Geometry of the tip of the global nilpotent cone

In this proposal we study the motion of Higgs bundles to develop new theories in diverse areas of theoretical physics and mathematics. Higgs bundles are two-dimensional relatives of the famous Higgs boson an elementary particle living in four-dimensional space-time responsible for endowing other particles with mass and recently found in the Large Hadron Collider in CERN, Geneva. The movements of Higgs bundles can be fully described as their mechanical system is completely solvable. The quantities which solve the motion of Higgs bundles were discovered by Hitchin in 1987 and form a so-called integrable system. This Hitchin integrable system is a prototype of all integrable systems in mathematical physics, which are important model systems to solve complicated dynamics in various contexts. Inside our configuration space of Higgs bundles there is a core a so-called global nilpotent cone which is like an iceberg. We study the Hitchin integrable system around the tip of this iceberg. The metaphorical iceberg has horizontal layers too, which can be scanned from the top. They are multisections of the Hitchin system. We look at the Hitchin system like an x-ray, descending from the top of the nilpotent cone. The main novelty in the proposal is the introduction of explicit algebraic structures so-called multiplicity algebras which give an algebraic measurement of the intersection of the horizontal layer with the iceberg. This multiplicity algebra is located at the single intersection point of the horizontal layer and the iceberg, and it has a surprisingly complex structure. We can compute it in certain cases and surprisingly find that it is the intersection ring of some Grassmannian space. The intersection ring is an algebraic topological quantity of a shape, which measures the intersection structure of high dimensional holes on our surface. The Grassmannian surface is the space of linear subspaces in a high dimensional linear space with an intriguing and well understood shape. Thus, we discover that the intersection between the horizontal layers of the x-ray and the iceberg are understandable in terms of the intersection of holes in certain high dimensional Grassmannian spaces. In turn, this gives explicit formulas for the Hitchin integrable system on these horizontal layers - solving the movement of Higgs bundles with explicit formulas. We find unexpected properties of these multiplicity algebras, which link together questions from representation theory of continuous group of symmetries, geometry of integrable mechanical systems, mirror symmetry in string theory and Langlands duality in number theory. Developing this circle of ideas will lead to new theories related to all the fields in theoretical physics and mathematics mentioned above.

In this proposal we study the motion of Higgs bundles to develop new theories in diverse areas of theoretical physics and mathematics. Higgs bundles are two-dimensional relatives of the famous Higgs boson an elementary particle living in four-dimensional space-time responsible for endowing other particles with mass and recently found in the Large Hadron Collider in CERN, Geneva. The movements of Higgs bundles can be fully described as their mechanical system is completely solvable. The quantities which solve the motion of Higgs bundles were discovered by Hitchin in 1987 and form a so-called integrable system. This Hitchin integrable system is a prototype of all integrable systems in mathematical physics, which are important model systems to solve complicated dynamics in various contexts. Inside our configuration space of Higgs bundles there is a core a so-called global nilpotent cone which is like an iceberg. We study the Hitchin integrable system around the tip of this iceberg. The metaphorical iceberg has horizontal layers too, which can be scanned from the top. They are multisections of the Hitchin system. We look at the Hitchin system like an x-ray, descending from the top of the nilpotent cone. The main novelty in the proposal is the introduction of explicit algebraic structures so-called multiplicity algebras which give an algebraic measurement of the intersection of the horizontal layer with the iceberg. This multiplicity algebra is located at the single intersection point of the horizontal layer and the iceberg, and it has a surprisingly complex structure. We can compute it in certain cases and surprisingly find that it is the intersection ring of some Grassmannian space. The intersection ring is an algebraic topological quantity of a shape, which measures the intersection structure of high dimensional holes on our surface. The Grassmannian surface is the space of linear subspaces in a high dimensional linear space with an intriguing and well understood shape. Thus, we discover that the intersection between the horizontal layers of the x-ray and the iceberg are understandable in terms of the intersection of holes in certain high dimensional Grassmannian spaces. In turn, this gives explicit formulas for the Hitchin integrable system on these horizontal layers - solving the movement of Higgs bundles with explicit formulas. We find unexpected properties of these multiplicity algebras, which link together questions from representation theory of continuous group of symmetries, geometry of integrable mechanical systems, mirror symmetry in string theory and Langlands duality in number theory. Developing this circle of ideas will lead to new theories related to all the fields in theoretical physics and mathematics mentioned above.