Tuesday, November 13, 2007

Berkovits, Vafa: proving AdS/CFT

Berkovits and Vafa have a paper that I choose to be the most interesting paper on the arXiv today. They may have made steps towards a full proof of the perturbative part of the most famous example of the AdS/CFT correspondence, namely between the N=4 gauge theory on one side and the AdS5 x S5 background of the type IIB string on the other side.

Some aspects of their construction are well-known to me but others are new. Among the aspects we have realized for some time, we can mention the fact that the relevant worldsheet description of the type IIB string is similar to the pure spinor language of Berkovits. It is an A-model which is a quotient of the U(2,2/4) supergroup and its maximal bosonic subgroup. Another aspect that I have believed for a few years is that the proof is based on a reduction of the two-dimensinoal worldsheet to Feynman diagrams, by "erasing" most of the information inside disk-shape regions included in the Feynman diagram.

What seems new, and what I am not able to fully check at this moment, is their statement that the proof could be analogous to the Ooguri-Vafa proof of a similar but topological Gopakumar-Vafa duality. The key feature that Berkovits and Vafa claim to be shared is that the worldsheet theory has a new Coulomb branch in this case much like it had in the topological case. The pieces of the worldsheet that are found in this Coulomb branch are interpreted as the faces of the Feynman diagrams while the boundaries between them are its propagators and vertices. As the 't Hooft coupling goes to zero, the regions found in the Higgs branch shrink and become the propagators and vertices.

Clearly, Cumrun Vafa is the first person who would be expected to suggest such a picture - in fact, I have heard such hints from him some time ago - and people who are less topological than he is, which surely includes your humble correspondent but most likely also all other humans on this planet, with a possible exception of Edward Witten and hypothetically also Marcos Mariňo (and with apologies to Aganagič, Gopakumar, Ooguri, Saulina, and others), face greater hurdles in trying to follow the details here.

At any rate, this Coulomb branch picture is different from what I wanted to do with the faces of the Feynman diagram most recently. I wanted to "integrate them out" by a Euclidean version of the AdS2/CFT1 correspondence. The faces would be interpreted as the Euclidean AdS2 whose geometry is a Poincaré disk but the partition sum over these disks as a function of the boundary conditions could be calculated by AdS/CFT methods, leading to a CFT1 theory that would match the action needed in the Schwinger parameterization of the N=4 gauge propagators. The vertices would then require a special treatment as additional singular places.

Now, is such a proof important? I surely don't think that we need a proof in order to be more certain about the correspondence because the correspondence is obviously true and only crackpots - such as those whose shoes are connected by a rope - doubt it at this moment. But I do think that such a proof would allow us to see the degrees of freedom in different descriptions of string theory from a more unified perspective. It would help us to see a little bit more why the degrees of freedom in different formulations of string theory look so different and how they can actually be directly transformed to each other.

And that would be certainly worth a memo.


  1. Hi Lubos,

    A comment RE Berkovits and Vafa, section 5.3 Twistorial Formulation p22.

    Has any HEP physicist looked at the math of electrical and mechanical engineers?
    There is a discussion of helical functions and helical subspaces which may relate to twistor theory.

    Hamish Meikle [MS EE, European Radar consultant], ‘A New Twist to Fourier Transforms’ with support by Maple Product Description

    A Mahalov, et al [applied math and ME, Cornell, UC-I], ‘Invariant helical subspaces for the Navier-Stokes equations’

  2. Dear doug, based on available sources, helical functions are just some sines and cosines (or complex exponentials) of arguments linear in variables.

    If that's the case, I think that most HEP physicists know it from the high school even if they don't use the name. Why do you exactly think it is relevant to twistor theory and how relevant?

    I think it is probably relevant but well-known. Twistor space representation of a field is a partial Fourier transform, indeed. But that means that your new proposed Fourier transform is either the same thing or wrong, doesn't it?


  3. Hi Lubos, I would agree that my” proposed Fourier transform” is essentially “the same“, although the perspective may be different. I think helical functions are relevant to twistor theory by providing a mechanism for information transformation among various gauges or scales.

