Witten on Yang Mills

Witten speaks at the latest Simons Center conference on mass gaps in Yang Mills theories. In four dimensions, he stresses that the role of the mass gap is to hide quantum phenomena at large distances, and that no simple perturbation theory can really see the mass gap. A theme of the conference seems to be a renewal of interest in constructive approaches, starting with tractable questions.

So the talk focuses on three dimensional theories, starting with the pure Yang Mills action for $R^3$. Here a mass gap exists on dimensional grounds, with $m = w g^2$ for $w$ some constant and $g$ the coupling. But what happens at long distances in three dimensions? Several approaches are briefly reviewed, including (i) a string tension Hamiltonian approach, and (ii) Feynman and Singer's Hamiltonian for the space $R^2$. In (ii) there is the usual (infinite dimensional) space of connections $A$ and gauge transformations $G$. Witten writes down the basic Hamiltonian

$H = g^2 \int \textrm{Tr} E^2 + \frac{1}{g^2} \int \textrm{Tr} B^2$

noting that the first term dominates in the limit $g^2 \rightarrow \infty$, and that the eigenvalues of $H$ should be basic numbers, like $\zeta (2)$.

But he appears to believe that a currently tractable approach to mass gaps would look at non-trivial CFTs, for the following reasons. First observe that (in three dimensions) if we add a few essentially massless fermions (like up and down quarks) there is no mass gap. Now we can add a Chern-Simons term, as a mass gap term, to the usual Yang-Mills action, where $m$ is proportional to $k$, the CS integral coefficient (the fourth dimension is irrelevant). If this (CS) theory is rigorously defined in three dimensions, there should be a theory for any $3$-manifold $M$ with any Riemannian metric. Scale the metric by a factor $t \rightarrow \infty$. That is, we imagine that human experiments are limited by the characteristic scale of the manifold $M$, and at long distances there is the TFT (IR limit). This is of course the context of Jones polynomials and knots.

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.

Higgsy Codes IV

The cubed root symbol $\omega$ may be thought of as a non zero element in the four element field. In the hexacode description of the miracle octad there are $3 \times 6$ arrays, that is, three length $6$ words in four letters. A key three word matrix is built from the $3 \times 3$ identity and the Fourier operator,
in analogy to the (gauge fixed) $2 \times 4$ arrays that create a Grassmannian Minkowski space in the $2 \times 2$ twistor formalism. Conway has a similar construction for the ternary code, known as miniMOG. The hexacode is used to specify five points from which a Steiner octad is created.

Higgsy Codes III

On Cullinane's page on the miracle octad generator we find a beautiful note by Robert Wilson, the Wilson of the new Leech lattice, for R. T. Curtis' 60th birthday in 2007. The note looks at Lie algebra root systems in terms of integral quaternions and octonions. He defines an order $24$ group

$\omega \equiv \frac{1}{2} (-1 + i + j + k)$

with elements $\{ \pm 1, \pm i, \pm j, \pm k \}$ and $\{ 1, \omega, \overline{\omega} \}$, and the neat relation $i^{\omega} = j$. This gives a root system for $D_4$ and is extended using norm $2$ elements to a root system for $F_4$. Of course, with the octonions, we get $E_8$. Morally, the octonions are responsible for all exceptional Lie algebras. The exceptional algebras $G_2$, $F_4$ and $E_8$ have orders $14$, $52$ and $248$, with respective prime factors

$7 = 1 + 2 + 2^2$
$13 = 1 + 3 + 3^2$
$31 = 1 + 5 + 5^2$.

Wilson spots the combinatorics of finite projective planes, but then notes we can also write the triality

$14 = 2 (2^3 - 1) = 16 - 2$
$52 = 2 (3^3 - 1) = 54 - 2$
$248 = 2 (5^3 - 1) = 250 - 2$

where, as kneemo keeps saying, these numbers appear in qutrit symmetry groups. Recall that $14$ also counts the vertices of the three dimensional associahedron, and $16$ the extension to crossing partitions, while $54$ is the dimension of the bioctonion algebra. Speaking of trialities, the largest Heegner number is also written as

$163 = 1 + 1 + 1 + 2^3 + 3^3 + 5^3 = 54 + 54 + 54 + 1$

where $2^3$ counts vertices on a parity cube, $3^3$ paths for the tetractys, and $5^3$ paths for $5$-valued states.

