field theory
spacetime, connections, topology, and the geometry of gauge fields
Field theory abandons the Galilean separation of absolute time and relative space, beginning instead with spacetime conceived as a topological and differentiable manifold. Classical fields appear as sections of vector or principal bundles and therefore possess global configurations that cannot be captured by local differential equations alone. Comparison of field values at distinct points requires a connection, which encodes the rules of parallel transport and measures the obstruction to path-independence through its curvature.
The Lorentz group is lifted to its double cover, the Spin group, so that fermionic fields transform as spinors. Dynamical equations and conservation laws for electromagnetism and gravitation then arise from action principles, gauge symmetry, and relativistic mechanics.
the program
The interplay of differential operators, cohomology, and global topology classifies the obstructions that separate distinct field sectors. Characteristic classes and cobordism theory show how curvature detects the nontriviality of bundles, while K-theory organizes vector bundles into a ring and links them to the analytical index of Fredholm operators. Spin geometry and the Atiyah-Singer theorem demonstrate that topology itself determines the analytical indices of natural differential operators.
The culminating geometric insight is that gauge fields are connections on principal bundles; their absolute minima in four dimensions - the instantons - are classified by twistor theory, which converts the nonlinear anti-self-dual Yang-Mills equations into problems of holomorphic geometry and, ultimately, into finite-dimensional algebraic equations via the ADHM construction.