String Theory returns to symmetry

The strings 2012 conference has finished and it is great to see that all the talks are online as slides and videos. Despite what you hear from some quarters, string theory is alive and progressing with many of the brightest young people in physics still wanting to do strings. Incredibly the next three strings conferences in Korea, US and India are already being organised. How many conference series have that many groups keen to organise them?

It has become a tradition for David Gross to give some kind of outlook talk at these conferences and this time he said there were three questions he would like to see answered in his lifetime

  • How do the forces of nature unify?
  • How did the universe begin and how will it end?
  • What is string theory?

The last of these questions is one he has been asking for quite a few years now. We know string theory only as a small set of perturbative formulations linked together by non-perturbative dualities. There has to be an underlying theory based on some unifying principle and it is important to find it if we are to understand how string theory works at the all-important Planck scale. This time Gross told us that he has heard of something that may answer the question. Firstly he now thinks the correct question to ask is “What are the underlying symmetries of string theory?” and he thinks that work on higher spin symmetries could lead to the answer. What is this about?

For about 16 years it has been known that an important element of quantum gravity is the holographic principle. This says that in order to avoid information loss is black holes, the amount of information in any volume of space must be bounded by the area of a surface that surrounds it in Planck units. This might mean that the theory in the bulk of spacetime is equivalent to a different theory on the boundary. How can that happen? How can it be that all the field variables in the volume of spacetime only carry an amount of information that can be contained on the surface. We can reason that measurement below the Planck length is not possible, but even then there should be at least a few valid field parameters for each plank volume of space. If the holographic principle is right there must be a huge amount of redundancy in this volumetric description of field theory. Redundancy can be taken to imply symmetry. Each degree of symmetry or dimension of the group Lie algebra tells us that one field variable is redundant and can be taken out by gauge fixing it. In gauge theories we get one set of redundant parameters for each point in spacetime but if the holographic principle is correct there must be a redundancy for almost every field variable in the bulk of spacetime and we will need it to be supersymmetry to deal with the fermions. I call this complete symmetry and I’ve no idea if anyone else appreciates its significance. It means that the fields of the theory are given by a single adjoint representation of the symmetry. This does not happen in normal gauge theories or in general relativity or even supergravity, but it does happen in Chern-Simons theory in 3D which can be reduced to a 2D WZW model on the boundary, so perhaps something is possible. Some people think that the redundancy aspect of symmetry means that it is uninportant. They think that the field theory can be reformulated in a different way without the symmetry at all. This is incorrect. The redundant nature of the local symmetry hides the fact that it has global characteristics that are not redundant. In holographic theories you can remove all the local degrees of freedom over a volume of space but you are left with a meaningful theory om the boundary.

If there is symmetry for every degree of freedom in the bulk then the generators of the symmetries must match the spin characteristics of the fields. Supergravity only has symmetries corresponding to spin half and spin one fields but it has fields from spin zero scalars up to spin two. String theory goes even further with higher excitations of the string providing an infinite sequence of possible states with unlimited spin. This may be why the idea of higher spin symmetries is now seen as a possible solution to the problem.

Surprisingly the idea of higher spin symmetry as a theory of quantum gravity is far from new. It goes back to the 1980s when it was founded by Vasiliev and Fradkin. It is a difficult and messy idea but recent progress means that it is now becoming popular both in its own right and as a possible new understanding of string theory.

There is one other line of development that could lead to a new understanding of the subject, namely the work on supersymmetry scattering amplitudes. Motl has been following this line of research which he calls the twistor mini-revolution for some time and has a nice summary of the conference talk on the subject by Nima Arkani-Hamed. It evolved partly out of the need to calculate scattering amplitudes for the LHC where people noticed that the long pages of solutions could be simplified to some very short expressions. After much thought these expressions seem to be about permutations and Grassmanians with things like infinite dimensional Yangian symmetry playing a big role. Arkani-Hamed believes that this is also applicable to string theory and could explain the holographic principle. The Grassmanians also link nicely to algebraic geometry and possibly work on hyperdeterminants and qubits.

