One of the most important recent developments in string theory is the discovery of integrability on both sides of the AdS/CFT correspondence. Nepomechie's main objective is to apply techniques of integrable quantum spin chains and integrable quantum field theories, which he has helped develop during the past two decades, to problems in string/gauge theory. He has already contributed to the understanding of exact bulk and boundary scattering matrices, which he plans to exploit to obtain further exact results. Alvarez will conduct research in quantum field theory and string theory. Alvarez is studying the extension of the elliptic genus to families of Dirac-Ramond operators. The families index theorem of Atiyah and Singer for the Dirac operator can be used in gauge theories to study the anomalies that spoil the quantization of the theory. The analog in string theory should be a families index theorem for the Dirac-Ramond operator. Even though we do not really know what is string theory, this families index theorem should give us some information about the structure of the theory beyond perturbation theory. This is an important attempt at trying to learn about non-perturbative aspects of string theory. The broader impacts of this research are as follows: this work may open new directions in string theory, mathematics, as well as condensed matter physics and statistical mechanics. Both Alvarez and Nepomechie will help provide scientific training to the next generation of American citizens, in particular to those from the Latino community in Miami. Their findings will be disseminated through the internet and scientific publications, as well as through seminars and international conferences. The PIs also give lectures at local public schools on science.
High-energy experiments (such as those performed at CERN) have until now confirmed that much of the matter and forces in Nature can be accurately described by the so-called Standard Model in terms of a small set of elementary "matter" particles (electrons, quarks, etc.) that interact through a small set of elementary "force" particles (photons, gluons, etc.). The theoretical framework underlying the Standard Model is Quantum Field Theory (QFT). The basic idea is that each type of elementary particle (either matter or force) is assumed to be 0-dimensional (i.e., point-like) and is described in terms of a corresponding field, in a way that is consistent with the principles of quantum mechanics and special relativity. Despite the success of QFT and of the Standard Model, there is still much work left for theorists to do. Indeed, there are important problems in QFT that still have not been solved. For example, the main tool in QFT is perturbation theory, which is applicable only if the interaction is weak. New tools are needed to handle strong interactions, such as the so-called color interaction between quarks. Moreover, the Standard Model cannot be exact, since it is not consistent with general relativity. At sufficiently high energies, the Standard Model will surely break down. Much of the effort in high-energy theoretical physics over the past 40 years has been devoted to formulating a theory that will supersede the Standard Model. It is generally believed that quantum field theory is not adequate for this task, so a more general framework is needed; and the best candidate so far is String Theory, in which elementary particles are regarded as 1-dimensional extended objects (instead of as points). This framework is consistent with both quantum mechanics and general relativity, and has sufficient structure to be phenomenologically viable. Alvarez and Nepomechie conducted research in QFT and String Theory. Nepomechie's work relates to the problem of strong interactions in QFT. He focused on certain QFTs that are not very realistic, but which can perhaps be solved exactly. Throughout the history of physics, exactly solvable models have played an important role -- for example, the Kepler problem (determining the orbit of the Earth about the Sun) that was solved by Newton, and the hydrogen atom (determining the electron energy levels) that was solved by Bohr and Schroedinger. The solutions of these QFTs may similarly play an important role in the theory of elementary particles. Remarkably, as discovered by Maldacena, these QFTs are equivalent to certain string theories, which therefore may also be exactly solvable. Nepomechie worked on developing and implementing techniques for solving these theories. He also lectured widely at international conferences, workshops and schools, and helped to train the next generation of scientists and engineers. There are many potential QFTs and String Theories that one can write down, but only a small subset are consistent with Quantum Mechanics and Relativity. Alvarez researches what are the criteria for a good theory. There are restrictions on the type and the number of particles, and this is very rigid in String Theory and more flexible in QFT. Interestingly, the known consistency criteria are related to the topology of the geometrical space of quantum fields that describe the theory. Think of a geometrical space as made of rubber; the topological properties of a space are those that remain the same if you stretch and distort the rubber. A rubber donut can be distorted to a coffee cup, with the hole becoming the handle and a small depression in the donut growing to become the coffee receptacle. No matter what you do, there will always be a hole. Alvarez has contributed to the development of these ideas, and continues to explore the criteria for consistency by studying the topology of QFTs and String Theory.
