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Edward Witten works on theoretical physics and its connections with geometry and topology. His Institute for Advanced Study profile identifies the researcher.^[ref]

His Monopoles and Four-Manifolds by Edward Witten used ideas from supersymmetric gauge theory to introduce a new description of smooth four-manifold invariants. The Seiberg–Witten equations replace a nonabelian instanton-counting problem with equations involving an Abelian gauge field and a spinor. The contribution demonstrates how a physical dual description can reveal a more tractable mathematical structure.^[ref]

On an oriented Riemannian smooth four-manifold with a Spin-c structure, let $W^{\pm }$ be the spinor bundles and $L=\det W^{+}$ their determinant line. The Seiberg–Witten equations couple a unitary connection $A$ on $L$ and a positive spinor $\psi \in \Gamma (W^{+})$:Here $D_A:\Gamma (W^{+})\to \Gamma (W^{-})$ is the Dirac operator, $F_A^{+}$ is the self-dual part of the curvature, and $q$ is the quadratic spinor map into imaginary self-dual two-forms. Numerical factors and signs in $q$ depend on the Clifford conventions; a perturbation adds a prescribed imaginary self-dual two-form.

The first equation is first-order and elliptic, while the curvature equation makes the coupled system nonlinear. Gauge transformations identify equivalent connection-spinor pairs. Solutions with $\psi =0$ are reducible and satisfy $F_A^{+}=0$ in the unperturbed system.

On closed four-manifolds, suitably perturbed moduli spaces yield the Seiberg–Witten invariant of a four-manifold. The abelian gauge equations arose from the monopole description of supersymmetric gauge theory and provide tools for distinguishing smooth four-dimensional geometry.^[ref]

For a closed oriented smooth four-manifold $X$ with $b_2^{+}(X)>1$, the Seiberg–Witten invariant of a four-manifold associates an integer-valued invariant, with a choice of homology orientation, to a Spin-c structure. It is obtained from the moduli space of solutions of the Seiberg–Witten equations modulo gauge transformations. When the expected dimension is zero, it is a signed count of solutions after a generic perturbation. Positive-dimensional cases require suitable insertions; when $b_2^{+}=1$, chamber data matters. These analytic counts can distinguish smooth structures and, through Taubes nonvanishing theorem, obstruct decompositions of symplectic manifolds of dimension four.^[ref]