AdS₃ x S³ (Un)twisted and Squashed, and an O(2,2;Z) Multiplet of Dyonic Strings

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We consider type IIB configurations carrying both NS-NS and R-R electric and magnetic 3-form charges, and whose near horizon geometry contains AdS₃ x S³. Noting that S³ is a U(1) bundle over CP¹ S², we construct the dual type IIA configurations by a Hopf T-duality along the U(1) fibre. In the case where there are only R-R charges, the S³ is untwisted to S² x S¹ (in analogy with a previous treatment of AdS₅ x S⁵.) However, in the case where there are only NS-NS charges, the S³ becomes the cyclic lens space S³/Zₚ with its round metric (and is hence invariant when p=1), where p is the magnetic NS-NS charge. In the generic case with NS-NS and R-R charges, the S³ not only becomes S³/Zₚ but is also squashed, with a squashing parameter that is related to the values of the charges. Similar results apply if we regard AdS₃ as a bundle over AdS₂ and T-dualise along the fibre. We show that Hopf T-dualities relate different black holes, and that they preserve the entropy. The AdS₃ x S³ solutions arise as the near-horizon limits of dyonic strings. We construct an O(2,2;Z) multiplet of such dyonic strings, where O(2,2;Z) is a subgroup of the O(5,5) or O(5,21) six-dimensional duality groups, which captures the essence of the NS-NS/R-R and electric/magnetic dualities.