In differential geometry, the Seiberg–Witten flow is a gradient flow described by the Seiberg–Witten equations, hence a method to describe a gradient descent of the Seiberg–Witten action functional. Simply put, the Seiberg–Witten flow is a path always going in the direction of steepest descent, similar to the path of a ball rolling down a hill. This helps to find critical points, called (Seiberg–Witten) monopoles, which solve the Seiberg–Witten equations. Illustratively, they are the points on the hill on which the ball can rest. The Seiberg–Witten flow is named after Nathan Seiberg and Edward Witten, who first formulated the underlying Seiberg–Witten theory in 1994.
Definition Let M {\displaystyle M} be a compact orientable Riemannian 4-manifold. Every such manifold has a spinc structure, which is a lift of the classifying map f : M → BSO ( 4 ) {\displaystyle f\colon M\rightarrow \operatorname {BSO} (4)} of the tangent bundle T M {\displaystyle TM} (hence so that T M ≅ f ∗ γ ~ R 4 {\displaystyle TM\cong f^{*}{\widetilde {\gamma }}_{\mathbb {R} }^{4}} is the pullback bundle of the oriented tautological bundle along it) to a continuous map f ^ : M → BSpin c ( 4 ) {\displaystyle {\widehat {f}}\colon M\rightarrow \operatorname {BSpin} ^{\mathrm {c} }(4)} (hence so that it factors over the map induced by the canonical projection Spin c ( 4 ) ↠ SO ( 4 ) {\displaystyle \operatorname {Spin} ^{\mathrm {c} }(4)\twoheadrightarrow \operatorname {SO} (4)} on classifying spaces). All possible spinc structures correspond exactly to the second singular cohomology H 2 ( M , Z ) ≅ [ M , BU ( 1 ) ] {\displaystyle H^{2}(M,\mathbb {Z} )\cong [M,\operatorname {BU} (1)]} . Because of the central identity:
Spin c ( 4 ) ≅ U ( 2 ) × U ( 1 ) U ( 2 ) ≅ { A ± ∈ U ( 2 ) | det ( A − ) = det ( A + ) } , {\displaystyle \operatorname {Spin} ^{\mathrm {c} }(4)\cong \operatorname {U} (2)\times _{\operatorname {U} (1)}\operatorname {U} (2)\cong \left\{A^{\pm }\in \operatorname {U} (2)|\det(A^{-})=\det(A^{+})\right\},}
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