Archive for May, 2007

Magic Motives

Speaking of orbifold Euler characteristics, let’s put the magic formula

$f(n) = \prod_{m=0}^{n – 1} \frac{m!}{(m + n)!}$

in terms of Euler characteristics. First, let $m = 2g – 2$ be the Euler characteristic of a closed surface of genus $g$. This already suggests allowing non-orientable surfaces to account for odd values of $m$. Then consider moduli spaces $M_{m,n}$ for $(n + 1)$ punctured surfaces. The orbifold Euler characteristic of such a space will be denoted by $E_{m,n}$. Using Mulase’s expression for $E_{m,n}$ and assuming it may be extended to the non-orientable case, one finds a natural definition for the moment coefficients of the form

$f(n) = \frac{1}{(n + 1)!} \prod_{m=0}^{n – 1} b_{m + 2} E_{m,n}^{-1}$

which is a product over surfaces of genus $g$ limited by $n$, and where $b_{i}$ is a Bernoulli number (for even $m$ these are defined in terms of zeta values for odd negative reals). One should take more care with the non-orientable factors, but this simple exercise shows that the zeta moments are naturally dependent on categorical invariants associated to complex moduli.

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M Theory Lesson 62

Last November, in the pre maths blogger days, we started with Mulase’s lectures on moduli spaces of Riemann surfaces. In particular, let us look once more at the $S_3$ action on the Riemann sphere $\mathbb{CP}^1$. The real axis is the equator, with the point $\frac{1}{2}$ sitting opposite the point at infinity. The dihedral action helps define a compactified form of the moduli space for the once punctured torus $M_{1,1}$ (elliptic curve), which was described by a glued region of the upper half plane, sitting above the unit circle. The j invariant gives the mapping from the 3-punctured Riemann sphere to the complex plane which respects the dihedral action on the equatorial triangle, and the torus orbifold is the quotient space.

The j invariant is used to obtain Grothendieck’s ribbon diagram from the inverse image of the interval $[0,1]$, so both sphere and torus moduli are essential in understanding the ribbon for the 3-punctured sphere. Recall that these are the only moduli of real dimension 2. In the six dimensions of twistors there are three complex moduli, namely $M_{0,6}$, $M_{1,3}$ and $M_{2,0}$ which have (respectively) orbifold Euler characteristics of $-6$, $- \frac{1}{6}$ and $- \frac{1}{120}$. Octonion analogues of such moduli will be easier to understand using n-operad combinatorics, because non-commutative and non-associative geometry is a tricky business.

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M Theory Lesson 61

The purpose of this post is mainly to invite Doug, one of our oldest companions, to comment here on the general direction of the Lesson series. Doug expressed an interest in doing so back in Lesson 59. So just this once, Doug can post a series of long comments here. I hope this will encourage him to set up his own blog. After all, anyone is free to do so!

In Lesson 59 we were looking at theta series for lattices. For the Leech lattice this involved the Ramanujan function, which is a Fourier series for the modular discriminant. Fourier coefficients for cusp forms were part of Conrey’s motivation in looking at even moments for the Riemann zeta function.

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Riemann Rave III

The aforementioned moment series (for the unitary ensemble) of Keating and Snaith is defined in their paper as

$f(n) = \prod_{j=0}^{n – 1} \frac{j!}{(j + n)!}$

for which one finds the first few terms

$1$, $\frac{2}{4!}$, $\frac{42}{9!}$, $\frac{24024}{16!}$, …

It is fun to find alternative expressions with fewer factorials because the size of terms is then more immediately apparent. For example,

$f(n) = \frac{n^{n – 1}}{(2n – 1)!} \prod_{j=0}^{n-2} (n^{2} – j^{2})^{j-n+1}$

for which one can check, say the third term

$f(3) = \frac{3.3}{} \frac{1}{9.9} \frac{1}{8} = \frac{6.7}{9!} = \frac{42}{9!}$

This shows up the dependence on the factorial of squares $n^2$, which is not so apparent in the original expression. Note that $24024=$ is similarly made of pieces that are needed for a factor of $16!$ in the 4th term. The nth term provides the coefficient of the moment $\int_{0}^{T} | \zeta (0.5 + it) |^{2n} dt$ of the Riemann zeta function, as discussed by Conrey et al in the paper that appeared around the time of Keating’s Vienna talk.

It seems that single prime factors miraculously appear in the numerators of $f(n)$. Moving on to $f(5)$ we find a numerator of $$ (assuming I did the sum correctly). So if we ignore powers of 2, the first few numerators are built from single prime factors, covering mainly primes between $2n$ and $n^{2}$.

