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I once tried reading John Baez's textbook on Algebraic QFT, but I got stuck on something which I didn't understand maybe because I didn't have a good grasp of Lie Algebras at the time.

I also tried reading Rudolf Haag's textbook but then he started mentioning Several complex variables.

It seems in order to understand texts on Constructive and Algebraic QFT one needs to know at least most if not all of analysis, algebra and topology and geometry.

Is there a shortcut?

:-)

This is my biased opinion on this , You might find it helpful to learn about the atyiah-segal axioms , Topological quantum field theory & Conformal field theory. I know of many places where this is discussed in a clear manner. See for example Dijkgraaf's lectures on strings and fields or Greg moore's lectures on two dimensional Yang mills and equivariant cohomology. Michael Atyiah's paper is also clear. and there are lectures by daniel freed. Unfortunately , I don't know much about constructing higher dimensional QFT in a mathematically rigorous fashion. However , I think that we have new and deep mathematical insight about QFT in the recent years. String theory , holography & research in the structures in scattering amplitudes are related.

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