Higgsy Codes II

One more pertinent fact about the number $28 = 27 + 1$: it counts the number of bitangents to a quartic curve, or pairs of lines from the $56$ lines on a (degree $2$) del-Pezzo surface. Thus it gives the number of (odd) theta characteristics. For us, $56 = 2 \times 28$ will always be the dimension of the FTS for the $3 \times 3$ octonion Jordan algebra, or the $56$ triangles of the genus $3$ Klein quartic, which has an automorphism group of order $168 = 3 \times 56$ $= 7 \times 24$, as there are $7$ sides on the heptagon tiles.

The Mathieu group $M_{24}$ can also be constructed using the Klein quartic symmetries, along with an extra permutation associated to the (Leech lattice's) small cubicuboctahedron, as mentioned recently by kneemo. If we can have this much fun with classical codes, just imagine how much fun we can have with braids!

Higgsy Codes

Dharwadker's four color theorem uses the Steiner system $S(5,8,24)$, otherwise known as the Witt design. To construct the Steiner system, one starts with the extended binary Golay code. This has $2^{12} = 4096$ length $24$ words, selected from all possible binary words by writing a list of length $24$ binary words so that each word differs from the previous ones in at least $8$ places. It turns out that the code words are partitioned

$4096 = 1 + 759 + 2576 + 759 + 1$

into words with no $1$-bits, words with $8$ $1$-bits, words with $12$ $1$-bits, $16$ $1$-bits and $24$ $1$-bits, respectively. The $729$ words with $8$ $1$-bits form the Steiner system, which is defined as a set of $24$ points, along with the $759$ special subsets of size $8$, called blocks, such that every $5$ points is contained in one of the blocks. The number $759$ is very interesting. It can be partitioned into

$759 = 1 + 280 + 448 + 30$

which counts, respectively, the number of subsets that intersect a given subset in $8$, $4$, $2$ or zero elements. That is, there are $30$ blocks with no intersection with a given block, and $729 = 27^2$ with a non empty intersection. We write $280 = 4 \times 70$, $448 = 16 \times 28$ and $30 = 30 \times 1$, where $70$, $28$ and $1$ are weights for the Reed-Muller code $RM(2,3)$. The $4$, $16$ and $30$ appear in the bottom row of the Pascal triangle below. The number 759 is the summit of this triangle, which counts the number of blocks that (i) contain a point set with $n$ elements and (ii) are disjoint from another set with $m$ elements. The union cardinality $m + n$ is fixed in each row. So there are $253$ occurrences of any given object, and a complementary $506$ blocks that do not contain it.

There are $42504$ ways to choose $5$ objects from $24$, and $42504 = 759 \times 56$. To obtain the $8$ bits, after choosing the first $5$, there are $969$ possibilities (namely $3$ out of $19$) and we see that $969 \times 759$ is indeed the binomial coefficient $B(24,8) = 735471$. Why the redundancy factor of $56$, the FTS dimension? This counts the number of ways of choosing $3$ objects from $8$, namely the $3$ that are omitted from the set of $5$.

The Mathieu group $M_{24}$ has a 729 dimensional permutation representation with suborbit lengths given by the $1$, $30$, $280$ and $448$. Similarly, the Mathieu group $M_{12}$ is associated to the extended ternary Golay code. The two binary and two ternary Golay codes, with lengths $11$, $12$, $23$ and $24$, are the only binary and ternary codes with suitably nice properties. There are $729 = 27^2$ words in the ternary code of length $11$, and the extended ternary code also forms a Steiner system, on $12$ objects. The $12$ objects are thought of as a projective line over the $11$ element field $F_{11}$ with the addition of $\infty$. The blocks are built using fractional linear transformations, which we know how to cover with Jordan algebra braids.

Cat Neutrinos

Recall that Ma's finite group $A_4$ is a subgroup of the permutation group $S_4$ on four letters. Thus it labels $12$ of the $24$ vertices of the permutohedron polytope in three dimensions, and contains the Klein four group (using $2143, 3412$ and $4321$). The reduction of the permutohedron to the associahedron, which itself exhibits triality, suggests a selection of $7$ vertices out of the $14$, which happens to be the number of points (or lines) in the Fano plane.

The neutrino literature tends to associate these finite groups directly with larger continuous symmetry groups, but this really misses the fundamental nature of arithmetic structure in emergent geometry. When the neutrino set is doubled to include mirror neutrinos, we can presumably characterise mixing with the full set of vertices on operad polytopes. Similarly, Fourier circulants can arise from a Fano star, or double tetractys. Since the tetractys indexes a much larger (secondary) polytope, based on the central point, the polytopes provide more information than the basic finite groups.

