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New Witten paper (geometric Langlands)

  1. Dec 10, 2006 #1


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    Gauge Theory, Ramification, And The Geometric Langlands Program
    Sergei Gukov, Edward Witten
    160 pages

    "In the gauge theory approach to the geometric Langlands program, ramification can be described in terms of 'surface operators,' which are supported on two-dimensional surfaces somewhat as Wilson or 't Hooft operators are supported on curves. We describe the relevant surface operators in N=4 super Yang-Mills theory, and the parameters they depend on, and analyze how S-duality acts on these parameters. Then, after compactifying on a Riemann surface, we show that the hypothesis of S-duality for surface operators leads to a natural extension of the geometric Langlands program for the case of tame ramification. The construction involves an action of the affine Weyl group on the cohomology of the moduli space of Higgs bundles with ramification, and an action of the affine braid group on A-branes or B-branes on this space."
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  3. Dec 10, 2006 #2


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    OK, Marcus, this one gets my attention. Cheers. I think the words to highlight are:

    after compactifying on a Riemann surface, we show that the hypothesis of S-duality for surface operators leads to a natural extension of the geometric Langlands program

    Last edited: Dec 10, 2006
  4. Dec 13, 2006 #3
    Hi Kea and Marcus:

    Are there any QLG papers which have an 'Index of Notation' [Appendix B] as found in this Gukov-Witten paper?
  5. Dec 13, 2006 #4


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    Lattice (in the sense of discrete group) theory papers, eg Borcherds, use to do it.
  6. Dec 13, 2006 #5
    Thanks arivero!

    A thought about Golay codes - could Nature [physics] use them in the phenomena of 'electromagnetic reconnection' [EMR]?

    In the context of category theory EMR may be like flop transitions [Yau] and orbifold rephrasing [B Greene]?
  7. Dec 13, 2006 #6


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    Dcase, I can not see the relationship, But again, I no not know what EMR is. Is it related to this paper or should it need a new thread?
  8. Dec 13, 2006 #7
    Hi arivero:

    I got lazy and should have posted these comments to your thread:
    strings, codes and phonemes at sci.physics.strings

    Golay codes are error correcting for use in anthropic communications.
    EMR may be error correcting for various torn manifolds.

    Is there a relationship?

    A - However, to avoid the posting delay, I will explain my EMR abbreviation for electromagnetic reconnection:

    1 - Subject Category: Plasma physics
    Magnetic merging in the MRX by Paul Hanlon
    Electromagnetic fluctuations within the heart of a controlled magnetic reconnection experiment could provide an explanation for the unusual rates observed, and provide another piece in the puzzle of how magnetic fields couple to plasmas.

    2 - The Mysterious Origins of Solar Flares By Gordon D Holman
    Researchers begin to understand how the dynamics of the solar magnetic field trigger titanic eruptions of the sun's atmosphere [SCIAM April 2006]

    3 - Probing the Geodynamo By Gary A Glatzmaier and Peter Olson
    Studies of our planet's churning interior offer intriguing clues to why the earth's magnetic field occasionally flips and when the next reversal may begin [SCIAM April 2005]

    4 - Letter: Nature 442, 457-460 (27 July 2006) | doi:10.1038/nature04925
    Electrical signals control wound healing through phosphatidylinositol-3-OH kinase- and PTEN by Min Zhao, [et al] and Josef M Penninger
    Wound healing is essential for maintaining the integrity of multicellular organisms. In every species studied, disruption of an epithelial layer instantaneously generates endogenous electric fields, which have been proposed to be important in wound healing1, 2, 3. The identity of signalling pathways that guide both cell migration to electric cues and electric-field-induced wound healing have not been elucidated at a genetic level. Here we show that electric fields, of a strength equal to those detected endogenously, direct cell migration during wound healing as a prime directional cue. Manipulation of endogenous wound electric fields affects wound healing in vivo. Electric stimulation triggers activation of Src and inositol–phospholipid signalling, which polarizes in the direction of cell migration. Notably, genetic disruption of phosphatidylinositol-3-OH kinase- (PI(3)K) decreases electric-field-induced signalling and abolishes directed movements of healing epithelium in response to electric signals. Deletion of the tumour suppressor phosphatase and tensin homolog (PTEN) enhances signalling and electrotactic responses. These data identify genes essential for electrical-signal-induced wound healing and show that PI(3)K and PTEN control electrotaxis.

    B - Golay codes [GC12 and GC24] and the bimonster: [especially Appendix A]
    The complex Lorentzian Leech lattice and the bimonster by Tathagata Basak
    Comments: 24 pages, 3 figures, revised and proof corrected. Some small results added. to appear in the Journal of Algebra
    Subj-class: Group Theory; Number Theory
    MSC-class: 11H56, 20F55
    We find 26 reflections in the automorphism group of the Lorentzian Leech lattice L over Z[exp(2*pi*i/3)] that form the Coxeter diagram seen in the presentation of the bimonster. We prove that these 26 reflections generate the automorphism group of L. We find evidence that these reflections behave like the simple roots and the vector fixed by the diagram automorphisms behaves like the Weyl vector for the refletion group.
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