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Transitions in the Web of Heterotic Vacua

Lara B. Anderson, James Gray, Burt Ovrut

TL;DR

This work links the geometry of stable holomorphic vector bundles on Calabi–Yau threefolds to the physics of four-dimensional heterotic vacua through stability walls in the Kähler cone. It develops a dual viewpoint: (i) four-dimensional effective field theory with anomalous U(1) D-terms that encode stability data, and (ii) explicit geometric deformations of bundles including rank-preserving and rank-changing transitions, notably Li–Yau type constructions. The authors classify the possible vacuum branches using Ext^1 data, Harder–Narasimhan/Gr(V) decompositions, and S-equivalence, and illustrate how branch structure can radically alter the low-energy spectrum while preserving or altering gauge symmetry. They then apply these ideas to count Donaldson–Thomas invariants and derive wall-crossing relations, including recursive methods to handle higher-rank bundles, thereby linking moduli space geometry to vacuum counting in string phenomenology.

Abstract

We analyze transitions between heterotic vacua with distinct gauge bundles using two complementary methods - the effective four-dimensional field theory and the corresponding geometry. From the viewpoint of effective field theory, such transitions occur between flat directions of the potential energy associated with heterotic stability walls. Geometrically, this branch structure corresponds to smooth deformations of the gauge bundle coupled to the chamber structure of Kähler moduli space. We demonstrate how such transitions can change important properties of the effective theory, including the gauge symmetry and the massless spectrum. Geometrically, this study is divided into deformations of the vector bundle which preserve the rank of the gauge bundle and those which change the rank. In the latter case, our results provide explicit solutions to a class of Li-Yau type deformation problems. Finally, we use the framework of stability walls and their effective theory to study Donaldson-Thomas invariants on Calabi-Yau threefolds.

Transitions in the Web of Heterotic Vacua

TL;DR

This work links the geometry of stable holomorphic vector bundles on Calabi–Yau threefolds to the physics of four-dimensional heterotic vacua through stability walls in the Kähler cone. It develops a dual viewpoint: (i) four-dimensional effective field theory with anomalous U(1) D-terms that encode stability data, and (ii) explicit geometric deformations of bundles including rank-preserving and rank-changing transitions, notably Li–Yau type constructions. The authors classify the possible vacuum branches using Ext^1 data, Harder–Narasimhan/Gr(V) decompositions, and S-equivalence, and illustrate how branch structure can radically alter the low-energy spectrum while preserving or altering gauge symmetry. They then apply these ideas to count Donaldson–Thomas invariants and derive wall-crossing relations, including recursive methods to handle higher-rank bundles, thereby linking moduli space geometry to vacuum counting in string phenomenology.

Abstract

We analyze transitions between heterotic vacua with distinct gauge bundles using two complementary methods - the effective four-dimensional field theory and the corresponding geometry. From the viewpoint of effective field theory, such transitions occur between flat directions of the potential energy associated with heterotic stability walls. Geometrically, this branch structure corresponds to smooth deformations of the gauge bundle coupled to the chamber structure of Kähler moduli space. We demonstrate how such transitions can change important properties of the effective theory, including the gauge symmetry and the massless spectrum. Geometrically, this study is divided into deformations of the vector bundle which preserve the rank of the gauge bundle and those which change the rank. In the latter case, our results provide explicit solutions to a class of Li-Yau type deformation problems. Finally, we use the framework of stability walls and their effective theory to study Donaldson-Thomas invariants on Calabi-Yau threefolds.

Paper Structure

This paper contains 17 sections, 201 equations, 3 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Review of Stability Walls and Kähler Sub-Structure
  3. Rank-Preserving Transitions and S-Equivalence Classes
  4. An Example
  5. Branch Structure and S-Equivalence Classes
  6. An Example of More Complicated Branch Structure
  7. Rank-Changing Transitions
  8. The Geometry of Rank-Changing Deformations
  9. Determining the Li-Yau bundle
  10. An Example
  11. More General Rank-Changing Deformations
  12. Donaldson-Thomas Invariants and Wall-Crossing Formulae
  13. An $SU(2)$ Example
  14. Donaldson-Thomas Invariants By Recursion
  15. Appendix: Extension Bundles, Isomorphisms and the Snake Lemma
  16. ...and 2 more sections

Figures (3)

  • Figure 1: An example of a stability wall associated with the bundle (\ref{['E7_eg']}) in Section \ref{['stab_wall_review']}. In the two chambers of the Kähler cone (shown at left) the bundle $V$, is respectively, stable/unstable. The non-trivial potential (shown at right) forces the system back into the supersymmetric region of moduli space.
  • Figure 2: An illustration of a stability wall connecting two bundles, $V_{1}$ and $V_{2}$. In the two chambers of the Kähler cone (shown at left), $V_1$ and $V_2$, respectively, are stable. The moduli spaces of the two stable bundles (shown at right) are connected at only at a point, corresponding to the shared zero of the two $Ext^1$ groups in (\ref{['branch_bundles2']}).
  • Figure 3: An example of a stability wall connecting multiple bundles $V_{i}$. The local moduli spaces of the stable extension bundles, shown above, are connected at only at a point, corresponding to the shared zero of the $Ext^1$ groups in (\ref{['ext_types']}) and (\ref{['graded_ext']}).