July 13, 2007

This Week’s Finds in Mathematical Physics (Week 254)

Posted by John Baez

In week254, learn about Witten’s new paper on 3d quantum gravity and the Monster group, mysterious relations between exceptional Lie superalgebras and the Standard Model of particle physics…

… and continue reading the Tale of Groupoidification.

Posted at July 13, 2007 12:45 PM UTC

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Re: This Week’s Finds in Mathematical Physics (Week 254)

In case people missed it, let me repeat my recommendation of Helgason’s (free!) book on the Radon Transform.

Posted by: David Corfield on July 13, 2007 1:24 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

…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…

But maybe that doesn’t count as ‘literature’.

Posted by: David Corfield on July 13, 2007 1:56 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

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!

Posted by: John Baez on July 13, 2007 4:01 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

John Baez wrote:

“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’.

[…]

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!

OK, admittedly this is not my field, so I don’t know references, but I thought it was a “standard” thing that the Hecke algebra associated to a (finite or affine) Weyl group (the q-deformation of the group algebra that you mention, aka the quotient of the braid group) is the same thing as a convolution algebra of bi-invariant functions on a (finite Chevalley or p-adic) group.

Although, admittedly, I think I learned this from a representation-theorist, not from a book.

Posted by: Thomas Nevins on July 13, 2007 7:32 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

Thomas Nevins wrote:

I thought it was a “standard” thing that the Hecke algebra associated to a (finite or affine) Weyl group (the $q$-deformation of the group algebra that you mention, aka the quotient of the braid group) is the same thing as a convolution algebra of bi-invariant functions on a (finite Chevalley or $p$-adic) group.

Yes, it is! But how many people, reading this, will think Hecke operator, and spot the analogy to the Hecke operators showing up in the theory of modular forms?

Not enough…

Did Hecke himself notice it, or not?

Posted by: John Baez on July 13, 2007 7:41 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

Hi John! Thanks for the great post.

I agree there is a cultural gap in the appreciation of what Hecke algebras are… a friend of mine in grad school, now a distinguished number theorist, was studying the “classical” Hecke operators and I was studying the others and for a long time we thought they were unrelated!

However, this picture is (as we learned) the standard picture in representation theory — and whether or not Hecke himself knew it, by the ’60s it was standard — it is part of the very definition of the Langlands program, see e.g. the Corvallis volumes. (And I imagine many more number theory grad students know this today than
a decade ago, thanks to the overwhelming successes of the Langlands program since Wiles, Taylor, etc.)

Namely for any group G and a subgroup K we can look at intertwining operators from the induced rep C[G/K] to itself. By Frobenius reciprocity this algebra is the endomorphisms of the functor of K-invariants. Applying it back to the induced rep we find it is the same as K\G/K, the subalgebra of the group algebra of bi-invariant functions. (As for what I mean by functions there are many candidates of a “function theory” depending on what kind of group we’re discussing, but the formal structure is always the same.)

The commutative Hecke algebras acting on modular forms are just this construction for K=the max compact of G= a p-adic group (well SL2(Qp) in the usual setting) — the commutativity is the theorem of Satake which Langlands reinterpreted and used to define the Langlands correspondence. Indeed irreducible representations of p-adic groups that have a K-fixed vector are the same as irreducible modules for this commutative Hecke algebra, and this is precisely how one assigns L-functions to automorphic representations.

For higher level modular forms, say for Γ0(p), we immediately run into the Iwahori–Hecke algebra, where we replace K by the Iwahori subgroup, version of a Borel. This is the one that relates to the braid group, as Iwahori–Matsumoto in the ’60s or ’70s already understood (they give its usual presentation, see also Borel’s ’70s Inventiones paper on Iwahori–Hecke algebras). In general in Langlands one uses such algebras for smaller and smaller K, ie more and more complicated Hecke algebras. But they are always understood as intertwiners of induced reps, or endomorphisms of spaces of K-invariants on general reps, or algebras of double cosets.

Of course we don’t need to go to p-adic groups to see this — if we look at any standard book on reps of finite groups of Lie type (e.g. by Carter or Digne–Michel etc) we’ll soon run into the finite Hecke algebra, C[B\G/B] where G=say GLn(Fq) and B = upper triangulars — it is presented as endomorphisms of the induced rep C[G/B], used to define unipotent reps of G, and calculated to be the q-deformation of group algebra of the Weyl group that is familiar from say quantum-groupy things.

