- Knot theory is important in understanding the physics of particles with fractional statistics: anyons or parafermions. These particles, which can exist in one or two dimensions have properties between fermions and bosons. The symmetric group is the symmetry of fermions and bosons, while the braid group from knot theory plays the same role for anyons.
- Knot theory is important in canonically quantised quantum gravity. Where knotted loop states provide a basis of solutions to the quantum gravity equations. This is described in the important loop representation of quantum gravity.
- Knot theory is closely related to quantum groups. These are a generalisation (or deformation) of classical Lie groups and are important in condensed matter theory, string theory and other physics. Knot theory seems to be very closely related to symmetry.
- Quantum groups are also used to construct Topological Quantum Field theories which can be used to find invariants of manifolds.
- closed strings which do not have any events in common commute or anticommute. This is important because it can be interpreted to mean that strings only interact when they touch.
- the algebra contains a subalgebra isomorphic to a Clifford algebra. It also has a homomorphism onto a Clifford algebra which is defined by stripping out the string connections. This is important because the algebra of creation and annihilation operators for fermions is also isomorphic to a Clifford algebra. This justifies the interpretation of this algebra as a model of discrete closed strings made from fermionic partons.
- The length two strings generate an orthogonal group acting on the vector space spanning events. The symmetric group permuting events exists as a subgroup of this. It follows that the symmetry of event-symmetric space-time is included in this supersymmetry.
- Is quantum gravity algebraic?, Louis Crane
- Symmetry in the Topological Phase of String Theory, Phil Gibbs
- What can we learn from the study of non-perturbative quantum general relativity?, Lee Smolin
- Strings. Loops, Knots and Gauge Fields, John Baez
- Is String Theory in Knots?, Phil Gibbs
- Symmetry in the Topological Phase of String Theory, Phil Gibbs
Part 8 of 9
Is String Theory in Knots?
Surely Not Knots
When I was a mathematics student at Cambridge back in 1980, I remember going to one of John Conway's popular lectures which he gave to the mathematics clubs. This one was about knot theory. Conway performed a series of tricks with bits of rope to demonstrate various properties of knots. A fundamental unsolved problem in knot theory, he told us, is to discover an algorithm which can tell when a loop of string is a knot or not.
It is possible to tie up closed loops of string into complicated tangles which can nevertheless be untied without cutting the string. But suppose I gave you a tangled loop of string. How could you determine if it could be untied?
Conway showed us a clever trick with groups which enabled him to determine that some knotted loops could not be untied, but there were others which were not classified in this way. Conway had generalised a polynomial invariant of knots first discovered by Alexandria many years ago. The Conway Polynomial was quite a powerful tool to distinguish some knots from others, but it could not separate all.
I remember thinking at the time that this was a piece of pure maths which would never have any useful applications apart from providing a way of proving that your boat can't slip its moorings, perhaps. Mathematicians delight in these kind of problems.
Ten years later a dramatic change had taken place. Knot theory now looked like it was going to have applications to solving quantum gravity and probably other problems in condensed matter theory. Louis Kauffman had even written a substantial book called Knots and Physics (World Scientific). Conway's Knot Polynomial had been generalised and the problem of classifying knots seemed all but solved.
I am not going to try to review all those things which have been found out recently about knots. Instead I refer you to Kauffman's book and John Baez: This weeks finds in Mathematical Physics which is where I learnt just about all of what little I know on the subject.
To summarise, I will list just a few points of interest here
From this point on things are going to get more technical and I am going to assume that the reader knows some maths.
From the Symmetric Group to the Braid Group
The principle of event symmetric space-time states that the universal symmetry of physics must have a homomorphism onto the symmetric group acting on space-time events. Now the symmetric group can be defined by the following relations among the transposition generators a1, a3, a3,...
ai2 = 1
The braid group is defined in the same way but with only the former relation. Put into words, this means that the braid group describes a symmetry where it does not matter in which order you exchange things but if you exchange two things then exchange them again you don't necessarily get back to what you had before.
There is a homomorphism from the braid group onto the symmetric group generated by the second relation. This means that the braid group is also a candidate for part of the universal symmetry according to the principle of event-symmetric space-time. In that case space-time events would behave like particles with fractional statistics.
Discrete String Theory
Now I will turn to another question. Are strings discrete? In string theory as we currently know it there is not much indication that string theory is discrete. Strings are described as continuous loops in space. However, there has been some interesting work by Susskind and others which does seem to suggest that string theory could be discrete. It may be possible to describe strings as objects made of small partons strung together. These partons would not exist as hard objects but can be conceptually subdivided and rejoined. They are points on the string which describe the topology of its interactions.
