Abstract
Classical single centered solutions of 1/4 BPS dyons in mathcal{N} = 4 theories are usually constructed in duality frames which contain non-trivial hair degrees of freedom localized outside the horizon. These modes are in addition to the fermionic zero modes associated with broken supersymmetry. Identifying and removing the hair from the 1/4 BPS index allows us to isolate the degrees of freedom associated with the horizon. The spherical symmetry of the horizon then ensures that index of the horizon states has to be positive. We verify that this is indeed the case for the canonical example of dyons in type IIB theory on K3 × T2 and prove this property holds for a class of states. We generalise this observation to all CHL orbifolds, this involves identifying the hair and isolating the horizon degrees of freedom. We then identify the horizon states for 1/4 BPS dyons in mathcal{N} = 4 models obtained by freely acting ℤ2 and ℤ3 orbifolds of type IIB theory compactified on T6 and observe that the index is again positive for single centred black holes. This observation coupled with the fact the 1/4 BPS index of single centred solutions without removal of the hair violates positivity indicates that there exists no duality frame in these models without non-trivial hair.
Highlights
Degrees of freedom with charge qi and dhair(qhair; {qi}) is the index of the hair carrying total charge qhair of a n-centered black hole whose horizons carry charges q1, · · · qn
These modes are in addition to the fermionic zero modes associated with broken supersymmetry
We identify the horizon states for 1/4 BPS dyons in N = 4 models obtained by freely acting Z2 and Z3 orbifolds of type IIB theory compactified on T 6 and observe that the index is again positive for single centred black holes
Summary
In the work of [5, 6] the hair modes of the 1/4 BPS dyonic black hole in type IIB theory compactified on K3×T 2 were constructed. 21 left moving bosonic modes, these arise from the deformations of the 21 anti-selfdual forms of type IIB on K3. These deformations involve 21 scalar functions folded with the 2 form dωT N on the Taub-Nut given by δHs = hs(v)dv ∧ dωT N , v = t + x5, s = 1, · · · 21. Observe that the partition function of the bosonic hair modes is given by (2.24) This is the partition function of the fundamental string in the N = 2 CHL orbifold of the heterotic theory with the zero point energy removed [19, 29]. Let us conclude by writing the general formula for the horizon states as
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