Universal hypermultiplet metrics

Abstract

Some instanton corrections to the universal hypermultiplet moduli space metric of the type-IIA string theory compactified on a Calabi-Yau threefold arise due to multiple wrapping of BPS membranes and fivebranes around certain cycles of Calabi-Yau. The classical universal hypermultipet metric is locally equivalent to the Bergmann metric of the symmetric quaternionic space SU(2,1)/U(2), whereas its generic quaternionic deformations are governed by the integrable SU(infinity) Toda equation. We calculate the exact (non-perturbative) UH metrics in the special cases of (i) the D-instantons (the wrapped D2-branes) in the absence of fivebranes, and (ii) the fivebrane instantons with vanishing charges, in the absence of D-instantons. The solutions of the first type preserve the U(1)xU(1) classical symmetry, while they can be interpreted as the gravitational dressing of the hyper-K"ahler D-instanton solutions. The second type solution preserves the non-abelian SU(2) classical symmetry, while it can be interpreted as a gradient flow in the universal hypermultiplet moduli space.Comment: 30 pages, LaTe

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“…The superfield description of the R + R 2 theory in the context of N = 1 and N = 2 supergravities have been considered by Ketov and collaborators [8,9,10]. However, in the N = 2 case, the proposed standard supergravity dual fails to describe the scalaron multiplet and the corresponding potential for the reasons outlined in this paper.…”

Summing up D-instantons in N=2 supergravity

“…In the (classical) supergravity approximation, their LEEA (or kinetic terms) are given by the NLSM whose structure is entirely fixed by duality: the NLSM target space is given by the two-dimensional non-compact homogeneous space SL(2, R)/U(1). In the full 'superstring theory' (including branes) the continuous symmetry SL(2, R) ∼ = SO(2, 1) ∼ = SU(1, 1) is generically broken to a discrete subgroup of SL(2, Z) [21], whereas the local NLSM metric may receive some non-perturbative (instanton) corrections [1,6,14,15,16].…”

“…In a generic case, the NS five-branes wrapped about the entire CY manifold also contribute to the UH quantum geometry -see e.g., refs. [1,14] for details. So our results apply when the five-brane instantons are absent or suppressed in the UH sector.…”

“…Because of the brane charge quantization, the classical symmetries of the UH metric are generically broken by the wrapped D2-branes and the solitionc five-branes wrapped about the entire CY [1]. However, when merely the D-instantons are taken into account, a continuous abelian symmetry of the UH moduli space may survive, while it also makes actual calculations possible [14].…”

“…The relation between the Bergmann parametrization (2.16), used in the mathematical literature, and the UH parametrization (φ, D, C,C), used in the physical literature [1,17,14], is given by 17) where the new complex variable C can be identified with a RR-scalar of UH, whereas another complex scalar S of UH is now given by (cf. eq.…”