I have to confess that as an undergraduate at Cambridge University in the late 1970s I was completely brainwashed into the idea that symmetry is the route to the underlying principles of nature. At the time the peak of this idea was supergravity and Stephen Hawking – who had just been inaugurated into the Lucasian chair at Trinity college – was one of its greatest advocate. When string theory took over shortly after, people looked for symmetry principles there too but without convincing success. It is true that there are plenty of symmetries in string theory including supersymmetry of course, but different sectors of string theory have different symmetry, so symmetry seems more emergent than an underlying principle. I think the generations of undergraduates after mine were given a much more prosaic view of the role of symmetry and they stopped looking out for it as a source of deep principles.

Due to my brainwashing I have never been able to get over the idea that symmetry will play a huge role in the final theory. I think that all the visible symmetries in string theory are remnants of a much larger hidden symmetry so that only different residual parts of it are seen in different sectors.  In the 1990s I developed my own idea of how infinite dimensional symmetries from necklace algebras could describe string theory in a pregeometric phase. The permutation group played a central role in those ideas and was extended to larger string inspired groups with the algebra of string creation operators generating also the Lie algebra of the symmetry. Now that I know about the importance of complete symmetry and higher spin symmetry I recognise that these aspects of the theory could also be significant. Perhaps it is just a matter of time now before string theorists finally catch up with what I did nearly twenty years ago 🙂

In any case it is good to see that there is now some real hope that the very hard problem of understanding string theory from the bottom up may finally have some hope of a solution. It will be very interesting to see how these ideas mature over the next few strings conferences.

16 Responses to String Theory returns to symmetry

  1. Leo Vuyk says:

    I am very happy with this symmetry thinking because:
    in my q-fff model
    Our MULTIVERSE as an entangled system of dual CP(T) symmetrical PINBALL MACHINES in RASPBERRY bubble shape.

  2. The holographic principle is something very deep but could it reduce to something even deeper? Not anything new but very familiar: general coordinate invariance (GCI) but applied in new context. Not for abstract space-times but for space-times identified as 4-D surfaces of some higher dimensional space. In this context the implications are much stronger than in general relativity.

    All begins from an attempt to do path integrals for 4-surfaces using some general coordinate invariant action principle – say generalization of gauge action for gauge field defined by projection of spinor connection. The attempt fails because of the incredible non-linearities. Solution: try to generalize Einstein’s geometrization program from classical physics to quantum physics. Loops spaces are encouraging example: the Kahler geometry is unique in this case and has maximal symmetries.

    Since space-time surfaces are “orbits” of 3-surfaces one should take them as basic objects and try to construct the geometry for this “world of classical worlds”. General coordinate invariance is however 4-D gauge symmetry and this requires that the action of 4-D diffeomorphisms on 3-surfaces is well-defined. The definition of the WCW geometry – infinite-D Kaehler geometry by simple physical arguments – must be such that its definition assigns to 3-surface a 4-D surface as a kind of Bohr orbit. So we have it : holography from GCI! Path integral is replaced with functional integral involving the analog of Gaussian damping in space-time regions of Euclidian signature and becomes mathematically well-defined. Classical physics inside space-time surface is needed for quantum measurement theory: this is why 3-D objects are not enough.

    One can go even further. GCI requires a choice of gauge. Zero energy ontology (ZEO) is key part of TGD and leads to the proposal that 4-surfaces inside causal diamonds (CD) define sub-WCWs. CD is defined as Cartesian product of intersection of future and past directed light-cone of M^4 and of CP_2. Its boundary consists two light-like 7-surfaces.

    Space-like 3-surfaces at the ends of CD are excellent candidates for a gauge choice. Also the light-like 3-surfaces at which the induced metric of space-time regions with Euclidian signature changes to Minkowskian are very natural gauge choices. If both are accepted, one obtains strong form of GCI: the intersections of these two kinds of 3-surfaces defining what I call partonic 2-surfaces together with their 4-D tangent space data code for physics. Basic objects are effectively 2-dimensional.

    The generalization of superconformal invariance to 3-D light-like surfaces which are metrically 2-dimensional plays a key role in the theory. This by the way fixes the dimension of space-time surface to four and Minkowski space factor of imbedding space must be 4-dimensional to obtain generalized conformal invariance at boundaries of CD where physical states are defined.

    For connections with the twistorialization program of Nima Arkani Hamed et al see the blog postings and article (see this, this, and this).

