Conic Symplectic Singularities

Symmetry is a fundamental concept in both mathematics and physics since most objects appearing in nature have a certain degree of symmetry. In mathematics, Lie theory is the study of continuous symmetries. Think, for instance, of the rotational symmetries of the earth, which can be thought of as a continuous family of symmetries. Symmetries are important because they allow for (often major) simplification of problems; in the case of the rotation symmetries of the earth, this leads to the notation of polar coordinates which is a useful simplification tool when studying physically motivated systems of differential equations. The set of all continuous symmetries of an object form a group (the Lie group) and its properties are largely governed by the associated Lie algebra. One of the cornerstone results of algebra, achieved in the mid 20th-century, is the Cartan-Killing classification of simple finite-dimensional complex Lie algebras. This classification forms the core of Lie theory, on which the rest of the theory is built. We propose a programme to generalize this classification result to all conic symplectic singularities. Such a classification would form the core of what is today commonly known as symplectic representation theory. As is oft quoted (and attributed to Okounkov), "symplectic representation theory is the Lie theory for the 21st century'' - we propose to pursue a Cartan-Killing classification for the 21st century. The notion of a symplectic singularity was introduced by Beauville, extending the notion of symplectic manifold to singular spaces. Not only have symplectic singularities proven to be an important class of singularities in algebraic geometry, but they have also come to form the core of geometric representation theory: in any class of examples, the starting point is always the symplectic singularity, from which one goes on to consider resolutions, deformations, quantizations... etc. More specifically, in geometric representation theory the focus is on conic symplectic singularities. These are symplectic singularities which are also affine cones for which the symplectic form has positive weight. We propose a programme to completely classify these conic symplectic singularities; this is made plausible by an extraordinary result of Namikawa, which says that these singularities are countable, up to isomorphism. We will introduce the concept of Hamiltonian Cox ring and associated Hamiltonian Cox space of a conic symplectic singularity and show that these Hamiltonian Cox spaces are Q-factorial and terminal conic symplectic singularities. This reduces the classification problem to that of classifying the much smaller class of Q-factorial terminal conic symplectic singularities. We will apply techniques (computations of Hilbert series) and constructions (Coulomb branches of 3d N=4 SUSY gauge theories) from mathematical physics to classify these conic symplectic singularities. We expect applications to the representation theory of quantizations of symplectic singularities and to the classification of crepant partial resolutions for these singularities.