Monstrous moonshine

I received an email from a fellow journalist last week with the following subject:

Banks wont recover even half of the Rs 4,000,000,000,000 bad loans of 37 companies

That number with all those zeroes is four trillion. It’s a large number. For example, there are only 30 billion stars in the Large Magellanic Cloud, only 100-400 billion stars in the Milky Way galaxy, only one trillion stars in the Andromeda galaxy.

There’s a famous quote attributed to Richard Feynman, the theoretical physicist, that very large numbers that used to be called astronomical should in fact be called economic. Why? Because at the time he said that, the US national deficit was $100 billion.

Four trillion, however, is a number bursting at its astronomical seam. In fact, consider the entire debt load of the world’s governments: an estimated $200 trillion. It’s not the sort of number we hear every day, in any context, and it’s not a number our cognition is equipped to easily fathom, at least not without notational assistance and some level of abstraction.

This brings me to an interesting anecdote my roommate, a physicist, once shared with me. It involves what’s called the Monster group. It’s a set of numbers organised according to certain rules such that it contains 8 × 1053 of them. Pause for a moment, let it sink in: the Monster group’s defining rules allow a very large number of numbers to be included, but not an infinite number of them.

How could something so large exist in nature without being all-encompassing at the same time?

This is why, even though the Monster group contains 10 billion times fewer numbers than the number of atoms in the universe, it is infinitely more interesting. While the world’s governments have been arbitrarily borrowing money such that the global debt is several multiples of the global GDP, while it is meaningless to try to understand the universe in terms of the vigintillions of atoms it holds, the Monster group – for all its mind-blowing vastness – is quite well-defined and meaningful.

To truly appreciate why this is so, we must start at what’s called a ‘finite group’.

Consider a set with five elements. The ‘permutation group’ of this set consists of all possible permutations of these five elements, which total 120. If the set had had six elements, then its permutation group would’ve had 720 elements. If the set had had seven elements, then its permutation group would’ve had 5,040 elements. And so forth.

Since the permutation group will always contain a finite number of elements, it’s called a finite group. Similarly, every mathematical groups that contains a finite number of elements can be classified as a finite group.

Now, in order to make sense of the different kinds of finite groups that are possible in mathematics, mathematicians came up with a classification scheme. They were able to categorise various finite groups according to their various mathematical properties. As a result, there are 18 families each comprising an infinite number of finite groups. Then there are 26 groups called sporadic groups, none of which can be fit under any of the 18 families.

The largest sporadic group is called the Monster group, holding 8 × 1053 numbers. And this is where it gets more interesting.

Meet ‘monstrous moonshine’.

Instead of fumbling with intricate mathematical concepts, I will defer at this point to a 2015 article in Quanta describing the idea (edited for brevity):

In 1978, the mathematician John McKay … had been studying the different ways of representing … the Monster group. … Mathematicians weren’t sure that the group actually existed, but they knew that if it did exist, it acted in special ways in particular dimensions, the first two of which were 1 and 196,883.

McKay … happened to pick up a mathematics paper in a completely different field, involving something called the j-function, one of the most fundamental objects in number theory. Strangely enough, this function’s first important coefficient is 196,884, which McKay instantly recognised as the sum of the monster’s first two special dimensions. …

John Thompson, a Fields medalist, … made an additional discovery. … The j-function’s second coefficient, 21,493,760, is the sum of the first three special dimensions of the Monster: 1 + 196,883 + 21,296,876. It seemed as if the j-function was somehow controlling the structure of the elusive Monster group.

The beautiful thing here is that, until the moments of McKay’s and Thompson’s discoveries, mathematicians had no reason to believe the Monster group and the j-function were even remotely related. However, there it was, hinting at a deep and mysterious connection between two distant branches of mathematics. This connection has come to be called monstrous moonshine.

In 1992, another mathematician named Richard Borcherds figured out the nature of this connection. Of all places, he found the answer lurking in string theory – the theory that imagines that “the universe has tiny hidden dimensions, too small to measure, in which strings vibrate to produce the physical effects we experience at the macroscopic scale” (to quote the same Quanta article).

In 2012, three physicists floated an even more bizarre idea: that apart from monstrous moonshine – which bridges group theory, number theory and string theory – there were 23 other moonshines establishing hitherto unknown links between mathematics and physics. This is called the umbral moonshine conjecture, and in 2015, scientists proved that they do exist.

What the hell is going on?

I will stop here, trusting that I’ve led you sufficiently far into a deep, but not bottomless, rabbit hole. 🙂