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$G_2$-manifolds from Diophantine equations

Jakob Moritz

TL;DR

This work establishes a duality between perturbatively flat vacua in type IIB flux compactifications and M-theory on $G_2$-manifolds, enabling the search for novel $G_2$-manifolds via Diophantine equations in flux quanta. It develops explicit toroidal PFVs, maps them through mirror symmetry to type IIA $SU(3)$-structure backgrounds, and then uplifts to M-theory as $G_2$-manifolds, embedding the flux data into geometric data. A key result is that warping corrections at large complex structure, though significant in IIB, become purely geometric and classically computable in the M-theory frame, offering a tractable route to control and understand these corrections. The paper also provides a general Calabi–Yau extension of PFVs, analyzes parametric control regimes across IIB, IIA, and M-theory, and discusses the strengthened singular bulk problem, outlining concrete paths to explicitly construct the corresponding $G_2$-manifolds and compute nonperturbative effects.

Abstract

We argue that perturbatively flat vacua (PFVs) introduced in \cite{Demirtas:2019sip} are dual to M-theory compactifications on $G_2$-manifolds, enabling the enumeration of potentially novel $G_2$-manifolds via solutions to Diophantine equations in type IIB flux quanta. Independently, we show that warping corrections to the effective action of type IIB flux vacua grow parametrically at large complex structure, and we demonstrate that these corrections can nonetheless be captured by a classical geometric computation in M-theory.

$G_2$-manifolds from Diophantine equations

TL;DR

This work establishes a duality between perturbatively flat vacua in type IIB flux compactifications and M-theory on -manifolds, enabling the search for novel -manifolds via Diophantine equations in flux quanta. It develops explicit toroidal PFVs, maps them through mirror symmetry to type IIA -structure backgrounds, and then uplifts to M-theory as -manifolds, embedding the flux data into geometric data. A key result is that warping corrections at large complex structure, though significant in IIB, become purely geometric and classically computable in the M-theory frame, offering a tractable route to control and understand these corrections. The paper also provides a general Calabi–Yau extension of PFVs, analyzes parametric control regimes across IIB, IIA, and M-theory, and discusses the strengthened singular bulk problem, outlining concrete paths to explicitly construct the corresponding -manifolds and compute nonperturbative effects.

Abstract

We argue that perturbatively flat vacua (PFVs) introduced in \cite{Demirtas:2019sip} are dual to M-theory compactifications on -manifolds, enabling the enumeration of potentially novel -manifolds via solutions to Diophantine equations in type IIB flux quanta. Independently, we show that warping corrections to the effective action of type IIB flux vacua grow parametrically at large complex structure, and we demonstrate that these corrections can nonetheless be captured by a classical geometric computation in M-theory.

Paper Structure

This paper contains 27 sections, 182 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Perturbatively flat vacua in Type IIB string theory
  3. GKP flux vacua
  4. Perturbatively flat vacua
  5. Toroidal Models
  6. An O3 orientifold of $T^6$
  7. Three-form fluxes
  8. 10d type IIB solution
  9. Mirror-dual type IIA solution
  10. NSNS sector
  11. RR sector
  12. M-theory lift
  13. Concrete solutions
  14. $\mathcal{N}=4$ solution
  15. $\mathcal{N}=2$ solutions
  16. ...and 12 more sections

Figures (1)

  • Figure 1: The regimes of parametric control in type IIB string theory (blue), type IIA string theory (red), and M-theory (green). To the upper right of the solid red lines, the superpotential is exponentially small. Near the solid red lines the exponential corrections to the superpotential become $\mathcal{O}(1)$. The dashed lines mark the transitions between controlled Type IIB and Type IIA description (with $\mathcal{O}(1)$ SYZ-fiber volume), and between Type IIA and M-theory (with $\mathcal{O}(1)$ string coupling).