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Two-groups! Pfeiffer scoops Freidel (maybe by a few days)

  1. Aug 22, 2007 #1


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    If I understand what Kea said here recently, Laurent Freidel and Aristide Baratin have a paper in the works about BF theory
    of 4D spacetime with matter put in as topological defects in the gravitational field (like places where space is somehow tangled)

    To do this one needs the theory of 2-groups----at least one needs the Poincaré two-group. So the Baratin-Freidel paper will have two-groups.
    Baez had a thread here at PF a year or so ago telling us about this. There are some talks about it at his website.

    So a Baratin-Freidel paper was expected about this, but, to add to the excitement, this paper from Hendryk Pfeiffer comes out

    Topological Higher Gauge Theory - from BF to BFCG theory
    Florian Girelli, Hendryk Pfeiffer, E. M. Popescu
    15 pages
    (Submitted on 22 Aug 2007)

    "We study generalizations of 3- and 4-dimensional BF-theory in the context of higher gauge theory. First, we construct topological higher gauge theories as discrete state sum models and explain how they are related to the state sums of Yetter, Mackaay, and Porter. Under certain conditions, we can present their corresponding continuum counterparts in terms of classical Lagrangians. We then explain that two of these models are already familiar from the literature: the SigmaPhiEA-model of 3-dimensional gravity coupled to topological matter, and also a 4-dimensional model of BF-theory coupled to topological matter."
    Last edited: Aug 22, 2007
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  3. Aug 22, 2007 #2


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    this will probably be an important paper. I should get the links to tutorials at Baez website.
    the codeword is "higher"
    higher gauge theory is gauge theory with two-groups instead of groups
    higher Yang-Mills etc etc.
    also has to do with "higher" category theory----two-categories instead of just categories.
    for a taste, look at Baez Schreiber paper about "two-connections on two-bundles".
    this is all structure you build using two-groups instead of groups.

    the interesting thing is that it is ACTUALLY NOT SILLY because if you want to realize matter as the KINKS in the gravitational field (i.e. the geometry) then you are FORCED to to use two-groups when you move up from 3D to 4D. Freidel showed that it works fine in 3D with just ordinary groups. but to move up to 4D you need to boost the idea of a group.

    The Baez tutorials will make it much easier to understand what is happening with the Pfeiffer et al and with the Freidel et al when it comes out.

    Meanwhiles lets take a look at the REFERENCES for the Girelli-Pfeiffer-Popescu paper

