important role in the discussion of black-holes. As it happens for all the other supergravity models, the quartic invariant of the symplectic vector of magnetic and electric charges:

is related to the entropy of the extremal black-holes, the latter being its square root. The origin of the quartic invariant is easily understood in terms of the symmetric tensor tij k . Using the SL(2, R)-invariant antisymmetric symbol εij we can construct an invariant order four polynomial in the tensor tij k by writing:

If we use the standard basis t111, t112, t122, t222, we rotate it with the matrix (9.2.58) and we identify the components of the resultant vector with those of the charge vector Q the explicit form of the invariant quartic polynomial is the following one:

9.2.7Fixed Scalars at BPS Attractor Points: The S3 Explicit

Example

In the case of BPS attractors we can find the explicit expression in terms of the (p, q)-charges for the scalar field fixed values at the critical point.

By means of standard special geometry manipulations the BPS critical point equation

can be rewritten in the following celebrated form which, in the late nineties, appeared in numerous research and review papers (see for instance [19]):

In all cases where the special Kähler manifold is a homogeneous symmetric space the above formula can be explicitly inverted yielding the fixed values of the scalar fields in terms of the charges. We present such a solution for the S3-model.

where I4(p, q) is the quartic invariant defined in (9.2.75).

By replacing the fixed values (9.2.79) into the expression (9.2.37) for the poten-

The above result implies that the horizon area in the case of an extremal BPS blackhole is proportional to the square root of I4(p, q) and, as such, depends only on the charges. The argument goes as follows.

Consider the behavior of the warp factor exp[−U ] in the vicinity of the horizon, when τ → −∞. For regular black-holes the near horizon metric must factorize as follows:

where rH is the Schwarzschild radius defining the horizon. This implies that the asymptotic behavior of the warp factor, for τ → −∞ is the following one:

In the same limit the scalar fields go to their fixed values and their derivatives become essentially zero. Hence near the horizon we have:

Since for extremal black-holes the sum of the above three terms vanishes (see (9.2.5)), we conclude that: