Dear Categorists - Here's the portion of "week224" that deals with category theory. Happy Holidays to everyone! Best, jb ....................................................................... Also available as http://math.ucr.edu/home/baez/week224.html December 14, 2005 This Week's Finds in Mathematical Physics - Week 224 John Baez This week I want to mention a couple of papers lying on the interface of physics, topology, and higher-dimensional algebra. But first, some astronomy pictures... and a bit about the mathematical physicist Hamilton! [...] Now for some mathematical physics that touches on higher-dimensional algebra. If you still don't get why topological field theory and n-categories are so cool, read this thesis: 13) Bruce H. Bartlett, Categorical aspects of topological quantum field theories, M.Sc. Thesis, Utrecht University, 2005. Available as math.QA/0512103. It's a great explanation of the big picture! I can't wait to see what Bartlett does for his Ph.D.. If you're a bit deeper into this stuff, you'll enjoy this: 14) Aaron D. Lauda and Hendryk Pfeiffer, Open-closed strings: two-dimensional extended TQFTs and Frobenius algebras, available as math.AT/0510664. This paper gives a purely algebraic description of the topology of open and closed strings, making precise and proving some famous guesses due to Moore and Segal, which can be seen here: 15) Greg Moore, Lectures on branes, K-theory and RR charges, Clay Math Institute Lecture Notes (2002), available at http://www.physics.rutgers.edu/~gmoore/clay1/clay1.html Lauda and Pfeiffer's paper makes heavy use of Frobenius algebras, developing more deeply some of the themes I mentioned in "week174". In a related piece of work, Lauda has figured out how to *categorify* the concept of a Frobenius algebra, and has applied this to 3d topology: 16) Aaron Lauda, Frobenius algebras and ambidextrous adjunctions, available as math.CT/0502550. Aaron Lauda, Frobenius algebras and planar open string topological field theories, available as math.QA/0508349. The basic idea behind all this work is a "periodic table" of categorified Frobenius algebras, which are related to topology in different dimensions. For example, in "week174" I explained how Frobenius algebras formalize the idea of paint drips on a sheet of rubber. As you move your gaze down a sheet of rubber covered with drips of paint, you'll notice that drips can merge: \ \ / / \ \ / / \ \ / / \ \ / / \ \_/ / \ / | | | | | | | | | | but also split: | | | | | | | | | | / _ \ / / \ \ / / \ \ / / \ \ / / \ \ / / \ \ In addition, drips can start: _ | | | | | | | | | | | | | | | | | | but also end: | | | | | | | | | | | | | | | | |_| In a Frobenius algebra, these four pictures correspond to four operations called "multiplication" (merging), "comultiplication" (splitting), the "unit" (starting) and the "counit" (ending). Moreover, these operations satisfy precisely the relations that you can prove by warping the piece of rubber and seeing how the pictures change. For example, there's the associative law: \ \ / / / / \ \ \ \ / / \ \ / / / / \ \ \ \ / / \ \/ / / / \ \ \ \/ / \ / / / \ \ \ / \ \ / / \ \ / / \ \_/ / \ \_/ / \ / \ / | | | | | | | | | | = | | | | | | | | | | | | | | | | | | | | | | The idea here is that if you draw the picture on the left-hand side on a sheet of rubber, you can warp the rubber until it looks like the right-hand side! There's also the "coassociative law", which is just an upside-down version of the above picture. But the most interesting laws are the "I = N" equation: \ \ / / | | | | \ \ / / | | | | \ \_/ / | | | | \ / | \ | | | | | \ | | | | | |\ \ | | | | | | \ \ | | | | | | \ \ | | | | = | | \ \ | | | | | | \ \ | | | | | | \ \| | | | | | \ | / _ \ | | \ | / / \ \ | | | | / / \ \ | | | | / / \ \ | | | | and its mirror-image version. So, the concept of Frobenius algebra captures the topology of regions in the plane! Aaron Lauda makes this fact into a precise theorem in his paper on planar open string field theories, and then generalizes it to consider "categorified" Frobenius algebras where the above equations are replaced by isomorphisms, which describe the *process* of warping the sheet of rubber until the left side looks like the right. You should look at his paper even if you don't understand the math, because it's full of cool pictures. Lauda and Pfeiffer's paper goes still further, by considering these paint stripes as "open strings", not living in the plane anymore, but zipping around in some spacetime of high dimension, where they might as well be abstract 2-manifolds with corners. Following Moore and Segal, they also bring "closed strings" into the game, which form a Frobenius algebra of their own, where the multiplication looks like an upside-down pair of pants: O O \ \ / / \ \ / / \ / | | | | | | | | | | O These topological closed strings are the subject of Joachim Kock's book mentioned in "week202"; they correspond to *commutative* Frobenius algebras. The fun new stuff comes from letting the open strings and closed strings interact. You can read more about Lauda and Pfeiffer's work at Urs Schreiber's blog: 17) Urs Schreiber, Lauda and Pfeiffer on open-closed topological strings, http://golem.ph.utexas.edu/string/archives/000680.html In fact, I recommend Schreiber's blog quite generally to anyone interested in higher categories and/or the math of string theory! ----------------------------------------------------------------------- Addendum: Here's what Urs Schreiber had to say about Frobenius algebras, modular tensor categories and string theory: John Baez wrote: [...] Following Moore and Segal, they also bring "closed strings" into the game, which form a Frobenius algebra of their own, where the multiplication looks like an upside-down pair of pants: [...] I would like to make the following general comment on the meaning of Frobenius algebras in 2-dimensional quantum field theory. Interestingly, _non_-commutative Frobenius algebras play a role even for closed strings, and even if the worldhseet theory is not purely topological. The archetypical example for this is the class of 2D TFTs invented by Fukuma, Hosono and Kawai. There one has a non-commutative Frobenius algebra which describes not the splitting/joining of the entire worldsheet, but rather the splitting/joining of edges in any one of its dual triangulations. It is the _center_ of (the Morita class of) the noncommutative Frobenius algebra decorating dual triangulations which is the commutative