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Vertex algebras from the Hull-Strominger system

Luis Álvarez-Cónsul, Andoni De Arriba de La Hera, Mario Garcia-Fernandez

Abstract

Motivated by the programme on mirror symmetry for non-Kähler manifolds, we construct representations of the $N=2$ superconformal vertex algebra associated to solutions of the Hull-Strominger system. The construction is via embeddings of the $N=2$ superconformal vertex algebra in the chiral de Rham complex of a string Courant algebroid. Our results require that the connection $\nabla$, one of the unknowns of the system, is Hermitian-Yang-Mills. Our main theorem proves that any solution of the Hull-Strominger system satisfying this condition has an associated $N=2$ embedding.

Vertex algebras from the Hull-Strominger system

Abstract

Motivated by the programme on mirror symmetry for non-Kähler manifolds, we construct representations of the superconformal vertex algebra associated to solutions of the Hull-Strominger system. The construction is via embeddings of the superconformal vertex algebra in the chiral de Rham complex of a string Courant algebroid. Our results require that the connection , one of the unknowns of the system, is Hermitian-Yang-Mills. Our main theorem proves that any solution of the Hull-Strominger system satisfying this condition has an associated embedding.
Paper Structure (26 sections, 59 theorems, 432 equations)

This paper contains 26 sections, 59 theorems, 432 equations.

Table of Contents

  1. Introduction
  2. Killing spinors, special coordinates and the Bismut Ricci form
  3. The Killing spinor equations
  4. Closed dilaton and special coordinates
  5. Bismut Ricci form and torsion bivector
  6. Canonical metrics on string algebroids
  7. Background on string algebroids
  8. Killing spinors on string algebroids
  9. Canonical metrics and the F-term condition
  10. The D-term equation and special frames
  11. Embeddings of $N=2$ superconformal vertex algebras
  12. Background on SUSY vertex algebras
  13. Local generators of supersymmetry
  14. Local embeddings
  15. Global embeddings
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $(\Psi,\omega,A)$ be a solution of the twisted Hull--Strominger system eq:twistedStromintro. Consider the associated string Courant algebroid $E$ and the chiral de Rham complex $\Omega^{\operatorname{ch}}_{E \otimes {\mathbb{C}}}$. Then the following expressions define global sections of $\Omega Furthermore, the sections $J$ and $H$ induce an embedding of the $N=2$ superconformal vertex algebr

Theorems & Definitions (137)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 2.6: Jordan
  • ...and 127 more