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On Folding Calabi-Yau Diagrams in M-theory Black Brane Scenarios

Adil Belhaj, Abderrahim Bouhouch

TL;DR

The paper develops a diagrammatic folding framework for tetra-quadric Calabi-Yau compactifications in M-theory to relate 5D black brane physics to simpler, lower-moduli-space compactifications. By encoding complete intersection data into CY diagrams and applying outer-automorphism foldings from groups such as Z2, Z2×Z2, Z3, and Z4, the authors derive mapped moduli and charge relations that reproduce known black hole and black string potentials on CP^1×CP^1×CP^2, CP^2×CP^2, CP^1×CP^3, and the CP^4 quintic. The approach yields explicit transformations of the Kähler moduli t_i and charges q_i (and similarly p_i) that implement the folding and reduce the effective potentials to established forms in the folded geometries. This provides a unified, graph-theoretic method to connect complex tetra-quadric vacua to well-understood lower-dimensional CY compactifications, with implications for stability analyses and model-building in 5D supergravity from M-theory.

Abstract

In this paper, we reconsider the study of five-dimensional supersymmetric black branes in the context of the M-theory compactification on a special Calabi-Yau manifold called tetra-quadric, being realized as complete intersections of homogenous polynomials in the projective space $ \mathbb{CP}^{1}\times\mathbb{CP}^{1}\times\mathbb{CP}^{1}\times\mathbb{CP}^{1}$. Combining colored graph theory and outer-automorphism group action techniques, we approach the tetra-quadric Calabi-Yau diagram leading to new features. Using a procedure referred to as folding, we show that M-theory black branes on the tetra-quadric Calabi-Yau manifold can be reduced to known compactifications with lower dimensional Kähler moduli spaces.

On Folding Calabi-Yau Diagrams in M-theory Black Brane Scenarios

TL;DR

The paper develops a diagrammatic folding framework for tetra-quadric Calabi-Yau compactifications in M-theory to relate 5D black brane physics to simpler, lower-moduli-space compactifications. By encoding complete intersection data into CY diagrams and applying outer-automorphism foldings from groups such as Z2, Z2×Z2, Z3, and Z4, the authors derive mapped moduli and charge relations that reproduce known black hole and black string potentials on CP^1×CP^1×CP^2, CP^2×CP^2, CP^1×CP^3, and the CP^4 quintic. The approach yields explicit transformations of the Kähler moduli t_i and charges q_i (and similarly p_i) that implement the folding and reduce the effective potentials to established forms in the folded geometries. This provides a unified, graph-theoretic method to connect complex tetra-quadric vacua to well-understood lower-dimensional CY compactifications, with implications for stability analyses and model-building in 5D supergravity from M-theory.

Abstract

In this paper, we reconsider the study of five-dimensional supersymmetric black branes in the context of the M-theory compactification on a special Calabi-Yau manifold called tetra-quadric, being realized as complete intersections of homogenous polynomials in the projective space . Combining colored graph theory and outer-automorphism group action techniques, we approach the tetra-quadric Calabi-Yau diagram leading to new features. Using a procedure referred to as folding, we show that M-theory black branes on the tetra-quadric Calabi-Yau manifold can be reduced to known compactifications with lower dimensional Kähler moduli spaces.
Paper Structure (14 sections, 49 equations, 7 figures, 1 table)

This paper contains 14 sections, 49 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. On complete intersection Calabi-Yau threefolds
  3. CY matrix configurations
  4. $\hbox{CY}$ diagrams
  5. Folding CY diagrams
  6. M-theory black branes from the tetra-quadric CY
  7. Black holes
  8. Black strings
  9. Folding tetra-quadric CY diagram in M-theory black brane scenarios
  10. $\mathbb{Z}_{2}$ folding procedure in M-theory scenario
  11. $\mathbb{Z}_{2}\times \mathbb{Z}_{2}$ folding procedure in M-theory scenario
  12. $\mathbb{Z}_{3}$ folding procedure in M-theory scenario
  13. $\mathbb{Z}_{4}$ folding procedure in M-theory scenario
  14. Conclusion

Figures (7)

  • Figure 1: Diagram of bi-cubic in $\mathbb{CP}^{2}\times \mathbb{CP}^{2}$.
  • Figure 2: $\mathbb{Z}_{2}$ folding of diagram of bi-cubic in $\mathbb{CP}^{2}\times \mathbb{CP}^{2}$.
  • Figure 3: Tetra-quadric CY diagram.
  • Figure 4: $\mathbb{Z}_{2}$ folding of tetra-quadric CY diagram.
  • Figure 5: $\mathbb{Z}_{2} \times \mathbb{Z}_{2}$ folding of the tetra-quadric CY diagram.
  • ...and 2 more figures