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Division Algebras, Supersymmetry and Higher Gauge Theory
, 2012
"... A dissertation is the capstone to a doctoral program, and the acknowledgements provide a useful place to thank the countless people who have helped out along the way, both personally and professionally. First, of course, I thank my advisor, John Baez. It is hard to imagine a better advisor, and no o ..."
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A dissertation is the capstone to a doctoral program, and the acknowledgements provide a useful place to thank the countless people who have helped out along the way, both personally and professionally. First, of course, I thank my advisor, John Baez. It is hard to imagine a better advisor, and no one deserves more credit for my mathematical and professional growth during this program. “Thanks ” does not seem sufficient, but it is all I have to give. Also deserving special mention is John’s collaborator, James Dolan. I am convinced there is no subject in mathematics for which Jim does not have some deep insight, and I thank him for sharing a few of these insights with me. Together, John and Jim are an unparalleled team: there are no two better people with whom to talk about mathematics, and no two people more awake to the joy of mathematics. I would also like to thank Geoffrey Dixon, Tevian Dray, Robert Helling, Corinne Manogue, Chris Rogers, Hisham Sati, James Stasheff, and Riccardo Nicoletti for helpful conversations and correspondence. I especially thank Urs Schreiber for many discussions of higher gauge theory and L∞superalgebras, smooth ∞groups, and supergeometry.
Division Algebras and Supersymmetry III
, 2011
"... Recent work applying higher gauge theory to the superstring has indicated the presence of ‘higher symmetry’. Infinitesimally, this is realized by a ‘Lie 2superalgebra ’ extending the Poincaré superalgebra in precisely the dimensions where the classical superstring makes sense: 3, 4, 6 and 10. In th ..."
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Recent work applying higher gauge theory to the superstring has indicated the presence of ‘higher symmetry’. Infinitesimally, this is realized by a ‘Lie 2superalgebra ’ extending the Poincaré superalgebra in precisely the dimensions where the classical superstring makes sense: 3, 4, 6 and 10. In the previous paper in this series, we constructed this Lie 2superalgebra using the normed division algebras. In this paper, we use an elegant geometric technique to integrate this Lie 2superalgebra to a ‘Lie 2supergroup ’ extending the Poincaré supergroup in the same dimensions. Briefly, a ‘Lie 2superalgebra ’ is a twoterm chain complex with a bracket like a Lie superalgebra, but satisfying the Jacobi identity only up to chain homotopy. Simple examples of Lie 2superalgebras arise from 3cocycles on Lie superalgebras, and it is in this way that we constructed the Lie 2superalgebra above. Because this 3cocycle is supported on a nilpotent subalgebra, our geometric technique applies, and we obtain a Lie 2supergroup integrating the Lie 2superalgebra in the guise of a smooth 3cocycle on the Poincaré supergroup. 1
QUANTUM SYMMETRIES, OPERATOR ALGEBRA AND QUANTUM GROUPOID REPRESENTATIONS: PARACRYSTALLINE SYSTEMS, TOPOLOGICAL ORDER, SUPERSYMMETRY AND GLOBAL SYMMETRY BREAKING
, 2011
"... Novel approaches to extended quantum symmetry, paracrystals, quasicrystals, noncrystalline solids, topological order, supersymmetry and spontaneous, global symmetry breaking are outlined in terms of quantum groupoid, quantum double groupoids and dual, quantum algebroid structures. Physical applicati ..."
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Novel approaches to extended quantum symmetry, paracrystals, quasicrystals, noncrystalline solids, topological order, supersymmetry and spontaneous, global symmetry breaking are outlined in terms of quantum groupoid, quantum double groupoids and dual, quantum algebroid structures. Physical applications of such quantum groupoid and quantum algebroid representations to quasicrystalline structures and paracrystals, quantum gravity, as well as the applications of the Goldstone and Noether's theorems to: phase transitions in superconductors/superfluids, ferromagnets, antiferromagnets, mictomagnets, quasiparticle (nucleon) ultrahot plasmas, nuclear fusion, and the integrability of quantum systems are also considered. Both conceptual developments and novel approaches to Quantum theories are here proposed starting from existing Quantum Group Algebra (QGA), Algebraic Quantum Field Theories (AQFT), standard and effective Quantum Field Theories (QFT), as well as the refined `machinery' of