M Theory Lesson 108

Well, it is fun playing with circulants, but as Carl points out we might want to worry about idempotency also. For 1-circulants of the form

XYY
YXY
YYX

the only determinant 1 idempotent is the identity matrix. Proof: since $Y = Y^2 + 2XY$ then $X^2 = - 2Y^2 - \frac{1}{2} Y + \frac{1}{2}$, and from the relation $Y = Y^2 + 2XY$ it follows that $X = - \frac{1}{2} Y + \frac{1}{2}$. Thus $- 2 Y^2 = 2 Y^2$ and so $Y = 0$. By symmetry, the circulant matrices given by $(X,Y,Z) = (0,1,0)$ and $(0,0,1)$ also satisfy both the determinant and idempotency constraints. If we assume that $Y = \overline{Z} = r e^{i \theta}$ and that $X$ is real, it follows that the discriminant

$1 - 8X (a - ib)$

must be a positive real so that $b = 0$ and $Y = Z$ is real, and thus once again the identity is the only solution.

M Theory Lesson 107

First let $Y = r e^{i \theta}$. Continuing our conversation on determinant cubics, for a complex circulant with $Z = \overline{Y}$ we can solve the cubic

$X^3 - 3 r^2 X + (1 + Y^3 + {\overline{Y}}^{3}) = 0$

for $X = X(Y, \overline{Y})$ with Chebyshev radicals, under certain restrictions on $Y$. In terms of the Chebyshev root function $C$ (omitting the subscript) the solutions are

$X_1 = r C(t) - \frac{1}{3}$
$X_2 = -r C(-t) - \frac{1}{3}$
$X_3 = r C(-t) - r C(t) - \frac{1}{3}$

where $-t = r^{-3} (1 + 2r cos (3 \theta))$. Note that for $r = 1$, $t = 0$ when $\theta = \frac{2 \pi}{9}$ and $C(t) = \sqrt{3}$. But for the case $r = 1$ (which might not be interesting since we have used the determinant to renormalise the matrix) this solution set makes sense only provided $cos 3 \theta < 0.5$ and then

$C(t) = 2 cos (\frac{cos^{-1} (0.5 t)}{3})$

Observe that the solution condition states that $\theta > \frac{\pi}{9}$. Now observe that $\frac{2}{9} < \frac{\pi}{9}$ so this method is not directly helpful in analysing the lepton type matrix with $r = 1$. On the other hand, $\frac{1}{2} > \frac{\pi}{9}$ so the neutrino type cubic has three real solutions for $X$, but only $X_1 = 1.5644$ is positive. For a general positive real determinant $D$, $t = D + 2 cos (3 \theta))$ and the solution condition says that $cos (3 \theta) < \frac{2 - D}{2}$ which is less restrictive if $D$ is small.

M Theory Lesson 106

Let $\omega$ be the primitive cube root of unity. The determinant of a $3 \times 3$ complex circulant

XYZ
ZXY
YZX

is given by

$(X + Y + Z)(X + \omega Y + \omega^{2} Z)(X + \omega^{2} Y + \omega Z)$
$= X^3 + Y^3 + Z^3 - 3XYZ$

On setting the determinant to 1 (another choice of normalisation) the inverse of the circulant takes the simple form

(XX - ZY)(ZZ - XY)(YY - XZ)
(YY - XZ)(XX - ZY)(ZZ - XY)
(ZZ - XY)(YY - XZ)(XX - ZY)

which is again a 1-circulant since matrix multiplication is closed for this set. For 2-circulants, on the other hand, inverses can be 2-circulant. An inverse of an idempotent will also be idempotent.

Search Terms

After telling Tommaso Dorigo that search terms for this blog were boring, it now turns out that AF is at least high on a google search for pizza + theory + cheers (with absolutely no qualification regarding any specific physics or mathematics). Who are all these people? Don't they know* that I give women a bad name? If you're looking for a great pizza recipe, folks, I'm afraid you came to the wrong place! Hmmm. How about seared venison, English spinach, Kalamata olives, a fine Kapiti goat's cheese and roast kumara on a thin crust. * This may have something to do with my lack of faith in the Dark Force ($\Lambda$), Higgs bosons or SUSY partners.

