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

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.

References in corpus (50)

• Fault-tolerant quantum computation by anyons
• Non-Abelian Anyons and Topological Quantum Computation
• Detecting topological order in a ground state wave function
• Quantum Glassiness
• Proposed experiments to probe the non-abelian ν=5/2 quantum Hall state
• Groupoids: unifying internal and external symmetry
• Mott--Hubbard transition vs. Anderson localization of correlated, disordered electrons
• Fermions, strings, and gauge fields in lattice spin models
• Photons and electrons as emergent phenomena
• Chiral structure of modular invariants for subfactors
• Solutions of the quantum dynamical Yang-Baxter equation and dynamical quantum groups
• Projective Construction of Non-Abelian Quantum Hall Liquids
• Quantum order from string-net condensations and origin of light and massless fermions
• Generalized Lie bialgebroids and Jacobi structures
• Exchange dynamical quantum groups
• Lectures on the dynamical Yang-Baxter equations
• Graph inverse semigroups, groupoids and their C*-algebras
• Critical behavior at Mott-Anderson transition: a TMT-DMFT perspective
• Fermions without fermi fields
• Quantum ether: photons and electrons from a rotor model
• Broken quantum symmetry and confinement phases in planar physics
• Weak Hopf Algebras I: Integral Theory and C^*-structure
• Some Possible Roles for Topos Theory in Quantum Theory and Quantum Gravity
• Geometrical measurements in three-dimensional quantum gravity
• A lattice bosonic model as a quantum theory of gravity
• New mean field theory of the tt't''J model applied to the high-Tc superconductors
• A Treatise on Quantum Clifford Algebras
• On q-analog of McKay correspondence and ADE classification of sl^(2) conformal field theories
• Quantum groupoids and deformation quantization
• A topos perspective on the Kochen-Specker theorem: I. Quantum States as Generalized Valuations
• Scaling behavior of an Anderson impurity close to the Mott-Anderson transition
• Symmetry Representations in the Rigged Hilbert Space Formulation of Quantum Mechanics
• A topos perspective on the Kochen-Specker theorem: II. Conceptual Aspects, and Classical Analogues:
• On quantum and parallel transport in a Hilbert bundle over spacetime
• Continuous representations of groupoids
• Detecting topological orders through continuous quantum phase transition
• A Topos Perspective on the Kochen-Specker Theorem: III. Von Neumann Algebras as the Base Category
• The Differential Form Method for Finding Symmetries
• Elliptic Quantum Group U_{q,p}(\hat{sl}_2), Hopf Algebroid Structure and Elliptic Hypergeometric Series
• Dynamical R Matrices of Elliptic Quantum Groups and Connection Matrices for the q-KZ Equations
• Geometric quantization of Hamiltonian actions of Lie algebroids and Lie groupoids
• A Topos Perspective on the Kochen-Specker Theorem: IV. Interval Valuations
• On spectral representation of coalgebras and Hopf algebroids
• Exact sequences of fibrations of crossed complexes, homotopy classification of maps, and nonabelian extensions of groups
• Weak C* Hopf Symmetry
• This Week's Finds in Mathematical Physics (51-100)
• Quantization of discretized spacetimes and the correspondence principle
• The Inhomogeneous Invariance Quantum Supergroup of Supersymmetry Algebra
• Hamiltonian Algebroids and deformations of complex structures on Riemann curves
• Compact Quantum Groupoids

Cited by in corpus (1)

• The Free Energy Principle drives neuromorphic development