There have been many amazing discoveries about superstring theory, but there are still some deep conceptual problems concerning the way it is formulated. The most profound of these is that string theory does not directly account for the equivalence principle. We know that superstring theory has gravitons and supergravity is therefore a component of the effective theory of strings at low energy. Supergravity is generally covariant and so incorporates ordinary general relativity with its equivalence principle. Thus string theory seems to include the equivalence principle, but the formulations we know are not generally covariant. There are versions which are Lorentz covariant but that is a long way short of the general covariance under all co-ordinate transformations. It is a little surprising and frustrating that this is the case and it may well be a key part of why we do not fully understand string theory.
The principle of event-symmetric space-time is the solution which I propose as a resolution of the superstring mystery. Event symmetry is a step beyond the diffeomorphism invariance of general covariance. If we can formulate string theory in a way which is event-symmetric we can leap frog over the conceptual hurdles.
Another reason which I already covered is the solution to Witten's puzzle. Topology change and the universal symmetry put together are difficult to reconcile without event symmetry.
The third reason is the presence of a very large symmetry of string theory beyond its Hagedorn temperature. It is not known what this symmetry is but it seems to reduce the effective number of degrees of freedom enormously. It is likely that there must be one dimension of symmetry to match each degree of freedom of the string. No mere gauge symmetry can achieve this but event symmetry is much larger than any gauge theory in quantum field theory.
Next I cite the important idea that strings can be considered as composites of discrete partons; particles bound together like beads on a necklace. Space-time too seems to have a discrete character. This picture may seem opposed to the usual formulation of strings as cords of continuous substance, yet it can explain many mysteries especially in the context of black holes. In that case it is easy to picture strings as loops connecting discrete points of space, and with such discreteness, event symmetry is easily imagined.
After that comes matrix models. String theory may ultimately be described by something like a model of random matrices whose rows and columns may index particles, colours of gauge symmetry or space-time events. Models on event-symmetric space-time also drive physics towards the dynamics of matrices. The matrix model which seems to contain the essence of M-theory can be interpreted in any of these ways, bringing event symmetry a step clearer. A unification of gauge symmetry and particle statistics was a prediction of the principle of event symmetry which soon after appeared as a feature of this matrix model.
Then there are the new S-dualities which reverse the roles of solitons and particles, or more generally, solitonic p-branes with fundamental p-branes. But string theory also has instantons, sometimes called (-1)-branes because they have one less dimension than particles which are 0-branes and two fewer than strings which are 1-branes. Instantons are excitations of a field which exist for an instant. Their importance in non-abelian gauge theories such as QCD has been known for many years and now they are playing a starring role in string theories too. In passing through a duality transformation the instanton must reverse its role with a fundamental (-1)-brane and what other character can that be than a space-time event? Like particles and any other p-brane instantons have statistics; a symmetry over their permutations. This symmetry must be dual to a corresponding symmetry of space-time events; event symmetry.
I have now given seven bits of evidence that event symmetry is a feature of string theory. Some of them are more convincing than others. None of them are absolutely conclusive. The final proof would be a version of string theory which explicitly exhibited event symmetry and which was equivalent to the familiar string theories. I cannot offer that yet, but I can describe some string inspired supersymmetries which appear to lead the way. These supersymmetries are especially elegant and, of course, they include event symmetry.
There have been many discoveries or near discoveries of new symmetry in string theory, but there is one which I found particularly inspirational. It was the string inspired symmetries of Michio Kaku. Symmetry is about groups so to discover a new symmetry all you really need is a way of defining an associative product with an inverse and a unit on whatever objects come to mind. So how might open ended strings be multiplied? Strings can interact by joining together at their ends so we could think about multiplying them in a similar way. Think of open strings as continuous paths through space starting at one point and ending at another. We will multiply them together by joining them together if the end of the first coincides with the start of the second, cancelling out the part where they join.
Take one string A starting at a point W passing through point X and ending at point Y and multiply it by another string B which starts at Y, passes back through X and ends at Z. B follows the same path in reverse as A took from X to Y. The product C=AB is then the path from W to Z passing through X and following the same path as A between W and X and the same path as B between X and Z. This product of strings is nicely associative, i.e. (AB)C = A(BC) but it fails miserably to make a group. It has no unit, no inverses and it only defines multiplication for strings which join together at their ends.
