Instanton corrections to the universal hypermultiplet and automorphic forms on SU(2, 1)

The hypermultiplet moduli space in Type IIA string theory compactified on a rigid Calabi-Yau threefold χ, corresponding to the "universal hypermultiplet," is described at tree level by the symmetric space SU(2, 1)/(SU(2) × U(1)). To determine the quantum corrections to this metric, we posit that a discrete subgroup of the continuous tree level isometry group SU(2, 1), namely the Picard modular group SU(2, 1;Z{double struck}[i]), must remain unbroken in the exact metric-including all perturbative and non-perturbative quantum corrections. This assumption is expected to be valid when χ admits complex multiplication by {double struck}[i]. Based on this hypothesis, we construct an SU(2, 1;{double struck}[i])-invariant, non-holomorphic Eisenstein series, and tentatively propose that this Eisenstein series provides the exact contact potential on the twistor space over the universal hypermultiplet moduli space. We analyze its non-Abelian Fourier expansion, and show that the Abelian and non-Abelian Fourier coefficients take the required form for instanton corrections due to Euclidean D2-branes wrapping special Lagrangian submanifolds, and to Euclidean NS5-branes wrapping the entire Calabi- Yau threefold, respectively. While this tentative proposal fails to reproduce the correct one-loop correction, the consistency of the Fourier expansion with physics expectations provides strong support for the usefulness of the Picard modular group in constraining the quantum moduli space.SCOPUS: ar.j