# Arcadian Functor

occasional meanderings in physics' brave new world

## Tuesday, April 29, 2008

Browsing jobs online, with no regard for location, I came across a fantastic opportunity at the Department of Physics in Cambridge: they need a new waitress, and lunch and overalls are provided! Actually, I have been spending a bit of time on a more exciting job application, which I submitted today. Even if I don't get the job, it was fun trying.

### M Theory Lesson 178

Recall that $2 \times 2$ spin matrices are associated with the quantum Fourier transform for $q = -1$. The Weyl rule $UV = - VU$ may be thought of as a square with paired edges marked $U$ and $V$, just like in the planar paths considered by Kapranov. In 3 dimensions one draws paths on a cubic lattice. The paths on a single cube form the vertices of one of our favourite hexagons. A simple braid on three strands is formed by composing two edges of this hexagon, which correspond to two faces on the cube. Since the Weyl edges $U$ and $V$ have become faces in 3D, this composition can represent fermionic spin, just as Bilson-Thompson said.

## Sunday, April 27, 2008

### Light Nostalgia

Louise Riofrio continues with excellent cosmology posts, and now Carl Brannen also weighs in on the subject. I was wondering what originally got me very interested in the subject of a varying $c$, and I decided it probably happened around 1995, when I spent a few months studying the early physics papers on quantum group fiber bundles.

I seem to recall that these papers were not particularly mathematically sophisticated, but one element stood out: whereas a classical principle bundle looks the same at every point, the deformation parameter in a quantum bundle may easily vary from point to point. Even in those days, people thought a lot about relating deformation parameters to $\hbar$. This was all just a mathematical curiosity, until it became clear that some tough (and extremely interesting) algebraic geometry, and other mathematics, lay at the bottom of it. (Of course, all roads led to category theory in the end).

Algebraic geometers love spaces with extra structure which varies from point to point. They talk about spectra (usually of rings) and we need not be afraid of these gadgets because they are naturally specified by a functor from a suitable category of algebras into a category of spaces. And it turns out that this functor is best understood from the point of view of a special topos, because the weird topologies that algebraic geometers like to use are neatly encoded by axioms of Grothendieck. (In fact, this is where the idea of a topos comes from in the first place).

At the time, I believe it was Zamolodchikov who advised me to ditch lattice gauge theory (which I was supposed to be doing) for something more interesting. In the end, I did give up the lattice gauge theory, but I can't say it was because I listened to anybody's advice. (And as it turns out, lattice gauge theory has actually done rather well over the last decade).

I seem to recall that these papers were not particularly mathematically sophisticated, but one element stood out: whereas a classical principle bundle looks the same at every point, the deformation parameter in a quantum bundle may easily vary from point to point. Even in those days, people thought a lot about relating deformation parameters to $\hbar$. This was all just a mathematical curiosity, until it became clear that some tough (and extremely interesting) algebraic geometry, and other mathematics, lay at the bottom of it. (Of course, all roads led to category theory in the end).

Algebraic geometers love spaces with extra structure which varies from point to point. They talk about spectra (usually of rings) and we need not be afraid of these gadgets because they are naturally specified by a functor from a suitable category of algebras into a category of spaces. And it turns out that this functor is best understood from the point of view of a special topos, because the weird topologies that algebraic geometers like to use are neatly encoded by axioms of Grothendieck. (In fact, this is where the idea of a topos comes from in the first place).

At the time, I believe it was Zamolodchikov who advised me to ditch lattice gauge theory (which I was supposed to be doing) for something more interesting. In the end, I did give up the lattice gauge theory, but I can't say it was because I listened to anybody's advice. (And as it turns out, lattice gauge theory has actually done rather well over the last decade).

## Saturday, April 26, 2008

### Return of the Jedi

There's only one thing to say to the next restaurant patron who thinks they need to add the change for me, or the next guy who thinks he needs to point out to me that physical theories have to agree with experiment: I'll be back. (Thanks to Backreaction for the picture)

## Thursday, April 17, 2008

### Ternary Geometry III

Topological field theory enthusiasts like extending the 1-categorical constructions to the world of 2-categories. A candidate source category is then a category of spaces with boundaries which themselves have boundaries. That is, the vertices are the objects, the edges the 1-arrows and surfaces 2-arrows. In the world of ternary geometry this brings to mind the three levels of the generalised Euler characteristics, which were seen as cubed root of unity analogues to the alternating signs that occur in the world of 2. Since the boundary of a boundary is not necessarily empty, it makes more sense to look at the cubic relation $D^3 = 0$ than the usual homological $D^2 = 0$ of duality. Since the latter arises from a fundamental categorical concept, namely monads, one would like to understand the ternary categorical construction. This is why M Theory looks at ternary structures such as Loday's algebras and higher dimensional monads.