  3. Yuri Danoyan says:

    Dear Phil
    What are you thinking about my naive broken metasymmetry idea?
    No one was collecting so many facts at first sight have nothing in common.
    See also posts.

  4. Brainwashing or not, a perfect symmetry must be the only role for a final physics theory. But, this perfect symmetry must also be a “simultaneously” broken symmetry.

  5. vmguptaphy says:

    As a pico-physicists, I can say that two out of three questions from David Gross are simply answered as below;

    How do the forces of nature unify?

    Currently in mainstream physics we have the question context in terms of concepts of Energy, Force and Inertia built-in to describe dynamics of nature.

    In PicoPhysics, we have affinity of space to possess Knergy (Another word of Matter that signifies non-distinction between Matter and Energy in terms of conservation sans neutralization) as the cause of dynamics in nature. All other forces are modeled out of this concept as embedded in unary law ‘Space contains Knergy’. As a result not only forces are unified, but exceptions such as gravitation, uncertainty principle & Quantum Physics, as well as theories connected with astronomical observations (Big Bang, CBR, expanding universe, black matter etc) are integrated with unary law or its corollaries.

    How did the universe begin and how will it end?

    The question evolves due to particular way reality of nature is conceptualized in science. In way it is a conflict that was created in the concept of conservation to explain gain in Kinetic Energy of projectile as it falls to earth. The hypothesis of potential energy to apply strict conservation (of matter) to energy without creating distinction between energy and charge, keeps conservation from answering the question.

    Otherwise, conservation as eternal presence of Energy (Knergy of Picophysics) in space makes the question redundant. The question transforms to how does universe as we know evolved out of simple distribution of Energy (Knergy) in space.

    PicoPhysics describe the process of evolution of elementary particles (matter) from Knergy as well as return to simply distributed Knergy in space. Knergy present thus continuously cycles around different local and complex distributions.

    Knergy (Simple distribution) -> Dark Energy -> Elementary Particle->Matter-> Astronomical Objects-> Cosmic Background Radiation-> Knergy (Simple distribution)-> Dark Energy

    So it is a continuous process, where-in at certain sections of space the matter is getting formed, and returned to virgin state.

    What is string theory?

    No answer. Picophysics has a particular state of Knergy in matter which looks like tin strings of energy moving with constant speed in a confined space. We may call them strings as only one dimension along the drift direction is significant.

    Thus PicoPhysics has some answers to questions posed by Mr David Gross.

    Vijay Gupta

  6. PSTJ Editor says:

    Read “The Higgs Boson and the Power of Consistency”

    And other parts of PSTJ Special Issue 3(9) “Great Triumph in 21st Century Particle Physics”

  7. Lubos Motl says:

    Someone wrote on my blog that Gross believes that the Vasiliev stuff may ignite the 3rd superstring revolution. A reason to look at the talk, after all, but only after I return from the pond swimming. 😉

    Is it the same discrete Fradkin who wanted to hire me as a replacement of Feynman who would chat about his discrete nonsense? It was fun to talk to him a few times at Harvard. Or was it a Fradkin I know from Rutgers who already visited the department to play the computer games only? 🙂

    I am going to carefully read your summary before David! 🙂

    • Philip Gibbs says:

      It is Fredkin, not Fradkin who is keen on the discrete physics. I cant tell if Fradkin is the other guy from Rutgers.

      Glad you are interested in this. I am sure you could say more useful stuff about it than me if you investigate. I agree with your recent string comments 😉

      • Lubos Motl says:

        Dear Phil,

        thanks for the synergy etc. Now I’ve read your text. Nice. (And Gross’ Flash video is opened in another tab and waiting.) I appreciate many things you are saying. For example, I agree that “gauge symmetry as a redundancy” doesn’t capture all the content of a gauge symmetry because it’s only redundancy locally and is able to leave global remnant generators that don’t have to annihilate the physical states, like the ADM mass etc.

        I am familiar with the picture of holography as gauge symmetry that eliminates all the degrees of freedom in the bulk and only keeps some on the boundary. If I haven’t found it independently, then I think it was Steve Shenker from whom I learned it first. But he may have gotten it from Lenny Susskind himself. I don’t remember too clearly.

