The theory of superstring with flux on non-Kahler manifolds and the complex Monge-Ampere equation

TL;DR

The work constructs smooth solutions to Strominger’s system on non‑Kähler, flux‑supported manifolds by wiring a holomorphic T^2‑bundle X over a K3 surface or complex torus S and solving a base complex Monge–Ampère equation for the warp factor u. The reduction hinges on a GP geometric model and Li–Yau Hermitian‑Yang‑Mills theory, yielding a Monge–Ampère type equation Δ(e^u − (α'/2) f e^{−u}) + 4α' det u_{iȷ}/det g_{iȳ} + μ = 0 with μ integral constraint and a small volume parameter A. The core contribution lies in a chain of a priori estimates (zeroth to third order) that establish ellipticity and regularity, enabling a successful continuity method to obtain existence of u and the HYM metric on the pulled‑back bundle, thereby producing explicit non‑Kähler Strominger solutions. This work bridges non‑Kähler geometry, string flux compactifications, and complex Monge–Ampère techniques, offering concrete first examples and a framework potentially extensible to higher dimensions and elliptic fibrations.

Abstract

The purpose of this paper is to solve a problem posed by Strominger in constructing smooth models of superstring theory with flux. These are given by non-Kahler manifolds with torsion.

Theorems & Definitions (36)

• Lemma 1
• Proposition 2
• Theorem 3
• Theorem 4
• Theorem 5
• Proposition 6
• Remark 7
• Proposition 8
• proof
• Lemma 9