As for the categorified version of this story, it is the foundation of the geometric Langlands program, and it is beautifully (though somewhat dauntingly) presented for example in the chapter Hecke Patterns (which should be a book of its own) in Beilinson–Drinfeld’s text on Quantization of Hitchin Hamiltonians. They explain
a very general version of the above story with spaces of functions replaced by categories of sheaves. For example all of the activity on actions of braid groups on categories and Khovanov homology etc. can be traced back to the categorification of the finite Hecke algebra that has been studied since Kazhdan–Lusztig, namely the category of sheaves on B\G/B. Again this category acts as intertwiners of the G-action on sheaves on G/B or various variants (e.g. category O, Harish–Chandra bimodules, etc. etc.) and gives rise to braid group actions on these categories. The way these algebras arise in geometric Langlands is precisely by correspondences (spans?) as you describe. I think this should be explained somewhere in Frenkel’s expository writing on geometric Langlands, but I’m not sure of the exact reference.

[So I guess I fail your reference challenge…sorry!]

Posted by: David Ben-Zvi on July 13, 2007 8:31 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

David Ben-Zvi wrote:

I agree there is a cultural gap in the appreciation of what Hecke algebras are — a friend of mine in grad school, now a distinguished number theorist, was studying the “classical” Hecke operators and I was studying the others and for a long time we thought they were unrelated!

Interesting. I’m mildly relieved that it’s not just Jim who had to discover these facts of life on his own. But, it’s basically just sad that more textbooks don’t start with a general introduction to Hecke operators, Hecke algebras, Radon transforms, etc., showing they’re all different faces of the same thing!

Namely for any group $G$ and a subgroup $K$ we can look at intertwining operators from the induced rep $C\left[G/K\right]$ to itself. By Frobenius reciprocity this algebra is the endomorphisms of the functor of $K$-invariants. Applying it back to the induced rep we find it is the same as $K$\$G/K$, the subalgebra of the group algebra of bi-invariant functions.

Yes, this is about the level of generality at which Jim and Todd and I like to work, since this is the level of generality at which it’s easiest to groupoidify the construction you just described.

I say “about” because we prefer the case where $G$ is finite, to avoid technicalities — and also because it’s good to consider a pair of subgroups $H,K\subseteq G$ and build intertwining operators $C\left[G/H\right]\to \left[G/K\right]$ from double cosets in $H$\$G/K$… and to groupoidify this construction.

if we look at any standard book on reps of finite groups of Lie type (e.g. by Carter or Digne–Michel etc) we’ll soon run into the finite Hecke algebra, $C\left[B$\$G/B\right]$ where $G=$ say ${\mathrm{GL}}_{n}\left({F}_{q}\right)$ and $B=$ upper triangulars — it is presented as endomorphisms of the induced rep $C\left[G/B\right]$, used to define unipotent reps of $G$, and calculated to be the $q$-deformation of group algebra of the Weyl group that is familiar from say quantum-groupy things.

This is the example Jim and Todd and I have been thinking about for a long time, and I’ll soon start discussing. I love this stuff, since I knew about quantum groups already, and this seems to ‘explain’ some aspects of them.

I really wish I understood the other examples you mention, related to the Langlands program. Indeed I wish I could download your whole brain! But I seem condemned to slowly struggle at learning many of the things you’re so fluently discussing here.

Posted by: John Baez on July 15, 2007 11:41 AM | Permalink | Reply to this

Witten numbers on OEIS; Re: This Week’s Finds in Mathematical Physics (Week 254)

A122505 Arises from energy spectrum of three dimensional gravity with negative cosmological constant, in analysis by Edward Witten.

An anonymous commenter on “Not Even Wrong” suggested several more terms in this integer sequence, which I have not yet been able to verify.

Posted by: Jonathan Vos Post on July 13, 2007 4:24 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

Here are some questions Christine Dantas raised on Physicsforums, and some attempted answers. Since they’re a bit technical, I prefer to discuss them here, where I think experts are more likely to help us out.

