Instanton Corrections to the Universal Hypermultiplet and Automorphic Forms on SU(2,1)
Résumé
The hypermultiplet moduli space in Type IIA string theory compactified on a rigid Calabi-Yau threefold, corresponding to the "universal hypermultiplet", is described at tree-level by the symmetric space SU(2,1)/(SU(2) x 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[i]), must remain unbroken in the exact metric - including all perturbative and non-perturbative quantum corrections. Based on this assumption, we construct an SU(2,1;Z[i])-invariant, non-holomorphic Eisenstein series and analyze its non-Abelian Fourier expansion. We show that its Abelian and non-Abelian Fourier coefficients exhibit the expected form of instanton corrections due to Euclidean D2-branes wrapping special Lagrangian submanifolds, as well as Euclidean NS5-branes wrapping the entire Calabi-Yau threefold. Relying on the construction of quaternionic-Kahler manifolds M via their twistor space Z_M, a CP^1 bundle over M, we conjecture that the exact contact potential on the twistor space of the universal hypermultiplet is given by the aforementioned Picard Eisenstein series.