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Chern-Simons, LQG black hole entropy, and central intertwiner

  1. Jul 26, 2010 #1

    marcus

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    Spin networks describing BH have a central node with label taken from an intertwiner Hilbertspace of high dimension (corresp. to hole entropy). Horizon area corresponds to the network links that pass through the horizon. The horizon Hilbertspace turns out to have the same high dimension (again corresponding to the hole entropy.) It was found in 2009 that Chern Simons theory enters usefully here---triggering publication of a followup paper by Krasnov and Rovelli.
    http://arxiv.org/abs/0905.3168
    Black hole entropy and SU(2) Chern-Simons theory
    Jonathan Engle, Karim Noui, Alejandro Perez
    4 pages Physical Review Letters, additional detail in longer http://arxiv.org/abs/1006.0634
    (Submitted on 19 May 2009)
    "Black holes in equilibrium can be defined locally in terms of the so-called isolated horizon boundary condition given on a null surface representing the event horizon. We show that this boundary condition can be treated in a manifestly SU(2) invariant manner. Upon quantization, state counting is expressed in terms of the dimension of Chern-Simons Hilbert spaces on a sphere with marked points. Moreover, the counting can be mapped to counting the number of SU(2) intertwiners compatible with the spins that label the defects. The resulting BH entropy is proportional to aH with logarithmic corrections Δ S=-3/2 log aH. Our treatment from first principles completely settles previous controversies concerning the counting of states."

    http://arxiv.org/abs/0905.4916
    Black holes in full quantum gravity
    Kirill Krasnov, Carlo Rovelli
    5 pages Classical and Quantum Gravity
    (Submitted on 29 May 2009)
    "Quantum black holes have been studied extensively in quantum gravity and string theory, using various semiclassical or background dependent approaches. We explore the possibility of studying black holes in the full non-perturbative quantum theory, without recurring to semiclassical considerations, and in the context of loop quantum gravity. We propose a definition of a quantum black hole as the collection of the quantum degrees of freedom that do not influence observables at infinity. From this definition, it follows that for an observer at infinity a black hole is described by an SU(2) intertwining operator. The dimension of the Hilbert space of such intertwiners grows exponentially with the horizon area. These considerations shed some light on the physical nature of the microstates contributing to the black hole entropy. In particular, it can be seen that the microstates being counted for the entropy have the interpretation of describing different horizon shapes. The space of black hole microstates described here is related to the one arrived at recently by Engle, Noui and Perez, and sometime ago by Smolin, but obtained here directly within the full quantum theory."

    Then in June 2010 a longer piece by Engle et al further developing the idea:
    http://arxiv.org/abs/1006.0634
    Black hole entropy from an SU(2)-invariant formulation of Type I isolated horizons
    Jonathan Engle, Karim Noui, Alejandro Perez, Daniele Pranzetti
    30 pages, 1 figure
    (Submitted on 3 Jun 2010)
    "A detailed analysis of the spherically symmetric isolated horizon system is performed in terms of the connection formulation of general relativity. The system is shown to admit a manifestly SU(2) invariant formulation where the (effective) horizon degrees of freedom are described by an SU(2) Chern-Simons theory. This leads to a more transparent description of the quantum theory in the context of loop quantum gravity and modifications of the form of the horizon entropy."
     
    Last edited: Jul 27, 2010
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  3. Jul 27, 2010 #2

    marcus

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    Feel free to comment. I will have some more to say about it, but this is enough for now.
     
  4. Jul 27, 2010 #3

    atyy

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    SU(2) Chern Simons pops up here too!

    http://arxiv.org/abs/1007.4234
    "For example, let us consider the d = 2 case. The class C and D TIs/TSCs are described by Chern-Simons theories with the Sp and O gauge groups, respectively ... then the gauge group on the D5- and Dq-branes are the same and it is SU(2) for class C ..."
     
  5. Jul 27, 2010 #4

    marcus

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    Here is a 1988 case of Chern-Simons popping up:
    ==quote==
    Actually, the general situation that must be studied is that in which possible Wilson lines on M are "cut" by Σ, as in the figure. In this case Σ is presented with finitely many marked points P1...Pk, with a G representation Ri assigned to each Pi (since each Wilson line has an associated representation). To this data -- an oriented topological surface with marked points, and for each marked point a representation of G -- we wish to associate a vector space.
    ==endquote==
    (Top of page 367 of http://projecteuclid.org/DPubS/Repo...ew=body&id=pdf_1&handle=euclid.cmp/1104178138 )

    This is the "Chern-Simons Hilbert space" that the Lqg people (Engle et al) are talking about. The LQG black hole horizon is a roughly spherical surface which is punctured by the links of the spin network. These punctures are labeled with G representations.
    Since G = SU(2), these labels are half-integers p.
    The representations (customarily called "spins" although it's not always the best way to think of them) each contribute a specific area. The total area is calculated by summing over all the punctures.
    What they are getting at here is relating the C-S Hilbert space to the entropy.
    I suspect that's already clear to you Atyy but I want to make it explicit in case anyone else is reading.
     
    Last edited: Jul 27, 2010
  6. Jul 27, 2010 #5

    atyy

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    Any links to these - speculating along the lines of http://www.emergentgravity.org/drupal/sites/default/files/EGIV_presentations/Vidal.pdf ?

    http://arxiv.org/abs/0907.0846
    Entropy of Isolated Horizons revisited
    Rudranil Basu, Romesh K. Kaul, Parthasarathi Majumdar

    http://arxiv.org/abs/1004.5487
    Schwarzschild horizon dynamics and SU(2) Chern-Simons theory
    Romesh K. Kaul, Parthasarathi Majumdar

    http://arxiv.org/abs/0706.3359
    Three-Dimensional Gravity Revisited
    Edward Witten

    http://arxiv.org/abs/0802.3231
    Topological Entanglement Entropy in Chern-Simons Theories and Quantum Hall Fluids
    Shiying Dong, Eduardo Fradkin, Robert G. Leigh, Sean Nowling
     
    Last edited by a moderator: Apr 25, 2017
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