Algebraic Geometry and String Theory

多纳吉9802456这个项目试图探索代数几何和高能物理之间最近发现和发展的一些深刻而美丽的相互关系。PI将研究三个问题，每个问题都涉及代数几何、代数可积系统、量子场论和弦理论的思想组合。其中，第一个是关于代数几何思想在弦对偶猜想中的应用，在此过程中提出了一个代数几何模空间的新描述；第二个是探索由弦思想驱动的可积系统的构造，这反过来又有望导致对某些超对称量子场论的新描述；第三个是根据弦理论的最新见解来研究代数几何中的一个问题。(1)椭圆原纤维上的主丛、谱覆盖和杂化/F-理论对偶。本文建议将前人在可积系统背景下得到的谱覆盖的结果应用于椭圆纤维簇上主丛的模空间的描述，并将由异质弦的紧化产生的模空间和超势与F-理论的结果进行比较。(2)Calabi-Yau vs.Seiberg-Witten可积系统。弦理论的思想根据Donagi和Markman构造的Calabi-Yau系统的简并给出了Seiberg-Witten可积系统的构造和分类。人们希望这反过来将允许构造缺失的短波系统，包括任意规范群的带有伴随物质的理论的那些系统。(3)几何朗兰兹猜想。基于“经典”可积系统的PI所导致的部分构造将基于量化类比与另一种构造进行详细的比较，并将探索这样一种可能性：某一“弦”系统提供了两种构造的正确的共同扩展，从而产生几何朗兰兹猜想所需的所有自同构层。这是对代数几何和弦理论边界的研究。代数几何是现代数学中最古老的部分之一，但在过去的25年里，它已经取得了革命性的成就。弦理论是统一自然四种基本力量的物理理论的最新和最令人兴奋的候选者。在过去的几年里，这两个领域进行了深入的互动，跨越了共同感兴趣的问题的广泛前沿。这些相互作用导致了这两个领域最近一些最令人兴奋的突破，并有望导致物理上的“万物理论”以及完全结合量子和弦现象的代数几何。

Donagi 9802456 This project attempts to explore some of the deep and beautiful interrelationships discovered and developed recently between algebraic geometry and high energy physics. The PI will study three issues, each involving a combination of ideas from algebraic geometry, algebraically integrable systems, quantum field theory and string theory. Of these, the first is concerned with applications of an algebro-geometric idea to a string duality conjecture, suggesting along the way a new description of an algebro-geometric moduli space; the second explores a construction of integrable systems which is motivated by stringy ideas and which is expected in turn to lead to new descriptions for certain supersymmetric quantum field theories; and the third investigates a problem within algebraic geometry in light of recent insights from string theory. (1) Principal bundles on elliptic fibrations, spectral covers, and Heterotic/F-theory duality. It is proposed to apply and adapt previous results on spectral covers, obtained in the context of integrable systems, to the description of the moduli space of principal bundles on an elliptically fibered variety, and to the comparison of both the moduli spaces and the superpotentials arising from compactifications of the Heterotic string with those coming from F-theory. (2) Calabi-Yau vs. Seiberg-Witten integrable systems. Ideas from string theory suggest a construction and classification of Seiberg-Witten integrable systems in terms of degenerations of the Calabi-Yau system constructed by Donagi and Markman. It is hoped that this in turn will allow the construction of the missing SW systems, including those of the theory with adjoint matter for an arbitrary gauge group. (3) The Geometric Langlands Conjecture. A partial construction due to the PI, based on "classical" integrable systems, will be compared in detail to another, based on a quantized analogue, and the possibility will be explored that a certain "stringy" system provides the correct common extension of both constructions, producing all the automorphic sheaves required by the geometric Langlands conjecture. This is research on the boundary of algebraic geometry and string theory. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. String theory is the newest and most exciting candidate for a physical theory unifying the four fundamental forces of nature. In the last few years, the two fields have interacted at great depth and across a broad frontier of problems of common interest. These interactions have led to some of the most exciting recent breakthroughs in both fields, and hold the promise of leading to a physical "theory of everything" as well as to an algebraic geometry which fully incorporates quantum and stringy phenomena.