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Complex/Symplectic Mirrors

Wu-yen Chuang, Shamit Kachru, Alessandro Tomasiello

TL;DR

The paper constructs a family of Type II vacua on complex non-Kähler and symplectic non-Kähler manifolds with SU(3) structure, extending the Polchinski–Strominger framework and revealing mirror pairs. By compactifying on Calabi–Yau threefolds with RR flux and analyzing conifold-type singularities, the authors show how new vacua arise when massless hypermultiplets appear at singular loci, leading to non-Kähler geometries via transitions. They systematically relate these vacua to non-Calabi–Yau extremal transitions and mirror symmetry, and interpret the results through a generalized Reid fantasy that envisions a connected configuration space of SU(3) structure geometries enriched by relative cohomology and massive modes. The work highlights a deep interplay between flux, geometry, and dualities, suggesting an overarching landscape of complex and symplectic non-Kähler vacua linked by extremal transitions and a unified effective potential description.

Abstract

We construct a class of symplectic non--Kaehler and complex non--Kaehler string theory vacua, extending and providing evidence for an earlier suggestion by Polchinski and Strominger. The class admits a mirror pairing by construction. Comparing hints from a variety of sources, including ten--dimensional supergravity and KK reduction on SU(3)--structure manifolds, suggests a picture in which string theory extends Reid's fantasy to connect classes of both complex non-Kaehler and symplectic non-Kaehler manifolds.

Complex/Symplectic Mirrors

TL;DR

The paper constructs a family of Type II vacua on complex non-Kähler and symplectic non-Kähler manifolds with SU(3) structure, extending the Polchinski–Strominger framework and revealing mirror pairs. By compactifying on Calabi–Yau threefolds with RR flux and analyzing conifold-type singularities, the authors show how new vacua arise when massless hypermultiplets appear at singular loci, leading to non-Kähler geometries via transitions. They systematically relate these vacua to non-Calabi–Yau extremal transitions and mirror symmetry, and interpret the results through a generalized Reid fantasy that envisions a connected configuration space of SU(3) structure geometries enriched by relative cohomology and massive modes. The work highlights a deep interplay between flux, geometry, and dualities, suggesting an overarching landscape of complex and symplectic non-Kähler vacua linked by extremal transitions and a unified effective potential description.

Abstract

We construct a class of symplectic non--Kaehler and complex non--Kaehler string theory vacua, extending and providing evidence for an earlier suggestion by Polchinski and Strominger. The class admits a mirror pairing by construction. Comparing hints from a variety of sources, including ten--dimensional supergravity and KK reduction on SU(3)--structure manifolds, suggests a picture in which string theory extends Reid's fantasy to connect classes of both complex non-Kaehler and symplectic non-Kaehler manifolds.

Paper Structure

This paper contains 11 sections, 35 equations, 1 figure, 1 table.

Table of Contents

  1. Complex and symplectic vacua
  2. Four--dimensional description of the vacua
  3. The singularities we consider
  4. Gauged supergravity analysis
  5. The field theory capturing the transition
  6. Geometry of the vacua
  7. Calabi--Yau extremal transitions
  8. Non--Calabi--Yau extremal transitions
  9. Vacua versus geometry
  10. The big picture: a space of geometries
  11. Details about an example

Figures (1)

  • Figure 1: Difference between compact and non--compact surgery: in the noncompact case (up), one loses an element in $H^1$ and one gains an element in $H^0$ (a connected component). In the compact case (down), one loses an element in $H^1$ again, but the would--be new element in $H^0$ is actually trivial, so $H^0$ remains the same. This figure is meant to help intuition about the conifold transition in dimension 6, where $H^0$ and $H^1$ are replaced by $H^2$ and $H^3$. We also have depicted various chains on the result of the compact transition, for later use.