What we are looking for is the stringy generalisation of gauge symmetry. The group elements of ordinary local gauge theories are described by a field, that is an element of the base group at each event in space-time.
For example, if we are talking about the U(1) gauge symmetry of the electromagnetic field there is an element of U(1) (i.e. a complex number of modulus one) at each event. In other words the gauge transformation is specified by a function f(X) from space-time events X to the complex numbers. The charged matter fields are gauge transformed by multiplying by this phase factor at each event with the accompanying gauge transformation of the electromagnetic field. To generalise this, think of events in space-time as possible points that a particle worldline can pass through. The stringy generalisation of a gauge transformation would be specified by a function f(A) from all possible string paths A to the complex numbers. A string path is just one of the path segments through space-time which we have already thought about. So what we are really looking for is a group of objects with a complex number assigned to each string.
Gauge transformations are multiplied together by on a simple event by event basis. If f(X) is one gauge transformation and g(X) is another, then the product h(X) is just,
For strings we do things a little differently like this,
The sum is over all pairs of strings A and B whose product according to the previous definition is C. For a complete field there would be an infinite number of such strings and the sum becomes a difficult to define integral, but we will not worry about this detail just yet.
This definition of string gauge fields actually includes ordinary particle field gauge transformations if a particle at X is identified with a zero length string which starts and ends at the same point X. A little thought will show that string fields which are non-zero only for such strings will multiply together in the same way as particle fields. Now we can also see that this multiplication has a group-like identity. It is the string field which is equal to one for every zero length string and zero for all others. Not all string fields have inverses for this multiplication but some do, and the set of those that do forms a group. This group is then what we will consider as the general gauge group for continuous open strings. It is essentially the symmetry which Kaku defined in 1988.
Of course we would need to define some model of string dynamics which was invariant under the action of this group. That is what Kaku tried to do with some success.
These open strings, however, are less interesting than closed strings, formed from closed loops. Indeed open string theory is incomplete without closed strings along side. Kaku tried to work out a version of gauge symmetry which also works for closed strings. It is not so easy. Closed strings can interact by coming together and joining where they touch to form a single loop, but if you multiply loops together by joining them in this way you do not get an associative algebra like we did by joining open strings at their ends. Kaku solved the problem by looking at the commutators of the product and defining a supersymmetry in a clever way, or at least he almost solved it. In fact there were cases which did not quite work out. The symmetry was flawed and sadly it never proved useful as a way to understand string gauge symmetry.
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 p-brane 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 two-dimensional string world sheets and three-dimensional space-time are more visible only because they stand out as a consequence of some as yet unknown quirk in the maths.
The shortest permissible strings have length 2 because they must have at least start and end points, even if these coincide at the same event. Strings can be any finite length from the 2 upwards.
These strings are taken as the defining basis of a vector space. This is just a way of saying that we are going to look at fields defined over these strings as we did for continuous fields. The field is a function from the set of all strings to the complex numbers. Those fields can be added, subtracted and multiplied by complex number constants like vectors, so we call the collection of fields over strings a vector space.
I define multiplication of strings where the end of one coincides with the start of the other by joining them together and summing over all possibilities where identical events are cancelled. If they do not meet it is convenient to define the product to be zero. e.g., using a dot for the product
From any such algebra a group can be formed simply by taking the subset of everything which has an inverse. This group could be the algebra of a symmetry of discrete open strings. Of course we would need to define some model of string dynamics which was invariant under the action of this group. This can be done in the same way as it is done for random matrix models. In fact, what I have defined is really just an extension of matrix algebra since the sub-algebra formed of strings of length two multiplies in the same way as N by N matrices.
A benefit of the discrete string version is that it is easy to go from the bosonic discrete open string to the supersymmetric version. Strings of even length are taken to be bosonic and strings of odd length are taken to be fermionic. 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 fermionic or bosonic partons at space-time events. The model is event-symmetric in the sense that the order in which the events are numbered is irrelevant, but the transformations of event symmetry which would permute the numbering of events are not a part of the symmetry algebra. This is a disappointing failure which means that string gauge symmetry and general covariance are not yet unified for open strings.
