Classical Continuum Limit of the String Field Theory Dual to Lattice Gauge Theory

We discuss the classical continuum limit of the string field theory dual to the $\mathrm{SU}(N)$ lattice gauge theory and investigate various fundamental phenomena in the continuum theory at the mean-field level. Our construction of the continuum theory is based on the concept of {\it area derivative}, which can be regarded as a generalization of the ordinary derivative $\partial/\partial x^\mu$ to operators acting on functional fields $\phi[C]$ on the loop space. The resultant continuum theory has a $\mathbb{Z}_N^{}$ $1$-form global symmetry, which originates in the $\mathbb{Z}_N^{}$ center symmetry in the gauge theory. We find that the confined and deconfined phases of the gauge theory are identified by the unbroken and broken phases of the $\mathbb{Z}_N^{}$ symmetry respectively by showing the Area/Perimeter law of the classical solution. In the broken phase, the low-energy effective theory is described by a $\mathrm{BF}$-type topological field theory and has a emergent $\mathbb{Z}_N^{}$ $(D-2)$-form global symmetry. The existence of the emergent symmetry is deeply related to $(D-2)$-dimensional topological configurations (i.e. center vortex for $D=4$), and we explicitly construct such a topological defect in the continuum theory. Finally, we also comment on the upper and lower critical dimensions of the gauge theory/string field theory.