Topology, Geometry and Physics

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This project will study the behavior of solutions to certain systems of partial differential equations that are thought to give information about the underlying structures of four dimensional spaces. The equations of interest came from high energy physics where they were introduced as very symmetric models for elementary particle interactions. Basic question to address with regards to solutions are these: Are there finitely many or infinitely many solutions for the specific applications? In either case, what do the solutions look like? And what are their implications with regards to four dimensional spaces? By way of background: The equations are far too complicated for a computer to solve (let alone to solve by hand). Because of this, almost all information about the solutions must be obtained via a detailed analysis of the equations themselves. With regards to questions about four dimensional spaces, the central issue here is to understand the list of possible four dimensional spaces. This last question is also relevant to physics by virtue of the fact that the observable universe has four dimensions--three spatial dimensions and then the time direction. Since the equations of interest come from high energy physics theory and are close cousins to others used there and in condensed matter physics, the analysis tools developed in this project may also find applications to questions in these other fields. There is also an important educational support aspect of this project that is meant to bring graduate students up to speed on the relevant geometry, topology and analysis.The research in this prject has three primary focus points. The first concerns the behavior of the solutions to the Kapustin-Witten equations on the product of a three-dimensional manifold and the half-line with singular boundary conditions at one end given by a knot in the three-manifold. The goal for this focus point is to see whether non-convergent sequences of solutions have pathologies that prevent solution counting definitions of knot invariants for knots in the three dimensional manifold. Two sources of trouble likely arise, one is a novel pathology from the boundary conditions, and the other due to the presence of Z/2 harmonic 1-forms. The second focus point concerns the algebraic implications of the structure of the solution space for these same Kapustin-Witten equations and for the Kapustin-Witten equations on the product of the three-manifold with the line: Can an algebraic invariant be defined from these spaces given the non-compactness issues and with the novelty of formally zero dimensional solution spaces? The third focus point concerns the significance of the afore-mentioned Z/2 harmonic 1-forms and, more generally, Z/2 harmonic sections of a spin bundle. Such Z/2 harmonic sections appear as renormalized limits of sequences of solutions to the Vafa-Witten equations on four dimensional manifolds and the multi-spinor Seiberg-Witten equations on manifolds of dimensions three and four. Specific topics and questions not withstanding, the long term goal of this project is to understand the structure of smooth, four dimensional manifolds.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.