Christine wrote:

Some basic questions. Why is k “an integer for topological reasons”? (k is a parameter that appears in a second term — a multiple of the Chern–Simons invariant of the spin connection — added to the action).

The Chern–Simons action $S$ is invariant under small gauge transformations (those connected to the identity by a continuous path), but changes by multiples of a certain constant $C$ under large gauge transformations. What shows up in path integrals is the exponentiated action $\mathrm{exp}\left(ikS\right)$ where $k$ is some coupling constant. The consequence is clear: $\mathrm{exp}\left(ikS\right)$ remains unchanged under large gauge transformations if and only if $\mathrm{exp}\left(ikC\right)=1$, meaning that $k$ has to be an integer multiple of $2\pi /C$.

If you set up all your normalization conventions nicely, $C=2\pi$, so $k$ has to be an integer.

This stuff is explained a bit more in my book Gauge Fields, Knots and Gravity, in section II.4, Chern–Simons Theory. Also see the end of section II.5.

In 3d quantum gravity, the consequence is that the cosmological constant can only take certain discrete values!!!

Further, what is “holomorphic” factorization? (A pointer to the basic literature on this will suffice).

I don’t really understand that term. It should be defined in Schelleken’s paper — this paper speaks of “meromorphic conformal field theories” instead of “conformal field theories with holomorphic factorization”, but they must be the same thing. However, I’m having a bit of trouble finding the precise definition! I just know a bunch of properties of these theories.

First, the central charge $c$ is an integer multiple of 24.

Second, as a consequence, the partition function is really a well-defined number, not just defined up to $\left(24/c\right)$th roots of unity. In other words, it’s “modular invariant”.

These two are very important in Witten’s paper.

Third, as another consequence, the Schwinger functions, otherwise known as “$n$-point functions”, are all well-defined meromorphic functions — that is, holomorphic except for poles. This is not so important in Witten’s paper, though.

Is it the only possible constraint?

Witten gives an argument that 3d quantum gravity has as its AdS/CFT dual a conformal field theory with $c=24k$ for some integer $k=1,2,3,...$ The main nice thing is that — modulo a certain big conjecture — Schellekens classified these conformal field theories for $k=1$.

He argues that the (naive) partition function ${Z}_{0}\left(q\right)$ differs from the “exact” $Z\left(q\right)$ by terms of order $O\left(q\right)$. Would this be correct for any $k$?

Yes, he argues this is true for any $k$. Then, around equation (3.13), he shows that this property, together with modular invariance of the exact partition function, completely determines the exact partition function! It’s a certain explicit polynomial in the $J$ function (which is the $j$ function minus 744).

He finds that for $k=1$ the Monster group is interpreted as the symmetry of 2+1-dimensional black holes. How sensitive is this result with respect to the value of $k$, and to respect to the other assuptions used in the derivation?

For $k=1$ he goes through Schelleken’s list of 71 conformal field theories with $c=24$ and picks the one that has the Monster group as its symmetries. He gives an argument for why this one is the right one, but it’s not airtight. I mention some problems in week254, and Jacques Distler has mentioned some more.

Witten doesn’t actually find the relevant conformal field theories with $c=24k$ for higher values of $k$. He just figures out their supposed partition functions. Since the coefficients of their partition functions are — just as in the $k=1$ case — dimensions of representations of the Monster group, it seems awfully plausible that these theories (if they really exist!) have the Monster group as symmetries.

However, this is something one would want to check. Nobody seems to know a $c=48$ theory with Monster group symmetries, for example.

Posted by: John Baez on July 13, 2007 6:20 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

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

Posted by: Christine Dantas on July 14, 2007 12:48 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

Does this new paper disprove Witten’s conjecture?

Re: This Week’s Finds in Mathematical Physics (Week 254)

I know there are known relations between exceptional objects, but is there any other theory outside of model theory which goes some way to explain systematically the relationship between classifications in terms of families and sporadic entities? I was speaking recently with Alexander Borovik of Mathematics Under the Microscope, and he mentioned some work along these lines of the model theorists, Lachlan and Cherlin.

