The impression I have is that the mathematical approach to quantum field theory can broadly be categorized into one that is more geometrical/topological, for example in gauge theories, and another that is more analysis oriented such as constructive or algebraic quantum field theory. I am curious about the overlap between these two approaches or do the two have separate goals?
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$\begingroup$ Geometric QFTs ignore lot of data because they start with classical ideas, (think of physics as a 2-category of physical theories, which are themselves categories, with states as objects, and evolution providing morphisms. We have quantum to classical "approximation" functors, and its adjoint the "quantization" functor, and cannot be equivalence of cats). AQFT ideas are purely quantum, the relativistic ideas are imposed on the objects (vN algebras and states), based on Haag-Kastler idea. If anything, I expect a relation of an equivalence class of AQFTs and a Geometric QFT. $\endgroup$– MrPajeetCommented Aug 21, 2023 at 23:39
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2$\begingroup$ You're observing a quite general phenomenon: looking for your lost keys under the street lamp, because there is light there. Any sufficiently complex research program is naturally attacked by different groups of people with different mathematical tools (differential geometry, algebraic geometry, representation theory, operator algebras, probability, ...), each one formulating their own simplifications and toy problems. TQFTs, CFTs, instantons, SUSY, AQFT, ... are all examples of such developments. At the moment, overlaps between these areas are unfortunately minimal. $\endgroup$– Igor KhavkineCommented Aug 22, 2023 at 9:23
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$\begingroup$ I think your classification is wrong, or at least under-specified. Gauge theory belongs to analysis. In topology and geometry, it has served as a source of ideas, but topology and geometry don't really provide tools for understanding gauge theory itself. $\endgroup$– user1504Commented Oct 20, 2023 at 20:49
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