Frobenius algebra describing the closed 2D TFT. One might wonder if it has any value to remember a non-commutative Frobenius algebra when only its center matters (in the closed case). The point is that the details of the non-commutative Frobenius algebra acting in the "interior" of the world sheet affects the nature of "bulk field insertions" that one can consider and hence affects the (available notions of) n-point correlators of the theory, for n > 0. This aspect, however, is pronounced only when one switches from 2D topological field theories to conformal ones. The fascinating thing is that even 2D "conformal" field theories are governed by Frobenius algebras. The difference lies in different categorical internalization. The Frobenius algebras relevant for CFT don't live in Vect, but in some other (modular) tensor category, usually that of representations of some chiral vertex operator algebra. It is that ambient tensor category which "knows" if the Frobenius algebra describes a topological or a conformal field theory (in 2D) - and which one. Of course what I am referring to here is the work by Fjelstad, Froehlich, Fuchs, Runkel, Schweigert and others. I can recommend their most recent review which will appear in the Streetfest proceedings. It is available as math.CT/0512076. The main result is, roughly, that given any modular tensor category with certain properties, and given any (symmetric and special) Frobenius algebra object internal to that category, one can construct functions on surfaces that satisfy all the properties that one would demand of an n-point function of a 2D (conformal) field theory. If we define a field theory to be something not given by an ill-defined path integral, but something given by its set of correlation functions, then this amounts to constructing a (conformal) field theory. This result is achieved by first defining a somewhat involved procedure for generating certain classes of functions on marked surfaces, and then proving that the functions generated by this procedure do indeed satisfy all the required properties. In broad terms, the prescription is to choose a dual triangulation of the marked worldsheet whose correlation function is to be computed, to decorate its edges with symmetric special Frobenius algebra objects in some modular tensor category, to decorate its vertices by product and coproduct morphisms of this algebra, to embed the whole thing in a certain 3-manifold in a certain way and for every boundary or bulk field insertion to add one or two threads labeled by simple objects of the tensor category which connect edges of the chosen triangulation with the boundary of that 3-manifold. Then you are to hit the resulting extended 3-manifold with the functor of a 3D TFT and hence obtain a vector in a certain vector space. This vector, finally, is claimed to encode the correlation function. This procedure is deeply rooted in well-known relations between 3-(!)-dimensional topological field theory, modular functors and modular tensor categories and may seem very natural to people who have thought long enough about it. It is already indicated in Witten's paper on the Jone's polynomial, that 3D TFT (Chern-Simons field theory in that case) computes conformal blocks of conformal field theories on the boundaries of these 3-manifolds. To others, like me in the beginning, it may seem like a miracle that an involved and superficially ad hoc procedure like this has anything to do with correlations functions of conformal field theory in the end. In trying to understand the deeper "meaning" of it all I played around with the idea that this prescription is really, to some extent at least, the "dual" incarnation of the application of a certain 2-functor to the worldsheet. Namely a good part of the rough structure appearing here automatically drops out when a 2-functor applied to some 2-category of surfaces is "locally trivialized". I claim that any local trivialization of a 2-functor on some sort of 2-category of surface elements gives rise to a dual triangulation of the surface whose edges are labeled by (possibly a generalization of) a Frobenius algebra object and whose vertices are labeled by (possibly a generalization of) product and coproduct operations. There is more data in a locally trivialized 2-functor, and it seems to correctly reproduce the main structure of bulk field insertions as appearing above. But of course there is a limit to what a _2_-functor can know about a structure that is inherently 3-dimensional. I have begun outlining some of the details that I have in mind here: http://golem.ph.utexas.edu/string/archives/000697.html This has grown out of a description of gerbes with connective structure in terms of transport 2-functors. Note that in what is called a _bundle_ gerbe we also do have a certain product operation playing a decisive role. Bundle gerbes can be understood as "pre-trivializations" of 2-functors to Vect: http://golem.ph.utexas.edu/string/archives/000686.html and the product appearing is one of the Frobenius products mentioned above. For a bundle gerbe the coproduct is simply the inverse of the product, since this happens to be an isomorphism. The claim is that 2-functors to Vect more generally give rise to non-trivial Frobenius algebras when locally trivialized. This is work in progress and will need to be refined. I thought I'd mention it here as a comment to John's general statements about how Frobenius algebras know about 2-dimensional physics. I am grateful for all kinds of comments. Here's the paper Urs refers to: 21) Ingo Runkel, Jens Fjelstad, Jurgen Fuchs and Christoph Schweigert, Topological and conformal field theory as Frobenius algebras, available as math.CT/0512076. ----------------------------------------------------------------------- Quote of the Week: Here's how you do it: First you're obtuse, Then you intuit, Then you deduce! - Garrison Keillor ----------------------------------------------------------------------- Previous issues of "This Week's Finds" and other expository articles on mathematics and physics, as well as some of my research papers, can be obtained at http://math.ucr.edu/home/baez/ For a table of contents of all the issues of This Week's Finds, try http://math.ucr.edu/home/baez/twfcontents.html A simple jumping-off point to the old issues is available at http://math.ucr.edu/home/baez/twfshort.html If you just want the latest issue, go to http://math.ucr.edu/home/baez/this.week.html