    [1] G. T. Horowitz: Exactly soluble diffeomorphism invariant theories. Comm. Math. Phys. 125, No. 3 (1989) 417–437,
    MR 1022521.
    [2] G. Ponzano and T. Regge: Semiclassical limit of Racah coefficients. In Spectroscopic and group theoretical methods in
    physics, ed. F. Bloch. North Holland Publications, Amsterdam, 1968, pp. 1–58.
    [3] V. G. Turaev and O. Y. Viro: State sum invariants of 3-manifolds and quantum 6j-symbols. Topology 31, No. 4 (1992)
    865–902, MR 1191386.
    [4] J. W. Barrett and B. W. Westbury: Spherical categories. Adv. Math, 143 (1999) 357–375, arxiv:hep-th/9310164,
    MR 1686423.
    [5] J. W. Barrett and I. Naish-Guzman: The Ponzano-Regge model and Reidemeister torsion (2006). Preprint
    [6] J. C. Baez: Higher Yang–Mills theory (2002). Preprint arxiv:hep-th/0206130.
    [7] H. Pfeiffer: Higher gauge theory and a non-Abelian generalization of 2-form electrodynamics. Ann. Phys. 308, No. 2
    (2003) 447–477, arxiv:hep-th/0304074, MR 2018680.
    [8] F. Girelli and H. Pfeiffer: Higher gauge theory — differential versus integral formulation. J. Math. Phys. 45, No. 10
    (2004) 3949–3971, arxiv:hep-th/0309173, MR 2095681.
    [9] J. C. Baez and U. Schreiber: Higher Gauge Theory: 2-Connections on 2-Bundles (2004). Preprint
    10] D. N. Yetter: TQFT’s from homotopy 2-types. J. Knot Th. Ramif. 2, No. 1 (1993) 113–123, MR 1209321.
    11] T. Porter: Topological quantum field theories from homotopy n-types. J. London Math. Soc. (2) 58, No. 3 (1998)
    723–732, MR 1678163.
    12] M. Mackaay: Spherical 2-categories and 4-manifold invariants. Adv. Math. 143, No. 2 (1999) 288–348,
    arxiv:math/9805030 [math.QA], MR 1686421.
    13] M. Mackaay: Finite groups, spherical 2-categories, and 4-manifold invariants. Adv. Math. 153, No. 2 (2000) 353–390,
    arxiv:math/9903003 [math.QA], MR 1770934.
    14] R. B. Mann and E. M. Popescu: Scalar and tensorial topological matter coupled to (2+1)-dimensional gravity: A
    Classical theory and global charges. Class. Quant. Grav. 23, No. 11 (2006) 3721–3746, arxiv:gr-qc/0511141, MR 2235430.
    15] J. C. Baez and A. D. Lauda: Higher-dimensional algebra V: 2-groups. Theor. Appl. Cat. 12 (2004) 423–491,
    arxiv:math/0307200 [math.QA], MR 2068521.
    16] H. Pfeiffer: 2-Groups, trialgebras and their Hopf categories of representations. Adv. Math. 212, No. 1 (2007) 62–108,
    arxiv:math/0411468 [math.QA].
    17] J. C. Baez and A. S. Crans: Higher dimensional algebra VI: Lie 2-algebras. Theor. Appl. Cat. 12 (2004) 492–538,
    arxiv:math/0307263 [math.QA], MR 2068522.
    18] J. H. C. Whitehead: On C1-complexes. Ann. Math. 41, No. 4 (1940) 809–824, MR 0002545.
    19] J. Cerf: Sur les diff´eomorphismes de la sph`ere de dimension trois ( 4 = 0). Lecture Notes in Mathematics 53. Springer,
    Berlin, 1968. MR 0229250.
    20] N. H. Kuiper: On ‘La nullit´e de 0(DiffS3)’ by J. Cerf. Mathematical Reviews No. 6641 a–d Vol. 33 (1963) 1126,
    MR 0198483.
    21] J. W. Alexander: The combinatorial theory of complexes. Ann. Math. 31, No. 2 (1930) 292–320, MR 1502943.
    22] T. Porter: Interpretations of Yetter’s notion of G-coloring: simplicial fibre bundles and non-abelian cohomology. J. Knot
    Th. Ramif. 5, No. 5 (1996) 687–720, MR 1414095.
    23] U. Pachner: PL homeomorphic manifolds are equivalent by elementary shellings. Europ. J. Combinat. 12, No. 2 (1991)
    129–145, MR 1095161.
    24] A. D. Lauda and H. Pfeiffer: State sum construction of two-dimensional open-closed Topological Quantum Field
    Theories. To appear in J. Knot. Th. Ramif. (2007). Preprint arxiv:math/0602047 [math.QA].
    25] J. F. Martins and T. Porter: On Yetter’s invariant and an extension of the Dijkgraaf-Witten invariant to categorical
    groups (2006). Preprint arxiv:math/0608484 [math.QA].
    26] L. Freidel and D. Louapre: Diffeomorphisms and spin foam models. Nucl. Phys. B 662, No. 1–2 (2003) 279–298,
    arxiv:gr-qc/0212001, MR 1984379.
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    (1989) 56–60, MR 1017394.
    28] R. B. Mann and E. M. Popescu: Higher Gauge Theory and Gravity in (2+1) Dimensions. Int. J. Mod. Phys. A, in
    press (2007). Preprint arxiv:gr-qc/0607076.
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  4. Aug 22, 2007 #3


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    some exerpts from Girelli-Pfeiffer-Popescu

    "Roughly speaking, in addition to the connection 1-form of conventional gauge theory which equips curves with holonomies in the gauge group G, higher gauge theory introduces a connection 2-form which can be used to equip surfaces with a new kind of surface holonomy, given by elements of another group H. More precisely, the algebraic structure that replaces the gauge group in higher gauge theory is a crossed module..."

    "...The present article is structured as follows. In Section II, we review the relevant algebraic tools involved in the description of higher gauge theory: 2-groups, crossed modules, Lie 2-algebras and differential crossed modules.

    In Section III, we define the discrete state sum models of topological higher gauge theory in dimensions d = 3, 4. A self-contained proof that these models are well defined, i.e. independent of the chosen triangulation, is contained in Appendix A. We then explain the relationship to Mackaay’s state sum in Appendix B.

    In Section IV, we present the continuum counterparts of our discrete models for the case of Lie groups and comment on their relationship with models known from the quantum gravity literature..."

    the basic insight (from a QG perspective) is simple:
    a CONNECTION something giving you an holonomy along a CURVE. you travel along a path or a loop and feel how the geometry changes as you go.
    this is the idea of LQG, the original loops of LQG were what you traveled along to explore the geometry and see if you were tilted some when you got back to start.

    but that is essentially a 3D spatial thing. Now suppose you want to connect two 3D geometries (each represented by holonomies along loops) by a 4D path.
    a spacetime is a history of how 3D geometry at one time evolves to be 3D geometry at a later time.
    TO DESCRIBE THAT YOU NEED HOLONOMIES ALONG SURFACES like the "cylindrical" surface that connects a loop at time one with a loop at time two.

    in the discrete language of spinnetworks and spinfoams----a spinnet is a labeled graph (a souped-up loop really) that explores the 3D geometry
    and a spinfoam is the 4D thing that connects two spin networks------so it has to have patches of SURFACE as well as the vertices and edges that a spinnetwork has. It takes a surface to connect two edges. In a linear holonomy a group is telling you how things change as you travel along a curve. If you want to connect two linear holonomies you need more than a group---you need a twogroup to tell you how things change as you slide across the connecting surface between the first linear holonomy and the second.

    Well, that is just a start. I should get the links to the Baez tutorials about this.
    Last edited: Aug 22, 2007
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