The n-Category Café

David wrote:

JB wrote:

…see if you can find any reference in the literature which admits that “Hecke algebras” are related to “Hecke operators”. It ain’t easy!

Just to get the obvious out of the way, Wikipedia has:

Algebras of Hecke operators are called Hecke algebras…

Interesting. But, it’s a red herring!

Here they are talking about certain commutative algebras generated by Hecke operators — all this taking place in the context of lattices and modular forms.

I was talking about a different bunch of Hecke algebras: the noncommutative algebras associated to a Coxeter group and a value of $q$! What almost nobody seems to admit is that these, too are generated by ‘Hecke operators’.

In fact, the Wikipedia article you cite tends to prove my point. Later it says:

Hecke algebras in general

For more details on this topic, see Hecke algebra

Other mathematical rings are called Hecke algebras, without the obvious link to Hecke operators. These include certain quotients of the group algebra of a braid group.

As far as I know, James Dolan was quite on his own as he gradually came to realize that these Hecke algebras do have an obvious link to ‘Hecke operators’ — as long as the latter are defined in sufficient generality. Later he came across one Russian book whose author seemed cognizant of this point. It’s actually obvious once you grok Hecke algebras and Hecke operators… but, practically nobody seems to admit it!

Dear John Baez,

Thanks a lot for the explanations, I’ll study them.

Over at my blog, I have linked the question of “holomorphic factorization” to the wikipedia article on the Weierstrass factorization theorem, in special, I was thinking about the section “Holomorphic functions can be factored” of that article. Please let me know whether you think that is a right pointer or not.

Best regards,
Christine

Does this new paper disprove Witten’s conjecture?

Dean wrote:

What is a good introduction to conformal field theory?

[…]

I am a complete beginner (an undergraduate mathematician).

That’s an interesting question.

I only started studying conformal field theory after having taken some grad courses in quantum field theory, differential geometry and so on. I was a grad student during the first string theory revolution, so I picked up a lot of conformal field theory from talks — all the bigshot mathematicians were busily learning it back then.

Later I polished my understanding by reading stuff like this mammoth yet readable tome:

• Phillippe Di Francesco, Pierre Mathieu, and David Senechal, Conformal Field Theory, Springer, Berlin, 1997.

For a quick summary including an annotated table of contents, try week124.

But, you’re in a different position! You’re getting started earlier in your career, and later in the development of the subject.

I bet most of the ‘complete beginners’ wanting to learn conformal field theory are undergraduate physicists wanting to learn string theory. For such people — and maybe even for you! — I’d recommend this book:

• Barton Zwiebach, A First Course in String Theory, Cambridge U. Press, Cambridge U. Press, Cambridge, 2004.

Don’t expect mathematical rigor, but do expect an introduction to the key ideas behind conformal field theory: quantum field theory, and the special features of the wave equation in 2d Minkowski spacetime and 2d Euclidean space, arising from conformal invariance.

What about online sources? Hmm… I’ll just use Google.

This seems like an effficient and fairly rigorous introduction to the ideas of conformal field theory. It even explains ‘holomorphic factorization’, which you’ll see I didn’t understand when reading Witten’s paper:

• Matthias R. Gaberdiel, An introduction to conformal field theory.

For a very nice gentle introduction to conformal field theory which explains operads, try this:

• Alexander A. Voronov, Mathematical Methods in Physics. See especially Lecture 1: conformal field theory.

Here’s a more comprehensive introduction, which conveys more of the richness of the subject:

• C. J. Efthimiou and D. A. Spector, A collection of exercises in two-dimensional physics, part 1.

There are probably lots more, and I hope other Café visitors list their favorites!

Warning: if you try to learn conformal field theory from a bunch of different books, you’ll be overwhelmed by the variety of different formalisms: operator product expansions, Segal’s axiomatic definition of conformal field theory, vertex operator algebras, chiral algebras, operads, and so on. It helps to bear in mind that these are just different ways of studying a bunch of examples of a concept from physics: a 2d quantum field theory that’s invariant under conformal transformations.

What is a good introduction to conformal field theory?

hep-th/9108028.

Possibly the first paper in the arxiv (submitted in 1988).

David wrote:

Presumably, you were alluding to:

the $n$-dimensional Fourier transform is the 1-dimensional Fourier transform of the Radon transform. (Helgason, p. 4)

Right — and I hope it’s clear why this is true. For example, the Fourier transform of a function

$$f : \mathbb{R}^3 \to \mathbb{C}$$

tells you how to express it in terms of plane waves. To see how much the plane wave $exp(i k( x))$ shows up in $f$, you need to take their inner product, which means integrating $f(x)$ times the complex conjugate of $exp(i k ( x))$ over all of space.

