Seiberg-Witten invariant

In mathematics , the Seiberg-Witten invariants are important invariants of differentiable 4-manifolds. Its applications include the proof of the Thom conjecture or the non-existence of metrics of positive scalar curvature , decompositions as a connected sum or symplectic structures on different 4-manifolds. Furthermore you can differentiate between different differential structures on topological 4-manifolds.

definition

Let be a compact , differentiable manifold with a Riemannian metric and a spin ^c structure with associated spinor bundles and determinant bundles . ${\displaystyle M}$ ${\displaystyle {\mathfrak {s}}}$ ${\displaystyle W^{\pm }}$ ${\displaystyle L}$

For a generic self-dual 2-form , the space of the solutions of the perturbed Seiberg-Witten equations is a compact, orientable manifold of dimensions ${\displaystyle \eta }$ ${\displaystyle {\mathcal {M}}}$

${\displaystyle i({\mathfrak {s}}):={\frac {1}{4}}(c_{1}(L)^{2}-2\chi (M)-3sign(M))}$.

The calibration group and its subgroup have an effect . The quotient space is one - main fiber bundle over . Be his Euler class . ${\displaystyle {\mathcal {G}}=Map(M,S^{1})}$ ${\displaystyle {\mathcal {G}}_{0}=\left\{u\in {\mathcal {G}}\colon u(x_{0})=1\right\}}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle {\mathcal {M}}/{\mathcal {G}}_{0}}$ ${\displaystyle S^{1}}$ ${\displaystyle {\mathcal {M}}/{\mathcal {G}}}$ ${\displaystyle e\in H^{2}({\mathcal {M}}/{\mathcal {G}};\mathbb {Z} )}$

If is odd, then the dimension of is an even number . Then you define ${\displaystyle b_{2}^{+}(M)-b_{1}(M)}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle i(L)=2d}$

${\displaystyle SW(M,{\mathfrak {s}};g,\eta ):=\int _{\mathcal {M}}e^{d}}$.

For this invariant does not depend on and and is called the Seiberg-Witten invariant . ${\displaystyle b_{2}^{+}(M)\geq 2}$ ${\displaystyle g}$ ${\displaystyle \eta }$ ${\displaystyle SW(M,{\mathfrak {s}})}$

properties

In the following, always be odd and . A cohomology class is called a base class if there is a spin ^c structure with and . ${\displaystyle b_{2}^{+}(M)-b_{1}(M)}$ ${\displaystyle b_{2}^{+}(M)\geq 2}$ ${\displaystyle c\in H^{2}(M;\mathbb {Z} )}$ ${\displaystyle {\mathfrak {s}}}$ ${\displaystyle c_{1}(L)=c}$ ${\displaystyle SW(M,{\mathfrak {s}})\not =0}$

• If is an orientation- preserving diffeomorphism , then is ${\displaystyle f\colon M_{1}\to M_{2}}$ ${\displaystyle SW(M_{1},f^{*}{\mathfrak {s}})=SW(M,{\mathfrak {s}}).}$
• The following applies to every base class . ${\displaystyle c}$ ${\displaystyle c\cdot c\geq 2\chi (M)+3sign(M)}$
• The following applies to the dual spin ^c structure ${\displaystyle {\mathfrak {s}}^{*}}$ ${\displaystyle SW(M,{\mathfrak {s}}^{*})=(-1)^{\frac {\chi (M)+sign(M)}{4}}SW(M,{\mathfrak {s}}).}$
• ${\displaystyle M}$ only has a finite number of base classes.
• If a metric has positive scalar curvature , then applies to all . ${\displaystyle M}$ ${\displaystyle SW(M,{\mathfrak {s}})=0}$ ${\displaystyle {\mathfrak {s}}}$
• If for compact, orientable, smooth 4-manifolds with , then applies to all . ${\displaystyle M=X\sharp Y}$ ${\displaystyle X,Y}$ ${\displaystyle b_{2}^{+}>0}$ ${\displaystyle SW(M,{\mathfrak {s}})=0}$ ${\displaystyle {\mathfrak {s}}}$
• If holds and the inequality holds for a spin ^c structure with , then is . ${\displaystyle b_{1}(X)=b_{2}^{+}(X)=0}$ ${\displaystyle {\mathfrak {s}}_{X}}$ ${\displaystyle c_{1}=c_{X}}$ ${\displaystyle c\cdot c-2\chi (M)-3sign(M)+c_{X}\cdot c_{X}+b_{2}(X)\geq 0}$ ${\displaystyle SW(M,{\mathfrak {s}})=SW(M\sharp X,{\mathfrak {s}}\sharp {\mathfrak {s}}_{X})}$
• For an embedded , compact, orientable area the gender applies to each base class . ${\displaystyle \Sigma \subset M}$ ${\displaystyle g(\Sigma )}$ ${\displaystyle 2g(\Sigma )-2\geq \Sigma \cdot \Sigma +\vert c\cdot \Sigma \vert }$ ${\displaystyle c}$
• If is a symplectic manifold with canonical spin ^c -structure , then is . ${\displaystyle M}$ ${\displaystyle {\mathfrak {s}}_{can}}$ ${\displaystyle SW(M,{\mathfrak {s}}_{can})=1}$

literature

• John Morgan : Lectures on Seiberg-Witten invariants , Lecture Notes in Mathematics, 1629 (2nd ed.), Berlin: Springer-Verlag, ISBN 3-540-41221-2
• Liviu Nicolaescu : Notes on Seiberg-Witten theory , Graduate Studies in Mathematics, 28, Providence, RI: American Mathematical Society, ISBN 0-8218-2145-8
• Alexandru Scorpan : The wild world of 4-manifolds , American Mathematical Society, ISBN 978-0-8218-3749-8

Web links

• Dietmar Salamon : Spin geometry and Seiberg-Witten invariants