arXiv 0803.33022008-03-23hep-thmath.AGmath.QA

超越拓扑理论的瞬子II

Instantons beyond topological theory II

E. Frenkel, A. Losev, N. Nekrasov

备注：: 125 pages, 9 figures

摘要：

本文是我们项目的第二部分，我们以“无限半径极限”（而不是自由场论的极限）为出发点，以一种新的方式描述了含有瞬子的量子场论。在这个极限下，理论大大简化了，因为所有观测量的关联函数，而不仅仅是拓扑（或BPS）观测量，都可以用瞬子组态的有限维模空间上的积分来明确计算。在第一部分（arXiv：hep-th/0610149）中，我们详细讨论了这种类型的一维（即量子力学）模型。在这里，我们分析了超对称的二维西格玛模型和四维杨-米尔斯理论，使用一维模型作为原型。我们超越了这些模型的拓扑（或BPS）部分，并将它们视为成熟的量子场论。我们详细研究了态空间，发现哈密顿量是不可对角化的，但有Jordan块。这导致了在关联函数中出现模糊。我们发现，我们的理论实际上是对数共形场论（这种类型的理论在凝聚态物理中很有意义）。我们定义喷气评估可观的，并详细考虑其关联函数。它们由全纯映射的模空间上的积分给出，推广了Gromov-Witten不变量.这些积分通常发散，需要正则化，导致西格玛模型的算子的复杂对数混合。四维杨-米尔斯理论也有类似的结构。

英文摘要：

The present paper is the second part of our project in which we describe quantum field theories with instantons in a novel way by using the "infinite radius limit" (rather than the limit of free field theory) as the starting point. The theory dramatically simplifies in this limit, because the correlation functions of all, not only topological (or BPS), observables may be computed explicitly in terms of integrals over finite-dimensional moduli spaces of instanton configurations. In Part I (arXiv:hep-th/0610149) we discussed in detail the one-dimensional (that is, quantum mechanical) models of this type. Here we analyze the supersymmetric two-dimensional sigma models and four-dimensional Yang--Mills theory, using the one-dimensional models as a prototype. We go beyond the topological (or BPS) sectors of these models and consider them as full-fledged quantum field theories. We study in detail the space of states and find that the Hamiltonian is not diagonalizable, but has Jordan blocks. This leads to the appearance of logarithms in the correlation functions. We find that our theories are in fact logarithmic conformal field theories (theories of this type are of interest in condensed matter physics). We define jet-evaluation observables and consider in detail their correlation functions. They are given by integrals over the moduli spaces of holomorphic maps, which generalize the Gromov--Witten invariants. These integrals generally diverge and require regularization, leading to an intricate logarithmic mixing of the operators of the sigma model. A similar structure arises in the four-dimensional Yang--Mills theory as well.

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英文摘要：

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