N = 2 $$ \mathcal{N}=2 $$ super-EYM coloured black holes from defective Lax matrices

Patrick Meessen,Tomás Ortín

doi:10.1007/jhep04(2015)100

Abstract

We construct analytical supersymmetric coloured black hole solutions, i.e. non-Abelian black hole solutions that have no asymptotic non-Abelian charge but do have non-Abelian charges on the horizon that contribute to the Bekenstein-Hawking entropy, to two SU(3)-gauged N=2 d= supergravities. The analytical construction is made possible due to the fact that the main ingredient is the Bogomol'nyi equation, which under the assumption of spherical symmetry admits a Lax pair formulation. The Lax matrix needed for the coloured black holes must be defective which, even though it is the non-generic and less studied case, is a minor hindrance.

Highlights

We construct analytical supersymmetric coloured black hole solutions, i.e. non-Abelian black hole solutions that have no asymptotic non-Abelian charge but do have non-Abelian charges on the horizon that contribute to the Bekenstein-Hawking entropy, to two SU(3)-gauged N = 2 d = supergravities
Seeing the interesting properties of the coloured black holes, it would be interesting to have more analytical examples; in the N = 2, d = 4 case we are quite fortunate as the main ingredient of their construction is the Bogomol’nyi equation and as was shown by Leznov & Saveliev in ref. [14], the SU(N ) Bogomol’nyi equation, under the assumption of spherical symmetry, is an integrable system related to the SU(N ) Toda molecule
Lax pair and after having identified the solutions needed to construct coloured black holes as corresponding to defective Lax matrices, these were constructed for the SU(3) Bogomol’nyi equation

Summary

Spherically symmetric solutions to the Bogomol’nyi equations

The derivation is best done using Hermitean generators, which means that we use the definitions. Where clearly ΨT = Ψ = Ψ† and an overdot means derivative with respect to r After this redefinition, the link to the Toda molecule can be established [17]: in terms of the components ψi of Ψ and fi of F , the above equations read ψi = fi2 , 2fi = fi Aijψj ,. The second step in Koikawa’s construction [15] is the definition of the new objects L, M from Ψ and F , which form a Lax pair, i.e. The existence of a Lax pair immediately implies that the quantities (“charges”) C(k) ≡ Tr Lk are constants of motion. Defining a coloured solution to the Bogomol’nui equation as one for which the colour charge is zero, i.e. P = 0, we see from eq (2.17) that s2i = i(n + 1 − i), so that S has no vanishing entries; this immediately implies by virtue of eq (2.18) that Φ∞ = 0. The solutions that we have constructed, automatically have this behaviour

Coloured solutions from defective Lax matrices

C-magic coloured black holes

Conclusions