Why are operads so closely connected to mathematical physics?

Mark Sapir's question inspired me to ask the question in the title. A lot of mathematicians who have done work related to mathematical physics (e.g Kontsevich, Stasheff, Getzler, Manin, etc.) have done work with operads, but I don't really grok why operads have anything to do with physics. I wonder if anyone has ideas about why there is such a close connection.

Added: Examples of how operads are used in physics are welcome, but just like in Mark Sapir's question, I am much more interested in a general reason that explains why they appear. Especially why multiple operads appear, not some particular operad. Note that operads can be defined in various categories (topological spaces, vector spaces, etc) and I'm most interested in operads in the category of vector spaces. But if there is an answer that covers more than one category, that would be very nice.

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String theory. Right? – Qiaochu Yuan Aug 9 2011 at 21:24
Your second sentences is missing something :) – Mariano Suárez-Alvarez Aug 9 2011 at 21:34
@Mariano: sorry, what is it missing? – Jim Conant Aug 9 2011 at 21:47
"A lot of mathematicians who have done work related to mathematical physics or, in fact, are mathematical physicists": there is one side of the or missing, no? – Mariano Suárez-Alvarez Aug 9 2011 at 22:07
Could it be something along the lines of the following? The universe likes to compose structures along graphs, e.g Feynman diagrams, and as Jeffrey Giansiracusa points out in his answer to Mark's question, cyclic and modular operads are especially suited to that situation. – Jim Conant Aug 10 2011 at 1:20
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I'm not a mathematical physicist, so parts of this may be wrong.

In quantum field theory, one encounters operators that are supported at points in spacetime, or at least are very local. For example (if we ignore uncertainty), one may have a "photon creation at $x$ with momentum $p$ and helicity $\xi$" operator, which changes the universe so that there is an extra photon at $x$ with momentum $p$ and helicity $\xi$. If you have encountered raising and lowering operators when studying the quantum mechanics of a harmonic oscillator, this is a generalization to the situation where you have harmonic oscillators at every point (and in interacting theories, the oscillators may coupled nonlinearly).

If we set up multiple operators supported at different points, there may be no canonical way to compose them. For example, if we have creations at spacelike separated points, any choice of time-ordering can be permuted by a boost. However, we can consider a parameter space of all possible ways to compose such operators, and we can consider how the compositions change when we continuously vary the positions of the creation events in spacetime. Furthermore, if we put a subset of such creations close to each other, we may view their composition as a single operator supported on the neighborhood, that depends on their relative position inside that neighborhood. Operads are precisely the objects that encode such families of composition laws and iterations.

One example where these arise is in the theory of vertex operator algebras, which (so I'm told) describe the chiral parts of conformal field theories. Given an compact complex algebraic curve and a vertex operator algebra, you can choose points with frames, and insert states (i.e., apply certain operators) at those points. There is a procedure by which you can construct from these data a space of chiral correlation functions, and this space will vary smoothly with the points and frames (and also with the complex moduli of the curve). If you put some points with frames close to each other and ignore the global geometry of the curve, the configurations of points with frames form a space in the framed $E_2$ operad, and the spaces of correlation functions are something like an algebra over the operad. This is a genus zero restriction of the compatibilities we see in conformal field theory, and can be strengthened to the statement (proved here under some extra hypotheses) that modules of a vertex operator algebra form a framed $E_2$ (in particular, braided) tensor category.

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It's not really possible to give a precise answer to this question, so I apologize for being vague here. One answer is because a lot of multiplications in physics are associated with moving two things close to each other and looking at the result as a single object. The archetype for this is the operator product expansion in quantum field theory. This naturally leads to thinking about a little n-spheres operad with various decorations. Most examples that come to mind right now really reduce to that.

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One significant connection to mathematical physics is through vertex operator algebras, used in conformal field theory. As algebraic objects, VOA's are fairly complicated, but the operations are conveniently packaged in geometric form, using an operad whose composition is given by sewings of punctured Riemann spheres. See this early paper by Lepowsky and Huang for information on this.

Edit: A more up-to-date account of the operadic story has been given by Huang's student Liang Kong, here.

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This does not really answer the question «*why* do operads show up in physics?», but gives an example of operads showing up in physics, no? – Mariano Suárez-Alvarez Aug 9 2011 at 21:42
I am not a mathematical physicist, so I am not best qualified to answer this question. But see this wonderful response from David Ben-Zvi, here: mathoverflow.net/questions/53988/… – Todd Trimble Aug 9 2011 at 21:55

This isn't really a full answer to your question, but I think there's a non-trivial amount of truth to it.

When you look at the subject of operads, what kind of diversity do you see?

I prefer the topological category so I'll list those in the topological category that come to my mind:

• Discs operads (equivalent to cubes)

• Framed discs operads (a semi-direct product of $SO_n$ with discs.

• Free product, tensor product operads...

• Cacti operads (different ones are equivalent to discs and framed discs....)

• McClure and Smith's operads, which are equivalent to cubes / cactus operads.

• Overlapping cubes operads (my terminology), which are equivalent to cubes operads.

• What people in the embeddings of manifolds community call the Kontsevich Operad, which is equivalent to cubes / discs operads.

• The operad I call "the splicing operad" in dimension $3$, is a free product of free operads with a semi-direct product of $O_2$ and the $2$-cubes operad...

Anyhow, if you look at the above, what do you see? Free operads (decorated trees) and cubes operads and various natural constructions. Moreover, some of these construction produce "more of the same" -- the tensor product of cubes is equivalent to cubes, for example.

An impression you could take out of this story is that operads perhaps have very few "core types" but a prolific number of variations on those "core types". One possible exception to this is the high-dimensional splicing operad may not be reducible to these "core types" but that is not clear at the moment.

So perhaps what you see isn't a physics thing, it's an operads thing?

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