# (How) is category theory actually useful in actual physics?

An answer to a recent question motivated the following question:

(how) is category theory actually useful in actual physics?

By "actual physics" I mean to refer to areas where the underlying theoretical principle has solid if not conclusive experimental justification, thus ruling out not only string theory (at least for the moment) but also everything I could notice on this nLab page (though it is possible that I missed something).

Note that I do not ask (e.g.) whether or not category theory has been used in connection with hypothetical models in physics. I've read Baez' blog from time to time over the decades and have already demonstrated knowledge of the existence of the nLab. I am dimly aware of stuff like (e.g.) the connection between between Hopf algebras and renormalization, but I have yet to encounter something that seems like it has a nontrivial category theoretic-component and cannot be expressed in some other more "traditional" language.

Note finally that I am ignorant of category theory beyond the words "morphism" and "functor" and (in my youth) "direct limit". So answers that take this into account are particularly welcome.

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Category theory serves as a formal setup with which one can organize ideas. Symplectic or Poisson manifolds are organized in a category, whose groups objects are the Lie-Poisson groups; the representations of various types of groups and of other type of symmetries organize themselves in categories, and using the language of category theory can be very helpful in expressing very complex ideas, and this is extremely helpful when dealing with complicated mathematical objects that physicists need to deal with; and so on and on. Is this "actual physics"? Well... Yes. –  Mariano Suárez-Alvarez Aug 7 '10 at 20:49
@Theo: For the purposes of this question, yes. –  Steve Huntsman Aug 7 '10 at 21:37
Poisson-Lie groups are group objects in the category of Poisson manifolds, and anyone who tells you otherwise is using the wrong definition of group object. The inverse map is always an anti-map when anti-maps make sense (e.g. Hopf algebras are group objects in algebra^op and the inverse map is an anti-algebra map). –  Noah Snyder Aug 7 '10 at 21:47
I realise that "actual physics" is defined in this question and hence there is a reasonable chance of answering objectively whether X is "actual physics". However I cannot but point out that in my already more than two decades as a mathematical physicist I have come across attempts to define (one could say, in fact, restrict) what Physics is more times than I care to remember. And in all this time I have come across but one satisfactory answer: "Physics is what physicists do." It may seem circular, but not completely, since the field is in constant evolution. –  José Figueroa-O'Farrill Aug 7 '10 at 23:56
@Kevin: Certain fractional Hall effect systems are now believed (based on not yet completely conclusive experimental evidence) to be described effectively by Chern-Simons theory. In particular, by su(2) at some small level. –  Noah Snyder Aug 8 '10 at 14:44

Fusion categories and module categories come up in topological states of matter in solid state physics. See the research, publications, and talks at Microsoft's Station Q.

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Theories of anyons are very much related to braided fusion categories and 2-d topological quantum field theories. You could look at the survey article rmp.aps.org/abstract/RMP/v80/i3/p1083_1 (this is the only recent survey article I know of for this field ... if you have any other suggestions, I'd like to see them as well), and there are also some popular articles linked from the Microsoft Station Q website above. –  Peter Shor Aug 7 '10 at 21:28
Continuing my comment ... depending on who you talk to, either the $5/2$ fractional quantum Hall effect has been experimentally proven to be generated by anyons, or has not quite been proven experimentally. Topological insulators are a recent experimental discovery with related behavior. So anyons/braided fusion categories/TQFT should meet your "actual physics" criterion. –  Peter Shor Aug 7 '10 at 21:40
@Peter: Thanks...it's funny, I had that RMP as something that I was planning to study but hadn't yet. I note that the authors pose--and then explicitly address--the question "Why is it necessary to invoke category theory simply to specify the topological properties of non-Abelian anyons?" –  Steve Huntsman Aug 7 '10 at 21:42
@Steve: Your quote just made me realize that I should have specified non-abelian anyons in my comment. Abelian anyons are much more widely realized experimentally, but they correspond to trivial examples of braided fusion categories, and thus you don't actually need category theory to understand them. I should probably also mention that the 12/5 fractional quantum Hall effect is also believed to be generated by non-abelian anyons, but the experimental confirmation is much farther away because it is experimentally harder to work with. –  Peter Shor Aug 7 '10 at 22:21
@Peter: Zhenghan Wang's CBMS monograph "Topological Quantum Computation" is now available from the AMS and might fit the bill as a recent survey--particularly chapters 6 and 8 deal with this subject. –  Eric Rowell Aug 8 '10 at 0:05

Categories (and higher categories) seem to be a good way of expressing the locality of the path integral in physics. In particular, it is the idea of gluing of local structures that is important. This line of thought leads to the axiomatization of (parts of) various QFTs, with the most success in topological and conformal field theories. This idea has its origins with Atiyah, Segal, Baez-Dolan, Freed and probably a ton of other people I'm forgetting. Braided fusion categories as in the previous answer are an example of this in three dimensions. Most recently, there's Lurie's classification of TQFTs in all dimensions in terms of $(\infty,n)$ categories.

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Jürgen Fuchs, Ingo Runkel and Christoph Schweigert have developed a complete treatment of Rational Conformal Field Theory based on algebra in braided tensor categories. They have applications to string theory as well as to statistical physics, most importantly to conformal defects and so-called Kramers-Wannier-dualities.

See "J. Fuchs, I. Runkel, C. Schweigert: TFT construction of RCFT correlators I-V " for the full story or, for a summary, Schweigert's ICM talk Categorification and correlation functions in conformal field theory.

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