Holonomies and Gauge Fields

Given a group N, we introduce N-connections on a graph, loop holonomies, and the associated bosonic and fermionic field. When the group is discrete, a connection induces a Galois covering. Loops and spanning forests on the cover are related to bosonic and fermionic fields which can be decomposed using group representation theory into fields interacting with the connection. We introduce the measure on connections given by the expectation of the product of holonomies of a loop ensemble and show that for high intensity and high killing rate this measure can approximate the Yang–Mills measure. Then we present an intertwining relation between merge-and-split generators on loop ensembles (which were introduced in Chap. 7 ) and Casimir operators on U(d)-connections. By adding a deformation part to the generator on loops, this result is extended to the Casimir operator modified in order to be self adjoint with respect to Yang–Mills measure. A consequence of this result is the Schwinger–Dyson equation obtained as an essential step in the proof of t’Hooft’s expansion for large d.