Geometry and Large N Limits in Quantum Hall States

• Applicant: Professor Dr. Semyon Klevtsov
• Subject Area: Mathematics
• Term: from 2017 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 376817586

Quantum Hall effect (QHE) is a remarkable phenomenon, where the precise quantization of Hall conductance occurs in materials with impurities. It was originally discovered in two-dimensional electron systems at low temperatures and in strong magnetic fields, and more recently in graphene, where it was reported even at room temperatures. It is well known that the quantization phenomena in QHE have essentially geometric origin. This project addresses a number of questions in geometric theory of the quantum Hall states, focusing mainly on mathematical aspects, with an eye for potential physics applications. The main goals of the project are (i) to define and construct the Quantum Hall states, such as Laughlin state and Pfaffian state on various geometric backgrounds (ii) to give a complete classification of the geometric adiabatic transport on the moduli spaces of Riemann surfaces for these states (iii) develop effective new methods and tools to study large N asymptotics of the partition functions, correlation functions, adiabatic curvatures for the QH states, (iv) uncovering new exciting connections of QH states to Liouville quantum gravity, Bergman kernels, Quillen theory and geometry. In order to reach these goals we propose new approaches to QH states, making use of recent methods in Kaehler geometry, asymptotic analysis of Bergman kernels, and quantum field theory techniques. As the outcome of the project we expect to uncover new exciting connections of the QH states with various aspects of modern geometry and physics. The results of the project will be of potential interest to the mathematical physics, Kaehler geometry, Bergman kernel, random matrix and QHE communities.

• DFG Programme: Research Grants