CBMS Conference: Topological and Geometric Methods in Quantum Field Theory NSF-CBMS Regional Conference in the Mathematical Sciences

Montana State University, Bozeman MT

Investigators

Abstract

A NSF-CBMS regional conference on "Topological and geometric methods in quantum field theory" will be held July 31- August 4, 2017 at Montana State University (Bozeman). Topological phases of matter are at the frontline of research in condensed matter physics and offer new possibilities in electronics and superconductors. Topological phases of matter - and more generally, quantum field theory - have elegant formulations in terms of pure mathematics, specifically from the fields of geometry (the study of objects via measuring lengths, areas, etc.) and topology (the study of more global quantity of objects which are intrinsic to the space and not dependent on lengths, etc.). Moreover, insight and imagination in physics has led to many mathematical advances in geometry and topology. Recently, mathematicians have undertaken a program to classify and exemplify all possible topological phases of matter. The main goal of the conference is to explain the interplay between physics and mathematics described above and to encourage further interaction and dialogue between condensed matter physicists, and geometers and topologists. In particular, the conference is aimed at young researchers (graduate students and postdoctoral fellows) in order to energize and educate a future generation. Further, it is an explicit goal of the conference to engage women and underrepresented mathematicians and physicists, as well as the research communities of the Northern Rocky Mountains, in these exciting developments. The conference will result in a monograph which will fill a gap in the academic literature concerning the interface of geometry and topology with physics. The classification of topological phases has been a hot topic in the last five years, and it's relationship to stable homotopy theory and bordism groups was realized early on. The recent work of Freed and others recognizes certain topological phases as invertible representations of bordism categories. Recent work of Galatius, Lurie, and others realizes the classification of such representations as an accessible computation in stable homotopy theory. Specifically, for instance, the groupoid completion of the bordism n-category is the (-n)-space of the Thom spectrum of the virtual negative of the universal rank n bundle. Understanding topological quantum field theory in terms of bordism categories has come a long way since its introduction by Atiyah in the 1980's. The consideration of 3d Chern-Simons theory in the 1990's made it clear that an extension to higher category theory was necessary. The subsequent development of higher category theory and derived geometry has ushered in a flourish of activity in topological quantum field theory, particularly since Lurie's inspired outline of the cobordism hypothesis in 2010. These developments are still underway and being consolidated by the mathematics community; this conference will go a long way in demonstrating their power and breadth. Further information can be found at the conference website: http://www.math.montana.edu/cbms/