The Universal Coefficient Theorem and Quantum Field Theory

1. Front Matter

   Pages i-xvi

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2. Introduction

   • Andrei-Tudor Patrascu

   Pages 1-15

3. Elements of General Topology

   • Andrei-Tudor Patrascu

   Pages 17-30

4. Algebraic Topology

   • Andrei-Tudor Patrascu

   Pages 31-38

5. Homological Algebra

   • Andrei-Tudor Patrascu

   Pages 39-52

6. Connections: Topology and Analysis

   • Andrei-Tudor Patrascu

   Pages 53-68

7. The Atyiah Singer Index Theorem

   • Andrei-Tudor Patrascu

   Pages 69-89

8. Universal Coefficient Theorems

   • Andrei-Tudor Patrascu

   Pages 91-106

9. BV and BRST Quantization, Quantum Observables and Symmetry

   • Andrei-Tudor Patrascu

   Pages 107-154

10. Universal Coefficient Theorem and Quantum Field Theory

    • Andrei-Tudor Patrascu

    Pages 155-198

11. The Universal Coefficient Theorem and Black Holes

    • Andrei-Tudor Patrascu

    Pages 199-236

12. From Grothendieck’s Schemes to QCD

    • Andrei-Tudor Patrascu

    Pages 237-257

13. Conclusions

    • Andrei-Tudor Patrascu

    Pages 259-261

14. Back Matter

    Pages 263-270

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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 ofview, 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.

• Quantum Field Theory and Topology
• Algebraic Topology for Physics
• Universal Coefficient Theorem
• Homological Algebra
• ER-EPR
• Algebraic Geometry
• Riemann-Roch Theorem
• Grothendieck's Work Applied in Physics

• Department of Physics and Astronomy, University College London Department of Physics and Astronomy, London, United Kingdom

  Andrei-Tudor Patrascu

With a surprisingly diverse research experience covering domains from molecular spectroscopy to theoretical high energy physics and from algebraic topology and geometry to homological algebra, Andrei Patrascu seeks to understand the origin of dualities in physics. After following graduate courses at internationally renowned institutions like the Ecole Normale Superieure in Paris, Rensselaer Polytechnic Institute in Troy, NY and University College London, the author, while also working on applied physics, initiated a new research program in mathematical physics with the goal of giving a homological algebraic interpretation to some of the most important results in theoretical physics: the AdS/CFT correspondence and the ER-EPR duality. This book shows how the ideas leading to this program took shape and may become a guide towards new results of this type.
        
 

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