The n-Category Café

Sorry, my typo. I meant 10^9 neurons in your gut. I recently discussed this at length with Christoph Koch, at Caltech.

When I was in undergrad Bio (Caltech 1968-1973) the textbooks said that there were roughly 10^10 (ten billion) neurons in the human brain. When I was doing my M.S. and PhD work in grad school (1973-1977) the textbooks said: “Of the 10 billion neurons in your brain, 100 billion are Glial Cells.”

To give a popularized citation: 10^9 neurons (especially Interstitial Cells of Cajal) in the plexus distributed between the two muscle layers in your stomach, small intestine, and large intestine, that being 1% as many neurons as in your brain, and many more than your spinal cord? Some people are intellectuals but only at the gut level. “Our understanding of virtue and vice, success and failure, has long been expressed in the language
of appetite, consumption, and digestion” [‘A Short History of the American Stomach’ by Frederick Kaufman, Harcourt, 2008, 206 pp.]

I can email a draft paper related to this to anyone who wants to be distracted.

The simple sine-Gordon model I was talking about is a model where ‘space’ (the ribbon) is 1-dimensional. It’s the simplest, most intuitively obvious theory with topological solitons.

You should really think of the sine-Gordon model as a theory where space is compactified to a circle and the field takes values in a circle. In other words, take a ribbon and attach the ends together. Then at any time the field is a continuous function

$$\phi : S^1 \to S^1$$

and the number of twists in your ribbon (usually called the ‘kink number’ in this game) is just the winding number. It’s a ‘topologically conserved charge’ — an integer that’s conserved for topological reasons. The word ‘charge’ is physics jargon here; if I were explaining this to kids I’d just call it the ‘number of particles’ or ‘number of twists’.

There are various generalizations; one of the most exciting is where space becomes 3-dimensional, and the field takes values in a 3-sphere. Again we compactify space, so at any time the field is a function

$$\phi : S^3 \to S^3$$

and the conserved charge is the ‘winding number’ arising from $\pi_3(S^3) = \mathbb{Z}$.

Of course this works with any number, not just 1 or 3. But the fun thing about 3 is that we live in 3 dimensional space, $S^3 \cong SU(2)$, and $SU(2)$ acts as a symmetry group in the simplest old-fashioned theory of protons, neutrons and pions! We now think of this $SU(2)$ as the approximate symmetry that acts on the space $\mathbb{C}^2$ whose basis vectors are the up and down quark. Protons, neutrons and pions are made of just these two quarks — the lightest quarks.

Anyway, there turns out to be a fairly successful — but phenomenological, not fundamental — model of protons, neutrons and pions in which the basic field at any moment in time is a map

$$\phi : S^3 \to S^3$$

It’s called the Skyrme model. The protons and neutrons are topological solitons in this model, just like twists in that ribbon!

So, you can tell the kids that there’s a theory where protons and neutrons are like twists in a ribbon… but try not to fool them into thinking this theory is ‘true’: it’s just pretty good for certain purposes.

Witten revived interest in the Skyrme model back when he was just starting to be Witten. He noticed that the fact $\pi_4(S^3) = \mathbb{Z}/2$ can be used to explain why protons and neutrons are fermions: the process of switching two of them gives a map $\phi : S^4 \to S^3$ that corresponds to the nontrivial element of $\mathbb{Z}/2$, and by adding a ‘Wess–Zumino–Witten term’ to the action of the Skyrme model, you can make this process give rise to a phase of $-1$ for purely topological reasons. Switching two fermions is supposed to give a phase of $-1$, you see.

Anyway, this is just the beginning of a long story, but probably more than enough for those students.

If you can dig up Witten’s original paper on this, it’s a lot clearer than anything I was easily able to find using Google. This paper probably dates back to the 70’s or 80’s, back when physicists were just starting to use homotopy groups.