Seiberg-Witten equations in all dimensions

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I will describe a generalisation of the Seiberg-Witten equations to a Spin-c manifold of any dimension. The equations are for a U(1) connection A and spinor \phi and also an odd-degree differential form b (of inhomogeneous degree). Clifford action of the form is used to perturb the Dirac operator D_A. The first equation says that (D_A+b)(\phi)=0. The second equation involves the Weitzenböck remainder for D_A+b, setting it equal to q(\phi), where q(\phi) is the same quadratic term which appears in the usual Seiberg-Witten equations. This system is elliptic modulo gauge in dimensions congruent to 0,1 or 3 mod 4. In dimensions congruent to 2 mod 4 one needs to take two copies of the system, coupled via b. I will also describe a variant of these equations which make sense on manifolds with a Spin(7) structure, or a G_2 structure.

The most important difference with the familiar 3 and 4 dimensional stories is that compactness of the space of solutions is, for now at least, unclear. I will outline a very speculative suggestion that if this problem can be overcome and the equations can be used to define an invariant then it might be related to the existence of certain geometric structures on the underlying manifold (much as in dimension 4, vanishing of the Seiberg-Witten invariant obstructs the existence of a symplectic structure). This is joint work with Partha Ghosh and, in the Spin(7) and G_2 setting, Ragini Singhal.

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