Employing D-brane moduli spaces as a unifying theme, this research program consists of three projects lying at the interface between algebraic geometry and string theory. The first is centered around a new mathematical construction (ADHM sheaf invariants) yielding new mathematical conjectures concerning local BPS invariants, quantum cohomologyof quiver varieties and black hole correspondence. The second project investigates the cameral structure and wallcrossing behavior of moduli spaces of Bridgeland stable objects on local surfaces emphasizing a connection between Donaldson-Thomas type invariants and black hole physics. The third proposes a systematic approach to correlators of surface operators in topological gauge theories via virtual cycles and localization for parabolic Higgs bundles.
Broader Impact: The interaction between mathematics and string theory has been a major driving force in modern mathematical physics. The research planned here aims at revealing new aspects of this interaction opening new directions of research in Donaldson-Thomas theory, quantum cohomology of quiver varieties as well as fundamental areas of theoretical physics such as black hole physics and string duality. This aims to stimulate the development of several areas of mathematics and physics by unraveling new connections between them. With respect to education, this research program offers multiple training opportunities for students of all levels as well as postdoctoral fellows, stimulating the development of a variety of technical and communication skills with a wide range of applicability beyond the academic environment.
This is an interdisciplinary project focused lying at the interface between physics and geometry. The primary research goal is to construct microscopic models of black holes employing geometric methods in the context of string theory. Black holes are singularities of the gravitational field predicted by Einstein's theory of relativity which have been recently confirmed by astronomical observations. From a theoretical point of view, black hole dynamics should be described by a quantum theory of gravity, which is a fundamental open problem in modern theoretical physics. String theory provides a theoretical model for black holes, which are viewed as extended objects vibrating in extra dimensions. A useful analogy to keep in mind would be the microscopic models in condensed matter phsyics where the magnetic materials are viewed as a system of interacting particles located at some lattice sites. Then the macroscopic parameters of the material are obtained as an average over the quantum states of its microscopic constituents. String theory provides similar microscopic models for black holes, employing extended vibrating objects called D-branes. In particular black hole entropy is determined by counting D-brane quantum states. This is a challenging problem since the complexity of such systems makes exact entropy computations very difficult. The present project addresses this problem employing cutting edge algebraic and geometric methods. A new class of mathematical models for black holes is developed studying D-brane dynamics in a certain geometric background. The foundational work includes a careful study of a phenomenon called wallcrossing, which encodes the response of the system under variations of geometric parameters. Again, as a familiar analogy, one may think of the variation of material parameters in condensed matter systems under changes in temperature, pressure etc. In the present context wallcrossing yields exact results for black hole entropy in situations lying well beyond the range of existing constructions. These results are the subject of the eight research articles listed below and one publication in conference proceedings. They have been also reported in talks given at many international conferences in mathematical physics. Although the primary goal is black hole physics, the models constructed in this project have remarkable consequences for certain areas of pure mathematics such as algebraic geometry and knot theory. Their main virtue is to reveal novel connections between previously unrelated areas, shedding new light on classical problems and opening new directions of research. This project also has a strong educational component, most results being obtained in collaboration with graduate students and postdocs. Given its interdisciplinary nature, it has created many training opportunities for physics graduate students in algebraic geometry, combinatorics and computer programming. In the course of completeing this work, they have acquired valuable analytic skills with a broad range of applicability in unrelated fields.