Mt Ruapehu

This post is just to share a nice photo from the news today. We are being warned to stay away from Mt Ruapehu's crater after Tuesday night's small eruption. One climber, who happened to be high on the mountain, almost died when a large rock crushed his legs. Fortunately nobody else was injured, despite the skifields being busy with the school holidays.

M is for Magic

As we have seen, Carl Brannen's QFT uses circulant matrices. By resetting a mass scale, one may renormalise a 1-circulant

XYZ
ZXY
YZX

by a constant $\lambda = \frac{1}{Y + Z - 2X}$ so that the resulting circulant obeys the condition $2X = Y + Z$. This turns the circulant into a magic square. For 2-circulants the condition is instead $2Z = X + Y$. Although perhaps not a very useful observation, it is certainly entertaining! The total number of $5 \times 5$ normal (ie. matrices built from the first few ordinals) magic squares was only computed in 1973, and the number of $6 \times 6$ ones is still unknown. There is only one $3 \times 3$ normal square, up to rotation and reflection.

A paper by A. Adler uses circulants to find an algorithm for generating higher order normal magic n-cubes, by playing with p-adic L functions. For $p = 3$, Adler constructs two cute normal magic cubes: a $3 \times 3 \times 3$ cube and a $27 \times 27 \times 27$ cube. I was further intrigued by this paper of Adler's, containing the conjecture that magic n-cubes always form a free monoid. It shows first that sets of magic squares contain prime squares, out of which all others are constructed, and then that generating functions built from cardinalities for magic cubes have the remarkable property of being everywhere divergent!

Blogging Cats

The categorical blogosphere is rapidly expanding, now with Jeff Morton's new blog, Theoretical Atlas, and the Infinite Seminar. Meanwhile Urs Schreiber reports on being mobbed by reporters in Croatia, as he attends the maths conference on Categories and Geometry. It sounds like a fantastic conference: Batanin, Cartier, Kontsevich, Leinster and many others will be there. And what looks like the first real categorical physics conference has just ended in Texas. Lastly, check out the hot catsters lectures on You Tube.

Bookworms

If I were a Springer-Verlag Graduate Text in Mathematics, I would be W.B.R. Lickorish's An Introduction to Knot Theory. I am an introduction to mathematical Knot Theory; the theory of knots and links of simple closed curves in three-dimensional space. I consist of a selection of topics which graduate students have found to be a successful introduction to the field. Three distinct techniques are employed; Geometric Topology Manoeuvres, Combinatorics, and Algebraic Topology. Which Springer GTM would you be? The Springer GTM Test

United We Stand

Today's quote comes from Tony Smith (who has some noteworthy string theory papers on the CERN server). To summarise an interesting conversation here with Matti Pitkanen, he said

If you map the twistor 5 dimensional hypersurface in CP3 to S5, then you would have a twistor version of my view of string segments as local WorldLines in the Feynman PathIntegral picture. Global WorldLines would follow, as [Michael Rios] said: ... the continuous string would be recovered as a long polymer of projective space line segments ...

Meanwhile Carl Brannen posts on the snuark mass interaction calculation, based on his version of the measurement algebra. On the cosmological side, Louise Riofrio continues to keep us updated.

M Theory Revision

While Carl Brannen is on the subject of 1-circulants, let us recall the simple hexagon with both tree labels and the corresponding permutations from $S_3$. Note that the black nodes naturally form a 2-circulant matrix

123
231
312

whereas the open nodes form a 1-circulant

213
321
132

The first matrix contains the identity along with the other two permutations appearing in Bilson-Thompson's ribbon diagrams. Such triple node sets must be idempotent if viewed as sources (or targets) since the source of a source (identity arrow) is just the source again. In ordinary 1-categories sources are usually single objects, but in 3-logos theory (or in looking at (co)operads as multicategories) they should be triples. If we add a central origin to the hexagon, turning it into a cube, the three black nodes could mark the three directions in Space. This puts a little cube at the target of the big cube, and Time is just what we see in a mirror.

