“(…) The answer to the Great Question…is…Forty-two(…)” said Deep Thought with infinity majesty and calm (frequently found quote from The Hitchhiker’s Guide to the Galaxy, Douglas Adams, London 1979)
Dear readers, yesterday, while I was editing, re-editing and writing my next blog posts found an extraordinary short paper by John Mc McKay and Yang-Hui He. You can found it here: http://arxiv.org/abs/1408.2083 Its title said it all: Moonshine and the Meaning of Life.
In that incredible and uncommon short paper (one of the shortest and most wonderful papers I have ever read) the authors hit with the connection and link between the Moonshine conjecture/theorem and the number 42. The number that, according to the cult geek book above, answers the Great Question, i.e., the Meaning of Life.
What is the Moonshine? Essentially, it is a striking link between two very different branches of mathematics: finite group theory and modular forms. In particular, the original moonshine conjecture applied to the biggest sporadic finite group, the Monster group M and the so called modular (invariant) form. That is the reason why sometimes it was dubbed as monstrous moonshine. The Monster group is the biggest of the sporadic finite groups, and it has about
elements, or equivalently
order, and that is about group elements!!!! Remarkably, note that the Avogadro constant is about , so the number of elements of the Monster is more or less a million of times the Avogadro constant SQUARED! Furthermore, note that the estimated number of protons in the observable Universe is around , therefore, the Monster group elements is really a big number…Not far away from that cosmological and fundamental number!
What about the quoted paper? It is truly simple in its hypotheses! In fact, the whole plan of the paper is to communicate the following two number theoretic identities (mathematical congruences):
by John McKay
by Yang-Hui He
What are those things? Well, firstly, let me introduce the elliptic modular form (sometimes called j-invariant) . This thing is some kind of mathematical object invariant under the group , the group of projective special linear transformations of 2×2 matrices with entries in the integers (). This object has a Fourier expansion given by the expression
and where, as the paper remembers, as , then is the nome for z. Furthermore, we have the theta series for the lattice
is the modular discriminant. The new congruences discovered by McKay and He are the Answer to the Great Question…42…The Meaning of Life… For a geek mathematician (physmatician) ! 🙂
Recall that the lattice vector
is truly unique, and it “lives” in the Lorentzian lattice known as in…26D spacetime dimensions (25 space-like, 1 time-like)! Indeed, it is an “isotropic Weyl vector” that allows us to construct (and build) the Leech lattice as . The remarkable Watson’s result, that the unique non-trivial solution to the equation
is a well-known mathematical result from the 20th century. The Moonshine conjecture is the connection between the Fourier expansion coefficients of the j(q)-invariant and the dimensions of the irreducible representations of the Monster group M. It arrived after this simple sum
35 years ago, in 1979, as the monstrous moonshine conjecture (now a theorem). That sparked idea revealed deep links and striking connections into two worlds: finite group theory (discrete in elements and origin) and modular forms (complex, continuous and holomorphic/automorphic forms/functions). The moonshine connection was finally proved by Richard Borcherds using tools from string theory, and his invention of the Vertex Operator Algebras (VOA) !
Recently, the so called Mock modularity appears in black hole state counting (DMZ) and in the computation of the elliptic genus of non-compact sigma models (Troost, Ashok, Eguchi, Sugawara). There is a simple physical explanation for the tension between holomorphy and modularity in these examples. Simply put, continuous spectrum is not holomorphic but discrete spectrum IS, and both are “mapped” somehow by certain kind of modular transformation that mixes these two types of spectra, more precisely, mock (modular) transformations do exist connecting the states of the discrete and the continuum worlds!
Thus, mathematics and physics have been connected again. Modular forms/automorphic functions (and their recent MOCK siblings) with finite groups, string theory and gravity (Witten gave. long ago, a talk relating certain Black Holes, the Monster group and the moonshine concept through quantum statistics and the partition function), and some generalized moonshine conjectures have arised in pure mathematics, in parallel sites where the old j(q)-invariant and the Leech lattice connections are essential. Current research is being done about the Umbral Moonshine (generalized moonshine), mock modular forms, and their relationships with the modern superstring theory/M-theory, specially with BPS states, branes and their “(quantum) counting procedure”. Some sophisticated mathematical objects called “shadows” are also introduced, but I am not going to discuss them today, after we have unveiled the Meaning of Life and the Answer to the Great Question… It is…
See you in another (hopefully so meaningful) blog post!!!!!!
P.S.: Read THE PAPER and references therein!!!!
P.S.(II): Update from 2019, the 42 decomposition into sum of three integer cubes reads
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