        One must be careful about overestimating such strategies, however. String field theory also has a string field Psi[x,b,c(sigma)] and a string-field gauge symmetry given by Lambda[x,b,c(sigma)] – albeit at a different ghost number. This stringy gauge symmetry may also kill almost all the degrees of freedom. Well, some of them are left. They’re the cohomology, of course.

        When I describe it in this way, you must admit that any BRST formulation of a gauge-like theory is a model of the complete symmetry. Almost all the states are killed but some of them are still left physics, namely the BRST cohomologies. In reality, the number of physical states still grows exponentially with the level and it has to so the slogan “almost all states are killed” is of course just marketing.

        I will try to approach it as constructively as I can but my estimate for the probability that Vasiliev’s theory is relevant for the formulation of string theory is rather low or very low at this moment. It seems to me to be a not-quite-full-fledged cousin of the stringy vacua (is that a full-fledged theory of quantum gravity with black holes and their correct qualitative Hawking-like features? I will revisit all these questions). Well, Gross’ talk and several papers I had wanted to read in detail, including those by Xi Yin, are waiting.

        Concerning symmetry group on events, I must have already told you. But in early 1996 (or did I change the date to a newer one myself?), I was excited by some web pages I found, see e.g.

        and click at other links. The Cyclotron Notebooks were written by a guy named “Phil Gibbs” and as you can see, I made a local pirate copy that I couldn’t have deleted for about 10 years because at some moment, they blocked this particular computer account in Prague (which wasn’t my most standard one, anyway).

        It was exciting but in 1996, I had to make lots of progress with my understanding of string theory – a scientifically intense year – because at the beginning, I was ready to believe that your speculation could have been a summary of a real paper that reformulates string theory in that way. At the end (Christmas) of 1996, I e.g. discovered matrix string theory (and wrote the heterotic matrix paper) so at that time, I already knew in quite some extreme detail that your page didn’t really have much to do with existing demonstrated string theory etc. 😉

        Nevertheless, your focus on permutations surely resembles the focus on permutations in matrix string theory. For a while, I had thought that you have influenced me but then I realized that I had been aware of the matrix string theory concept since 1992, well before the duality revolution, and it was just waiting for an appropriate mathematical formalism where it could have been inserted.

        The paradigm going back to 1992 – the first afternoons in Fall 1992 when I could study the library with Nuclear Physics B right when I came to Prague as a freshman – was that it was extremely unnatural for string field theory – I meant Green-Schwarz light-cone gauge string field theory, my preferred formulation of string theory at least in the 1990s – to define the operator for the number of strings N. That’s because 1 string differs from 2 strings just by some global topology so there should exist a more unified way to encode the values of x(sigma) of all the points along the string whether sigma has the topology of one circle or two circles. Of course, matrix string theory was ready to realize this old dream. And the crossover interaction of closed strings is a rule for composition of permutations so in the more satisfactory formulation, it must be represented by a permutation of some degrees of freedom that remember the strings’ positions. It could have been made meaningful within matrices with S_N embedded into the U(N) group. There’s no natural “number of strings” operator over there. So your Cyclotron Notebook could have at most reinforced my interest in permutations but it was fun, anyway…


      • Lubos Motl says:

        OK, after some time by refreshing myself concerning the developments in the Vasiliev theory research, I tend to think that it is a point or limit in some classes of string theory backgrounds, it is a quantum gravity of a sort, and that the black holes are probably there but have some unusual properties – that may nevertheless justified by other unusual properties of these backgrounds. Holography may perhaps emerge as some sort of symmetry breaking in those theories but the whole bulk emerges at the same moment so it’s no along the lines of your “everything in bulk is redundant” paradigm, Phil.

        It’s fun work and I am inclined to spend lots of time with it, too.

        But I don’t believe it may lead to a 3rd revolution or answer “what is string theory”. At most, it’s just another background with some symmetry, perhaps an unusual or large one. But the symmetry gets broken everywhere else, much like other symmetries that are relevant in some corners of the stringy configuration space, so the role of this symmetry is at most “constructive” and the relevance fades away more generally on the configuration space.

