Research in Novel Symmetries of Quantum Field and String Theory

该奖项资助了纽约州立大学石溪分校的尼基塔·涅克拉索夫教授的研究。所有已知的基本粒子的相互作用都可以用称为规范对称的数学对称性来描述。基本粒子的标准模型就是用这种规范对称来描述的。在物理系统中，通常有几个基态来描述系统（例如，冰点处的冰和水有两个基态），规范对称也不例外。本研究将研究不同系统的不同基态，包括规范理论。最近人们已经认识到，一些系统的基态可以与非常不同的系统的基态密切相关。例如，凝聚态系统中的相变与规范理论中可能的基态密切相关；强相关电子系统的基态与量子信息理论等中的基态有关。数学物理中的这种协同研究提供了将看起来非常不同的物理系统联系起来的可能性。在过去，明显不同的系统之间的联系产生了巨大的影响。朗道和金兹堡对超导性的研究导致了标准模型的希格斯机制，黎曼ζ函数的纯数学研究使人们更好地理解了素数，这对密码学产生了重大影响，弦理论的数学研究使人们更好地理解了强相互作用粒子（如质子和介子）的性质。该奖项旨在支持各种时空维度规范理论的理论研究和教育，与强相关电子系统和相变理论相关的统计物理模型，以及与研究强耦合现象和基本小尺度时空性质相关的弦理论模型。这个项目的独特之处在于所调查的一系列问题的广泛影响。它涉及到描述基本粒子相互作用的规范理论，在欧洲核子研究中心的大型强子对撞机实验和世界上其他机器上研究的那种，以及具有不同程度超对称的规范理论，用于测试对现象学重要思想的理论实验室；在不同实验室研究的准一维和二维强相关电子系统，一方面与量子信息有关，另一方面与超导性有关；代数几何和拓扑性质的数学问题，泛函分析和薛定谔算子及其非厄米类似物谱的研究；量子力学的基础和量化的艺术；拓扑弦和拓扑量子引力。所有这些问题的统一主题是将在特定超重力背景下计算的规范/弦理论的非微扰配分函数简化为统计力学模型。统计模型的相关函数与无穷维代数结构的生成器的矩阵元素有关。计划使用上述所有研究领域的技术和方法。具体来说，计划推广量子场论的Ward恒等式和Dyson-Schwinger恒等式，以纳入非微扰现象。目的是利用这些恒等式，建立规范理论，至少是超对称规范理论，至少在低能区具有一种新型的对称性，这是二维共形场理论的常规对称性的变形。利用这些对称性，有望在大统一理论的真空领域甚至弦紧化中发现新的几何和代数结构。该研究处于理论物理多个领域和数学多个领域的交叉点，为培养研究生提供了良好的环境。

This award funds the research of Professor Nikita Nekrasov of SUNY Stony Brook. All known interactions of elementary particles are described by mathematical symmetries called gauge symmetries. The Standard Model of elementary particles is described by such a gauge symmetry. In physical systems there are often several ground states which describe the system (ice and water at the freezing point has two ground states, for example), and gauge symmetries are no exception. This research will study the different ground states of various systems, including gauge theories. It has recently been realized that the ground states of some systems can be related closely to ground states of very different systems. For example, phase transitions in condensed matter systems are closely related to the possible ground states of gauge theories; ground states of strongly correlated electron systems are related to those in quantum information theory, etc. This synergistic research in mathematical physics offers the possibility of connecting what appear to be very different physical systems. Connections between apparently different systems have had great impact in the past. The study of superconductivity by Landau and Ginzburg led to the Higgs mechanism of the Standard Model, the purely mathematical study of the Riemann zeta function led to a better understanding of prime numbers and this has had major impact in cryptography, and the mathematical study of string theory has led to a better understanding of the properties of strongly interacting particles such as protons and pions.This award is for support of theoretical research and education on gauge theory in various space-time dimensions, models of statistical physics relevant for strongly correlated electron systems and the theory of phase transitions, and models of string theory relevant for studies of strongly coupled phenomena and the nature of space-time at fundamentally small scales. The unique nature of this project is the broad impact of the cluster of questions being investigated. It touches upon the gauge theories describing the interactions of fundamental particles, the kind studied at the LHC experiment at CERN and at others machines all over the world, and the gauge theories with various degrees of supersymmetry, theoretical laboratories for testing ideas of importance for phenomenology; the quasi-one and two-dimensional strongly correlated electron systems studied in different laboratories in connection with quantum information on the one hand and superconductivity on the other hand; mathematical questions of algebraic geometric and topological nature, functional analysis and the study of spectra of Schroedinger operators and their non-hermitian analogues; foundations of quantum mechanics and the very art of quantization; topological strings and the topological quantum gravity.The unifying theme in all these problems is the reduction of the non-perturbative partition function of gauge/string theory computed in a specific supergravity background to a statistical mechanical model. The correlation functions of the statistical model are related to matrix elements of generators of some infinite dimensional algebraic structure. It is planned to use the techniques and methods of all the fields of research mentioned above. Specifically, it is planned to generalize the Ward and Dyson-Schwinger identities of quantum field theory to incorporate non-perturbative phenomena. The intention is to establish, using these identities, that the gauge theories, at least supersymmetric gauge theories, possess a new type of symmetry, at least in the low-energy regime, which is a deformation of conventional symmetries of two dimensional conformal field theories. Using these symmetries it is hoped to find new geometric and algebraic structures in the landscape of vacua of grand unified theories and possibly even string compactifications. The research lies at the interface of several domains of theoretical physics and several domains of mathematics and provides a good environment for training graduate students.