Dirac Geometry and Moduli Spaces

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Dirac Geometry and Moduli Spaces.******Moduli spaces of flat bundles over surfaces have been the subject of intensive investigation in mathematics and physics. On the physics side they appear in conformal field theory, gauge theory and string theory. Mathematically they arise in a variety of contexts in symplectic geometry, algebraic geometry, knot theory, and representation theory. The closely related moduli spaces of `polygonal linkages' play a role in geometric mechanics and robotics. ******Beginning with the work of Atiyah-Bott and Witten in the 1980s, the moduli spaces have been studied using techniques from Poisson geometry, and specifically the theory of momentum maps. (In physics, Poisson manifolds arise as classical limits from quantum theories, while momentum maps are the generators of symmetries.) While this has led to deep insights into the structure of the moduli spaces, interesting open questions remain. ***Over the past few years, Dirac geometry has emerged as a new tool in the study of moduli space problems. Introduced in the 1990s, Dirac geometry is a far-reaching generalization of Poisson geometry. Its original purpose was to provide a geometric framework for classical mechanical systems with constraints, but it turned out to have a wide range of applications in mathematics and physics. It was also found to be the appropriate setting for the finite-dimensional approach to moduli spaces via non-linear momentum maps, the so-called quasi-Hamiltonian spaces. From the physics perspective, Dirac geometry enters the theory of D-branes in string theory as well as aspects of mirror symmetry.***This research will contribute to the theory of moduli spaces and group-valued momentum maps from the perspective of Dirac geometry. Its expected applications are in mathematical physics, for example the theory of D-branes and the theory of infinite-dimensional Hamiltonian systems, as well as in areas of pure mathematics such as representation theory or index theory. The project includes a number subtopics that will be well-suited as thesis projects for Ph.D. students.**