Here’s the opening paragraph of a paper by Cherlin – Sporadic homogeneous structures:

When classification results are enlivened by the appearance of uninvited guests in the form of “sporadic” objects, those who take an interest in these interlopers may be tempted to account for them in various ways, possibly by viewing them as coming from infinite (perhaps even continuous) families of more general objects which may be natural from some broader point of view. In pure model theory, Lachlan’s classification theory for finite homogeneous relational structures provides a relatively well understood illustration (or “toy model”, if you will) of this sort of thing. This theory, which will be reviewed below, provides an infinite number of classification theorems of a general character for combinatorial structures with rich automorphism groups, parametrized by certain bounds on the complexity of the structures. Any finite structure will actually appear at some stage in one of these classifications, and may well occur as a sporadic structure initially; in the long run, every sporadic structure winds up belonging to a family parametrized by numerical invariants; at any given stage, only finitely many structures occur as sporadics; and finally, one will never “move beyond” the sporadics: we will always encounter new structures making their appearance as (temporarily) sporadic structures.

Posted by: David Corfield on July 13, 2007 7:38 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

Dear Professor Baez

What is a good introduction to conformal field theory? It seems like it’s an area where many interesting areas of mathematics and physics meet!

Are there any references (especially online) that you would recommend? I am a complete beginner (an undergraduate mathematician).

Posted by: Dean on July 14, 2007 1:24 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

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.

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.

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:

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

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

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.

Posted by: John Baez on July 14, 2007 8:35 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

Ah, I see now that you point to Gaberdiel’s paper (for an explanation about holomorphic factorization). Thanks!

Christine

Posted by: Christine Dantas on July 14, 2007 12:52 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

What is a good introduction to conformal field theory?

I was on vacation when this dscussion here was going on. Otherwise I would have said that, to my mind, a very good place to start when wondering about what CFT (and actually QFT in general, too) is is introductory texts by Ingo Runkel, like this and then after that this. And if one still feels the need of more details, try this.

Posted by: Urs Schreiber on August 17, 2007 2:06 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

What is a good introduction to conformal field theory?

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

Re: This Week’s Finds in Mathematical Physics (Week 254)

Thanks for the reference, Thomas! But what I’m really hoping is that you’ll comment on this statement in week254:

Alas, E(3|8) gets the hypercharges of some fermions wrong. Larsson seems to say this problem also occurs for E(3|6), which would appear to contradict what Kac claims — but I could be misunderstanding.

Posted by: John Baez on July 18, 2007 9:39 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

You write:

Using hyperplanes is nicer, because it gives a very simple relationship between the Radon transform and the Fourier transform. But never mind – that’s not the point here!

Presumably, you were alluding to

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

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

Oh dear, I’ve probably laid the basis for some awful puns. Who examines whether linear algebra can be groupoidified? The Spanish Inquisition.

Posted by: David Corfield on July 16, 2007 12:17 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

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

$f:{ℝ}^{3}\to ℂ$

tells you how to express it in terms of plane waves. To see how much the plane wave $\mathrm{exp}\left(ik\left(x\right)\right)$ shows up in $f$, you need to take their inner product, which means integrating $f\left(x\right)$ times the complex conjugate of $\mathrm{exp}\left(ik\left(x\right)\right)$ 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 $\mathrm{exp}\left(ik\cdot x\right)$ 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 $x\in {ℝ}^{3}$ and a plane wave with momentum $k\in \left({ℝ}^{3}{\right)}^{*}$ 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 $\mathrm{exp}\left(ik\left(x\right)\right)$ 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!

Posted by: John Baez on July 17, 2007 9:27 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

Would this composition be allowable because the matrix for the Radon transform is seen as having ‘phase’ values? Hmm, that doesn’t seem right. But there must be some kind of compatibility between the rigs appearing in the matrices to be composed.

In this case I guess they’re both $ℂ$-valued.

Maybe more compatibility issues will open up with composition involving more general Fourier-like situations, e.g., matrix-valued matrix entries of the group element and representation relation.

Posted by: David Corfield on July 17, 2007 10:56 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

Hmm, that doesn’t seem right.

Right, that ain’t right.

In this case I guess they’re both $ℂ$-valued.

Right, in normal math all these kernels are just $ℂ$-valued. You’re goading me to replace $ℂ$ by a category, and I’m hoping the category of ‘phased sets’ will do the job. A one-element phased set is just a phase…

… and that’s precisely what the plane wave $\mathrm{exp}\left(ik\left(x\right)\right)$ hands us when we give it a position $x\in V$ and a momentum $k\in {V}^{*}$.

