Theory Update 72

Recall that the three stranded ribbon diagrams are elements of either $B_3$ or $B_6$. We consider the special elements of $B_6$ that group the six strands into pairs, to form ribbons. Today's diagram shows how the three positive generators of a $B_4$ segment of $B_6$ can create a twisted ribbon that crosses over a straight ribbon segment.

Thus a neutral neutrino braid might be represented by the length $12$ $B_6$ word

$( \sigma_1 \sigma_2 \sigma_3 )^2 ( \sigma_{3}^{-1} \sigma_{4}^{-1} \sigma_{5}^{-1} )^2$

In this example, the ribbon twist is undone by canceling both the central $\sigma_{3} \sigma_{3}^{-1}$ standard $B_6$ term and the end crossings, $\sigma_1$ and $\sigma_{5}^{-1}$, leaving a word

$\sigma_2 \sigma_3 \sigma_1 \sigma_2 \cdot \sigma_{4}^{-1} \sigma_{5}^{-1} \sigma_{3}^{-1} \sigma_{4}^{-1}$

where a block of form $\sigma_2 \sigma_3 \sigma_1 \sigma_2$ represents one flat ribbon crossing another in $B_4$. That is, all flat ribbon diagrams are generated by these cyclic blocks of $B_n$ generators.

Theory Update 55

For the ribbon braid group $B_3$, we can twist ribbons and also perform braiding within the three strands of $B_3$. One cyclic set of generators is shown.
The generators $(12,23,31)$ are those given by the trefoil quandle. The ribbon strands are labeled $1$, $2$, $3$. We see that two qutrits may be used to label the generators, under the correspondence $1 = XX$ and $12 = \{ XY , YX \}$. In other words, the three ribbons are specified by the letters $X$, $Y$ and $Z$.

Observe that the trefoil quandle rule is now naturally associated to braids of the form $\sigma_{1} \sigma_{2}^{-1} = (12)(23)^{-1}$ $= 12 3^{-1} 2^{-1}$, which are used to specify the Bilson-Thompson particle spectrum. Since qutrits and triality are used to specify braid information, a $3 \times 3$ Koide mass matrix now lives in an exceptional bioctonion Jordan algebra, as do the neutrino and CKM mixing matrices.

Theory Update 38

In Khovanov homology the bit values, $0$ and $1$, are associated to smoothings of a knot crossing. The trefoil knot in the braid group $B_2$ has three crossings. Let us label these three bits $X$, $Y$ and $Z$. A Bilson-Thompson particle braid is an element of the braid group $B_3$, although its ribbon strands may be either twisted or untwisted. Two crossings on a ribbon strand represent a $1/3$ down quark charge.

Cohomology Revisited

Modern physicists and mathematicians like to draw diagrams with strings and ribbons. An object in a category is often depicted by a strand with an arrow (as in (a)). Recognising that an arrow represents two colours, we might like to use three colours instead, as in (b). A tensor product of objects is given by placing their strands side by side. In (c) we thicken the strand to a ribbon, which allows twists in our diagram networks. But why stop there? We might as well use the three dimensions at our disposal and thicken the ribbon backwards, as in (d). Finally, taking multiple coloured beams, we can form braided networks from them.

One of the simplest diagrams in a $1$-dimensional category is a triangle. With beams we could draw this:
This is Penrose's triangle. He uses it to discuss the first cohomology group, where the numbers are interpreted as positive distances. Elements of a cohomology group are classes of cocycles, and in one dimension a category theorist would draw the cocycle condition as a triangle
where the $d_{ij}$ are distances of observation. We have a coboundary when the cycle composition results in the number $1$, where an inverted arrow corresponds to taking the reciprocal positive number. That is, each $d_{ij}$ is a ratio $x_{i}/ x_{j}$. Reciprocation is the multiplicative analog of a minus sign for addition. In a higher dimensional category, more complicated polytopes are used to specify cocycle conditions. When these polytopes are canonical axioms for a suitable class of categories, the cohomology should be universal.

Auld Lang Syne

As Kitaev will tell you, quantum codes are about things like ribbon categories. Ribbons and knots, knots and ribbons.

Back in the mid '90s, everyone was into quantum field theories with knots. I was supposed to be working in standard lattice QCD, in a Physics department, but being enamoured of knots I instead found myself lost amongst the deserted stacks of the mathematics library, and chatting to mathematicians like Bai-Ling Wang and Alan Carey. At that time, classical geometry was still boss, and everywhere I looked there were manifolds, manifolds, manifolds. The mathematicians weren't at all bothered by all the manifolds, but as a physicist who haunted the library stacks, I was looking for something more, let's say, background independent. Alan gave me some excellent advice, and told me to work on my thesis and forget about the knots (but naturally, the advice was bound to be ignored).

