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Minimal flux Minkowski classification

Niall T. Macpherson, Alessandro Tomasiello

TL;DR

This work develops a minimal, symmetry-driven framework to classify Minkowski_4 vacua in type IIA with N=2 supersymmetry and SU(2) R-symmetry, translating the SUSY constraints into a network of PDEs on a four-dimensional base. By exploiting an identity structure and pure spinor formalism, the authors organize solutions into ramified branches labeled by the internal spinor data (a1,a2,b) and systematically reduce the problem to tractable PDEs and Bianchi identities; this unifies a variety of known brane intersections and AdS limits within a single class. They explicitly connect to intersecting brane systems, and demonstrate how AdS7, AdS6, and AdS5 backgrounds arise as specializations or duals, including recovering the massive AdS7 solutions. The results also yield promising compact Minkowski_4 vacua with localized D-branes and O-planes, illustrating the potential for explicit, fully backreacted backgrounds in string phenomenology and holography.

Abstract

We classify Minkowski$_4$ solutions in type IIA supergravity, with N=2 supersymmetry and an SU(2) R-symmetry of a certain type. Many subcases can be reduced to relatively simple PDEs, among which we recover various intersecting brane systems, and AdS$_d$ solutions, $d=5,6,7$, and in particular the recently found general massive AdS$_7$ solutions. Imposing compactness of the internal six-manifold we obtain promising solutions with localized D-branes and O-planes.

Minimal flux Minkowski classification

TL;DR

This work develops a minimal, symmetry-driven framework to classify Minkowski_4 vacua in type IIA with N=2 supersymmetry and SU(2) R-symmetry, translating the SUSY constraints into a network of PDEs on a four-dimensional base. By exploiting an identity structure and pure spinor formalism, the authors organize solutions into ramified branches labeled by the internal spinor data (a1,a2,b) and systematically reduce the problem to tractable PDEs and Bianchi identities; this unifies a variety of known brane intersections and AdS limits within a single class. They explicitly connect to intersecting brane systems, and demonstrate how AdS7, AdS6, and AdS5 backgrounds arise as specializations or duals, including recovering the massive AdS7 solutions. The results also yield promising compact Minkowski_4 vacua with localized D-branes and O-planes, illustrating the potential for explicit, fully backreacted backgrounds in string phenomenology and holography.

Abstract

We classify Minkowski solutions in type IIA supergravity, with N=2 supersymmetry and an SU(2) R-symmetry of a certain type. Many subcases can be reduced to relatively simple PDEs, among which we recover various intersecting brane systems, and AdS solutions, , and in particular the recently found general massive AdS solutions. Imposing compactness of the internal six-manifold we obtain promising solutions with localized D-branes and O-planes.

Paper Structure

This paper contains 31 sections, 187 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Ansatz
  3. Pure spinors
  4. Classification
  5. The $a_2=b=0$ case: $M_6=S^2\times T^2\times M_2$
  6. $F_0=0$ case
  7. $F_0\neq 0$ case
  8. The $a=0$ case: $M_6=S^2\times S^1\times M_3$
  9. $F_0=0$ Ansatz
  10. $g=0$ Ansatz
  11. The $a_1=b=0$ case
  12. $H_1=0$ Ansatz
  13. $\lambda_2=0$ Ansatz
  14. The $a_1=0$, $b\neq 0$ case
  15. $\partial_{x_1}^2(e^{-4A})=0$ Ansatz
  16. ...and 16 more sections

Figures (1)

  • Figure 1: A summary of the classification in this section. The interior of the triangle represents the generic case where none of the parameters vanish, and the sides and vertices represent the various particular cases; recall $a_1\equiv {\rm Re} a$, $a_2 \equiv {\rm Im} a$. The shading of the sides is meant to suggest which limits reproduce the cases represented by the vertices. For example, if one takes the equations for the $a_1=0$ case (lower side) and takes the $a_2\to 0$ limit, one recovers the equations of the $a=0$ case, while the same is not true if one takes the $b\to 0$ limit.