If you haven't found something strange: Seiberg Witten

Consider the equations

$D_A \psi =0$
$F_A^+=\frac{1}{4}\langle e_i e_j \psi, \psi \rangle e^i \wedge e^j$

Where do these come from? How about the action for a gauge theory. One such action is $S(A,\psi)=\int_X \mid \nabla_A \psi \mid^2 + \mid F_A^+ \mid^2 + \frac{k}{4} \mid \psi \mid ^2 + \frac{1}{8} \mid \psi \mid^4$

What we want to do is understand the space of solutions to these equations. To help do that perturb the second equation by a $i \eta$ term so that the solutions are irreducible i.e. $\psi \neq 0$ .

After some index theorem calculations you get that the expected dimension of this moduli space is $c_1(\sqrt{L})^2 - \frac{2\chi+3\sigma}{4}$ which is the index of the deformation complex and then identifying the tangent space with the first homology of said complex.

The moduli space can also be proven to be oriented.

We can now define our invariant for 4 manifolds. If the expected dimension is below zero, we assign that manifold 0 because there are generically no solutions to the Seiberg-Witten equations. If it is zero, then we expect a moduli space consisting of a finite number of points which each have signs. We add them up and get a number.

The physics of this is involved in describing massless magnetic monopoles on the 4-manifold.

http://www.its.caltech.edu/~matilde/swcosi.pdf

If you haven't found something strange

Tuesday, December 20, 2011

Seiberg Witten

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