Abstract:

Putting continuum QFT (not just TQFT) on the lattice is important for both fundamental understandings and practical numerics. The traditional way of doing so, based on simple intuitions, however, does not admit natural definitions for general topological operators of continuous-valued fields---one such example is the long standing problem in lattice QCD of lacking a natural definition for Yang-Mills instantons.

In this lecture series, I will explain a more systematic way to relate continuum and lattice QFT, based on higher categories and higher anafunctors, so that the topological operators in the continuum can be naturally defined on the lattice. The idea, though formulated formally, is physically very intuitive---we want to effectively capture the different possibilities of how a lattice field may interpolate into the continuum, so the higher categories that are employed to study higher homotopy theory should be naturally involved. Via this formalism, we resolve the long standing problem of defining instanton (as well as Chern-Simons term) in lattice Yang-Mills theory, in terms of multiplicative bundle gerbes. Moreover, when the fields become discrete, our formalism can recover the Dijkgraaf-Witten and Turaev-Viro theory, so we hope this formalism to be a good starting point towards (in the very long term) a comprehensive categorical understanding of QFT that encompass both continuous and discrete degrees of freedom, applicable both to IR and to UV.
Reference: 2406.06673