I am very excited about all the ideas surrounding quantum field theories (with or without the use of infinity categories). When I look around I see the following interesting topics:
- Extended TQFTs in Lurie sense.
- General QFTs in the sense of Costello-Gwilliam, using factorization algebras and relations to the work of Ayala-Francis et al of Factorisation homology.
- Various constructions related to character varieties (Ben-Zvi et al?), homological mirror symmetry and other applications of TQFT.
- Explicit constructions of manifold invariants using TQFTs that assign various tensor categories (or representation categories of various Hopf algebras) to a circle or a point.
- Applications to physics by explicitly constructing Lagrangian or Hamiltonians like Chern Simons, and Kitaev's lattice models.
Additionally, connections to conformal field theory and supersymmetry would be fantastic as well. Apparently there is a "superalgebraic geometric" point of view of supersymmetry (Freed, Kapranov?), and it would be cool to know how this fits in with the above points of view.
I know basics of infinity category theory (just definitions of various models like complete Segal spaces, quasi categories, dg categories, and so on) and I am comfortable with basic category theory (Riehl's book), basic algebraic topology (Hatcher), differential geometry and homological algebra. However, I have no knowledge of algebraic geometry and derived geometry.
I have read Joachim Kock's book on TQFT for dimension 2, Schommer-Pries thesis for 2-extended TQFT and I have been reading Lurie's sketch of the proof of cobordism hypothesis. I have read Hiro Lee Tanaka's introduction to factorisation homology and that was really cool. I have also been looking at various video lectures due to Ayala and Francis on factorisation homology. I have also read Kitaev's seminal paper that constructs a modular tensor category of anyon data starting from a lattice Hamiltonian.
Basically, I want to know what would be the shortest path to the theory of QFTs. Ideally, I would like an order that goes from very general abstraction (that unifies different points of view) to specific computations. I would appreciate books, survey articles or even a path through research papers. I have been looking around for sometime and I can't find a gentle entry point into these topics.