## Wednesday, April 16, 2008

## Sunday, April 13, 2008

### Purple

Today's pretty picture, from the University of Bristol website, is a convergent beam electron diffraction pattern.

## Saturday, April 12, 2008

## Thursday, April 10, 2008

### Achilles and the Tortoise

Zeno of Elea's lost book is said to have contained 40 paradoxes concerning the concept of the continuum. The paradoxes are mostly derived from the deduction that if an interval can be subdivided, it can be subdivided infinitely often. As an Eleatic, Zeno subscribed to a philosophy of unity rather than a materialist and sensual view of reality. This led to greater rigour in mathematics, since more emphasis was placed on logical statements than on physical axioms laid down arbitrarily on the basis of (inevitably deluded) experience.

Most famously, the paradoxes discuss Time as a continuum. If we have already laid out in our minds a notion of classical motion through a continuum, the infinite subdivisibility of Time must follow. But note the introduction here of a separation between object and background space. To the Eleatics, this is the source of the problem, not the mathematical necessity of infinity itself. By placing a fixed finite (relative to the observer) object in a continuum, we have allowed ourselves to ask questions about its motion which are physically unfeasible.

But the resolution comes not from concrete physical axioms about an objective reality, based as they are on the very prejudices that lead to paradoxes in the first place. Rather, it comes from refining the mathematics until its definitions are capable of quantitatively describing the physical problem correctly. We have known this for thousands of years, but do many physicists really appreciate this today?

Most famously, the paradoxes discuss Time as a continuum. If we have already laid out in our minds a notion of classical motion through a continuum, the infinite subdivisibility of Time must follow. But note the introduction here of a separation between object and background space. To the Eleatics, this is the source of the problem, not the mathematical necessity of infinity itself. By placing a fixed finite (relative to the observer) object in a continuum, we have allowed ourselves to ask questions about its motion which are physically unfeasible.

But the resolution comes not from concrete physical axioms about an objective reality, based as they are on the very prejudices that lead to paradoxes in the first place. Rather, it comes from refining the mathematics until its definitions are capable of quantitatively describing the physical problem correctly. We have known this for thousands of years, but do many physicists really appreciate this today?

## Wednesday, April 09, 2008

### Knot Monkey

Carl has been playing with knots that cover a sphere. Rather, when a piece of cord or wool is used, its substantial thickness allows a covering of a sphere with a small finite number of crossings.

In the mathematical world, ideal knots are drawn with an infinitely thin line. Such lines can still fill a sphere (a la Thurston) but monkey knot curves with crossings are more interesting in the context of M theoretic quantum information, and it would take some (kind of) infinite number of crossings to properly fill out a sphere. But basically, the monkey knot is a set of Borromean rings in three dimensions (or Borromean ribbons). The rings form a 6 crossing planar diagram. Note that if the outer 3 crossings are smoothed, one obtains a trefoil knot from the centre of the rings (along with a separate unknotted loop). I can't help wondering what this means.

In the mathematical world, ideal knots are drawn with an infinitely thin line. Such lines can still fill a sphere (a la Thurston) but monkey knot curves with crossings are more interesting in the context of M theoretic quantum information, and it would take some (kind of) infinite number of crossings to properly fill out a sphere. But basically, the monkey knot is a set of Borromean rings in three dimensions (or Borromean ribbons). The rings form a 6 crossing planar diagram. Note that if the outer 3 crossings are smoothed, one obtains a trefoil knot from the centre of the rings (along with a separate unknotted loop). I can't help wondering what this means.

## Tuesday, April 08, 2008

### M Theory Lesson 177

Note that an intersection on the triangle plane arrangement becomes a square face on the cube. A (directed) cone from the top vertex will pick out the central horizontal edge of the cube, with the central point of the hexagon at one end representing the triangle. Observe that the number of edges in corresponding diagrams (planar arrangements to graphs) remains unchanged, whereas faces become vertices and vertices become faces. That is, this is a kind of Poincare duality.

## Sunday, April 06, 2008

### M Theory Lesson 176

In Ben-Zvi's notes of recent work by Ben Webster et al (which he calls the cutting edge of mirror symmetry math) there is this diagram of a triangular arrangement of planes and its associated graph. The vertices represent the 7 regions of the Euclidean space and the edges an adjacency via an edge segment. Notice how this looks like a centered hexagon, or one side of a cube. This is a kind of Cayley graph. The permutations of four letters (which label the vertices of the permutohedron) also give a cubical Cayley graph. Koszul duality is about the correspondence between intersections of the planes and cones emanating from such points in the plane arrangement.