait for the incandescent turn signals to arrive which should work as intended with a mechanical flasher unit