Theory Update 127

Dolan's triality studies conformal field theories at $c = 24 = 8 \times 3$, where an $8$ is associated to the qubit edge of a tetractys. It should not surprise us then that massive theories contain trialities using the centre of the tetractys. As it happens, the four largest Heegner numbers all equal $1$ mod $6$.

These are the numbers $d$ that generate near integers using the j-invariant value

$j( \frac{1 + \sqrt{-d}}{2} ) = k^3$

for $k$ an integer. The number $k^3$ always has a factor of $27 \times 8$. The Euler primes that generate the Heegner numbers are supersingular. Observe that the largest Heegner number uses the dimension of the bioctonion Jordan algebra.
Strangely, if we restrict to the massless dimensions, such as $24$ in $27$, we obtain the following.

M Theory Lesson 360

Mitchell directs our attention to a recent paper by prominent M theorists, relating the associahedra to braids. Wow, who would have thought of that. What a good idea!

OPERA Act IV

A search of stringy maths tachyon papers brought to light this fantastic 2004 paper, with tachyons appearing when legs cross in a $(p,q)$ web diagram.
Looks familiar, doesn't it? These webs are supposed to describe stringy brane configurations, but as usual we will think of them as toric variety diagrams. The paper discusses getting rid of tachyons, but do we need to?

Theory Update 115

As number 26 suggests, the Riemann zeta value $\zeta (-3)^{-1} = - 120$ is associated to the four dimensional quaternions, whereas $\zeta (-7)^{-1} = - 240$ is associated to the octonions. The three parallelizable spheres give the triplet $( \zeta(-1), \zeta(-3), \zeta(-7) )$. Under the functional switch $s \mapsto 1 - s$ of zeta arguments, we obtain the triplet $(2,4,8)$ of dimensions for the corresponding number fields. In categorical scattering theory, we often note that $\zeta (n)$ is associated to $n+3$ particles, giving the triplet $(5,7,11)$ of primes for the exceptional Galois groups.

In the moduli triplet of twistor dimension, the genus $2$ orbifold characteristic was $\zeta (-3)$. The number $120$ counts the vertices of the three dimensional permutoassociahedron, which is obtained by turning each vertex of the $24$ vertex permutohedron into a pentagon. Hence, $120 = 24 \times 5$. In the complex case we have the string regulator, $\zeta (-1)^{-1} = -12$, like the $12$ pentagons that tile the Riemann sphere for the complex moduli space $M_{0,4}$, or the $12$ sides of the planar permutoassociahedron.

And if my calculator hand is working, real arguments give nice real numbers like $\zeta (26) = 1.0000000149$.

Theory Update 106

With renormalisation, a basic circulant may be written as a matrix $A$ with integral entries. Let us consider powers, such as
More generally, with coefficients $n$ and $n - 1$ we obtain

For example, when the initial coefficient pair is $(3,2)$ the sequence of powers gives coefficient pairs $(j, j - 1)$ that sum to $5^{m}$. That $3$ is odd corresponds to the fact that $5 = 1 \textrm{mod} 4$, since we write $n = 2k + 1$ and the sum is then $4k + 1$. The matrix sequence demonstrates the obvious fact that $5^m = 1 \textrm{mod} 4$ for all $m$, as the first coefficient is always odd.

A number theorist will tell you that an odd prime $p$ equal to $1 \textrm{mod} 4$ satisfies $p = a^2 + b^2$ for unique $a$ and $b$. The integer ratio $a/b$ comes from $(2k)!$. For $p = 5$, $a$ and $b$ are usually found from the Gaussian integer factorization $5 = (2 + i)(2 - i)$, where $a = 2$ and $b = 1$. Each Gaussian factor may be expressed as another basic two crossing braid, as in where the factors also come from the Jordan product. The number $x$ (with $x^2 = 1$) is omitted here, since it always appears in the same places. Note that the square of one such factor multiplies (up to signs) the first coefficient by $3$ and the second by $2$, as in the matrix above. How nice to see products of basic $B_3$ configurations expressing simple arithmetic rules.

Nadler at KITP

The mathematician David Nadler is at KITP, talking about work with David Ben-Zvi on three dimensional topological field theories. The talk starts with the now familiar statement: there is a six dimensional field theory out there that contains all of mathematics. This must be apparent to the classical geometers and representation theorists who work on the Langlands correspondence!

