The n-Category Café

When you try to quantize 10-dimensional supergravity theories, you are led to some theories involving strings. These are fairly well understood, because the worldsheet of a string is 2-dimensional, so string theories can be studied using 2-dimensional conformal quantum field theories, which are mathematically tractable.

When you try to quantize 11-dimensional supergravity, you are led to a theory involving 2-branes and 5-branes. People call it M-theory, because while it seems to have magical properties, our understanding of it is still murky — because it involves these higher-dimensional membranes. They have 3- and 6-dimensional worldsheets, respectively. So, precisely formulating M-theory seems to require understanding certain quantum field theories in 3 and 6 dimensions. These are bound to be tougher than 2d quantum field theories… tougher to make mathematically rigorous, for example… but even worse, until recently people didn’t know what either of these theories were!

In 2008, Aharony, Bergman, Jafferis and Maldacena figured out the 3-dimensional theory: it’s a supersymmetric Chern–Simons theory coupled to matter in a way that makes it no longer a topological quantum field theory, but still conformally invariant. It’s now called the ABJM theory. This discovery led to the ‘M2-brane mini-revolution’, as various puzzles about M-theory got solved.

The 6-dimensional theory has been much more elusive. It’s called the (0,2) theory. It should be a 6-dimensional conformal quantum field theory. But its curious properties got people thinking that it couldn’t arise from any Lagrangian — a serious roadblock, given how physicists normally like to study quantum field theories. But people have continued avidly seeking it, and not just for its role in a potential ‘theory of everything’. Witten and others have shown that if it existed, it would shed new light on Khovanov duality and geometric Langlands correspondence! The best introduction is here:

• Edward Witten, Geometric Langlands from six dimensions, 2009.

Around 2008-9 we had several exchanges with Minhyong Kim here at the Café, in particular on his views of approaching number theory from a homotopic perspective, in particular in the post Kim on Fundamental Groups in Number Theory. (See also the threads Afternoon Fishing and The Elusive Proteus.)

I even recall proposing a polymath project based on his ideas in Galois Theory in Two Variables. Something physics-like was in the air, and this seemed a good location with two mathematical physicists as hosts, John having extensively written on number theory in This Week’s Finds.

Nothing came of that, but it’s interesting to see Minhyong is very much in the news these days, including in a popular article in Quanta magazine, Secret Link Uncovered Between Pure Math and Physics.

The Quanta article has Minhyong saying:

“I was hiding it because for many years I was somewhat embarrassed by the physics connection,” he said. “Number theorists are a pretty tough-minded group of people, and influences from physics sometimes make them more skeptical of the mathematics.”

Café readers had an earlier alert from an interview I conducted with Minhyong, reported in Minhyong Kim in The Reasoner. There he was prepared to announce

The work that occupies me most right now, arithmetic homotopy theory, concerns itself very much with arithmetic moduli spaces that are similar in nature and construction to moduli spaces of solutions to the Yang-Mills equation.

Now his articles are appearing bearing explicit names such as ‘Arithmetic Chern-Simons theory’ (I and II), and today, we have Arithmetic Gauge Theory: A Brief Introduction.

In the old days, when mathematics journals were all published on paper, there were hard budgetary constraints on the number of pages available in any issue, so long papers were naturally a much harder sell than short ones. But now that the primary means of dissemination of papers is electronic, this should no longer be the case. So journals that still impose draconian page constraints (I’m looking at you, CS conference proceedings), or reject papers because they are too long, are just a holdover from the past, an annoyance to be put up with until they die out.

At least, that’s what I used to believe.

I want to write a few posts (which means at least one!) on things I’ve done around SageMath, not necessarily about SageMath, but using that as a springboard. Here I’ll say how I used it to help visualization – for both the students and me! – in the differential geometry course I’ve been teaching this semester.

SageMath (formerly SAGE) is a computer algebra system like Mathematica, Maple and MATLAB. However, unlike those other systems, it doesn’t start with an ‘M’. More importantly though, it is an open source project which, amongst other things, provides a unified ‘front-end’ for many other pieces of open source mathematical software such as Maxima, PARI and GAP. Having been using Maple since I was a PhD student, I started to make the switch to SageMath a couple of years ago, which was not that easy as the biggest problem with SageMath is the lack of introductory material and documentation, although this is now definitely improving, see for instance the book Mathematical Computation with SageMath, originally available in French.

