Thematically organized into five parts, this article covers topics in the following areas.

1.

An introduction to the Gopakumar-Vafa approach to superstring theory. Supersymmetric terms in the effective potential that are supersymmetric cannot be written as integrals over all of superspace. Gopakumar and Vafa proposed that the interaction terms could be understood by considering a supersymmetric background obtained by turning on an anti-selfdual graviphoton field.

2.

The background in five dimensions and its supersymmetry. Minimal supergravity is described by bosonic fields forming the metric tensor and a \(U(1)\) gauge field \(V\). Its field strength \(dV\) describes the graviphoton.

3.

A Schwinger-like calculation for massive BPS states in five dimensions. Only the leading order approximations to the particle action are considered and couplings of higher dimension are ignored.

4.

The Schwinger calculation with fields. One aim here is to compute the one-loop effective action for a 4-dimensional BPS hypermultiplet in a background given by the graviton field.

5.

D4-branes and the Ooguri-Vafa formula as an opern-string analog of the Gopakumar-Vafa formula. It is pointed out that one should worry about infrared effects.

Three Appendices describe the supergravity conventions in dimensions 11, 5, and 4, the derivation of the 5-dimensional BPS superparticle action, and address the question whether short-range interactions will affect the Ooguri-Vafa formula. There are fifty-six published papers that address topics close to the abstract mathematical modeling used in the present article which now provides a very useful reference for everyone interested in modern field theory.