If the partons can be subdivided then they must be permitted to have fractional statistics. They must live on the string world sheet. The statistics of a whole loop of string would be the sum of the fractional statistics of its partons and would be an integer or half integer so that the string can live in three dimensional space.
If space-time is event-symmetric and we wish to consider event-symmetric string field theory, then a discrete string approach is essential. The partons of the string can be tied to the events through which the string passes. It will be permitted to pass through space-time events in any order it likes. In this way strings can tie together the events of space-time and provide an origin of topology in an otherwise unstructured event-symmetric universe.
If strings are formed from loops of partons with fractional statistics then it seems natural to allow them to be knotted. We should look for ways of describing this algebraically in an event-symmetric string theory.
String theorists are now also turning to higher dimensional membrane theories. If strings can be made of partons then surfaces, or 2-branes, can be made from strings. The process could continue ad infinitum. Space-time itself might be viewed as a membrane built in this way. There may be structures of all dimensions in physics. The 2 dimensional string world sheets and 3 dimensional space-time are more visible only because they stand out as a consequence of some as yet unknown quirk in the maths.
Event-Symmetric Open String Theory
As a first step I have constructed new types of super-symmetry inspired by the concept of discrete event-symmetric strings. The first of these is for open strings.
Imagine space-time as a large number N of discrete events which are arbitrarily numbered 1, 2, ... , N. An open ended string will be defined simply by the sequence of events it passes through. An example would beA general string of length 4 might be written
a, b, c, d are variables for the events the string passes through.
Strings can be any finite length starting from the null string of length zero which will be denoted by a pair of empty parentheses, ().
These strings are taken as the defining basis of a vector space.
I define multiplication of strings by joining them together and summing over all possibilities where identical events are cancelled. e.g., using a dot for the product
1234.4351 = 12344351 + 123351 + 1251
The null string acts as an identity and it can be checked that the multiplication is associative. What I have defined then, is an infinite dimensional unital associative algebra.
The associativity is not entirely trivial. To check that,
there are two main cases to consider which can be represented like this
fig 8.1
Here V, W, X, Y, Z are arbitrary bits of strings which are concatenated together to form the three complete strings.
Given any associative algebra it is possible to define the Lie algebra by defining the Lie product as the anticommutator of the algebra.
The Jacobi identity follows directly from the associativity.
The Lie algebra can be regarded as the generators of the symmetry of the discrete open string theory. The way I have defined it is inspired by a description of symmetry algebras for topological strings due to Michio Kaku for continuous strings.
A benefit of the discrete string version is that it is easy to go from the bosonic discrete open string to the supersymmetric version. The associative algebra is graded by parity of the length of strings. I.e. the product of two even length strings or two odd length strings is a sum of even length strings, while the product of an odd and an even is odd. It follows that a Lie superalgebra can be defined using the graded commutator.
where the + sign is chosen if both A and B have odd parity, and the - sign is chosen otherwise.
This describes a rather simple sort of string theory which does not do very much except have super-symmetry. The interpretation is that these are open strings made of discrete fernionic or bosonic partons at space-time events. The model is event-symmetric in sense that the order in which the events are numbered is irrelevant.
Event-Symmetric Closed String Theory
Can we do the same thing with discrete closed strings? Kaku had attempted this with his topological formulation of universal string theory so why not?
What is needed is a Lie superalgebra defined on a basis of closed discrete cycles. It actually took me quite a lot of investigation before I discovered the correct way to do this. I will spare you the details of how I found it and go directly to the result even though this will make it look like I have just pulled the answer out of a hat. As we shall see, the properties of the closed discrete string superalgebra are much more promising than for the open version.
Start with a set E of N events. Write sequences of events in the same way as for the open strings.
To introduce closed loops we define permutations on these sequences. The permutation can be shown as arrows going from each event to another (or itself). An example would look like this,
fig 8.2
The permutation is composed of cycles. In the example there are two cycles, one of length 2 and one of length 4. But the order of the events across the page is also important.
As before these objects form the basis of a vector space. An associative algebra is defined on these objects by simply taking multiplication to be concatenation of two of these objects together. The empty sequence is a unit for this algebra.
A more interesting algebra is formed by factoring out a set of relationships among these elements. The relations are defined in the following diagram.
fig 8.3
This says that the order of two events can be interchanged keeping the loop connections intact. The sign is reversed and if the two events are the same an extra reduced term must be included. To get a complete relation the ends of the string in these diagrams must be connected to something. If they are just joined together the following two equations can be formed,
fig 8.4
The first shows the cyclic relationships for a loop of two events. The second is the anti-commutation relation for two loops of single events.