Anyway, a standard gadget in the knotty manifold world was a framed knot, where one thickened a knot into a kind of twisted torus so that it sat in a nice way in a larger space. By drawing lines along the torus, one could get ribbon diagrams. For example, quantum mathematicians liked symplectic spaces, such as $T^{*}S^3$, the cotangent bundle of the three dimensional sphere $S^3$. Such a six dimensional space has three dimensional Lagrangian submanifolds, good for putting knots in, and with which mathematicians can do some knot wizardry.

In time, the trend moved towards combinatorial alternatives. The only constant was the universal obsession with knots and ribbons. Six strands for three ribbons. I could no longer afford to listen to mathematics lectures in Princeton or Canberra, and nobody was willing to supervise my thesis proposal on a lovely $7$ dimensional Chern-Simons theory, so I went to Wanaka and bumped chairs and served chips to hungry skiers, and worried instead about the bottletops that my flatmate insisted on leaving all over the floor.

Theory Update 28

The PeSla discusses how the six quarks fit onto a categorical cube, with charge represented by path length. The zero charge source stands for neutrinos and the charge $3$ target for the charged leptons. Edges are directed by the addition of one twisted ribbon piece to a three strand particle diagram.

Happy New Year.

Theory Update 49

Let us replace the Pauli unknot arcs with the suggestive letters $i$, $j$ and $k$. Now we can carefully draw the unknot in three dimensions, making sure that each straight line segment follows a $45$ degree line with respect to a chosen set of axes. Thus one arc moves down the $45$ degree line, one moves up the $45$ degree line, and one moves along the $135$ degree line.

If the knot arcs were thickened to ribbons, the twisting moves would look like ribbon twists, one for each of the three planes in space, as in this paper.

Theory Update 118

Recall that Furey's Fano plane mixes the charged leptons and neutrinos, as in the projection of a charge cube on the $27$ qutrit paths.

The $21$ Fano paths remind us of the (triality labeled) 21 edges in the three dimensional associahedron polytope. Recall also how 16 paths (out of $27$) are placed on the associahedron. Space really wants to start out being three dimensional. Ribbon charge is specified using the quark path mixtures of the quandle plane. An $XXX$ charged lepton path obtains a unit charge from $1/3 + 2/3$. The six cyclic permutations of the neutrino source vertex themselves represent the hexagon of $6$ paths from source to target on a cube. The neutrino hexagon appears in the honeycomb dual to the tetractys brane, and may thus be considered as a surface of last scattering.

Old Symmetry

As in the tetractys qutrit path count, the hexagonal symmetries of 20th century physics include a doubling of points at the centre of the hexagon. Although clearly discrete, such hexagons are usually interpreted in terms of the continuum representation theory.

Observe how the quantum numbers $q$ and $s$ take values $-1$, $0$, $1$. The charge $q = \pm 1$ is now associated to a triplet of ribbon twists. Similarly, rotation of a Furey hexagon (Fano plane) gives color, and a tripled hexagon gives generations. There are four index hexagons in the tetractys: three at the corners and one in the centre.

One easily triples the state count on the baryon octet, accounting for quarks. This octet count now agrees with the tetractys.

Theory Update 93

Repetition is tiring, but what else can one do. So once again: why are there no zombie superpartners or fairy fields?

The stringers' extension of classical symmetries to supersymmetric analogues assumed that quantum gravity hinged on convenient gauge theory techniques, learned from the ordinary model for QFT. The ad hoc extension of multiplets from representation theory does not at all address the basic question that 20th century QFT poses: why representation theory and why these representations? When spacetime is emergent, there are no points to attach symmetries to! And emergent localization begs a redefinition of particle, since the old definition utterly hinges on manifest locality. This has always been obvious.

But more troublesome for stringers is probably the fact that an alternative ribbon spectrum exists, in a framework that can derive M Theory.

Twistors Again

Last year at Oxford, I was fortunate enough to attend a number of interesting String Theory seminars in the mathematics department. Actually, the seminars were all about twistor theory, because the local string theorists said that twistor theory was where all the interesting stuff was happening. A typical seminar would begin with a T duality transformation into a twistor space, and after this neither string theory nor the usual local formulation of the Standard Model made any appearance whatsoever.

Recall that algebraic aspects of twistors include the relation between its geometry and Jordan algebra. As kneemo points out, the Koide mass matrices may be viewed as elements of a Jordan algebra. Similarly, the $R_2$ factors used to build mixing matrices may be associated to Jordan algebras.

