Q: Will sun fade the indigo? I noticed my indigo was a lot darker when I first placed it on the line to dry.
A: Yes, sun will naturally fade indigo. However, it takes days, weeks and months of UV exposure to remove substantial amounts of color. It sounds like you're describing the process of lightening that naturally happens when the piece dries out. Wet indigo dyed textiles always appears darker than the same fabric when dry. I recommend two or three more dips in the indigo vat AFTER you see the fabric reach the shade that you want.
Re: actual sun fading of indigo dyed textiles : In John Marshall’s book Singing The Blues (which I highly recommend as a practical guide to working with indigo), he published photos of fabrics that have been exposed to comparable amounts of sunlight. One piece of fabric had been treated with a soy-milk after bath, the other received no such treatment. The fabric which received a layer of soy milk following dyeing was appreciably better at resisting the UV fading effects of the sun. I’m now firmly in support of dipping all my own indigo dyed textiles in a solution of soy milk (I’ll make my own from whole soy beans if I have the time, but will also use pure, no-sugar-or-flavor-added bottled soymilk) after their pH balancing rinse and allowing the soy to dry on the fabric. I won’t wash the item for a month to allow the soy to properly cure onto the fabric.