Communications in Number Theory and Physics

Volume 4 (2010)

Number 1

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

Pages: 187 – 266



Ling Bao (Fundamental Physics, Chalmers University of Technology, Göteborg, Sweden)

Axel Kleinschmidt (Physique Théorique et Mathématique, Université Libre de Bruxelles, Belgium)

Bengt E.W. Nilsson (Fundamental Physics, Chalmers University of Technology, Göteborg, Sweden)

Daniel Persson (Fundamental Physics, Chalmers University of Technology, Göteborg, Sweden)

Boris Pioline (Laboratoire de Physique Théorique et Hautes Énergies, Université Pierre et Marie Curie, Paris, France)


The hypermultiplet moduli space in Type IIA string theorycompactified on a rigid Calabi–Yau threefold $\mc{X}$,corresponding to the “universal hypermultiplet,” isdescribed at tree level by the symmetric space$SU(2,1)/(SU(2)\times U(1))$. To determine the quantumcorrections to this metric, we posit that a discretesubgroup of the continuous tree level isometry group$SU(2,1)$, namely the Picard modular group$SU(2,1;\mbb{Z}[i])$, must remain unbroken in the exactmetric –- including all perturbative and non-perturbativequantum corrections. This assumption is expected to bevalid when $\mc{X}$ admits complex multiplication by$\mbb{Z}[i]$. Based on this hypothesis, we construct an$SU(2,1;\mbb{Z}[i])$-invariant, non-holomorphic Eisensteinseries, and tentatively propose that this Eisenstein seriesprovides the exact contact potential on the twistor spaceover the universal hypermultiplet moduli space. We analyzeits non-Abelian Fourier expansion, and show that theAbelian and non-Abelian Fourier coefficients take therequired form for instanton corrections due to EuclideanD2-branes wrapping special Lagrangian submanifolds, and toEuclidean NS5-branes wrapping the entire Calabi–Yauthreefold, respectively. While this tentative proposalfails to reproduce the correct one-loop correction, theconsistency of the Fourier expansion with physicsexpectations provides strong support for the usefulness ofthe Picard modular group in constraining the quantum modulispace.

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