Some time ago, we discussed the relevance of the three moduli spaces of twistor dimension (namely $M(0,6)$, $M(1,3)$ and $M(2,0)$) with a total of $12$ degrees of freedom. Euler characteristics and the index theorem for the six point sphere then tell us that the number of generations must be $3$. This is a ternary analogue of the pair $M(0,3)$ and $M(1,1)$ of the Grothendieck tower, an idea associated to the ribbon graph papers of Mulase et al.

Theory Update 74

The fun representation for braids in $B_3$ now has two forms for the neutrino braid. First, using $\omega$ and $\overline{\omega}$ for a positive or negative crossing, we obtain the matrix shown. But the mirror particle representation on the left results in a dual diagonal matrix $(1, \overline{\omega}^2, \omega^2)$. This forces $\overline{\omega} = \omega^2$, which says that $\omega$ is a cubed root of unity.

As we know, the cubed root limits the allowed number of twists in a $B_2$ section of a braid. The cubed roots thus also label the charges $\{ 0, \pm 1 \}$ of particle ribbon diagrams. We have now described the $Z$ boson diagonal in $B_3$, rather than in $B_6$. Instead of three separated ribbons, we use the strand groupings $(1,2)$, $(2,3)$ and $(3,1)$ as holders for a twist.

Theory Update 103

Three loops in the Hopf fibration form the $(3,3)$ torus knot. This is one of three links of three components that are described by six crossings. The other two are the Borromean rings and the loop chain. For the torus knot, where all three components are linked together, breaking one loop does not unlink the other two loops. Links are often used to discuss quantum entanglement. Note that another way to draw the torus knot is as a three stranded boson ribbon graph, with double twists on each strand in the directions $(+--)$. There is a simple $B_3$ representation for this knot, given by $B^{-1} A B^{-1} A^{-1} B A^{-1}$ in the generators $A$ and $B$.

Theory Update 130

Let's get back to the basic reason that String Theory is obviously wrong. As neatly summarised in Furey's paper, we can list all particle states using the normed division algebras from triality: the reals, complex numbers, quaternions and octonions. Let us call this large collection of numbers $RCHO$. Note that dimensionally, $RCHO$ resembles $O \cdot O$, octonions with octonion coefficients. M theorists like kneemo have been studying Jordan algebras over such large collections.

Consider $J_{3}(O \cdot O)$, the algebra of $3 \times 3$ matrices over $O \cdot O$. Since $O \cdot O$ has (real) dimension $64$, $J_{3}(O \cdot O)$ has dimension $216$, from three copies of $64$ off the diagonal and $3 \times 8$ along the diagonal. Note that $216 = 2^3 3^3$, the information dimension for three qubits and three qutrits. The simple prime factorization allows us to parse matrix components in many ways, for both $2 \times 2$ and $3 \times 3$ matrices. For instance, we could use a $3 \times 3$ pseudo-algebra built with off diagonal copies of $J_{3}(C \cdot O)$, the $54$ dimensional bioctonion algebra of the doubled tetractys. The diagonal would then require objects of dimension $18$, twice the dimension for the two qutrit hexagon path diagram.

Compare this with particles. There are $54$ left and right handed leptons and quarks, including color and generation. Then $216 = 4 \times 54$, allowing for fermions, antifermions and mirror versions. This full list is easily expressed in terms of ribbon diagrams with three strands. Fourier supersymmetry creates all bosonic states. No fairies or zombies appear anywhere.

Theory Update 121

As noted in McElrath's 2008 paper, gravitational condensates involving neutrinos must also consider the electroweak bosons. From the braid diagrams, one obtains the mirror sector by flipping all twisted ribbon charges. The neutrinos, mirror neutrinos and photons are the only particles invariant under this process. The $Z$ boson, although neutral, consists of a colored triplet of $(+,-,0)$ ribbons. This triplet is the Fourier dual of the right handed leptons. Observe that a $Z$ boson is the only braid object composing with its mirror braid to form a massless photon, a dual to the left handed neutrino.

Twistor Motives III

This paper by Gulotta contains the dessins algorithm (of course the stringers like to talk about dimers and AdS/CFT). In the previous example, there were two columns of the $2 \times 6$ array that formed the $2 \times 2$ matrix shown here.
This diagram illustrates a typical replacement of four paths on a torus, of winding number $\pm 1$, by two paths with opposite winding numbers, defining the columns of the matrix. For general $2 \times m$ arrays one can replace single intersection points, four intersection points (as drawn here), and so on. Since the twisting of curves must avoid further intersections, there is a very limited number of moves and it is fairly easy to construct a diagram from a given array. Torus dessins (ribbon graphs) from secondary polytopes! Grothendieck would love it.