    Helical 3D functions may be more powerful than elliptical 2D functions?

    When twistor theorists speak of “maximal” helical violations [MHV], my perspective is to think of extrema theory.

    A relatively short version of my circumstantial idea synthesis or integration via insight [?] into the rigor of others; obviously my discussion reveals a BA math, MD interpretation of biophysics and bioengineering speculative understanding of electromagnetism, gravity and other physics concepts:

    a- Richard Bellman [PhD math Princeton] is credited with control theory and the Hamilton-Jacobi-Bellman Theorem [HJB]. [To paraphrase: rigorous, it is not even precise, but the insight is a good principle.]

    b- Tamar Basar [EE] and Geert Han Olsder [math] wrote a SIAM classic on dynamic noncooperative game theory with the perspective of pursuit evasion games using HJB.

    c- Tropical Algebras, especially Max-Plus, greatly simplify this mathematics.

    d- Robotics applies this mathematics to medical prostheses and interplanetary travel and robotics, coordinating electrical and mechanical engineering.

    e- When I asked Terence Tao how the von Neumann Minimax Theorem may relate to the von Neumann Algebras, he replied that this may be done through degree mapping.

    f- The Nobel biography of Paul AM Dirac reveals that he had no degree in physics, but a BS in EE and PhD in math. [This led me to read more EE literature.]

    g- Charles Proteus Steinmetz [Edison-GE chief EE, 1890s] demonstrated that electromagnetism is associated with complex helices through phasor equations based upon Grassmann Algebra. Both Marconi and Einstein visited Steinmetz who was one of the great applied mathematicians of his time. [To paraphrase: everything that I am, I owe to mathematics.]

    h- Erwin Schrödinger used Clifford Algebra to formulate his Wave Equation in Mechanics. He also asked: “What is Life”

    i- Gabriel Kron, ‘Electric Circuit Models of the Schrodinger Equation‘, Physical Review v67, n1-2, 1-15 January 1945, relates electrical and mechanical models.

    j- MOD(n) functions are helical, although commonly viewed as planar loops as in clocks.

    k- Loops of 2D elliptical functions viewed from an axial perspective may easily be 3D helical functions when viewed also from a sagittal and coronal architectural perspective..

    l- Newtonian Relativity: Newton’s nearly circular elliptical equations for planetary orbits about the sun are correct only when the sun is considered relatively motionless [background independent]; the orbits become helical when the sun is allowed to orbit the galactic center [background perturbation].

    m- Newtonian mechanics are an extension of ballistics.

    n- Euler was an expert on ballistics or trajectory curves.

    o- Rifling [helix] improves the accuracy of ballistics by providing spin information.

    p- Other noted informational aspects of the helix include musical pitch space, nucleic acids and proteins and helical wired solenoids [electromagnetism].

    q- The ionic nucleic acids and proteins may have solenoid like activity.

    r- Borcherd’s string-D and Witten coupling [11D of M-theory] may be trajectory helical strings.

    s- The Richard Feynman ‘path integral formulation’ may be equivalent to a ‘trajectory integral formulation’

    t- An helical trajectory would appear to be constrained to 3-spatial-D except to a potential flatlander inhabitant [1D].

    u- The helix is consistent with the Randall and Arkani-Hamed concepts of “curled-up” unseen dimensions except when in the spectrum of visible light.

    v- The helix is consistent with the oscillating motion of zitterbewegung of quantum mechanical particles, general relativity celestial bodies and David Hestenes’ kinematic motion concept.

    w- Urs Wiedemann in the last slide of his Strings ‘07 talk: “Our task: find catenary”.

    x- The catenary is a curve between two fixed point which likely is an helix between two moving points in a manner related to the transformation of a catenoid into an helicoid.

    y- This PhysicsWorld illustration associates the helix and torus.

    z- Unfortunately, I have no smoking gun evidence; but it should be easier to unify gravity with EM using engineering transformations between electrical and mechanical systems?