Strings in Spring

How long is a piece of string? With this phrase a good friend of mine likes to describe my research to new acquaintances. This once led a prominent string theorist (who happened to be guided by my friend on a glacier trip) to pop into the department here to visit me, only to find, no doubt to his great dismay, that I wasn't actually working on string theory. Of course I eagerly told him I was working on category theory, but as this was several years ago his eyes immediately glazed over and he took the first opportunity of continuing on his holiday.

But how long is a piece of string? String theorists say that strings are tiny, and this is why we cannot immediately see stringy effects in our surroundings. If I pick up a handful of sand from the beach, it contains lots of tiny bits of string, which in principle have a length determined by a coordinate system set up by me, the observer. But special relativity tells us that an alien spacecraft flying by the beach will determine lengths differently to me, in principle all the way down to the so-called Planck scale. Can we actually look at stuff at this scale? Well, no. In fact, things are seen to be made of atoms at a length scale many orders of magnitude greater than this. We could look at the subatomic particles if we collided two grains of sand at roughly the speed of the alien spacecraft (let's say) and our measurements of these new particles would be imprinted on a template (perhaps water molecules, or silicon wafers) built from everyday low energy stuff from our world.

Since we lack the resources to reach the scale of the strings themselves, we can never measure their length. What type of observer could measure their length? Given the complexity of the mere SM zoo, the recording template would have to be capable of registering enormous quantities of information, such as that contained in galaxies, or clusters of galaxies. What does it mean to say that an observer on this scale sees a lot of strings? One would naturally guess that such an observer sees things made out of galaxies and stars, and maybe stuff that we can't see. Oh, all right, string theory does have U duality, but I'm not aware of any papers which actually take this idea seriously since it means giving up the idea that everything is made of strings. Can anybody provide me with links?



P.S. The fact that other (more category theoretic) approaches to unification are making contact with reality might also have something to do with my lack of enthusiasm for string theory, although stringy mathematics is nice.

Transcendence II

In the last post Carl Brannen (who has derived the number of particle generations in his latest blog post) said he had once played with the value $\zeta (3)$ and found that it could not be a simple rational multiple of $\pi^3$. In his classic paper [1] on MZVs, Hoffman mentions a number of conjectures based on a consideration of the zeta function as a map into the reals from a subalgebra of the noncommutative polynomial ring $\mathbb{Q}[x,y]$ on two letters (with both ordinary product and a shuffle type product), namely the subalgebra of polynomials of the form $\mathbb{Q}.1 + xTy$ for any $T$.

The conjecture states that the quotient of this algebra by the kernel of $\zeta$ is a simple polynomial algebra on some set of Lyndon words. If true, it would imply in particular that $\zeta (3)$ cannot be a rational multiple of $\pi^3$. Hoffman shows that $\zeta (x^{m} y^{n})$ can always be written as a simple combination of Riemann zeta values $\zeta (i)$ for $i \geq 2$. The theorem amounts to showing that, for real numbers $s$ and $t$ lying in the basic simplex (bounded by the line $s + t = 1$)

$\sum_{m,n} s^m t^n \zeta (x^{m} y^{n}) = 1 - \frac{\Gamma (1 - s) \Gamma (1 - t)}{\Gamma (1 - s - t)}$

where M theorists will recognise the Euler beta function that appears in the Veneziano amplitude.

[1] M. E. Hoffman, J. Alg. 194 (1997) 477-495

M Theory Lesson 105

An argument $(n_1 , n_2 , n_3 , \cdots , n_k )$ of an MZV is a list of elements $n_i \in \mathbb{N}$. Observe that this is just a 2-ordinal in the sense of Batanin, as shown by the 2-level tree. The total number of leaves $n_1 + n_2 + \cdots + n_k$ is the weight of the MZV and the number $k$ is the depth. These are important parameters for characterising MZVs.

