Geometric Partial Differential Equations from Unified String Theories

Abstract

An informal introduction to some new geometric partial differential equations motivated by string theories is provided. Some of these equations are also interesting from the point of view of non-Kähler geometry and the theory of non-linear partial differential equations. In particular, a survey is given of joint works of the author with Teng Fei, Bin Guo, Sebastien Picard, and Xiangwen Zhang.

Symplectic geometric flows

“…This flow coincides with the special case of the Anomaly flow[PPZ18b,PPZ18a] with slope parameter α ′ = 0, and "Type IIB flow" is a more precise name, since there is no longer a gauge field and no Green-Schwarz Anomaly cancellation mechanism[Pho20,FPPZb].…”

Special solutions to the Type IIA flow

“…In recent years, various geometric flows have been introduced as potential tools to solve the non-linear partial differential equations related to supersymmetric compactifications of string theories [9,10,11,20,21,22]. In this paper, we focus on the Type IIA flow introduced by Fei-Phong-Picard-Zhang in [10] to study the system of equations for the Type IIA string considered in [27].…”

Special Lagrangian cycles and Calabi-Yau transitions

“…The study of non-Kähler Calabi-Yau geometry in theoretical physics was initiated by Strominger [85], and has since grown into an active area of research; see e.g. [3,25,26,29,30,31,34,35,48,71] for examples, see [2,6,5,13,22,40,41,52] for developments in string theory, and see [27,28,36,38,39,58,65,72,73,74,75,84,88,90,91] for research programs in this area.…”