Topology, The Orbifold Delta Function, And 3 - D Gauge Analysis

All the measurements in the world do not balance one theorem by which the science of eternal truths is actually advanced. March 14, 1824. On Mathematics. As quoted in Carl Friedrich Gauss: Titan of Science (1955) by Guy Waldo Dunnington. p. 360.

In this rough-draft post, I will derive the 3-dimensional topological gauge 'part' of the action of supergravity. In my last post, I derived the orbifold Delta function

Deriving The Orbifold Delta Function Via Effective 4 - Dimensional Kähler Action(s)

Beauty is the test: there is no permanent place in the world for ugly mathematics. ~ G. H. Hardy!

In this stage-setting post, I will derive, via effective 4-D string-theoretic actions, the orbifold Delta function, to be used in upcoming posts on flux compactification analysis. Theoretically, 4-dimensional effective action(s) can be derived via S-Matrix analysis and one can then easily write massless field actions that yield the correct scattering amplitudes. However, a better and geometrically more elegant way is to derive the Beta functional equations of motion and analyze the action(s) in that setting: in fact, that is the only method that yields us the crucial and philosophically deep orbifold Delta function. Let me set all the super-charged fields to zero with large discrete symmetry: