The study of vector bundles is a fundamental problem in algebraic geometry. Their connections with physics and representation theory were pioneered in the work of Penrose and Atiyah. They played central roles in Donaldson theory and Seiberg-Witten theory. From 1995, physicists working in string theory have speculated many surprising but deep results concerning vector bundles, Gromov-Witten theory and their interplay with representation theory. In the past few years, elegant relations among moduli spaces of vector bundles (including Hilbert schemes of points and curves), Donaldson-Thomas theory and Gromov-Witten theory have been revealed. In this project, Professor Qin intends to study several problems concerning moduli of vector bundles, Donaldson-Thomas theory and Gromov-Witten theory in the general context of algebraic geometry and its interplay with representation theory and string theory. The main tools are localized virtual fundamental classes, techniques of vertex algebras, quantum cohomology, and stable bundles on surfaces and three folds.
Algebraic geometry studies geometric objects described by polynomial equations. It has been at the central stage of recent confluence between mathematics and physics. Many of these interactions have led to profound improvement in the understanding of both mathematics and physics. Professor Qin's research helps to strength these interactions.
Intellectual Merit: Gromov-Witten theory and Donaldson-Thomas theory play essential roles in string theory in high energy physics. The moduli space of Gieseker semistable torsion-free sheaves (including Hilbert schemes) over a surfaces plays important roles in gauge theory from physics. The PI studied Donaldson-Thomas theory and its correspondence with Gromov-Witten theory, and investigated moduli of stable bundles on surfaces and their interplay with representation theory and Gromov-Witten theory. The PI computed the Donaldson-Thomas invariants for two types of Calabi-Yau 3-folds. These invariants are associated to the moduli spaces of rank-2 Gieseker semistable sheaves. The PI proved a formula for Behrend's $ u$-functions when torus actions exist, and used it to obtain the generating series of the relevant Donaldson-Thomas invariants in terms of the McMahon series. The results might shed some light on the wall-crossing phenomena of Donaldson-Thomas invariants. The PI verified that the Cohomological Crepant Resolution Conjecture holds for the Hilbert-Chow maps from the Hilbert schemes of points on a surface to the symmetric products of the same surface. There are two main ideas in the proof. The first one is to use the representation theoretic approach proposed by the PI and Weiqiang Wang which involves vertex operator techniques. The second is to prove certain universality structures about the 3-pointed genus-0 extremal Gromov-Witten invariants of the Hilbert schemes by using the techniques of Jun Li which deals with the degree-0 Donaldson-Thomas invariants of 3-folds. Broader Impact: The PI published the outcomes in professional journals, presented the outcomes in professional meetings, and supervised Ph.D. students. In addition, the PI mentored postdocs, and actively participated in departmental seminars.
