QUANTUM SYMMETRIES- FUSION RULES, BRAIDS, AND INDICATORS
Abstract
Symmetries are ubiquitous in mathematics and theoretical physics, where the notion of a group is introduced to describe such symmetries. In recent decades, certain quantum objects have appeared whose symmetries are better described by group-like objects called tensor categories. Examples include subfactors, quantum groups, Hopf algebras, and topological phases of matter. These mathematical objects have applications in a diverse range of settings across mathematics and physics, including quantum invariants of links and 3-manifolds, representation theory, condensed matter physics, topological phases of matter, and quantum information. This project is dedicated to the algebraic manifestations of quantum symmetries, more specifically from the point of view of homological algebra, representation theory, category theory, and noncommutative algebraic geometry, in terms of tensor triangular geometry. Many subprojects are suitable for undergraduates interested in quantum computation, who will be involved in the research. The goal of this project is to provide theoretic foundations and practical techniques to study quantum symmetries that arise from tensor triangular geometry and to seek their applications in different fields of mathematics and theoretical physics. The research project will focus on 1) ring theoretic and homological properties in tensor triangular geometry; 2) homological Frobenius-Schur indicators in tensor triangular geometry; 3) reduction modulo p technique in tensor triangular geometry and its application in support variety theory; 4) Dixmier-Moeglin equivalence in tensor triangular geometry and its application in noncommutative algebraic geometry; and 5) deformation quantization in tensor triangular geometry and its applications in theoretical physics. Homological methods via tensor triangular geometry provide an abstract platform to transpose ideas and techniques between various areas of mathematics and theoretical physics. Our proposal will strengthen classical algebraic methods by integrating various algebro-geometric arguments where quantum symmetries play a crucial role to make possible solutions of many open problems in mathematics and theoretical physics.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Mar 07, 2023
- Source ID
- FA95502210272
Entities
People
- Xingting Wang
Organizations
- Air Force Office of Scientific Research
- Howard University
- United States Air Force