        Of course, when one analyzes my argument, it may be generalized to a broader reason why I don’t really believe that a symmetry may be the “holy grail” providing us with a unified perspective on all of string theory. A symmetry is something that holds in an environment only and the further we are from it, the more it is broken and the more it is irrelevant for the key physical objects and phenomena.

        Morever, this wishful thinking that the symmetry of the Vasiliev background would tell us everything about all of string theory seems to be another round of the string field theory wishful thinking all over again. Despite those dreams, string field theory turned out to be as irrelevant for the truly strongly coupled questions in string theory as any other perturbative approach. Similar limitations seem to be inherent here. There doesn’t seem to be a reason why the higher-spin formulation of string theory if it exists, whatever it is, would remain exact arbitrarily far from the “Vasiliev point”.

        It’s probably a great motivator if those great folks are thinking that this research may lead to much far-reaching consequences than the analysis of a strange exceptional background of (extended?) string theory but I still think it’s sensible to remain realistic and not overlook arguments in the opposite direction. Just to be sure, I think that David Gross was the only one who really suggested this far-reaching consequences of the higher-spin-theories research.


      • Philip Gibbs says:

        Thank you for your assessment. It is very useful.

        I cant help following the logiv that if the holographic principle is correct and if the theory in the bulk is described by a theory with field variables (or string variables) defined over the volume of space, then most of those variables must be redundant. Is there some sense in which such redundancy implies a symmetry? Redundancy and symmetry seem almost synonymous to me. If so then it is inescapable that the bulk theory must have a “complete symmetry” to remove the bulk of the variables. Is there any other way that the holographic principle can work? If not then this idea would be little more robust than wishful thinking, but I cant argue rigorously for it.

        It is useful to know that the Vasiliev-Fradkin theory does not seem to work by providing such a complete symmetry. It was an old idea and there could be other variations of higher spin symmetry theories where it does work that way. It will be interesting to see what Gross is saying a year or two from now.

      • Lubos Motl says:

        Dear Phil, I don’t consider redundancy and symmetry to be the same thing. Redundancy means that some degrees of freedom may be chosen arbitrarily e.g. by a gauge transformation so they don’t carry new information.

        Symmetry is an actual operation you use to map one classical configuration (classically) or one state in the Hilbert space (in QM) to another allowed but different state. This is an OK description for global symmetries (or global parts of the gauge symmetries) but not for localized “in the region of the bulk” gauge symmetries because configurations related by gauge transformations are viewed as physically equivalent, the same ones (classically) and, equivalently, the states must be invariant under these gauge symmetries i.e. annihilated by the generators (in QM).

        A redundancy actually reduces the number of physical degrees of freedom; a symmetry doesn’t, although a symmetry may be helpful to solve problems etc.

        I find your opinion that the “complete redundancy” must be behind the holographic principle naive. You really assume that quantum gravity may be written as an explicitly local theory where fields depend on space. Maybe in that case, it could follow that the theory has to be topological in the bulk in some sense. However, we know that quantum gravity isn’t strictly local. For example, some nonlocalities surely manifest themselves during the black hole evaporation.

        Moreover, you heavily underestimate the complexity of the “holographic code”. Even if you find a way to encode a region in the qubits on the surface, it is an extremely complicated code in general. It surely can’t be the case that for finite length surfaces, you just get some simple local theory on the surface that describes the interior. It couldn’t work. A simple local theory can’t contain so carefully hidden extra dimensions or other miracles.

  8. Yuri Danoyan says:

    “Gradually comes understanding that the symmetry group is most important that we today can learn about nature.I like now to say something that, until the end of what is not sure, but that could well become a reality, namely: all that we need apart from quantum mechanics to describe the physical picture of the world, is to determine the group symmetry of nature.”(Steven Weinberg)
    “Elementary Particles and the Laws of Physics: The 1986 Dirac Memorial Lectures Richard” P. Feynman,Steven Weinberg

  9. Yuri Danoyan says:

    Qui nimium probat, nihil probat
    One who proves too much, proves nothing

  10. Yuri Danoyan says:

    Only way of decreasing redundancies of symmetry to annihilate difference between global and local symmetries

  11. vmguptaphy says:

    If symmetry is invariance to change of parameter (and hence give rise to a conservation principle), does the redundancy means the reality is composed of inseparable reality pair.

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