Now onto visualization, here is a static picture of a catenoid surface.

Beneath the fold I’ll explain two ways in which you can use SageMath to embed a rotatable model of the surface in a webpage. All being well, in the main body of the post you should be able to play with the catenoid yourself.

This is, in some sense, a follow-on from one of the first posts I wrote here on using blender for creating 3d models of surface diagrams, nearly eight years ago.

In the comments last time, a conversation got going about $p$-adic entropy. But here I’ll return to the original subject: entropy modulo $p$. I’ll answer the question:

Given a “probability distribution” mod $p$, that is, a tuple $$\pi = (\pi_1, \ldots, \pi_n) \in (\mathbb{Z}/p\mathbb{Z})^n$$ summing to $1$, what is the right definition of its entropy $$H_p(\pi) \in \mathbb{Z}/p\mathbb{Z}?$$

In 1995, the German geometer Friedrich Hirzebruch retired, and a private booklet was put together to mark the occasion. That booklet included a short note by Maxim Kontsevich entitled “The $1\tfrac{1}{2}$-logarithm”.

Kontsevich’s note didn’t become publicly available until five years later, when it was included as an appendix to a paper on polylogarithms by Philippe Elbaz-Vincent and Herbert Gangl. Towards the end, it contains the following provocative words:

Conclusion: If we have a random variable $\xi$ which takes finitely many values with all probabilities in $\mathbb{Q}$ then we can define not only the transcendental number $H(\xi)$ but also its “residues modulo $p$” for almost all primes $p$ !

Kontsevich’s note was very short and omitted many details. I’ll put some flesh on those bones, showing how to make sense of the sentence above, and much more.

Here’s a draft of a little thing I’m writing for the Newsletter of the London Mathematical Society. The regular icosahedron is connected to many ‘exceptional objects’ in mathematics, and here I describe two ways of using it to construct $\mathrm{E}_8$. One uses a subring of the quaternions called the ‘icosians’, while the other uses Du Val’s work on the resolution of Kleinian singularities. I leave it as a challenge to find the connection between these two constructions!

(Dedicated readers of this blog may recall that I was struggling with the second construction in July. David Speyer helped me a lot, but I got distracted by other work and the discussion fizzled. Now I’ve made more progress… but I’ve realized that the details would never fit in the Newsletter, so I’m afraid anyone interested will have to wait a bit longer.)

You can get a PDF version here:

• From the icosahedron to E₈.

But blogs are more fun.

An adjunction is a pair of functors $f:A\to B$ and $g:B\to A$ along with a natural isomorphism

$$A(a,g b) \cong B(f a,b).$$

Question 1: Do we get any interesting things if we replace “isomorphism” in this definition by something else?

• If we replace it by “function”, then the Yoneda lemma tells us we get just a natural transformation $f g \to 1_B$.
• If we replace it by “retraction” then we get a unit and counit, as in an adjunction, satisfying one triangle identity but not the other.
• If $A$ and $B$ are 2-categories and we replace it by “equivalence”, we get a biadjunction.
• If $A$ and $B$ are 2-categories and we replace it by “adjunction”, we get a sort of lax 2-adjunction (a.k.a. “local adjunction”)

Are there other examples?

Question 2: What if we do the same thing for multivariable adjunctions?

A two-variable adjunction is a triple of functors $f:A\times B\to C$ and $g:A^{op}\times C\to B$ and $h:B^{op}\times C\to A$ along with natural isomorphisms

$$C(f(a,b),c) \cong B(b,g(a,c)) \cong A(a,h(b,c)).$$

What does it mean to “replace ‘isomorphism’ by something else” here? It could mean different things, but one thing it might mean is to ask instead for a function

$$A(a,h(b,c)) \times B(b,g(a,c)) \to C(f(a,b),c).$$

Even more intriguingly, if $A,B,C$ are 2-categories, we could ask for an ordinary two-variable adjunction between these three hom-categories; this would give a certain notion of “lax two-variable 2-adjunction”. Question 2 is, are notions like this good for anything? Are there any natural examples?

Now, you may, instead, be wondering about

Question 3: In what sense is a function $A(a,h(b,c)) \times B(b,g(a,c)) \to C(f(a,b),c)$ a “replacement” for isomorphisms $C(f(a,b),c) \cong B(b,g(a,c)) \cong A(a,h(b,c))$?

But that question, I can answer; it has to do with comparing the Chu construction and the Dialectica construction.