All Questions
Tagged with gauge-theory 4-manifolds
8 questions
5
votes
0
answers
185
views
Nullity of a self-dual connection
I consider Yang-Mills theory in the critical dimension $4$ on a $SU(2)$-bundle with positive Chern-Class. It is well known that self-dual connections ($*F=F$) are minimizers of the Yang-Mills ...
- 914
3
votes
1
answer
172
views
Do we have $d(P_{+}(\omega\wedge \theta))=d_{+}\omega\wedge \theta-\omega\wedge d_{+}\theta$ on a self-dual manifold?
Let $(X, g)$ be a smooth, oriented, Riemannian 4-diemnsional manifold. Let $\Lambda^2$ denote the bundle of 2-forms on $X$. Then the Hodge-star operator decomposes $\Lambda^2$ into the space of self-...
- 9,019
6
votes
1
answer
317
views
"Neck cutting" and why gauge theory doesn't work on homotopy 4-spheres
I attended a talk recently and the speaker said, essentially, that gauge theory invariants are expected to never be able to detect exotic 4-spheres because they always vanish, for a reason related to ...
- 187
12
votes
1
answer
522
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Does $(S^1\times S^3)\#(S^1\times S^3)\#(S^2\times S^2)$ admit a symplectic form?
This is a crosspost from this MSE question from a year ago.
Consider the smooth four-manifold $M = (S^1\times S^3)\#(S^1\times S^3)\#(S^2\times S^2)$. Does $M$ admit a symplectic form?
If $\omega$ ...
- 19.3k
5
votes
0
answers
179
views
Curvature of the line bundle $\mathcal{O}(2)$ on the twistor space
Let $M^4$ be a closed Riemannian manifold and $Z:=S\big(\Lambda^2_+(M)\big)$ denote the twistor space of $M,$ i.e., the sphere bundle of the self-dual 2-forms on $M$. Now at a point $(m,J)\in Z$ the ...
- 954
4
votes
1
answer
185
views
Self-dual differential on $4$-manifold with boundary
Let $(M,g)$ be an oriented compact Riemannian $4$-manifold with boundary $\partial M$.
Let $a\in \Omega^1$ such that $*a\big|_{\partial M}=0$, i.e. $a(\nu)=0$ on $\partial M$, where $\nu$ denotes the ...
- 1,915
12
votes
3
answers
4k
views
About Donaldson-Kronheimer's book on four dimensional manifold
Recently, I read Donaldson-Kronheimer's Geometry of Four Manifolds. It seems that the book requires a lot of background. I had a really hard time digesting the content. Do we have other textbooks ...
- 407
8
votes
1
answer
2k
views
Relation of SW and Donaldson Invariant
My question is:
I am request for the reference that Is there any relationship between the Seiberg-Witten Invariant and Donaldson's Invariant? Or the relationship between Seiberg-Witten Moduli Space ...
- 703