I think the 'best' book in your scenario is "Geometry, Topology, and Physics" by Nakahara. I put this in quotes because I don't think it teaches the physicists true rigorous math (it just skims the surface with a bunch of terminology and only hands you the tools needed to compute some things in physics)... I've tried conversing with all my theoretical physics friends after they learn Algebraic Topology and Differentiable Manifolds through this text, and it's torturous. But this is just an opinon. Anyway, the textbook talks about the math that is linked to the physics, so it could benefit you to hear about the physics in this text (there is a lot!). It's considered ''mandatory'' to be on every theoretical physicist's shelf.
Other than that text, take out any textbook on String Theory (such as the classic one by Green, Schwarz, Witten). There is also the text Quantum Field Theory for Mathematicians, but I don't know how much it will be able to relate to what Witten is doing... I don't think any textbook can help you with what Witten is doing haha. Only main source seems to be the referenced papers.