(For those how haven’t been keeping score: a ‘phased set’ is a set, each of whose elements is labelled by a unit complex number. Jeffrey Morton showed that the combinatorics of Feynman diagrams for perturbed harmonic oscillators can be nicely understood using spans of phased groupoids, so it’s tempting to do something similar for the Fourier transform. Once we’ve got the Fourier transform and the harmonic oscillator firmly in hand, the rest of physics is sure to follow. )

Posted by: John Baez on July 17, 2007 12:28 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

John Baez writes:

- within 2) Edward Witten …
“By the way: I’m working in Planck units here …”

- Yet refers in 4) “Richard Borcherds, online papers …”

On his blog ‘Mathematics and physics’, Borcherds had this post 16 May 2007, “Planck units (uselessness of).”

I tend to agree with Borcherds, but I am not as elegant.

- speed of light c may be closer to infinity than 1

- Planck’s constant h and the gravitational constant G are closer to 0 than 1, yet are so far apart in magnitude that it almost becomes ‘surreal’.

Is Borcherds correct or mistaken?

Posted by: Doug on July 17, 2007 4:08 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

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 $\hslash$ 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 $c=G=\hslash =1$.

(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 $c=C=\hslash =1$.)

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!)

Posted by: John Baez on July 17, 2007 8:33 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

Thank you for taking the time to respond to my question.

I certainly agree that Borcherds is more correct than I.

However, I still do not understand how physics allows for the condition:

c=G=h=1, since:

a - is this not a comparison of unity identities?
I can understand unity identites [I_x] such as I_c, I_G and I_h, but how can they be set equal to the anthropic I_1 when c is of power 10+, while G and h are of power 10-.

b - does c=1=(1/c). 1/c is at least on the same power side of a unity identity.
Is this a paradox that either c or 1/c could be used, apparently?

Posted by: Doug on July 17, 2007 4:24 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

Doug, how does physics allow for a foot to be 12 inches and also a troy pound to be 12 troy ounces? Isn’t saying both conversion factors are 12 comparing those constants?

When you realize what’s silly about that argument, you’ll know what’s wrong with your question (a).

Then apply the same reasoning to (b): why aren’t c and 1/c the same thing, even if c gets the numerical value 1?

Posted by: John Armstrong on July 17, 2007 5:09 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

Doug - I feel your pain. I remember when I entered physics grad school, and I first encountered the equation

(1)$c=1.$

I was completely baffled for a long time! I think I had a similar problem to you. My paradox went like this : if $c=1$ then we can prove that $1=2$! Because if $c=1$, then

(2)$299792458=1.$

Let $x$ be the real number such that

(3)$\left(299792458{\right)}^{x}=2.$

Thus

(4)$c=1⇔{c}^{x}={1}^{x}⇔2=1.$
Posted by: Bruce Bartlett on July 17, 2007 5:30 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

I like how you felt the need to deduce 2 = 1 to demonstrate the contradiction. Merely having $299792458=1$ was not shocking enough.

This reminds me of Bertrand Russell, who showed how to deduce a contradiction from the statement “I am the pope”, as follows: “The pope and I are two; if I am the pope, then the pope and I are one, therefore 2 = 1.”

(By the way, this is one of the few proofs I know that’s valid except when given by the pope.)

Posted by: John Baez on July 18, 2007 8:21 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

I had heard a slightly different version of this story:

Bertrand Russell told someone that if you assume something false like 1+1=1, then you can prove anything. The other person challenged him to prove that he was the pope, so he replied, “I am one. The pope is one. Therefore the pope and I are one.”

Posted by: Scott Carnahan on July 18, 2007 4:23 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

Posted by: John Baez on July 18, 2007 9:04 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

Seriously though, here’s why I sympathize with Doug’s issues with setting $h=c=G=1$ (here $h$ should be read “h bar”). By the way, the mere fact that I had/have trouble properly understanding this shows why I was never cut out to be a physicist!

When I first encountered it, I informed my physics professors of my confusion, and they frowned, mumbled something about “choosing to measure things in terms of the speed of light”, and pointed me to some physics texts.

