### Mathematical Phantoms

#### Posted by John Baez

A ‘mathematical phantom’ is a mathematical object that doesn’t exist in a *literal* sense, but nonetheless acts as if it did, casting a spell on surrounding areas of mathematics. The most famous example is the field with one element. Another is Deligne’s *S _{t}*, the symmetric group on $t$ elements, where $t$ is

*not a natural number*. Yet another is G

_{3}, a phantom Lie group related to G

_{2}, the automorphism group of the octonions.

What’s *your* favorite mathematical phantom? My examples are all algebraic. Does only algebra have enough rigidity to create the patterns that summon up phantom objects? What about topology or combinatorics or analysis? Okay, G_{3} is really a creature from homotopy theory, but of a very algebraic sort.

Last night I met another phantom.

A while back David Corfield interrogated me about the mathematical phantom called $SL(2,\mathbb{O})$, which — if this made sense! — would consist of $2 \times 2$ matrices with determinant 1 having *octonion entries*. Since the octonions are nonassociative, any direct attempt to describe such a group runs into a brick wall, and yet there are many reasons to *want* such a group, to complete various patterns in mathematics. A lot of people say $SL(2,\mathbb{O})$ should be the group $Spin(9,1)$. See for example this, and the many references therein:

- Tevian Dray, John Huerta and Joshua Kincaid, The magic square of Lie groups: the 2×2 case.

This makes a lot of things work. But is it the final answer?

Going slightly crazy from the boredom of coronavirus-induced lockdown, I’ve been browsing the arXiv while watching TV with my wife at night. I’m not sure that’s a good idea, but it’s more fun than social media. Last night I ran into this:

- Nigel Hitchin, SL(2) over the octonions.

He’s unsatisfied with usual idea that $SL(2,\mathbb{O})$ is $Spin(9,1)$, so he’s proposing a new candidate. Hitchin doesn’t like the usual idea because $Spin(9,1)$ is 45-dimensional and the space of $2 \times 2$ matrices of octonions is only 32-dimensional. His own preferred candidate has what might seem like an even worse problem: *it’s not even a group!* But it has some interesting properties and he makes a good argument for it.

I was planning to explain this and say more about how Hitchin’s work connects to another mathematical phantom: ‘octonionic twistors’. But I realize now that I’m not ready. So, I’ll think about this more, and turn the present blog post into a request: *please tell me about your favorite mathematical phantoms!*

## Phantoms

Does the zeroth power of a real number count as an example? How about the factorial of 0?

In other words, do mathematical phantoms usually turn out to have at least one valid “solidification” —a consistent definition when seen from a more general point of view, such as “in the limit” or “in agreement with a set of axioms”? That would also allow things like those infinite series that “converge” to values of the Riemann zeta function; we really mean that the definition of that series is taken to be the output of the function.