What is the cable that connects the logic board to the display in a Macbook Air?
The takeapart guide on iFixit sells replacement parts, but expect them to be pricey. The Air is a very, very delicate machine so be warned that getting it put back together takes more grace than getting it apart.

Worst case, you could buy an entire display assembly if you feel you are doing more damage than good by proceeding with this repair. Three cables go into the Air's display assemblyL


Wi-Fi antenna
Backlight power
LVDS cable


I presume you nicked the last one - it has the widest connector to the logic board. The real problem is getting inside the Air's display assembly as I haven't seen that documented anywhere publicly available. That cable might be part of the LCD panel and might be something you could try to buy used from a liquid damaged or physically damaged Air. Here's where the question lacks details to know why you were in there to begin.

What part was being replaced that caused you to need a razor blade on this repair? Are you going inside the display assembly in the first place?

Don't rule out taking it in to the store in a static safe back with your tail between your legs. You will then know if they can fix it by quoting you an out of warranty part cost plus labor to fix things - both what was originally wrong and the things that have now gone not right.