When I looked up the said physics texts, I found that they took the equation “h = c = G =1” quite seriously, and they tried to give a physical rationale for it, along the lines of “we are always free to change units from the meter stick in Paris to the length of time taken for the hydrogen atom to make a transition”.

This made me more confused! I tried to imagine explaining to my mother that what she thought was a meter was really the square root of a second per kg!! To see this observe that if h = 1 then

(1)$6.626068x{10}^{34}\left[{m}^{2}\mathrm{kg}/s\right]=1\left[\mathrm{dimensionless}\right]$

thus we must have

(2)$\left[m\right]=\sqrt{\left[s\right]/\left[\mathrm{kg}\right]}$

Gulp! Lol.

Nowadays I understand this more pragmatically as follows : we’re not setting anything equal to anything else. We’re just banking on the fact that we will always know beforehand what the units of a physical calculation should come out as. Since [c], [G] and [h] are independent functions of the fundamental units (m, kg, s), we can then uniquely insert the appropriate factors of c, G and h back into the result of our calculation, in order to come out with the correct units.

But on a philosophical level, that’s doesn’t seem the same to me as “setting $h=G=c=1$”.

Anyhow, it’s clear that after all these years I still don’t understand it !

Posted by: Bruce Bartlett on July 18, 2007 6:07 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

When I first encountered it, I informed my physics professors of my confusion, and they frowned, mumbled something about “choosing to measure things in terms of the speed of light”, and pointed me to some physics texts.

Luckily now the $n$-Café exists to help solve all such mysteries:

As I guess every regular reader recalls we had a long and detailed discussion of these issues in Dimensional Analysis and Coordinate Systems.

The punchline is: a choice of unit is a choice of isomorphism.

For instance a choice of unit of length is a choice of isomorphism of any line in the real world with the real numbers.

So the issue with units is precisely the one as that with isomorphisms, in general: we need not single out any single one, and indeed every such choice is arbitrary. But if we want we can single out one.

Posted by: Urs Schreiber on July 21, 2007 2:25 PM | Permalink | Reply to this

E(3|6)

Alas, E(3|8) gets the hypercharges of some fermions wrong. Larsson seems to say this problem also occurs for E(3|6), which would appear to contradict what Kac claims, but I could be misunderstanding.

On page 17-18 of math.QA/9912235, we have

Theorem 3. [KR] The complete list of irreducible degenerate E(3|6)-modules is as follows (m, b in Z+):
(0m, b,-b-2/3m-2), (0m, b, b-2/3m), (m0, b,-b + 2/3m), (m0, b, b+2/3m+2).

In particular, the list contains (01, 0, Y), where Y = -8/3, Y = -2/3.

If we now look in Table 1 on page 19, we see that the second value fits d_R, but Kac uses the value Y = 4/3 for u_R, which is physically correct but does not fit in the list in theorem 3. Similarly, (10, 0, Y) is in the list if Y = 2/3, Y = 8/3, whereas \tilde u_L has Y = -4/3.

Re: E(3|6)

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.

Posted by: John Baez on July 18, 2007 12:10 PM | Permalink | Reply to this
Weblog: The n-Category Café
Excerpt: The last session of Recent Developments in QFT in Leipzig was general discussion, which happened to be quite interesting for various reasons....
Tracked: July 22, 2007 7:30 PM

Re: This Week’s Finds in Mathematical Physics (Week 254)

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

This post gets very close to our interests too.

Posted by: David Corfield on July 25, 2007 2:00 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

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’.

Posted by: John Baez on July 25, 2007 6:12 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

So: is the monstrous conformal field theory known to be an RCFT? If so, what 3d TQFT does it give? Could this TQFT be the 3d quantum gravity theory Witten is seeking?

As far as I know, all these 71 theories of central charge 24 are rational. Even better, they are special in that they have not only finitely many isomorphism classes of irreps for the chiral algebra – but just a single one!

I can maybe provide better references later on, but right now, being a bit of in a hurry, I can point you to

See the paragraph on top of p. 11.

Actually, I have this paper here because I wanted to ask you all another question: this paper mentions a famous little formula which crucially involves the groupoid cardinality of the groupoid of lattices of a given “lattice genus”.

I bet this groupoid cardinality here must come from something like a finite path integral, as in Dijkgraaf-Witten thery. But it’s not entirely clear here in which context that would be.

