Sorry this is all a bit condensed, but people seem interested in(adsbygoogle = window.adsbygoogle || []).push({});

this mass gap question so here are some more thoughts.

Knots and Topos

---------------

In one of his papers on the Temperley-Lieb algebra, Lickorish says

that linear skein theory is useful rather than profound. But like

all magically useful things, it turns out that there is perhaps

depth here after all and that is in the realisation that skein

theory is about the topos theoretic issue of where the numbers

live, because in the topos approach even numerical invariants must

know their context.

The objects of the Temperley-Lieb category are ordinals

represented by dots in the plane. An arrow is a diagram of half

loops and lines. The algebra is given by formal sums over the

complex numbers up to planar isotopy such that the skein relations

of the Kauffman bracket (for A) hold.

The empty diagram on the right hand side of the second relation

has a one dimensional span = C. For a given value of A, this is

where the numerical invariant resides. The number 1 corresponds to

the empty diagram. This follows from the functorality of the

invariant. Similarly, the polynomial ring C[x] arises from the

diagram of a loop in an annulus, where the background 3sphere has

been replaced by a torus.

Loops are labelled with a spin j, representing j strands of the

loop. A ring morphism from C to C[x] arises from the insertion of

a tangle into a torus and the loop j is represented by x^{j} in

the commutative polynomial ring.

The Kauffman bracket becomes a full isotopy invariant under the

addition of a factor (- A)^{- 3 w} where w is the writhe of a link

diagram, which may be used to represent a framing for a knot.

A positive knot is one represesented by a braid diagram with only

positive crossings. For such knots, clearly the writhe is the same

as the total number of crossings. A deep connection between

Feynman diagrams and positive knots has been established by

Broadhurst, Kreimer and now Connes and Marcolli.

For a generalised zeta function (MZV) d is the depth, given by

(n-1) for a representation in B_{n}, and w = s_{1} + s_{2} + ... +

s_{d}.

The genus of a knot is defined to be the smallest integer g such

that the knot is the boundary of an embedded orientable surface in

S^{3}. One has 2 g = w - d.

Ribbon tree diagrams are instances of templates, or branched

surfaces with semiflows, which appear in dynamical systems theory.

The Lorenz attractor template is the pasted ribbon diagram with 2

holes. A positive knot embedded in this template is defined by a

braid word in two generators x_{0} and x_{1}, corresponding to the

two branch cuts of the surface. For example, the (4,3) torus knot

may be represented by the word x_{0}(x_{1}x_{0})^{3}.

There is a template on four letters which is universal in the

sense that all links may be embedded in it. This is very

interesting, because the letters x_{i} become quantum number

indexes appearing below when we consider the problem of solving

coherence conditions.

Now the three dimensionality of weak associativity relies on the

existence of non-trivial 3-arrows in the category in question. In

a braided monoidal category, these are essentially supplied by the

intertwiner maps. However, there is a subtlety to note here. The

tetrahedron treats U(VW) and (UV)W as a single edge. The component

of the weak nerve that distinguishes these objects is a 'parity

square'.

Considering a face as a puncture is what characterises the notion

of a qubit in the application of braids to quantum computers (see

reference below).

Let r = 5 set a value for A. The state space of a quantum computer

built on k qubits is given by the association of C^{2} \otimes k

to a disc with 3k marked points, on which there is naturally an

action of the braid group B_{3k}. Thus the computational power of

the computer translates directly into the number theoretic notion

of depth d introduced above.

If one believes that a higher dimensional analogue of area

operator should describe mass generation, then it is interesting

that such operators are related to sets of qubits, or rather

quantum gates, or intuitively an 'atom of logic'. Maybe the way in

which this idea naturally falls out of topos theoretic

considerations gives it a compelling physical significance.

Let q = A^{4}. Let a, b and c be complex variables such that the

Fermat equation a^{r} + b^{r} = c^{r} holds. The quantum plane

operators satisfy x^{r} = y^{r} = - 1. These ingredients are used

by Kashaev and Faddeev (see ref below) to construct solutions to

the Mac Lane pentagon. Spectral conditions are imposed in order to

set a 'coefficient' for the pentagon to 1. From Joyce's work,

however, it is clear that this parameter may have some physical

significance.

The point is that generalised spin labels are derived from the

requirement of a solution for the coherence conditions. This

replaces their selection by hand in other approaches to quantum

gravity.

It is well recognised that the usual m numbers are related to the

curvature of a puncture on the horizon in the LQG description of

black hole entropy. In a tricategorical context they label edges,

so the single arrow braided monoidal categories cannot hope to

capture the physical origin of mass.

The Parity Cube in Tricategories

--------------------------------

The objects of a tricategory T are labelled p, q etc. For each

pair of objects p and q there is a bicategory T(p,q). The

internalisation of weak associativity is a pseudonatural

transformation 'a'. The parity cube is labelled by bracketed words

on four objects, which may be loosely thought of as the

bicategories of particles. That is, one particle contains the

potential of all its interactions, and this information must be

mathematically realised.

The Mac Lane pentagon lives on 5 sides of the cube. The new top

face of the cube is the premonoidality deformation (of Joyce),

which closes the horn. This square appears as a piece of data for

a trimorphism, that is a map between tricategories, so that it

naturally appears in 4-dimensional structures.

The relation of this q to the braiding in the quantum group case

means that: if one believes (a) the deformation parameter q is

associated to the existence of a Planck scale somehow, and (b) one

has cohomological mass generation, then one really ought to

conclude that the appearance of a mass gap is associated with

quantumness.

The tetracategorical breaking of the hexagon amounts to the

breaking of topological invariance as described by Pachner moves

for 4D state sums based on fixed triangulations (spin foams). This

view thus explains why four dimensional gravity is not topological

in the usual sense of the word.

Higher dimensional categories are essential in exactly

characterising interactions between larger ensembles of particles,

since the Gray tensor product is dimension raising.

Some useful REFERENCES

------------------------

J. Roberts

Skein theory and Turaev-Viro invariants

Topology 34, 1995 p.771-787

L.D. Faddeev R.M. Kashaev

Quantum Dilogarithm

Mod. Phys. Lett. A9,5: 1994 p.427-434

A. Grothendieck

Pursuing Stacks

available at http://www.math.jussieu.fr/~leila/mathtexts.php [Broken]

C. Mazza V. Voevodsky C. Weibel

Notes on Motivic Cohomology

available at http://www.math.uiuc.edu/K-theory/0486

A. Goncharov

Volumes of hyperbolic manifolds and mixed Tate motives

http://arxiv.org/abs/alg-geom/9601021

J. Stachel

Einstein from B to Z

Birkhauser, 2002

Ross Street

Categorical and Combinatorial Aspects of Descent Theory

available at www.maths.mq.edu.au/~street/DescFlds.pdf

L.H. Kauffman M. Saito M.C. Sullivan

Quantum Invariants of Templates

available at www2.math.uic.edu/~kauffman/Papers.html

R.F. Williams

The universal templates of Ghrist

available at www.ma.utexas.edu/~bob/ghrist.ps

M. Freedman M. Larsen Z. Wang

A modular functor which is universal for quantum computation

http://arxiv.org/abs/quant-ph/0001108

Best regards

Kea

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# On knots, categories and mass gaps

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