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Instantons, Hypermultiplets and the Heterotic String

Nick Halmagyi, Ilarion V. Melnikov, Savdeep Sethi

TL;DR

This work builds a comprehensive web of dualities linking hypermultiplet corrections in N=2 theories across heterotic and type II constructions. By mapping world-sheet instantons on K3 to brane instantons wrapping special Lagrangian cycles in Type IIA via a heterotic analogue of the c-map and orientifold/F-theory limits, it provides two concrete routes to compute hypermultiplet F-terms and clarifies when instanton contributions survive due to zero-mode counting. A key outcome is a proposed topological bound b1(L) ≥ g for SLags dual to genus-g world-sheet instantons, tying geometric complexity to world-sheet data, and a localization phenomenon that reduces the sum over SLags to a structured subset. The framework also connects vector- and hypermultiplet corrections through heterotic-heterotic duality and emphasizes the role of BV/Borcea-Voisin constructions in organizing SLags across dual frames, with implications for exact HM couplings and potential new dimensions from wrapped NS5-branes.

Abstract

Hypermultiplet couplings in type IIA string theory on a Calabi-Yau space can be quantum corrected by D2-brane instantons wrapping special Lagrangian cycles. On the other hand, hypermultiplet couplings in the heterotic string on a K3 surface are corrected by world-sheet instantons wrapping curves. In a class of examples, we relate these two sets of instanton corrections. We first present an analogue of the c-map for the heterotic string via a dual flux compactification of M-theory. Using this duality, we propose two ways of capturing quantum corrections to hypermultiplets. We then use the orientifold limit of certain F-theory compactifications to relate curves in K3 to special Lagrangians in dual type IIA compactifications. We conclude with some results from perturbative string theory for hypermultiplet F-terms and a conjecture about the topology of brane instantons.

Instantons, Hypermultiplets and the Heterotic String

TL;DR

This work builds a comprehensive web of dualities linking hypermultiplet corrections in N=2 theories across heterotic and type II constructions. By mapping world-sheet instantons on K3 to brane instantons wrapping special Lagrangian cycles in Type IIA via a heterotic analogue of the c-map and orientifold/F-theory limits, it provides two concrete routes to compute hypermultiplet F-terms and clarifies when instanton contributions survive due to zero-mode counting. A key outcome is a proposed topological bound b1(L) ≥ g for SLags dual to genus-g world-sheet instantons, tying geometric complexity to world-sheet data, and a localization phenomenon that reduces the sum over SLags to a structured subset. The framework also connects vector- and hypermultiplet corrections through heterotic-heterotic duality and emphasizes the role of BV/Borcea-Voisin constructions in organizing SLags across dual frames, with implications for exact HM couplings and potential new dimensions from wrapped NS5-branes.

Abstract

Hypermultiplet couplings in type IIA string theory on a Calabi-Yau space can be quantum corrected by D2-brane instantons wrapping special Lagrangian cycles. On the other hand, hypermultiplet couplings in the heterotic string on a K3 surface are corrected by world-sheet instantons wrapping curves. In a class of examples, we relate these two sets of instanton corrections. We first present an analogue of the c-map for the heterotic string via a dual flux compactification of M-theory. Using this duality, we propose two ways of capturing quantum corrections to hypermultiplets. We then use the orientifold limit of certain F-theory compactifications to relate curves in K3 to special Lagrangians in dual type IIA compactifications. We conclude with some results from perturbative string theory for hypermultiplet F-terms and a conjecture about the topology of brane instantons.

Paper Structure

This paper contains 30 sections, 111 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. BPS couplings in four dimensions
  3. BPS couplings in three dimensions
  4. Plan and summary
  5. Special Lagrangian Submanifolds
  6. Basic definitions and properties
  7. Special Lagrangian cycles in $T^2$
  8. sLags in $K3$
  9. sLags in 3-folds
  10. Some sLags in $T^6$
  11. Product sLags in Borcea-Voisin three-folds
  12. A Heterotic/Heterotic Duality
  13. The basics of heterotic/type I on $K3 \times T^2$
  14. Heterotic/M-theory duality
  15. The basic duality in seven dimensions
  16. ...and 15 more sections

Figures (2)

  • Figure 1: The type IIA $\leftrightarrow$ heterotic duality sequence
  • Figure 2: The exchange of lattices for the case of $(1,1)$$B$-field.