FQFT and cohomology
Types of quantum field thories
Quantum field theory is the general framework for the description of the fundamental processes in physics as understood today. These are carried by configurations of fields under the generalized rules of quantum mechanics, therefore the name. Notably the standard model of particle physics is a quantum field theory and has been the main motivation for the development of the concept in general.
Historically quantum field theory grew out of attempts to combine classical field theory in the context of special relativity with quantum mechanics. While some aspects of it are understood in exceeding detail, the overall picture of what quantum field theory actually is used to be quite mysterious. There are two main approaches for axiomatizing and formalizing the notion:
functorial quantum field theory: FQFT – this encodes a quantum field theory as an assignment of spaces of quantum states to patches of codimension 1, and of maps between spaces of states – the time evolution operator – to cobordisms between such patches.
Both these approaches try to capture the notion of a full quantum field theory. On the other hand, much activity in physics is concerned with perturbative quantum field theory. This is a priori to be thought of as an approximation to a full quantum field theory akin to the approximation of a function by its Taylor series, but not the least because it is often the only available technique, the tools of perturbative quantum field theory are to some extent also taken as a definition of quantum field theory.
The gap for instance between the formal study of the AQFT axioms and physics as done in practice by physicists had to a large extent been due to the fact that AQFT had little to say about perturbative quantum field theory. But recently this has been changing. See perturbative quantum field theory for more.
Recent times have seen major progress in understanding these axiomatizations and connecting them to the structures studied in physics (see the references below), but still the number of interesting phenomena in quantum field theory that physicists handle semi-rigorously and that are waiting for a fully formal understanding is large.
Even though quantum field theory has been around for decades and has been very successful both as a phenomenological model in experimental physics as well as a source of deep mathematical structures and theorems, from a mathematical perspective it is still to a large extent mysterious, though recently much progress is being made.
There are essentially two alternative approaches for formalizing quantum field theory and making it accessible to mathematical treatment:
(Other structures which are used to define quantum field theories, such as vertex operator algebras are now more or less understood to be special cases of these two approaches. See there for details.)
Both AQFT and FQFT involve the language of category theory and higher category theory. In fact, a couple of important higher categorical structures were motivated from and first considered in the context of quantum field theory. For instance
In early 1990s A-∞ categories first appeared in works of Kontsevich and Fukaya on the categorical description of twisted sigma models what is then used in the formulation of homological mirror symmetry.
the description of higher background gauge fields very much motivated the development and study of differential cohomology, which like all notions of cohomology is intrinsically about (∞,1)-topos theory.
There are some indications that such higher categorical structures, such as those appearing in groupoidification, are essential for clarifying some of the mysteries of quantum field theory, such as the path integral. While this is far from being clarified, this is what motivates research in higher categorical structures in QFT.
Ours is the age to figure this out.
A short introduction to different aspects of QFT usually covered in a first course is this:
Gerald B. Folland, Quantum field theory: A tourist guide for mathematicians, Math. Surveys and Monographs 149 (ZMATH)
A standard textbook written from the perspective of effective field theory is
An extensive compilation of material on QFT aiming for mathematical precision is
A discussion of aspects of QFT with an eye towards applications in string theory and aimed at mathematicians (though requiring more of a physicist’s mindset than many pure mathematicians will find themselves in) is
Differential geometric and topological aspects (e.g. connections to index theorems and moduli spaces) are emphasized in
An indication of the modern foundational picture of quantum mechanics is attempted in
Albert Schwarz, Quantum field theory and topology, Grundlehren der Math. Wissen. 307, Springer 1993.
Texts on supersymmetric QFT include