Since the relationship can be used to order the events as we wish, it is possible to reduce every thing to a canonical basis which is a product of ordered loops. A more convenient notation without the connections shown is then introduced.
fig 8.5
This notation allows the relations to be written in a way similar to those of the open strings, but now the cyclic relations mean that they must be interpreted as closed loops.
The algebra is again graded by parity and so generates an interesting supersymmetry. As far as I know this infinite dimensional supersymmetry has never been studied by mathematicians. It is possible that it can be reduced to something well known but until this is demonstrated I will assume that it is original and interesting.
Here are a few important properties of the discrete closed string algebra which did not apply to the open string algebra.
A String made of anyons?
It is almost certainly incorrect to model strings as loops of fermions. They must have some continuous form. To achieve this in an event-symmetric framework it will be necessary to replace the fermions with partons having fractional statistics which can be divided, i.e. anyons.
Defining creation and annihilation operators for anyons is not a simple matter. Various schemes have been proposed but none seem ideal. However, here we have the advantage that our anyons are strung together. The statistics and symmetries of anyons must be described by knot theory.
The commutation relations used to generate the closed string algebra will remind anyone who knows about knot polynomials of Skein relations. This suggests a generalisation may be possible if the string connections are replaced by knotted cords which can be tied. These could be subject to the familiar Skein relations which define the HOMFLY polynomial.
fig 8.6
In the special case where q=1 and z=0 this relation says that string can pass through itself. This is what we have for the strings which join the fermions. The crucial question is, are there generalisations of the parton commutation relations which are consistent with the general Skein relation? If there are then I have not found them yet.
The ladder of dimensions
Louis Crane, working towards algebraic quantum gravity with category theory, has found a way to construct a ladder of algebras of increasing dimension. In string theory there is evidence that membranes and space-times of various different dimensions play important roles. It is possible to go down the scale of dimensions by compactifying space-times. From M-theory in 11 dimensions or F-theory in 12 dimensions it is possible to construct the important critical string theories in 10 dimensions.
I shall now demonstrate a simple dimensional ladder construction which generalises the discrete fermion string symmetry. This construction may explain why structures of so many different dimensions are important in string theory.
The fermionic operators which are strung together in the discrete string model form a Heisenberg Lie superalgebra when the strings are removed. The universal enveloping algebra of this is then a Clifford algebra. I would like to repeat the string construction starting from a general Lie superalgebra. To keep things simple I will begin with just an ordinary Lie algebra A.
As before, the elements of the Lie-algebra can be strung together on strings but this time the commutation relations will look like this,
fig 8.7
I have introduced the possibility that the strings can join and separate. This kind of thing is best understood in the context of category theory. The Lie algebra elements are connected to themselves by a functor which can be shown as a network of strings which join and separate like this,
fig 8.8
The commutation relations of fig 8.7 can be shown to be consistent with the Jacobi relations provided the functors satisfy the following associativity relationship,
fig 8.9
and also the similar coassociativity relationship upside down.
In this way we can take out Lie algebra A and generate a new Lie algebra L(A). The process can be generalised to a Lie superalgebra. In the case where A is a Heisenberg superalgebra there is a homomorphism from L(A) onto the discrete string algebra which I defined previously. So this process can be regarded as a generalisation.
The interesting thing to do now is look at what happens if we apply the L ladder operator to the string algebra. This can be visualised by circling the discrete strings around the network so that they are replaced with tubes. The interpretation is that we generate a supersymmetry algebra as string world sheets. The ladder operator can be applied as many times as desired to generate higher dimensional symmetry algebras. Furthermore. There is always a homomorphism from L(A) back onto A. This makes it possible to apply the ladder operator an infinite number of times to generate a single algebra which contains all the previous ones.
This last observation raises some interesting mathematical puzzles. The algebra formed by applying the ladder operator an infinite number of times will have the property that it is isomorphic to the algebra formed by applying the ladder operator to itself. It is certainly of interest to ask whether this situation actually arises after just a finite number of steps of the ladder. Would it be too daring to conjecture that the algebra becomes complete after only 26 steps in the ordinary Lie algebra case and 10 steps in the supersymmetric case?
To progress further it will be necessary to study more general categories like those defined by Skein relations. Mathematical physicists are looking at ways to construct n-categories by stepping up a ladder of dimensions. The symmetries I have described here could be a related to such structures. The hope is that a full theory of quantum gravity and string theory can be constructed algebraically in such a fashion.
Related Reading
This page was last updated 15 May 1996.