The Basics of Non-Invertible Symmetries and Dualities - Lecture III

Explore advanced concepts in theoretical physics through this comprehensive lecture that delves into the fundamental principles of non-invertible symmetries and their relationship to dualities in quantum field theory. Learn how these mathematical structures extend beyond traditional group symmetries to encompass more general categorical symmetries that cannot be inverted in the usual sense. Discover the profound connections between non-invertible symmetries and various duality transformations in condensed matter systems and high-energy physics. Examine specific examples and applications where these symmetries emerge naturally, including topological phases of matter and conformal field theories. Understand the mathematical framework underlying these concepts, including fusion categories and their role in characterizing non-invertible symmetry operations. Investigate how these symmetries constrain physical theories and lead to new insights about phase transitions and critical phenomena. Analyze the implications for understanding quantum many-body systems and their exotic phases, particularly in two-dimensional systems where these symmetries are most naturally realized. Gain insight into cutting-edge research directions in theoretical physics where non-invertible symmetries are opening new avenues for understanding strongly correlated quantum systems and their universal properties.