Aside: Carl Brannen tells us that in his next few blog posts he will run through the snuark derivation of the Koide mass formula for leptons, and also a formula that describes masses for baryon and meson resonances, numbering in the several hundred. That sounds kind of interesting to me. Long live blogging.

Check out this conference title!

Also in the news: it looks like the Martians have finally resorted to biowarfare to kill off the human plague on Earth.

Connes Lecture 7

Connes begins lecture 7 with a few remarks which are also stressed in the preface of the new book with M. Marcolli, namely that there are two very difficult but related problems behind his work and they are (i) the structure of spacetime and (ii) the prime numbers, or so called arithmetic site (recall that a site is a category equipped with a Grothendieck topology).

The lecture then begins with an analogy between the primes and elementary particles in QFT, which both form composites. Yesterday we saw that in higher topos theory this analogy might actually make some mathematical sense. But Connes discusses QFT, not simply quantum mechanics, so the composition of particles need not preserve particle number. Ordinary primes (in $\mathbb{N}$) are, as usual, cardinalities of sets. The Cartesian product of two sets results in the product of cardinalities, so any $n \in \mathbb{N}$ is represented by a Cartesian product of sets with a prime number of elements (we could always take, say, Saharan grains of sand, if we insist on constructing the sets in question). For basic quantum mechanical logic, Cartesian product is replaced by tensor product, but QFT requires a tensor product capable of shifting dimensions. This is the (higher categorical) Gray tensor product of Crans.

M Theory Lesson 104

Another way to see why a map $Q \rightarrow P$ between cubes and permutohedra is natural is to consider the parity cube labelled by bracketings of four objects, which is the way it naturally appears in the tricategorical axioms. The lower left vertex represents an object $A \otimes B \otimes C \otimes D$ with no specified bracketing. A first choice of bracketing leads to an unambiguous diagram, but further choices result in multiple diagrams. The target vertex is the full hexagon of levelled trees for the symmetric group $S_3$. Pentagons appear by forgetting the level information of the front top right vertex. The front face is also the deformation of the pentagon as represented by five sides of the parity cube.

Observe that this relates a three dimensional cube to a two dimensional hexagon, and is thus more like a homological connecting morphism (decategorification) than a Loday map, which operates in a fixed dimension. Conversely, the hexagon poset may be decomposed as shown to form the cube in one dimension higher.

M Theory Lesson 103

Changing topics somewhat: in a classical topos a natural number object enables Peano's axioms for arithmetic to hold in the internal logic. This object $N$ is equipped with a successor function $s: N \rightarrow N$ and a zero object $0: 1 \rightarrow N$. But quantum mechanical logic doesn't belong in a classical topos. In the category Vect of vector spaces, for example, the terminal object $1$ might be replaced by the number field $K$ and the subobject classifier by the qubit object $K \oplus K$.

So what happens to arithmetic? Now Vect is actually a higher dimensional structure, being a symmetric monoidal category. Thus we expect to do a lot more decategorifying before we can count, and this process could take us through a topos like Set in which arithmetic makes sense. For example, how do we count the dimension of a (finite dimensional) vector space? We usually take a basis set and then use arithmetic in Set to count its elements. The logical analogue of dimension in Vect would really be an arrow $K \rightarrow K^{\infty}$ which picks a one dimensional space from $K^{\infty}$. Thus numbers really look like quantum states, and the so-called collapse of the wavefunction would be a simple categorical transition of numbers between different logics.

M Theory Lesson 102

In the 2005 lecture 6 Alain Connes points out that although his framework predicts physical couplings that match $SU(5)$ unification, it achieves this without the addition of extra fields or supersymmetry arguments. Towards the end of the lecture he summarises the situation: the problem is to combine (1) the renormalisation theory (Hopf algebras, motivic Galois group) and (2) the geometric setup from NCG operator theory, in such a way that running geometries (on different scales) are possible. Connes proposes a functional integral over geometries which is spectral in nature, and in fact looks like a matrix model. The question is, what sort of constraints should be applied to geometries? It is made clear that this is a challenge to physicists: his spectral action principle is a statement about the nature of observables, but insufficient in itself to guide, as Connes puts it, the merging of motives and NCG.