Nadler thinks about compactifying ($N=4$ SYM) gauge theory in four dimensions on a circle, in one of two distinct ways, both leading to a three dimensional topological field theory based on $2$-categories of modules. Let us look at the $\chi_{LG}$ case, that is the character theory for a loop group. Here $G$ is our (semisimple) gauge group and $B$ a Borel subgroup.

As a functor, $\chi_{LG}$ assigns a certain $2$-category of (highest weight) module categories to a point, and then character information to an $S^1$. Geometric Langlands is about surface operators in a four dimensional theory, and these have become one dimensional operators in the three dimensional theory. The right category to consider turns out to be what string theorists would call the monoidal category of $A$ branes in $T^{*}(K)$ where $K = I \backslash LG / I$ for $I$ an Iwahori subgroup. If we think of $LG$ as matrices with Laurent series entries, then $I$ is the set of matrices with power series entries such that the constant term is in $B$, the upper triangular matrices. The end aim is to understand $S$ duality (between $A$ and $B$ models) in this rigorous context, using a Grothendieck-Springer resolution for ordering the eigenvalues of a matrix. It depends on an elliptic curve associated to the maximal torus for the Langlands dual of $G$.

One humorous aspect of the talk was Nadler's insistent apology for not knowing the physics jargon. At one point, he even explained his hesitency to use the word point, at which point the physicists said the word point was quite all right (clearly, these are not categorical physicists).

Theory Update 101

In this paper, the category theorist Michael Batanin explains how the groupoid of braid groups may be viewed as a collection of ordinals, and discusses equivalences between categories of braided and other operads. As usual, an $n$-ordinal is a rooted tree with $n$ levels. Things are most interesting for small values of $n$.

For example, when $n = 2$ an ordinal is given by a string $(n_1,n_2, \cdots, n_k)$ of ordinary ordinals, as in the arguments of a generalised zeta value. The $n = 2$ case already defines a large class of polytopes, including the $1$-ordinal associahedra.

Theory Update 100

Kostant's $e_8$ algebra is clearly about qubits and qutrits. One considers the qubit cube $Z_{2}^{5}$, corresponding to the field $F_{32}$ with $32$ elements. This sits inside the $2 \times 2$ matrix group $SL(2,F_{32})$, conveniently in the form which is used for braids in emergent geometry. A $32$ dimensional information space could be a plane over the bioctonions. Triality also appears, because $248 = 192 + 56$.

Theory Update 99

Recall that a $2$-ordinal is a pair $(m,n)$ of ordinary $1$-ordinals, denoted by a two level tree with $m + n$ leaves. A one level tree with $m$ leaves is usually used as a symbol for the associahedron of real dimension $m - 2$. That is, the vertices of an associahedron are given by a binary planar tree obtained by expanding the higher valent nodes until all nodes are trivalent. A one level tree can be turned into a two level tree by adding a single root edge as a base level. In this way, there are two sets of associahedra that naturally occur in the $2$-ordinals, one for $m$ and one for $n$. The other $2$-ordinals denote more complicated polytopes, whose vertices are similarly obtained. Since the associahedra tile the real points of the moduli spaces for Riemann spheres with $k$ marked points, and the associahedra are also used to count terms in real $N = 4$ SYM scattering amplitudes, the $2$-ordinals fill in the gaps, much like the complex numbers fill in the gaps between the real and imaginary axes in the plane.

Theory Update 98

The magic square of number fields is a $4 \times 4$ square $M_{ij}$ with the $16$ entries given by pairs $(A(i),A(j))$ of number fields, for $A(k)$ one of the real, complex, quaternion or octonion fields. Thus the bioctonions correspond to $M_{24}$ or $M_{42}$. People who love Lie algebras will tell you that this entry gives the algebra for the group $E_6$. The fundamental representation of this group has complex dimension $27$, just like the dimension of the three qutrit path space for a $3 \times 3$ Jordan algebra.

This and other coincidences are not surprising, because qudit path spaces are responsible for the emergence of all classical spaces and their symmetries. With the octooctonions at $M_{44}$ we obtain the $248$ dimensional $E_8$ Lie algebra. There are many ways to partition the integer $248$, such as Kostant's nice way, with $248 = 31 \times 8$. When higher categories allow us to twist classical logic, a number field is not just a boring set of things that can be added and multiplied.