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Gossip, Gossip

Those naughty experimentalists, such as Tommaso Dorigo and Charm, have been spreading rumours about a D0 discovery involving an excess of 4 bottom quark jet events. Of course, this has reignited the Higgs boson gossip, which must be starting to sound like a boy crying wolf even to those who actually (mistakenly) think that such a particle exists. One begins to suspect that these naughty children are just playing games to get a laugh out of all the crazy responses that pop up in the media in no time at all.

Naturally, we are all ears if D0 decides to tell us something Interesting and Statistically Significant, as they say.

In other gossip: the rumour that the Riemann hypothesis has been shown to be false has now extended beyond the blogosphere. One of my office mates heard it from a colleague in a casual conversation!

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Light Metal Life

Universe Today reports on the new discovery from Texas of a system with two Jupiter like planets orbiting the metal poor star HD 155358. The metallicity suggests the star is about 10 billion years old.

The astronomers were surprised to find planets in such a system, because heavy elements are needed to assist core accretion in this model of planet formation. The alternative model suggests that instabilities cause a breakup of the disc, but the mechanism for this is unclear. “Having this process happen to form not just one, but two, planets around a star that had so little solid material available for planet-building is quite remarkable,” Cochran said. This appears to be yet another confirmation of one of Louise Riofrio’s predictions. Small dark matter objects may aid core accretion without the need for heavy elements.

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Riemann Rave II

It may take a while to read through Gregory Moore’s work on Arithmetic and Attractors, but it looks very interesting. He also has another paper on black hole entropy with S. D. Miller, who has a handy page on L-functions and apparently a keen interest in the Riemann zeroes.

Note that this work is from a string theory perspective and so does not consider, for example, a varying speed of light or an $\hbar \rightarrow 0$ which appears both as a cosmological horizon constraint in Riofrio’s cosmology and also in the semiclassical analysis of random matrix ensembles done by Snaith and Keating.

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Riemann Rave

The most intriguing physical anecdote in du Sautoy’s The Music of the Primes is the one about Keating’s talk, entitled “Random matrix theory and some zeta-function moments”, at the Vienna meeting in 1998.

Many mathematicians were dubious that the physicists could tell them anything beyond their observations on the statistics of Riemann zeroes and quantum chaos on surfaces. Yet another fine bottle of wine had been put forward, which Keating collected in Vienna after his talk. Nina Snaith and Keating had found a formula for generating the numbers known as zeta moments. The sequence begins 1,2,42,24024. The last number had only just been found by mathematicians, and yet it fell out of the physicists’ formula when they looked at it just before Keating’s talk. Links to the papers by Keating and Snaith, and also to Snaith’s thesis, are available on Watkin’s page.

Later, Keating went to the library in Goettingen to look up Riemann’s original notes on both the zeta zeroes and hydrodynamics. He made the two requests to the librarian, but was handed only one pile of papers. Riemann had been working on both problems at the same time.

More recently, Carl and Michael have been making solid progress on understanding quantum black holes in M Theory. It would be really very interesting if the quantum chaos of such black holes had something to do with the Riemann zeroes. Oh, wait. It already does, because one uses 3×3 Hermitean matrices in a matrix theory setting.

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M Theory Lesson 60

I wish I had come across the work of Andrew Hodges before! Check out the papers:

1. Twistor diagram recursion
2. Helicity-independent formalism

Unbelievably, when I google MHV and operad, which I do periodically, I still get no hits (other than here). There are thousands of hits for MHV. What is everybody doing?

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Riemann Revisited III

OK, even if the Hypothesis turns out to be false, that was hilarious, but seriously now … one promising route to the Riemann Hypothesis is in fact to show that it is undecidable with the standard axioms. It is hard to imagine how this could be done. Even if the zeta zeroes were completely re-characterised in terms of higher categorical invariants, in a way that seemed utterly natural and compelling, that does not imply that we must look at the zeta function that way. Well, mathematically that is. Physically speaking, we only care about the zeta function in so much as it can describe measurable quantities.

But the zeta function soon gets swamped by its generalisations, inflating the difficulty of the problem. These are certainly physically relevant. Recall, for example, Brown’s paper for MZVs, which contains a 1-operad computation of Veneziano amplitudes. M Theory cannot avoid considering this construction outside set theory.

Update: Thanks to K. L. Lange for further interesting remarks about the possibility that Pati has inadvertently made progress on showing that RH is unprovable within standard analysis.

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