An Explicit Categorical Construction of Instanton Density in Lattice Yang-Mills Theory

Since the inception of lattice QCD, a natural definition for the Yang-Mills instanton on lattice has been long sought for. In a recent work (arXiv:2406.06673), the author showed the natural solution has to be organized in terms of bundle gerbes in higher homotopy theory / higher category theory, and introduced the principles for such a categorical construction. To pave the way towards actual numerical implementation in the near future, nonetheless, an explicit construction is necessary. In this talk, I will provide such an explicit construction for SU(2) gauge theory, with technical aspects inspired by Luscher's 1982 geometrical construction. I will show that the latter is in a suitable sense a saddle point approximation to the full categorical construction. This construction also allows for a natural definition of lattice Chern-Simons-Yang-Mills theory in three spacetime dimensions. Finally, I will briefly discuss the generalization to SU(N).