LOG#154. Moonshine and 42: THE PAPER.

Posted on by September 8, 2019

“(…) 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)

170px-Ultimate_Hitchhikers_Guide_front hitchhikeomnibus-blog h2g2-01 copy douglas_adams___the_htch_hikers_guide_to_the_galaxy

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 j(q) 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

(1)   \begin{equation*} N=2^{46}\cdot 3^{20}\cdot 5^9 \cdot 7^6 \cdot 11^2 \cdot 13^3 \cdot 17 \cdot 19 \cdot 23\cdot 29 \cdot 31 \cdot 41 \cdot 47 \cdot 59 \cdot 71\end{equation*}

elements, or equivalently

(2)   \begin{equation*}N=808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000\end{equation*}

order, and that is about N\approx 8\cdot 10^{53} group elements!!!! Remarkably, note that the Avogadro constant is about N\approx 6\cdot 10^{23}, 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 10^{80}-10^{83}, 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):

(3)   \begin{equation*} \boxed{\displaystyle{\left(\sum_{m=1}^{24} c_m^2\right)\mbox{mod} 70\equiv 42}}\end{equation*}

by John McKay

(4)   \begin{equation*}\boxed{\displaystyle{\left(\sum_{m=1}^{24} \tau_m^2\right)\mbox{mod} 70\equiv 42}}\end{equation*}

    by Yang-Hui He

What are those things? Well, firstly, let me introduce the elliptic modular form (sometimes called j-invariant) j(q). This j(q) thing is some kind of mathematical object invariant under the group PSL(2,\mathbb{Z}), the group of projective special linear transformations of 2×2 matrices with entries in the integers (\mathbb{Z}). This object has a Fourier expansion given by the expression

j(q)=\dfrac{E_4(q)^3}{\Delta (q)}=\displaystyle{\sum_{m=-1}^\infty c_mq^m=\dfrac{1}{q}+744+196884q+21493760q^2+\ldots}

and where, as the paper remembers, as z\rightarrow i\infty, then q=e^{2\pi i z} is the nome for z. Furthermore, we have the theta series for the E_8 lattice

E_4(z)=1+240\displaystyle{\sum_{n=1}^\infty\sigma_3(n)q^n}

with

\sigma_3(n)=\displaystyle{\sum_{d\vert n}d^3}

and

\Delta (q)=q\displaystyle{\prod_{n=1}^\infty\left(1-q^n\right)^{24}=\sum_{m=-1}^\infty\tau_m q^m=q-24q^2+252q^3-1472q^4+4830q^5-\ldots}

 \Delta (q) 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

\omega=(0,1,2,\ldots,24:70)

satisfying

\displaystyle{\sum_{n=0}^{24}n^2=70^2}

is truly unique, and it “lives” in the Lorentzian lattice known as II_{25,1} 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 \omega_\perp/\omega. The remarkable Watson’s result, that the unique non-trivial solution to the equation

\displaystyle{\sum_{p=1}^np^2=m^2}

is

(n,m)=(24,70)

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

196884=196883+1

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…

42cluster 42dontpanic 42entropytext_quotes_the_hitchhikers_gu_1600x1200 42everything 42finitegroups 42Notebook2008134

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

    \[X^3+Y^3+Z^3=42\]

where

    \[X = -80538738812075974,\;\; Y = 80435758145817515,\;\; Z = 12602123297335631\]

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LOG#154. Moonshine and 42: THE PAPER. — 2 Comments

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