These lectures are highly recommended to physicists. Don't expect to understand all the mathematical gobbledygook, but try to take in the big picture, which is fantastically conveyed. Note that more recent work removes much of the arbitrariness of the original NCG formulation of the SM, but still puts the number of generations in by hand.

Old Times, Ol' Timers

Backreaction has reminded me of the time I worked at Parkes telescope back in 1989, just before the Voyager Neptune encounter. Marissa Shaw and I were running the night shift (radio telescopes can run 24/7). NASA was there laying extra cables for the Voyager encounter. Many readers would be amazed at the banks of computers that lined the control room: their processing power wouldn't beat a modern laptop, and tracking data was backed up with handwritten logbooks.

But despite such primitive technology, Voyager was out there, exploring the solar system. It was possible because the physics was simple, allowing a focus on the essential effort of strategy and engineering. Are we really supposed to believe that, in a mere 20 years, the laws of physics have suddenly become ridiculously complicated? Physics is the science of building simple tools for the computation of useful physical quantities, such as particle masses.

Transcendence

Recall that the spectrum of the MZV algebra is somehow embedded in the Grothendieck-Teichmuller group, which Bar-Natan has defined in terms of non-associative braids. Today God Plays Dice discusses MZVs and the conjecture that all zeta values are transcendental numbers.

This suggests a real dichotomy between algebraic (in fact integral) arguments and non-algebraic values. On the other hand, only $\pi$ seems to crop up, which is not surprising given the appearance of volumes of spherical polytopes. If $\pi$ represents the sum of angles of a Euclidean triangle

$\pi = \theta_1 + \theta_2 + \theta_3$

then integral powers of $\pi$ look like homogeneous symmetric polynomials in three variables. For even $n$, the basic zeta values always contain a factor of $\pi^n$. (Do non-commutative zeta algebras contain a concept of non-commutative angle?)

M Theory Lesson 101

In their paper on trialgebras, Loday and Ronco begin with two operations, left $\dashv$ and right $\vdash$, which can form triple products in eight possible ways as indicated by the vertices of the cube. Note that the three directions correspond to (i) flip first operation (ii) flip second operation or (iii) reorder operations. A third operation $\perp$ is introduced as an interval lying between left and right. For example, $\perp$ may be the associator between left association and right association on three objects, represented by planar binary trees with three leaves.

Loday's later paper on stuffles (also called quasi-shuffles) discusses the MZV algebra in terms of trialgebras. It gives a trialgebra structure to $T(A)$, the tensor algebra for a (commutative) algebra $A$. For the stuffle product $\ast$ the operations left and right are given by

$x \ast y = x \dashv y + x \vdash y + x \perp y$
$ax \dashv by = a(x \ast by)$ with $1 \dashv x = 0$ and $x \dashv 1 = x$
$ax \vdash by = b(ax \ast y)$ with $1 \vdash x = x$ and $x \vdash 1 = 0$

so the left (right) operation has a right (left) unit. The $\perp$ operation is thought of as a deformation of shuffle product. The algebra determined by $\perp$ (with $x \perp y = y \perp x$) and $\dashv$ (with $x \vdash y = y \dashv x$) is a commutative example of trialgebra, leading to the idea that non-commutative zeta values may be studied in the context of trialgebras and their operads. Loday also shows that the functor from commutative algebras to commutative trialgebras of this type has a right adjoint which forgets the operation $\dashv$. Thus this type of algebra forms a natural triple with commutative and associative algebras.

