Nonperturbative Corrections to 4D String Theory Effective Actions from $\mathrm{SL}(2,\mathbb{Z})$ Duality and Supersymmetry

We find the $\mathrm{D}(-1)$- and D1-brane instanton contributions to the hypermultiplet moduli space of type IIB string compactifications on Calabi-Yau threefolds. These combine with known perturbative and world sheet instanton corrections into a single modular invariant function that determines the hypermultiplet low-energy effective action.

Figure 1

Dualities relating vector and hypermultiplet moduli spaces arising in CY compactifications of type II strings. The moduli spaces for vector and hypermultiplets are denoted by ${\mathcal{M}}_{\mathrm{VM}}$ and ${\mathcal{M}}_{\mathrm{HM}}$, respectively, with their dimensions below in terms of the Hodge numbers $h^{1,2}$ and $h^{1,1}$ of the CY $X$ or its mirror $\tilde{X}$ with $h^{1,1}(\tilde{X})=h^{1,2}(X)$. The lines below indicate the kind of quantum corrections as well as some of the dualities that relate the various moduli spaces. Mirror symmetry is the statement that ${\tilde{M}}_{\mathrm{VM}/\mathrm{HM}}^B={\mathcal{M}}_{\mathrm{VM}/\mathrm{HM}}^A$, while the $c$-map determines ${\mathcal{M}}_{\mathrm{HM}}^{B/A}$ at string tree level in terms of ${\mathcal{M}}_{\mathrm{VM}}^{A/B}$.