Algebraic Geometry and String Theory

Donagi 9802456 这个项目试图探索一些深刻而美丽的 相互关系 发现 最近在代数几何和高能物理之间发展起来。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的应用弦对偶猜想，建议 沿着 新的 描述 代数几何的 模空间;第二种是探索可积系统的构造，这种构造是由弦的思想所激发的，并期望反过来导致某些新的描述。 超对称 量子场论;第三个研究了代数几何中的一个问题，根据弦理论的最新见解。 (1)椭圆纤维化的主丛，谱覆盖，和杂合/F-理论对偶。 建议应用和适应以前的结果，光谱 覆盖，在可积系统的背景下获得的，以描述的模空间的主丛的椭圆纤维品种，并比较的模空间和超势所产生的紧化的杂合弦与那些来自F-理论。(2)卡拉比-丘对塞伯格-威滕 可积 系统.弦理论的观点表明， 建设 利用Donagi和Markman构造的Calabi-Yau系统的退化，对Seiberg-Witten可积系统进行了分类. 希望这反过来将允许缺失的SW系统的建设，包括那些理论与伴随物质的任意规范群。（三） 的 几何 朗兰兹 猜想 一 基于“经典”可积系统的PI的部分构造，将与基于量子化模拟的另一部分构造进行详细的比较，并且将探索某个“弦”系统提供正确的公共扩展的可能性 两 建筑， 生产 几何朗兰兹猜想所要求的所有自守层。 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，但在过去的四分之一个世纪里，它已经有了革命性的发展。 弦理论是统一自然界四种基本力的物理理论的最新和最令人兴奋的候选者。 在过去几年中，这两个领域在共同关心的广泛问题上进行了深入的互动。 这些相互作用导致了最近在这两个领域的一些最令人兴奋的突破，并有望导致一个物理的“万物理论”以及一个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.