Posted by: Urs Schreiber on July 27, 2007 11:06 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

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

The partition function of a rational 2d CFT, i.e. the number $Z\left(\tau ,\overline{\tau }\right)$ which it assigns to the conformal torus $T=ℂ/\left(ℤ×\tau ℤ\right)$ of modulus $\tau$ is generally of the form $Z=\sum _{i,j}{Z}_{ij}\phantom{\rule{thickmathspace}{0ex}}{\chi }_{i}\left(\tau \right)\phantom{\rule{thickmathspace}{0ex}}{\chi }_{j}\left(\tau {\right)}^{*}\phantom{\rule{thinmathspace}{0ex}},$ 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 $\mathrm{exp}\left(-\tau L\right)$ ${\chi }_{i}\left(\tau \right)={\mathrm{Tr}}_{i}\left(\mathrm{exp}\left(-\tau L\right)\right)$ 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. $Z=f\left(\tau \right)h\left(\overline{\tau }\right)\phantom{\rule{thinmathspace}{0ex}}.$

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 $Z={Z}_{11}\phantom{\rule{thickmathspace}{0ex}}{\chi }_{1}\left(\tau \right)\phantom{\rule{thickmathspace}{0ex}}{\chi }_{1}\left(\tau {\right)}^{*}\phantom{\rule{thinmathspace}{0ex}}.$

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

Posted by: Urs Schreiber on August 1, 2007 9:22 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 254)

I notice that what I said above (I, II) might look like being in contradiction to what Witten is talking about. It is not. I should maybe clarify:

what is rational – in that it even has just a single irrep, up to isomorphism – is the “fully extended” chiral algebra of those $\left(c=24k\right)$-theories. Meaning, it’s the chiral algebra which contains not just the Virasoro algebra, but also all the “currents”, i.e. all the generators of symmetries which are around.

This big chiral algebra has just a single irrep. But that single rep decomposes into plenty of irreps of the Virasoro algebra, which sits as a subalgebra inside the full thing.

It’s the dimensions of these Virasoro reps which, as far as I understand this matter, Witten is talking about.

By the way, the fact that there is just a single irrep of the full algebra means that its modular representation category is in fact isomorphic to $\mathrm{Vect}$! This means that the Reshitikhin-Turaev 3d TFT built from this is more or less trivial.

How can that be? Where is all the interesting physical information?

The thing is this: by the FRS construction the modular tensor category $C$ together with the 3d TFT it gives rise to allows to construct a 2-dimensional rational CFT only up to the information contained in a certain isomorphism, which identifies the abstract vectors that the TFT assigns to a given worldsheet (such that the sewing constraints of a 2d QFT are satisfied) with the actual function which assigns a number, the correlator, to each complex structure on that surface.

The latter is sometimes referred to as the “complex analytic” information, while the former is the “combinatorial topological one”.

In the case of the holomorphic $\left(c=24k\right)$ CFTS, the entire information about the CFT hence sits in that “complex analytic” part.

Posted by: Urs Schreiber on August 6, 2007 2:55 PM | Permalink | Reply to this

Hector Blandin, Rafael Diaz; Re: This Week’s Finds in Mathematical Physics (Week 254)

Regrading the Tale of Groupoidification, see the wonderful paper, which references John Baez very early:

http://arxiv.org/pdf/0708.0809
Hector Blandin, Rafael Diaz, Compositional Bernoulli numbers, 6 Aug 2007.

I added 14 sequences to the OEIS from tables in that paper, to kick things off there:

A132092 Numerators of Blandin-Diaz compositional Bernoulli numbers (B^sin)_3,n.
http://www.research.att.com/~njas/sequences/A132092

through

A133005 Denominators of Blandin-Diaz compositional Bernoulli numbers C_2,n.
http://www.research.att.com/~njas/sequences/A133005

This is very pretty, tying together Species, combinatorics, formal power series, and so much more.

Posted by: Jonathan Vos Post on August 10, 2007 7:31 PM | Permalink | Reply to this
Read the post The Canonical 1-Particle, Part II
Weblog: The n-Category Café
Excerpt: More on the canonical quantization of the charged n-particle for the case of a 1-particle propagating on a lattice.
Tracked: August 15, 2007 11:44 AM