M Theory Lesson 100

On page 127 of their book [1], Lambek and Scott discuss the unifying properties of topos theory, as seen from the perspective of type theory. The three traditional mathematical philosophies in question are intuitionism, Platonism and formalism (in the sense of concrete symbolism). They note that a fourth philosophy is somewhat neglected, namely neologicism: the idea that all mathematics can be formulated in terms of logic. Strict logicism would require deriving the properties of the natural numbers from more foundational principles.

But isn't that just what the physics has been telling us we must do? Lambek and Scott take the viewpoint that categorical logic treats logic pragmatically, as a part of ordinary mathematics, thus refuting logicism. Should logos theory refuse to accept this point of view? Recall that in M theory any useful logical statement has a physical meaning pertaining to a given experiment and its constraints. The numbers associated with logical statements (or rather diagrams) also take on a physical meaning (loosely speaking, a collection of measurements). Thus both logic and number theory should be derived from the physical principles of measurement, expressed as a higher dimensional topos theory.

There is a chicken-and-egg objection: that the formalism of higher toposes requires the mathematics of logic before acquiring a physical interpretation. However, since physics is undoubtedly guiding the axioms in question, this argument appears weak. Ironically, logicism in M Theory would be the ultimate in reductionism, despite the intuition of emergent schemes on different scales.

[1] J. Lambek and P.J. Scott, Introduction to higher order categorical logic, Cambridge U.P. 1986

Cool QFT Links

Matilde Marcolli provides links to You Tube lectures on QFT by Alain Connes and herself. Meanwhile Carl Brannen continues his series of posts on snuarks in QFT. Thanks to David Speyer, here is an interesting paper on lattice congruences and Hopf algebras.

Warning: The first Marcolli lecture begins with Lagrangian densities and actions and it isn't possible (at least on my monitor) to see the blackboard or overheads clearly. The first Connes lecture is a bit easier to see and an outstanding overview of the situation in renormalisation theory.

Spring Break

Those skis just accidentally fell into my hands, so I guess I'll have to take a couple of days off and head down south to my real home. Back soon. This is a photo from the town of Wanaka.

M Theory Lesson 99

Given an element $D$ of the descent algebra of $S_n$, let $Z(D)$ be the maximal permutation whose signature is $D$. Considering the natural order on the cube and permutohedra vertex sets, they become poset categories and $Z: Q_n \rightarrow P_n$ is a functor. In this paper it is shown that $Z$ has both a left and right adjoint. Moreover, the other pieces of the triangle

$Q_n \rightarrow A_n \rightarrow P_n$

also have left and right adjoints, giving a triple (cyclic) double adjunction involving these polytopes. This amazing categorical relation is a generalisation of an ordinary adjunction between just two categories. Since the associahedra were given as a 1-operad it is natural to try to view this triple as a 3 dimensional structure. Can we extend the use of vertices and edges to faces in a 2-category replacement for posets?


Aside: Regarding the Hopf algebras here, P. Cartier provides the clearest exposition. MZVs appear on page 65.

M Theory Lesson 98

Define the signature of a permutation to be the sequence of signs of differences. For example, for $(132)$ in $S_3$ the signature is $(+-)$. For any $S_n$ there are $2^{n - 1}$ signature types, forming a parity cube. Consider formal sums of elements in a signature class. For example, $(132) + (231)$ is such a sum. In 1976, Solomon [1] showed that the product of two such sums is always a linear combination (over $\mathbb{N}$) of signature sums. A simple example from $S_3$ is

$(--)(+-) = (321)[(132) + (231)] = (231) + (132) = (+-)$

This is called the descent algebra. In [2] Loday and Ronco define a Hopf algebra of binary trees which uses this algebra. M theorists will be more familiar with the alternative polytopes to cubes, namely associahedra and permutohedra. The vertices of these polytopes are related to the sequences

$P_n \rightarrow A_n \rightarrow Q_n$

where $Q_n$ is a basis for the Solomon descent algebra. That is, the associahedra sit in between the permutation labellings of trees with distinct levels and the cubes. For $S_3$ this yields the usual pentagon diagram. The direct sum of all group algebras $k S_n$ may be given a Hopf algebra structure, with the descent algebra as a sub Hopf algebra. Using the sequences above, it is shown that the associahedra also have a (graded) Hopf structure.

