The Universal Coefficient Theorem and Quantum Field Theory: A Topological Guide for the Duality Seeker
This thesis describes a new connection between algebraic geometry, topology, number theory and quantum field theory. It offers a pedagogical introduction to algebraic topology, allowing readers to rapidly develop basic skills, and it also presents original ideas to inspire new research in the quest for dualities. Its ambitious goal is to construct a method based on the universal coefficient theorem for identifying new dualities connecting different domains of quantum field theory. This thesis opens a new area of research in the domain of non-perturbative physics—one in which the use of different coefficient structures in (co)homology may lead to previously unknown connections between different regimes of quantum field theories. The origin of dualities is an issue in fundamental physics that continues to puzzle the research community with unexpected results like the AdS/CFT duality or the ER-EPR conjecture. This thesis analyzes these observations from a novel and original point of view, mainly based on a fundamental connection between number theory and topology. Beyond its scientific qualities, it also offers a pedagogical introduction to advanced mathematics and its connection with physics. This makes it a valuable resource for students in mathematical physics and researchers wanting to gain insights into (co)homology theories with coefficients or the way in which Grothendieck's work may be connected with physics.
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2 Elements of General Topology
3 Algebraic Topology
4 Homological Algebra
Topology and Analysis
6 The Atyiah Singer Index Theorem
7 Universal Coefficient Theorems
8 BV and BRST Quantization Quantum Observables and Symmetry
9 Universal Coefficient Theorem and Quantum Field Theory
10 The Universal Coefficient Theorem and Black Holes
11 From Grothendiecks Schemes to QCD
Appendix A Some Relevant Proofs
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The Universal Coefficient Theorem and Quantum Field Theory: A Topological ...
No preview available - 2016
adjoint anomalies appears associated axioms black hole BRST bundle chain complex Chern choice of coefficients classes coefficient groups coefficient structure cohomology group commutation connection considered construction covariant curve defined definition derived diagrams differential dimensional divisor duality elements elliptic encoded entanglement equations equivalent example exists extension finite formulation functor gauge transformation given global Grothendieck groups in co)homology harmonic oscillator Hence homological algebra homology homomorphism integral ISBN isomorphism Leibniz Leibniz algebras Lemma Lie algebra manifold Math mathematical Mayer–Vietoris meromorphic function metric module morphisms non-trivial notion objects observables obtain open sets operator Patrascu perturbative Phys physical polynomial quantization quantum field theory quantum gravity quantum mechanics qubit R-module renormalization Riemann surface ring short exact sequence simplexes spacetime Springer string theory subset subspace symmetry topological space torus trivial universal coefficient theorem zero