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, one of authors 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 paper we provide such an explicit construction for $SU(2)$ gauge theory, with technical aspects inspired by L\"{u}scher's 1982 geometrical construction. We will see how the latter is in a suitable sense a saddle point approximation to the full categorical construction. The generalization to $SU(N)$ will be discussed. The construction also allows for a natural definition of lattice Chern-Simons-Yang-Mills theory in three spacetime dimensions.