Towards Integrability of Topological Strings
Towards Integrability of Topological Strings
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走向拓扑弦的可积性
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通讯作者：
Jasmine Rhamie
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作者：
L. Rogers;S. Hallam;Jacquelene Shaw;Jasmine Rhamie
通常，物理学中最有趣的问题与动力学有关。系统如何随时间演化，平衡态是什么，已知的结构是如何动态生成的。关于动力学有很多有趣的问题。但在下面还有更基本的运动学问题，我们应该用什么变量来以最自然的方式描述系统。显然，运动学结构的选择先于动力学的讨论。寻找适当的结构显然不是一件容易的事。有迹象表明，我们不知道弦理论中的正确变量。因此，最重要的问题是揭示相应的潜在的运动学结构。从这个角度来看，描述我们世界的弦理论是非常复杂的，在这种情况下，要破译什么是正确的变量可能是极其困难的。问题是，从时空的角度来看，这些理论是高度非局部的。在弦几何中，如果有曲线通过这些点，则这些点被认为是接近的。因此，从某种意义上说，所有的观点都是一致的，不清楚什么是描述这种情况的正确方法。显然这不是一个缺点，而是理论的一个优点，它清楚地表明，弦理论不是关于光滑流形或任何类似标准几何的东西。不幸的是，在这种情况下说些有意义的话并不容易。因此，我们应该寻找更容易处理但又不平凡的弦理论例子。拓扑弦理论似乎给出了一组合适的例子。主要优点是自由度的大幅减少。最常讨论的拓扑弦理论有两种类型：A型和B型。在数学术语中，第一个是与辛几何和Gromov-Witten不变量和第二个是描述在复杂的几何霍奇结构的变化。在这个描述中有一些冗余，因为流形M上的A型理论可以等价于另一个流形M上的B型理论（如果我们用适当的推广来代替流形的概念，我们得到A型和B型拓扑弦的等价性）。A型拓扑弦本质上是非局部的。如果有一条全纯曲线穿过M上的点，则认为它们彼此接近。由于全纯映射的刚性，这种情况介于局部理论和“完全”非局部弦理论之间。特别是它是很难得到显式公式的GromovWitten不变量的一般流形。B型拓扑弦理论是最简单的情况，从这个角度看，它是一种局域量子场论。所以有人可能会怀疑，它对我们理解弦背后的基本自由度没有任何帮助。幸运的是，情况似乎并非如此。这些理论的公式化在适当的变量导致了理论的急剧简化。它成为一个具有二次作用泛函的“自由”理论。这种现象是好的。
Usually the most interesting questions in Physics are connected with the dynamics. How the system evolves in time, what are the equilibrium states, how the known structures are dynamically generated. There are a lot of interesting questions about the dynamics. But underneath there are more fundamental kinematic questions what variables should we use to describe the system in the most natural way. Obviously the choice of the kinematic structure precedes the discussion of the dynamics. And the search for the appropriate structure is obviously not an easy task in general. There are indications that we do not know the right variables in the string theory. Thus the most important problem is to uncover the corresponding underlying kinematic structure. From this point of view the string theories supposedly describing our world are very complicated and it might be extremely difficult to decipher what are the right variables in this case. The problem is that the theories are highly non-local from the point of view of the space-time. In the string geometry the points are considered to be close if there is a curve passing through them. So in a sense all points coincide and it is not clear what is the right way to describe this situation. Obviously this is not a disadvantage but actually an advantage of the theory it clearly indicates that string theory is not about smooth manifolds or anything like standard geometry. Unfortunately it is not easy to say something meaningful in this situation. Thus one should look for more manageable yet non-trivial examples of string theories. It seems that the appropriate set of such examples is given by the topological string theories. The main advantage is the huge reduction of the degrees of freedom. Most commonly discussed topological string theories are of two types Type A and Type B. In mathematical terms the first one is connected with the symplectic geometry and Gromov-Witten invariants and the second one is described in terms of the variations of the Hodge structure in complex geometry. There is some redundancy in this description because Type A theory on a manifold M can be equivalent to Type B theory on another manifold M̃ (if we substitute the notion of the manifold by an appropriate generalization we get the equivalence of Type A and Type B topological strings). The Type A topological strings are intrinsically non-local. The points on M are considered to be close to each other if there is a holomorphic curve passing through them. Due to the rigidity of the holomorphic maps this case is intermediate between local theories and ”fully” non-local string theories. In particular it is difficult to get the explicit formula for GromovWitten invariants for a generic manifold. Type B topological string theory is the most simple case from this point of view it is a local quantum field theory. So one might suspect that it could give us nothing to help to understand the fundamental degrees of freedom behind the strings. Fortunately it seems not the case. The formulation of these theories in appropriate variables leads to a drastic simplification of the theory. It becomes a “free” theory with quadratic action functionals. This phenomena is well