Topological Phases of Quantum Lattice Systems and Higher Berry Classes - Lecture 1

Explore recent advances in defining the space of gapped lattice systems and their implications for classifying gapped phases of matter in this 1 hour 34 minute lecture by Anton Kapustin from Caltech. Delve into the evolution of understanding topological phases, from the initial belief in Topological Quantum Field Theory (TQFT) classification to the current focus on Short-Range Entangled (SRE) phases and their potential relation to invertible TQFT. Examine the cobordism conjecture and its suggestion that SRE phases are classified by homotopy groups of certain Omega-spectra. Discover how this implies infinite-dimensional spaces of SRE systems carry cohomology classes generalizing the Berry curvature. Learn about the construction of "higher Berry classes" and their equivariant versions, including Hall conductance and nonabelian analogs. Gain insights into the key role of differential graded Frechet-Lie algebra in infinite-volume lattice systems. This lecture, presented at the Institut des Hautes Etudes Scientifiques (IHES), offers a deep dive into cutting-edge concepts in quantum lattice systems and topological phases of matter.