9.2.7.1 An Explicit Example of Exact Regular BPS Solution

This general mechanism can be illustrated with an explicit example of exact regular solution of the S3 model. The key identifier of the solution is its vector of electromagnetic charges that in our chosen example is the following one:

The corresponding explicit solution is given below and depends on three parameters p, q and κ which yields the value of the imaginary part of the scalar field at radial infinity, namely at τ = 0.

The Metric

The interested reader can verify that the expressions displayed above for all the fields fulfill the variational equations (9.2.6) of the σ -model and hence are bonafide solutions of the supergravity field theory.

which makes sense only as long as pq3 > 0. Inserting (9.2.86) into (9.2.75) we see that pq3 = I4. Hence we conclude that this solution is indeed BPS as expected. The horizon area is:

9.2.8The Attraction Mechanism Illustrated with an Exact Non-BPS Solution

Next we illustrate the attraction mechanism with an explicit example of exact regular non-BPS solution of the S3 model. The vector of electromagnetic charges of the considered solution differs from that of the above BPS-example only by means of a sign, namely it is:

The corresponding explicit solution which is given below depends on four parameters p, q and κ, ξ which yield the values of both the imaginary and the real parts of the scalar field at radial infinity, namely at τ = 0.

The Metric The metric is defined by the function U for which the integration techniques of the σ -model yield the following expression:

exp U (τ ) = κ3/4/ −q3κ3τ 3 − q3κξ 2τ 3 + 3q2κ5/2τ 2 + 3q2√κξ 2τ 2

The Electromagnetic Fields The explicit form of the two field strengths appearing in the S3 model is completely determined by (9.2.14). It suffices to know the magnetic charges (p1, p2) = (0, p), the Taub-NUT charge n = 0 and the derivatives