The main objective of this project is to study the geometry of moduli spaces of sheaves on low dimensional varieties and reveal possible connections of the subject with other fields of mathematics. The celebrated Gromov-Witten/Donaldson-Thomas correspondence is a conjectural correspondence between the integrals of characteristic classes of the universal ideal-sheaf over the Hilbert scheme of curves in the threefold and integrals over the moduli space of maps into the threefold. The conjecture is proven for toric threefolds and the lowest degree nontrivial characteristic classes. The project proposed is intended to extend the known results beyond the toric case and study the case of higher degree characteristic classes. Another part of the project is devoted to studying the topology of the Hilbert scheme of points on singular planar curves. A conjectural formula was proposed by the PI and V. Shende for the Poincare polynomials of the Hilbert scheme in terms of link invariants of the links of the singularities of the curve. To prove the formula is one of the goals of the project.
Moduli spaces of sheaves are mathematical models for the field theories. Mathematical physics predictions, such as duality between the field theories and string theory or path integral formulas for knot invariants, can be translated into fascinating mathematical conjectures. The goal of this project is to find a proof of some of these conjectures.
The main objective of the project is to study the geometry of moduli spaces of sheaves on low dimensional varieties and reveal possible connections of the subject with other fields of mathematics. The celebrated Gromov-Witten/Donaldson-Thomas correspondence is a conjectural correspondence between the integrals of characteristic classes of the universal ideal-sheaf over the Hilbert scheme of curves in the threefold and integrals over the moduli space of maps into the threefold. In simple terms, the conjecture is a mathematical manifestation of duality between two models for unified field theory: the first model assumes that the elementary particles have the field theoretic nature (thus we have to take into account the infinite dimensional group of symmetries corresponding to a change of phase), the second model assumes that the elementary particles are strings. Since both theories model the same mathematical phenominas, the measurements of of the same physical quantaties match. But since the two models are so different from mathematical point of view, the match between the measurements is a very non-trivial mathematical statement. The goal of the project is to explore the mathematical aspects of this theory. The conjecture is proven for toric threefolds and the lowest degree nontrivial characteristic classes. The project aims to extend the known results beyond the toric case and study the case of higher degree characteristic classes. PI advannces in this direction consits of the precise formula that relates the higher degree characteristic classes. Another part of the project is devoted to studying the topology of the Hilbert scheme of points on singular planar curves. A conjectural formula was proposed by the PI and V. Shende for the Poincare polynomials of the Hilbert scheme in terms of link invariants of the links of the singularities of the curve. One of outcomes of the project is proof of the conjectural formula for the large class of curves. Moduli spaces of sheaves are mathematical models for the field theories. Mathematical physics predictions, such as duality between the field theories and string theory or path integral formulas for knot invariants, can be translated into fascinating mathematical conjectures. Intellectual merit: The Gromov-Witten/Donaldson-Thomas correspondence reveals miraculous properties of the Gromov-Witten theory of threefolds. The objects which appear in the study of this correspondence are often of a combinatorial nature. We expect that these objects havea representation theoretic meaning, the elucidation of which will lead to newdevelopments in representation theory. As an example, the Nakajimaconstruction of the cohomology of the Hilbert scheme of points on asurface has a major impact on representation theory and algebraicgeometry. The theory of quantum cohomology (Gromov-Witten theory)of the Hilbert scheme of points on the surface provides a natural extension of the Nakajima construction, but its implications stillneed to be explored. The discovered connection between quantum knot invariants and the geometry of plane singular curves could alsofurther illuminate previously discovered links between knot theory and gauge theory. Broader impact: The interplay between string and gauge theory is one of the pillars of modern theoretical and mathematical physics. The Gromov-Witten/Donaldson-Thomas correspondence is the mathematical manifestation of String/Gaugetheory duality, and advances in GW/DT theory may initiate new progress in theoretical and mathematical physics.Also, recently discovered connections between Donaldson-Thomastheory, noncommutative geometry and the theory of melting crystals give hope thatdevelopments in Donaldson-Thomas theory may inspire advances instatistical mechanics. PI Alexei Oblomkov wrote several computer programs for evaluating geometric invariants. These programs can help other researchers to explore various aspects of enumerative geometry. Last year the PI mentored a senior thesis student who wrote athesis on spin Hurwitz numbers, and he plans to use NSF support to work with REU students on the combinatorial aspects of the GW/DT conjecture and on further development of the abovementioned software.