[1] L. Solomon, J. Alg. 41 (1976) 255-268
[2] J-L. Loday and M.O. Ronco, Adv. Math. 139 (1998) 293-309

Spring Arrives II

I really don't want to clutter up the blog with sketchy slides, but here's one more for a stunning spring day.

Quote of the Month

Copied straight out of the much talked about Strings article:

M Theory Lesson 97

F. Chapoton has a wonderful set of slides on operads and combinatorics. On slide 42 he explains how a symmetric operad $P$ (ones with a permutation action) in the category Set may be thought of as a Joyal species functor into Set (recall that species came up in Abel sum theory) along with an arrow $P.P \rightarrow P$ giving composition. Note that this is just how multiplication looks like for a monad.

On slide 62 we see a novel way of looking at the expression $J = 1 + 2J + J^2$, which arose in the blogrolling discussion. Chapoton's solution for $J$ is a series of binary trees which has a kind of inverse with respect to an interesting tree operad, called OverUnder, due to A. Frabetti. This inverse is the alternating series Notice the analogy with Rota's Mobius inversion, where alternating series naturally arise. It would be nice to understand alternation better, as Euler characteristics are defined using such series. In fact, there is a map from the OverUnder operad to the Associative operad which takes a sum of trees to the Abel series that collects sets of trees together. In this paper Frabetti looks at trees for QED renormalisation.

Spring Arrives

Spring is greeted, as usual, with a severe snowstorm across the South Island. Looks like I'll need to pull those skis out of the garage. Fortunately we are comfortably seated in a warm lecture hall and the beer will be served inside the department. Just time for one slide today from my talk on operads:

M Theory Lesson 96

Non-commutative polynomials are built out of words in an alphabet of letters. The simplest case is two letters $X$ and $Y$. If we imagine, as pointed out in this helpful paper, that the letter $X$ represents $\frac{dx}{x}$ and the letter $Y$ represents $\frac{dx}{1 - x}$ then an MZV may be expressed as an iterated integral

$\zeta (s_1, s_2, \cdots , s_k) = \int_{0}^{1} X^{s_1 - 1} Y X^{s_2 - 1} Y \cdots X^{s_k - 1} Y$

in dimension $d = \sum s_i$. Given any alphabet whatsoever, a shuffle is the operation sending words $V$ and $W$ to the sum of permutations on letters which preserve the order of letters within the words. Thus MZVs obey a shuffle product.

A two letter alphabet also arises in the Lorenz template representation of knots, where words in the two holes describe a knot or link. In fact, MZV algebras were initially studied in relation to the homfly polynomial as formulated by Kontsevich.

M Theory Lesson 95

The classic paper on Koszul duality for operads is [1]. It is very difficult for a physicist to follow, but let us make a note on a remark on page 3. They interpret the duality for operads in terms of a dualising functor hom(-,Lie) where Lie is the operad associated to Lie algebras.

This is like the behaviour of schizophrenic objects, meaning in this case the object Lie. A more classical example is the object U(1) underlying Pontrjagin duality.

[1] V. Ginzburg and M. Kapranov, Duke Math. J. 76, 1 (1994) 203-272

Spring Conference

At the annual departmental conference this week I will be giving yet another introduction to category theory for physicists! The talk is on Tuesday afternoon, and this time I'll be focusing on how combinatorics associated to operads can be very useful. All welcome.

A Day Out

A building nor'west breeze means a fine winter's day in Kaikoura, so I headed up the coast with three friendly astronomers. We did our best to avoid the seal colony, but there were over 200 seals, with many pups, blocking our path around the coast. Unable to detour over crumbling cliffs, we were forced to run the gauntlet of a few angry, but fortunately lazy, fellas. After a late burger and chips in town we stopped off for scallops and mussels (in garlic) at a caravan by the beach, before heading back home.