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Local descriptions of the heterotic SU(3) moduli space

Hannah de Lázari, Jason D. Lotay, Henrique Sá Earp, Eirik Eik Svanes

TL;DR

The paper analyzes the local moduli problem for the heterotic SU(3) (Hull–Strominger) system on compact 6-manifolds by introducing a deformation operator barD on a doubled extension bundle Q. It proves that barD^2 = 0 is equivalent to anomaly cancellation, defines finite-dimensional deformation and obstruction spaces H^{0,1}_{\bar{D}}(Q) and H^{0,2}_{\bar{D}}(Q), and shows these spaces are isomorphic, yielding a zero-dimensional expected moduli space near a solution. A Dolbeault-type theorem links the analytic cohomology to Čech cohomology via an explicit isomorphism with a holomorphic volume form Ω, enabling algebraic methods to study deformations. The authors illustrate the framework with Calabi–Eckmann and Iwasawa manifold examples, computing the relevant cohomology groups and demonstrating rigidity in one case and a nontrivial 11-dimensional moduli in the other. These results illuminate the local structure of heterotic moduli, provide tools for counting or classifying solutions, and connect deformation theory with Serre duality in a non-Kähler setting.

Abstract

The heterotic $SU(3)$ system, also known as the Hull--Strominger system, arises from compactifications of heterotic string theory to six dimensions. This paper investigates the local structure of the moduli space of solutions to this system on a compact 6-manifold $X$, using a vector bundle $Q=(T^{1,0}X)^* \oplus {End}(E) \oplus T^{1,0}X$, where $E\to X$ is the classical gauge bundle arising in the system. We establish that the moduli space has an expected dimension of zero. We achieve this by studying the deformation complex associated to a differential operator $\bar{D}$, which emulates a holomorphic structure on $Q$, and demonstrating an isomorphism between the two cohomology groups which govern the infinitesimal deformations and obstructions in the deformation theory for the system. We also provide a Dolbeault-type theorem linking these cohomology groups to Čech cohomology, a result which might be of independent interest, as well as potentially valuable for future research.

Local descriptions of the heterotic SU(3) moduli space

TL;DR

The paper analyzes the local moduli problem for the heterotic SU(3) (Hull–Strominger) system on compact 6-manifolds by introducing a deformation operator barD on a doubled extension bundle Q. It proves that barD^2 = 0 is equivalent to anomaly cancellation, defines finite-dimensional deformation and obstruction spaces H^{0,1}_{\bar{D}}(Q) and H^{0,2}_{\bar{D}}(Q), and shows these spaces are isomorphic, yielding a zero-dimensional expected moduli space near a solution. A Dolbeault-type theorem links the analytic cohomology to Čech cohomology via an explicit isomorphism with a holomorphic volume form Ω, enabling algebraic methods to study deformations. The authors illustrate the framework with Calabi–Eckmann and Iwasawa manifold examples, computing the relevant cohomology groups and demonstrating rigidity in one case and a nontrivial 11-dimensional moduli in the other. These results illuminate the local structure of heterotic moduli, provide tools for counting or classifying solutions, and connect deformation theory with Serre duality in a non-Kähler setting.

Abstract

The heterotic system, also known as the Hull--Strominger system, arises from compactifications of heterotic string theory to six dimensions. This paper investigates the local structure of the moduli space of solutions to this system on a compact 6-manifold , using a vector bundle , where is the classical gauge bundle arising in the system. We establish that the moduli space has an expected dimension of zero. We achieve this by studying the deformation complex associated to a differential operator , which emulates a holomorphic structure on , and demonstrating an isomorphism between the two cohomology groups which govern the infinitesimal deformations and obstructions in the deformation theory for the system. We also provide a Dolbeault-type theorem linking these cohomology groups to Čech cohomology, a result which might be of independent interest, as well as potentially valuable for future research.
Paper Structure (25 sections, 35 theorems, 252 equations)

This paper contains 25 sections, 35 theorems, 252 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Double extensions and self-duality
  4. The heterotic SU(3) or Hull--Strominger system
  5. The deformation operator
  6. Preliminaries
  7. The operator barD
  8. The adjoint operator barD*
  9. Ellipticity
  10. Serre duality
  11. Double extensions and exact sequences
  12. Simplified case
  13. General case
  14. Examples
  15. Calabi--Eckmann S3xS3
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $X$ be a compact 6-manifold endowed with a solution of the heterotic ${\rm SU}(3)$ system which we denote by $s_0$. There exist vector spaces $\mathcal{I}$ and $\mathcal{O}$ of the same finite dimension, an open set $U\ni 0$ in $\mathcal{I}$ and a smooth map $\pi:U\to \mathcal{O}$ with $\pi(0)=0

Theorems & Definitions (105)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 95 more