All Questions
Tagged with gauge-theory ag.algebraic-geometry
15 questions
3
votes
0
answers
147
views
Generalizing the Narasimhan–Seshadri theorem
There is a theorem that (stable, topologically trivial) holomorphic $G$-bundles are in one-to-one correspondence to flat $K$-bundles (with the appropriate corresponding condition), where $K$ is the ...
- 563
1
vote
0
answers
93
views
Sufficient condition for moduli space of slope-stable bundles to be non-empty
I'm trying to find to a statement concerning the non-emptiness of the moduli space of slope stable vector bundles over a Kähler surface in the literature.
Let $X$ be a Kähler surface. Let $\mathscr{M}(...
- 11
4
votes
0
answers
352
views
Moduli spaces of rank 2 stable bundles over curves as projective varieties
Let $\Sigma$ be a Riemann surface of genus $g$. Let's consider the moduli space of rank $2$ stable vector bundles with determinant $L$ such that $\deg(L)$ is odd. Denote this space by $\mathcal{M}_{\...
- 915
10
votes
1
answer
648
views
Gauge theory on schemes
Gauge theories are traditionally formulated in space-time, superspace, manifolds, or supermanifolds.
Are there formulations of gauge theory in the context of algebraic geometry (e.g. defining fields ...
- 11.8k
6
votes
0
answers
228
views
Infinite-dimensional manifold $ΩG$ of loops and the complex projective $n$-space
In a review seminar today, I heard the speaker takes the below for granted, but I have no enough background
"Yang-Mills (YM) instantons in 4D can be naturally identified with (i.e. have the same ...
- 10.5k
1
vote
1
answer
297
views
compact manifold as a hyperkahler quotient of an infinite-dimensional affine space
Is it possible to obtain K3 (or any other compact
hyperkahler manifold) with its hyperkahler structure
as a hyperkahler quotient of an infinite-dimensional affine
quaternionic vector space with an ...
- 9,237
6
votes
0
answers
255
views
Correspondence between $O(n)$-instantons on $S^4$ and holomorphic bundles over $\mathbb{P}^3$
Theorem 2.9 on page 49 of Atiyah's Geometry of Yang-Mills Fields states that there is a natural correspondence between $U(n)$-instantons on $S^4$ and holomorphic vector bundles of rank $n$ over $\...
- 161
9
votes
0
answers
661
views
Hartshorne's Conjectures about Algebraic Bundles?
In 1978 Hartshorne published a list of 26 open problems about algebraic bundles on projective spaces [Hart], proceeding from an Oxford conference organized by Atiyah.
I understand that many of these ...
user100272
3
votes
1
answer
606
views
Gottsche Nakajima Yoshioka define a weird slant product
In their article Instanton counting and Donaldson invariants the authors define the slant product for $\beta \in H_i(X)$ (where $X$ is a manifold) as following.
Let $P \to X$ and SO(3) bundle and $M(...
- 587
3
votes
1
answer
347
views
Transformation between two conventions of Hitchin equation
Recall that for a given Riemann surface $\Sigma$ Hitchin's self-duality equation consists of a complex rank $r$ vector bundle $E$ (with degree 0 for simplicity), a connection $d_A: \Omega^k(\Sigma, E) ...
- 367
1
vote
0
answers
580
views
On the Hitchin fibration
I will refer to Simpson's "Higgs bundles and local systems".
Proposition 1.4:
When $X$ is a smooth projective variety, one can build up the moduli space $\mathcal{M}(X,r)$ of rank $r$ Higgs ...
- 111
2
votes
0
answers
197
views
computation with Hilbert scheme of $n$ points on $\mathbb C^2$ [closed]
How can we show that
$$\sum_{n = 0}^\infty q^n \operatorname{char}_T S^n(\mathbb C[x,y])=
\prod_{p_1,p_2\geq 0}\frac{1}{1-t_1^{p_1}t_2^{p_2}q}$$
where $\operatorname{char}_T V$ denotes the character ...
- 533
33
votes
4
answers
3k
views
What is the precise statement of the correspondence between stable Higgs bundles on a Riemann surface, solutions to Hitchin's self-duality equations on the Riemann surface, and representations of the fundamental group of the Riemann surface?
I am trying to find the precise statement of the correspondence between stable Higgs bundles on a Riemann surface $\Sigma$, (irreducible) solutions to Hitchin's self-duality equations on $\Sigma$, and ...
- 21k
2
votes
1
answer
597
views
gauge theory construction of vector bundles on singular varieties
This is sort of a follow-up to:
Gauge theory construction of moduli of vector bundles
If I have a complex compact algebraic curve with at worst nodal singularities, is there an analytic description ...
- 33
7
votes
3
answers
2k
views
Gauge theory construction of moduli of vector bundles
Given a smooth projective complex variety $X$, instead of using Mumford's GIT to construct the moduli of rank $n$ topologically trivial vector bundles, we can also take the gauge theory approach.
To ...