It is shown that abelian Higgs vortices on a hyperbolic surface M can be constructed geometrically from holomorphic maps f : M → N , where N is also a hyperbolic surface. The fields depend on f and on the metrics of M and N . The vortex centres are the ramification points, where the derivative of f vanishes. The magnitude of the Higgs field measures the extent to which f is locally an isometry.Witten's construction of vortices on the hyperbolic plane is rederived, and new examples of vortices on compact surfaces and on hyperbolic surfaces of revolution are obtained. The interpretation of these solutions as SO(3)-invariant, self-dual SU (2) Yang-Mills fields on R 4 is also given. * N.S.Manton@damtp.cam.ac.uk † N.A. Rink@damtp.cam.ac.uk

We study pure Yang-Mills theory on Σ × S 2 , where Σ is a compact Riemann surface, and invariance is assumed under rotations of S 2 . It is well known that the self-duality equations in this set-up reduce to vortex equations on Σ. If the Yang-Mills gauge group is SU(2), the Bogomolny vortex equations of the abelian Higgs model are obtained. For larger gauge groups one generally finds vortex equations involving several matrix-valued Higgs fields. Here we focus on Yang-Mills theory with gauge group SU(N )/ZN and a special reduction which yields only one non-abelian Higgs field.One of the new features of this reduction is the fact that while the instanton number of the theory in four dimensions is generally fractional with denominator N , we still obtain an integral vortex number in the reduced theory. We clarify the relation between these two topological charges at a bundle geometric level. Another striking feature is the emergence of non-trivial lower and upper bounds for the energy of the reduced theory on Σ. These bounds are proportional to the area of Σ.We give special solutions of the theory on Σ by embedding solutions of the abelian Higgs model into the non-abelian theory, and we relate our work to the language of quiver bundles, which has recently proved fruitful in the study of dimensional reduction of Yang-Mills theory.

No abstract

The rapid rise in demand for training large neural network architectures has brought into focus the need for partitioning strategies, for example by using data, model, or pipeline parallelism. Implementing these methods is increasingly supported through program primitives, but identifying efficient partitioning strategies requires expensive experimentation and expertise. We present the prototype of an automated partitioner that seamlessly integrates into existing compilers and existing user workflows. Our partitioner enables SPMD-style parallelism that encompasses data parallelism and parameter/activation sharding. Through a combination of inductive tactics and search in a platform-independent partitioning IR, automap can recover expert partitioning strategies such as Megatron sharding for transformer layers.