Vacuum States of the Heterotic String

弦理论被认为是一种能够描述所有已知的自然力的理论，并为寻找与量子力学一致的引力理论这一令人尊敬的问题提供了一个解决方案。近似地说，我们所观察到的世界符合这个理论的真空。弦理论承认许多这样的真空状态，而最有可能描述观察到的世界的一类是所谓的“异质真空”。分析这些真空需要运用复杂的数学工具，特别是代数几何工具。如果历史有任何指导意义的话，将这些数学工具与从物理学中得出的观察结果相结合，不仅会导致物理学的重大进步，也会导致数学的重大进步。从弦理论中衍生出来的一个重要数学见解的例子就是镜像对称。我们观察到，在一个受限的弦真空中，它们以“镜像对”的形式出现。这样做的结果是，某些既重要又神秘的数学量可以用一种直接的方式计算出来。我们感兴趣的是，异质真空是更广泛的真空，一个重要的问题是，镜像对称推广到什么程度，以及它是如何作用于更广泛的真空的。在更精确的描述中，杂散真空空间是（X，V）对的参数空间，其中X是Calabi-Yau流形，V是X上的稳定全纯向量束。这个空间是代数和几何中的主要研究对象。弦理论告诉我们，它受制于量子修正。理解这些修正的本质是本提案的关键研究问题，我们在理解上的任何进展都将对数学和物理学产生重要影响。到目前为止，人们已经普遍认识到弦理论和几何是密切相关的，彼此之间有很多值得学习的地方，然而这种关系在异质弦中还相对未被探索。这一事实，加上最近的事态发展表明，长期存在的问题最近已变得可以处理，意味着现在是重新审视异质性真空几何形状的时候了。

String theory is believed to be a theory capable of describing all the known forces of nature, and provides a solution to the venerable problem of finding a theory of gravity consistent with quantum mechanics. To a first approximation, the world we observe corresponds to a vacuum of this theory. String theory admits many of these vacuum states and the class that is most likely to describe the observed world are the so-called `heterotic vacua'. Analysing these vacua requires the application of sophisticated tools drawn from mathematics, particularly from algebraic geometry. If history is any guide, the synthesis of these mathematical tools with observations drawn from physics will lead not only to significant progress in physics, but also important advances in mathematics. An example of such a major insight in mathematics, that arose from string theory, is mirror symmetry. This is the observation that within in a restricted class of string vacua, these arise in `mirror pairs'. This has the consequence that certain mathematical quantities, which are both important and otherwise mysterious, can be calculated in a straightforward manner. The class of heterotic vacua, of interest here, are a wider class of vacua, and an important question is to what extent mirror symmetry generalises and how it acts on this wider class.In a more precise description, the space of heterotic vacua is the parameter space of pairs (X,V) where X is a Calabi-Yau manifold and V is a stable holomorphic vector bundle on X. This space is a major object of study in algebra and geometry. String theory tells us that it is subject to quantum corrections. To understand the nature of these corrections is the key research problem in this proposal and any advance in our understanding will have a important impact in both mathematics and physics. By now it is widely understood that string theory and geometry are intimately related with much to be learned from each other, yet this relationship is relatively unexplored in the heterotic string. This fact, together with recent developments that indicate that longstanding problems have recently become tractable, means that the time is right to revisit the geometry of heterotic vacua.