Membrane instantons from mirror symmetry

TL;DR

This work uses mirror symmetry to determine and sum a class of membrane instanton corrections to the hypermultiplet moduli space in type IIA on Calabi–Yau threefolds, by relating them to D1 and D(-1) instantons in type IIB. The corrections are captured by a single function in projective superspace and, under the nonperturbative mirror map, are fixed by the Euler characteristic $\chi_E$ and the genus-zero Gopakumar–Vafa invariants of the mirror CY, enabling an all-orders $g_s$ completion. The results are encoded either in the tensor potential $\chi$ or the superspace density $\mathcal{L}$, with explicit expressions for D2-brane membrane instantons wrapping A-cycles, and show consistency with known IIB results in the conifold limit. This framework paves the way to a more complete nonperturbative understanding of hypermultiplet moduli spaces in $N=2$ string vacua and highlights the role of topological invariants in constraining quantum corrections.

Abstract

We use mirror symmetry to determine and sum up a class of membrane instanton corrections to the hypermultiplet moduli space metric arising in Calabi-Yau threefold compactifications of type IIA strings. These corrections are mirror to the D1 and D(-1)-brane instantons on the IIB side and are given explicitly in terms of a single function in projective superspace. The corresponding four-dimensional effective action is completely fixed by the Euler number and the genus zero Gopakumar-Vafa invariants of the mirror Calabi-Yau.

Figures (2)

• Figure 1: Prospective duality chain for determining the full quantum LEEA of type II strings compactified on a generic CY $X$, and its mirror partner $Y$. In the vector multiplet sector there are $\alpha'$ corrections that appear on the IIA side only and can be obtained via mirror symmetry; they comprise worldsheet loop and instanton corrections. The c-map transfers these into the IIB hypermultiplet sector. In addition, there is a one-loop $g_s$ correction, in the figure denoted by $1\ell$, determined in RSV. Imposing $\mathrm{SL}(2,\mathbb{Z})$ invariance produces the nonperturbative corrections arising from D1-brane and more general $(p,q)$-string instantons as well as D($-1$) instantons Robles-Llana:2006is. The latter naturally combine with the perturbative $\alpha'$ and $g_s$ corrections. As shown in this paper, applying mirror symmetry to these corrections gives rise to the $A$-cycle D2-brane instanton contributions on the IIA side. Though beyond the scope of this paper, one may now continue to employ various dualities that should in principle produce all possible quantum corrections: using electromagnetic (e/m) duality to impose symplectic invariance will give the $B$-cycle D2-brane instantons. Mirror symmetry will map these to the as of yet unknown D3- and D5-brane instanton corrections on the IIB side. Another application of $\mathrm{SL}(2,\mathbb{Z})$ duality then will give rise to pure NS5-brane and D5--NS5 bound state instantons. Finally, applying mirror symmetry one last time will produce the NS5-brane corrections on the IIA side.
• Figure 2: Massless matter spectrum arising in CY compactifications of type II strings. The moduli spaces for vector and hypermultiplets are denoted by ${\cal M}_{\rm VM}$ and ${\cal M}_{\rm HM}$ respectively, with their real dimensions below in terms of the Hodge numbers $h_{p,q}$ of the CY. The last line indicates possible quantum corrections to the respective sector. Here $\alpha^\prime$ and $g_s$ denote corrections from the worldsheet conformal field theory and in the (four-dimensional) string coupling constant, respectively. Note that the vector multiplet sector of the type IIB compactification is classically exact.