This is an expanded written version of a plenary talk delivered at ICMP 2021. We describe some rigorous results in quantum field theory that have been obtained in recent years, with particular emphasis on those results on relative entropies in the setting of conformal field theory. These results are motivated in part by recent work of physicists which, however, depends on heuristic argumentsâ€”such as introducing cutoff and using path integrals and replica trickâ€”which are hard to justify mathematically. Our main technical tools are from the theory of operator algebras, such as modular theory and the theory of subfactors. A discussion about open problems is presented at the end the paper.

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