Theory Update 97

Everyone knows that it takes three dimensions to tie a knot. Modern knot theory is very important in mathematical physics, where invariants for knots are typically computed using interesting quantum field theories. A simple knot, or link, is formally a set of closed loops in a background space.

The easiest way to think of a flat Euclidean space as a collection of lines is to take an uncountable set of parallel lines. For a sphere, one could use an uncountable set of parallel loops, with two pointlike loops at the poles, but there is a nice way of creating certain spheres entirely out of loops, known as a Hopf fibration. For example, the three dimensional sphere may be thought of as flat three dimensional space with an extra point at infinity. In the diagram, the point at infinity closes the line into a loop. Another loop of the Hopf fibration is linked to this large loop, as shown.

The reason this works is that the three dimensional sphere can be embedded as a unit sphere in a four dimensional flat space, which is given the structure of a complex number plane $P = \{ z_1, z_2 \}$. The Hopf fibration is a map

$h(z_1, z_2) = (2 z_1 \overline{z}_{2} , |z_1|^2 - |z_2|^2)$

on $P$. The three dimensional sphere maps to a two dimensional sphere, such that each point on this $2$-sphere comes from a loop. We think of the $2$-sphere as the set of complex numbers plus a point at infinity. There are also Hopf fibrations for other number fields, namely the reals, quaternions and octonions. For the octonions, one maps a $15$ dimensional sphere to the $8$ dimensional one.

In the compex number case, there is a way to create loop fibrations for higher dimensional complex spaces. For the octonionic plane, however, one cannot take the $31$ dimensional sphere and obtain a fibration. But like Peano before us, we would like to build other spaces out of simple curves. We could start with other systems of numbers, like the bioctonions.

Daniel's Comment

Recently, Daniel made a comment that I accidentally deleted. He was reading Carl Brannen's paper on spin path integrals, and he noticed that the spin eigenvalues $\pm 1/2$ were obtained from the infinite series

$1 + 1 + 1 + 1 + \cdots = - \frac{1}{2} = \zeta (0)$
$1 - 1 + 1 - 1 + \cdots = \frac{1}{2} = \eta (0)$

which are explained on that old post at the Everything Seminar. Here $\zeta$ is of course the Riemann zeta function and $\eta$ the Dirichlet eta function. These closely related functions both have a functional relation responsible for extending the function to the complex plane.

In constructive number theory, this will be the typical fashion in which quantum numbers arise. The initial $\pm 1$ may be interpreted as the bosonic spin eigenvalues in the steps that create the emergence of fermionic spin. However, we should not forget the supersymmetry between fermions and bosons in the mathematical formulation of emergent geometry. For example, we know that

$1 = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots = \sum \frac{1}{2^n}$
$-1 = 1 + 2 + 4 + 8 + \cdots$

is one means of obtaining a bosonic eigenvalue from a fermionic one. Two eigenvalues may be collected into a matrix:

Higgs Bundles and Arithmetic

Now that fairy fields are unofficially dead, we can resurrect the word Higgs in a mathematical context! At the latest KITP conference, the mathematician Hausel gives a mesmerising talk about moduli spaces of Higgs bundles, encompassing a theorem for the Langlands correspondence. His appearance at a physics conference is explained by a stringy description of his conjecture on polynomial invariants for moduli spaces.

The stringy connection creates what they call (quote) quantum motivic Pandharipande-Thomas invariants, associated to D6 branes wrapping a Calabi-Yau $3$-fold. But Hausel seemed happier sticking to Chern-Simons theories with a finite gauge group, which is to say that the talk focuses on $GL_{n}(F)$, for $F$ a finite field, and the character variety given by $2g$ matrix generators for a curve of genus $g$, in terms of a root of unity $\textrm{exp}(2 \pi i d/n)$ for $d$ coprime to $n$. For $g = 1$ the Stone von Neumann theorem tells us what this variety is, namely the cotangent bundle of $\textrm{Jac}(C)$. By fancy algebraic geometry, the character variety is like the moduli space of semistable rank $n$ degree $d$ Higgs bundles.

Higgs bundles go back to Hitchin (who now seems to like twistors and octonions). The technical part of the talk stresses the importance of finding a suitable filtration for cohomology. On the character variety side, mixed Hodge polynomials $H(X,q,t)$ give Hausel's $E$ polynomials $E(X,q)$, which have nice motivic properties, meaning that one just ends up counting points for varieties over finite fields.

Perhaps the physics of the real world actually has more to do with the simple arithmetic picture than the stringy one?