But this can be calculated if we know the integrals of $f$ over planes — the planes on which the plane wave $exp(i k\cdot x)$ is constant.

If the Fourier transform could be given a span-ish kind of interpretation, this would presumably be a form of composition of spans.

Yes! So far it’s just a composition of integral kernels relating three types of `figures’:

• points
• planes
• plane waves

The $3$-dimensional Fourier transform takes a function of points and turns it into a function of plane waves.

The Radon transform takes a function of points and turns it into a function of planes.

So, we can write the 3d Fourier transform as the Radon transform followed by a transform that takes a function of planes and turns it into a function of plane waves — essentially a 1d Fourier transform.

So, we’re composing integral kernels. The only thing that makes it tricky to interpret this as composition of spans is that the relation between a point $x \in \mathbb{R}^3$ and a plane wave with momentum $k \in (\mathbb{R}^3)^*$ is not a “yes or no” sort of thing like whether a point lies on a plane — it takes values in phases! That’s what $exp(i k(x))$ is all about.

All the more reason to expect that Jeffrey Morton’s ‘phased sets’ will take over the universe someday.

Oh dear, I’ve probably laid the basis for some awful puns.

I’ve been waiting to use those puns for a long time. I already talk about things like “the spanish interpretation of quantum mechanics”.

But I’m waiting for a good time to say “Nobody expects the spanish inquisition!”

It doesn’t make sense to say the speed of light is ‘closer to infinity than 1’, or Planck’s constant or the gravitational constant are ‘close to 0’, or that they are ‘far apart in magnitude’, unless you’re comparing them to something else.

For civil engineering, he is certainly right. For objects on the human scale it makes sense to measure mass in units on the order of kilograms, time in units on the order of seconds, and length in units on the order of meters. In these units $c$ looks very large, $G$ looks small, and $\hbar$ looks smaller.

For quantum gravity, one is interested in things that move near the speed of light, where quantum and effects are important. So here, it simplifies life immensely to take $c = G = \hbar = 1$.

(It often simplifies life more, in 4 dimensions, to take $8 \pi G = 1$. But, I wanted to make it clear that we’re talking about units where $c = C = \hbar = 1$.)

For other problems, other units are handy. This is why chemists like angstroms, planetary physicists like astronomical units, astronomers studying stars in our galaxy like parsecs, and cosmologists like megaparsecs. What’s big and what’s small depends on what you’re interested in!

(Borcherd’s complaints about Planck units were completely different, and much more interesting: he points out that in a theory of quantum gravity we expect G to be a running coupling constant, so it’s observed value at low energies is not a fundamental thing!)

Okay, thanks for the clarification! I’m too wimpy to check these calculations right now, but maybe someday — it’s nice at least knowing the precise issue at stake.

David wrote:

They’re categorifying Hecke algebras over at the Secret Blogging Seminar, using Soergel bimodules.

Yes, I saw that. Interesting! But we’re gonna do it in a much more conceptual, fun manner. And, our construction will not only be ‘sorgenfrei’, it’ll be ‘soergelfrei’.

One more comment on why these CFTs have just a single rep:

The partition function of a rational 2d CFT, i.e. the number $$Z(\tau,\bar \tau)$$ which it assigns to the conformal torus $$T = \mathbb{C}/(\mathbb{Z}\times \tau \mathbb{Z})$$ of modulus $\tau$ is generally of the form $$Z = \sum_{i,j} Z_{i j} \; \chi_i(\tau) \; \chi_j( \tau)^* \,,$$ where the sum is over all irreps of the chiral vertex operator algebra, where $Z_{i,j}$ are some coefficients, and where the “characters” $\chi_i$ are the trace of the conformal propagator $\exp(- \tau L)$ $$\chi_i(\tau) = \mathrm{Tr}_i ( \exp(- \tau L))$$ in the $i$-th irrep of the chiral algebra.

Some people call the existence of this formula already “holomorphic factorization”, since it expresses the partition function as a sum of products of a holomorphic and an anti-holomorphic function of the complex parameter $\tau$.

But more strictly, and that’s how Witten is using it in his paper, one might say that we have holomorphic factorization if the partition function is a product of one holomorphic with one anti-holomorphic function. $$Z = f(\tau) h(\bar \tau) \,.$$

So this happens if (I think if and only if) there is just a single irrep of the chiral algebra, since then the above formula collapses to $$Z = Z_{1 1} \; \chi_1(\tau) \; \chi_1(\tau)^* \,.$$

So it’s the assumption of holomorphic factorization which makes Witten arrive at this famous family of rational CFTs with central charge $c = 24 k$.