Mathematical Sciences: Conformal Field Theory and Its Generalizations

9627782扎克曼拟议的研究涉及数学物理的一般领域。此前，这位研究人员致力于二维量子场论，澄清了量子上同调理论中代数结构的各个方面。研究人员计划进一步探索这一点，还包括更多的几何方面，如镜像流形和Calabi-Yau理论。另一个子项目涉及使用双环共循环的量子场论的四维推广。最后，研究者打算通过微分同胚群的几何来考虑保形场论和理想流体力学之间的可能联系。量子场论是在试图给出自然力的统一公式的背景下产生的。至少从纯数学的角度来看，这一理论的二维“共形”表述相对容易理解，目前的提议试图进一步研究二维模型，以期将弦理论应用于基本粒子的现象学。

9627782 Zuckerman The proposed research lies in the general area of mathematical physics. Previously, the investigator worked on two-dimensional quantum field theory, clarifying various aspects of the algebraic structure in quantum cohomology theory. The investigator plans to further pursue this, incorporating also the more geometric aspects such as mirror manifolds and Calabi-Yau theory. Another subproject deals with four-dimensional generalizations of quantum field theory using double loop cocycles. Finally, the investigator intends to consider possible links between conformal field theories and ideal hydrodynamics via the geometry of diffeomorphism groups. Quantum field theory arises in the context of attempting to give a unified formulation of basic forces of nature. Two-dimensional 'conformal' formulation of this theory is relatively well-understood, at least from a purely mathematical standpoint, and the present proposal seeks to further study the two-dimensional model with a view towards applying string theory to the phenomenology of elementary particles.