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SIGMA, 2009, Volume 5, 051, 70 pages (Mi sigma397)  

Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review

Ion C. Baianua, James F. Glazebrookba, Ronald Brownc

a University of Illinois at Urbana-Champaign
b Eastern Illinois University
c School of Computer Science, University of Bangor, Dean Street, Bangor Gwynedd LL57 1UT UK

Abstract: A novel algebraic topology approach to supersymmetry (SUSY) and symmetry breaking in quantum field and quantum gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromodynamics, nonlinear physics at high energy densities, dynamic Jahn–Teller effects, superfluidity, high temperature superconductors, multiple scattering by molecular systems, molecular or atomic paracrystal structures, nanomaterials, ferromagnetism in glassy materials, spin glasses, quantum phase transitions and supergravity. This approach requires a unified conceptual framework that utilizes extended symmetries and quantum groupoid, algebroid and functorial representations of non-Abelian higher dimensional structures pertinent to quantized spacetime topology and state space geometry of quantum operator algebras. Fourier transforms, generalized Fourier–Stieltjes transforms, and duality relations link, respectively, the quantum groups and quantum groupoids with their dual algebraic structures; quantum double constructions are also discussed in this context in relation to quasi-triangular, quasi-Hopf algebras, bialgebroids, Grassmann–Hopf algebras and higher dimensional algebra. On the one hand, this quantum algebraic approach is known to provide solutions to the quantum Yang–Baxter equation. On the other hand, our novel approach to extended quantum symmetries and their associated representations is shown to be relevant to locally covariant general relativity theories that are consistent with either nonlocal quantum field theories or local bosonic (spin) models with the extended quantum symmetry of entangled, ‘string-net condensed’ (ground) states.

Keywords: extended quantum symmetries; groupoids and algebroids; quantum algebraic topology (QAT); algebraic topology of quantum systems; symmetry breaking, paracrystals, superfluids, spin networks and spin glasses; convolution algebras and quantum algebroids;

DOI: https://doi.org/10.3842/SIGMA.2009.051

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Bibliographic databases:

ArXiv: 0904.3644
MSC: 81R40; 16W30; 22A22; 81R50; 43A05; 46L10
Received: November 25, 2008; in final form April 9, 2009; Published online April 23, 2009
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Citation: Ion C. Baianu, James F. Glazebrook, Ronald Brown, “Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review”, SIGMA, 5 (2009), 051, 70 pp.

Citation in format AMSBIB
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\by Ion C.~Baianu, James F.~Glazebrook, Ronald Brown
\paper Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A~Review
\jour SIGMA
\yr 2009
\vol 5
\papernumber 051
\totalpages 70
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\crossref{https://doi.org/10.3842/SIGMA.2009.051}
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