All Questions
12 questions
2
votes
0
answers
169
views
Understanding the Seiberg-Witten equations in dimension $3$
I am trying to understand the dimensional reduction of Seiberg-Witten equations from dimension $4$ to $3$, more specifically my concern is about ellipticity of the new equations in dimension $3$ under ...
- 954
3
votes
1
answer
306
views
Moduli space of flat connection over homology 3-sphere
I'm trying to understand the space of flat connections of the trivial $\mathrm{SU}(2)$-bundle over a closed, oriented homology three-sphere (for the purpose of understanding the instanton Floer ...
- 1,908
3
votes
1
answer
507
views
Local coordinates of one form on a principal bundle
I am reading "Natural and Gauge Natural Formalism for Classical Field Theory" by Lorenzo Fatibene and I am really confused by his definition of a connection in local coordinates.
Let's say ...
- 212
1
vote
0
answers
373
views
About irreducible connection
The irreducible connection is a connection whose holonomy group is just $G$ (let us just assume the base space $X$ is just connected). Otherwise, it is called reducible if the holonomy group $H_A$ ...
- 2,821
5
votes
1
answer
411
views
Curvature as infinitesimal holonomy 2
This question may be seen as a follow up of this original question. I'm learning Cheeger-Simons differential characters (reading Differential Characters of Bär and Becker).
If I understand correctly, ...
- 6,825
9
votes
0
answers
231
views
Is there a contact instanton connection on the tangent bundle of the 5-sphere?
A well-known example of a contact manifold is $S^5$, arising from it being a circle bundle over $\mathbb{CP^2}$. This is somewhat related to the reduction of the structure group from $SO(5)$ to $SU(2)$...
- 35.5k
2
votes
1
answer
271
views
Canonical connection on $\mathcal{A}\times X$
Let $E\rightarrow X$ be a vector bundle and let $\mathcal{A}$ denote the space of connections on $E$. Pulling back $E$ by the second projection we obtain a vector bundle $\mathbb{E}=p_2^*E\rightarrow ...
- 6,813
2
votes
0
answers
47
views
Solving equations of motion of holomorphic BF theory - pure gauge in complex coordinates
In this paper by Bailieu and Tanzini, aspects of holomorphic BF theory are presented.
Holomorphic BF theory on a four dimensional Kahler manifold is discussed from page 5, and on page 8 the ...
- 1,155
3
votes
2
answers
653
views
When is the action of the gauge group on the space of connections free?
Let $G$ be a compact Lie group. Let $\mathcal{A}$ be the space of connections on the principal trivial $G$-bundle $G\times \mathbb{R}^4$ possibly with some growth condition (to specify it is a part of ...
- 5,824
4
votes
1
answer
412
views
Symmetries of non-Riemannian curvature tensor
The curvature tensor, $R_{ab}{}^c{}_d$, can be obtained from a connection which not necessarily is a metric connection.
By construction it is antisymmetric in the first two indices, since roughly ...
- 356
4
votes
2
answers
419
views
What is the space for the coefficients of the connection 1-form of a connection in a vector bundle?
Let $E\to X$ is a a (smooth real) vector bundle with structure group some Lie group $G$. Suppose we have a (linear) connection $\nabla$ on $E$.
Is it true that if $A$ is the connection 1-form of ...
- 25.3k
1
vote
0
answers
177
views
Obstruction to this gauge choice of the connection of a vector bundle
Let $M$ be a compact manifold with a nowhere-vanishing vector field $R$. Consider principal $G$-bundle $P$ over $M$, and $\mathcal{A}$ being the space of irreducible connections.
Let me denote a ...
- 857