  We construct in this article a new realization of quantum geometry, which is
obtained by quantizing the recently-introduced flux formulation of loop quantum
gravity. In this framework, the vacuum is peaked on flat connections, and
states are built upon it by creating local curvature excitations. The inner
product induces a discrete topology on the gauge group, which turns out to be
an essential ingredient for the construction of a continuum limit Hilbert
space. This leads to a representation of the full holonomy-flux algebra of loop
quantum gravity which is unitarily-inequivalent to the one based on the
Ashtekar-Isham-Lewandowski vacuum. It therefore provides a new notion of
quantum geometry. We discuss how the spectra of geometric operators, including
holonomy and area operators, are affected by this new quantization. In
particular, we find that the area operator is bounded, and that there are two
different ways in which the Barbero-Immirzi parameter can be taken into
account. The methods introduced in this work open up new possibilities for
investigating further realizations of quantum geometry based on different
vacua.