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 started by writing down strings of events just like for open strings, but if they are closed strings the starting point should not matter. For example a loop which went through the events numbered 2, 5, 3, 4 and 1 returning back to 2 can also start and return at any other of the five events, so long as it went round in the same cyclic order. This is signified by equations such as this,
What about odd length strings, were they to be excluded? The answer was not difficult to guess as with open strings the odd length loops could be considered as fermionic. The commutators for fermionic variables must be replaced with anti-commutators where the minus sign is changed to a plus sign. These define a supersymmetry algebra in place of a classical symmetry. This was a very satisfying result. I had found myself forced to use supersymmetry for closed strings even before I had begun to think about any dynamics, or anomalies or any of the things which were usually used to justify supersymmetry in string theory.
There was one other satisfying result. The way the strings of length two commuted with all other strings was exactly what was required to define a rotation matrix acting on the vector space where events correspond to axis. A rotation can be used to permute axis, in other words, event symmetry must be part of the symmetry algebra I had discovered. This seemed to happen only by chance, if the signs had needed to be different, or it had been necessary to cancel out even length bits of string instead of odd length bits, this would simply not have worked. Yet I had had no choice in the matter. It was a sign that I was doing something right. It meant that if I built a model of strings with this supersymmetry algebra, it would have space-time symmetries unified with internal gauge symmetry; something that had never been achieved with string theories in continuum space-time.
I wanted to know if the supersymmetry algebra I had discovered was already known to mathematicians. The way the relations worked out was rather mysterious. Usually when you find something like this there turns out to be some deeper explanation of why it exists. Anything I could turn up might help me understand what to do next.
In 1995 a strange coincidence helped me out. I saw a paper about the role of Borcherds algebras in superstring theory. Borcherds was a name I recognised. The algebras had been discovered by an old friend of mine. I had become aquatinted with Richard Borcherds at high school when we used to participate in mathematics competitions. In fact Richard and I had been the joint winners of the 1978 British Mathematical Olympiad. We had both been in the same British team for the International Mathematical Olympiads two years running and then we knew each other at Cambridge University.
However, we had very different tastes in what kind of maths we liked. Richard was definite that he wanted to do pure maths, whereas I was becoming interested in mathematical physics. It was a bit of a surprise to discover 15 years later that Richard had made his name from a discovery about string theory, but he had approached the subject as a pure mathematician to study its symmetry. He had found a rigorous way to define an infinite-dimensional supersymmetry algebra of string theory which was of interest to mathematicians.
I sent an e-mail to Richard with an explanation of my super-symmetry algebra. I knew that they were not the same thing but perhaps there would be a relation between them. I was a little surprised when Richard quickly replied to tell me that my algebra did not quite work. He had found a particular case which failed to satisfy the Jacobi identity. In fact he too had already looked at Kaku's definitions of superstring gauge theory and had found that they were flawed. He easily found a similar fault in my discrete string versions.
Fortunately, as so often happens, the flaw itself gave the clue to how it should be repaired. I had to extend my algebra to include more than one loop at a time, and I had to allow them to interact by touching at more than one point of contact so that two loops which could come together and split into two others. At first it seemed like this was going to be even harder to define but I found that actually there was a conceptually simpler way to do it. This new way would give further clues about what the algebra meant.
Start with a set E of N events. Write sequences of events in the same way as for the open strings
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 now formed by factoring out a set of relationships among these elements. The relations are defined in the following diagram.
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,
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.
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 associative and it is consistent to consider combinations of loops with an odd total number of events as fermionic, and with an even number of events as bosonic. So again this generates a supersymmetry using the appropriate commutator and anti-commutators. 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.
It may be that string theory has to be formulated in the absence of space-time which will then emerge as a derived property of the dynamics. Another interpretation of the event-symmetric approach which embodies this is that instantons are fundamental. Just as solitons may be dual to fundamental particles instantons may be dual to space-time events. Event-symmetry is then dual to instanton statistics. In that case a unification between particle statistics and gauge symmetry follows on naturally from the principle of event-symmetry. It is encouraging that this unification also appears in the matrix model of M-Theory.
The final string theory may be founded on a mixture of geometry, topology and algebra. The dual theory origins of string theory hide a clue to an underlying algebraic nature. In dual theories the s-channel and t-channel amplitudes are supposed to be equal. At tree level, in terms of Feynman diagrams this means that,
This diagram could also be distorted to look like this,
This figure is familiar to many mathematicians who recognise it as a diagrammatic representation of the associative law,