Research in Novel Symmetries of Quantum Field and String Theory

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

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.