Questions tagged [gauge-theory]

Gauge theory in physics and mathematics refers to a field theory whose fields include principal bundles with connection.

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12 votes
1 answer
663 views

Reference request: Gauge natural bundles, and calculus of variation via the equivariant bundle approach

Let $P\rightarrow M$ be a principal fibre bundle with structure group $G$, $F$ a manifold and $\alpha: G\times F\rightarrow F$ a smooth left action. There is an associated fibre bundle $E\rightarrow ...
1 vote
0 answers
61 views

Spin(7)-instanton

Let $M$ be a Spin$(7)$-manifold with a spin-bundle $S=S_+\oplus S_-$. There's an obvious connection on $S$ which comes from lifting the Levi-Civita connection. And it induces a connection on the ...
2 votes
1 answer
408 views

A question about the book "the geometry and dynamics of magnetic monopoles"

In chapter 2 of the book "The geometry and dynamics of magnetic monopoles", by M.F. Atiyah and N.J. Hitchin (the chapter is called "Geometry of the monopole spaces"), it is written:...
2 votes
0 answers
106 views

Changing the sign of the moment map in the Seiberg Witten equations

The Seiberg-Witten equations on a closed four manifold $$ D_A \varphi = 0, F_A^+ = \mu(\varphi) $$ are elliptic (up to gauge transformations), and so the equations $$ D_A \varphi = 0, F_A^+ = -\mu(\...
1 vote
0 answers
101 views

Does a gauge-invariant Caccioppoli inequality hold?

(I previously asked this question on Math.SE but got no responses after two weeks.) Let $V \Subset U$ be domains in a Riemannian manifold $M$, and $W := U \setminus \overline V$. If $u: U \to \mathbb ...
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 ...
2 votes
0 answers
138 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 ...
15 votes
2 answers
2k views

Every 4-manifold has a $\operatorname{Spin}^c$ Structure

$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}$I'm having trouble understanding the proof given in Morgan's The Seiberg–Witten Equations and Applications to the Topology of Smooth Four-...
1 vote
0 answers
99 views

Is this a correct description of the BPS monopole of charge $1$?

I am reading the book "The Geometry and Dynamics of Magnetic Monopoles", by M.F. Atiyah and N.J. Hitchin, and I got to this part: "... let $H$ be the Hopf line bundle over $S^2$ and ...
1 vote
1 answer
223 views

Spin connection vs. Cartan connection

I am studying the tetradic Palatini formalism of general relativity. In this formalism, one usually considers a manifold $M$, which is either non-compact or compact with Euler-characteristic $\chi(M)=...
3 votes
1 answer
162 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-...
3 votes
1 answer
179 views

Hyperkähler structure of framed instantons over $\smash{\overline{\mathbb{C}P}}^2$

When underlying $4$ manifold is compact and hyperkähler, the philosophy of infinite dimensional moment map tells us that its instantons moduli space is also hyperkähler. I'm curious about the ...
2 votes
1 answer
514 views

On Corlette's paper Flat G-bundles with canonical metrics

I am now reading Kevin Corlette's paper: Flat G-bundles with canonical metrics, JDG, 1988 Link I think I do understand the core part of its argument, i.e. finding the harmonic metric via heat equation....
1 vote
0 answers
123 views

Heat kernel coefficients for Laplacian in instanton background

The heat kernel coefficients $b_{2k}(x,y)$ of the covariant Laplacian in an $SU(2)$ instanton background (for simplicity let's say $q=1$ topological charge, so the 't Hooft solution) on $R^4$ is ...
7 votes
0 answers
308 views

Existence of Yang-Mills connection

My question is about what we know, in dimension $4$, about the loss of compactness of Yang–Mills connections with $L^2$-bounded curvature. My background is more analytical than geometrical and it is ...
6 votes
1 answer
297 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 ...
1 vote
0 answers
106 views

Instantons on the 4-sphere with respect to other Riemannian metrics

It is known that the moduli space of $\text{SU}(2)$ instantons of charge $1$ on $S^4$ is diffeomorphic to the five-ball $B^5$ if $S^4$ is endowed with the round metric. Question: what does the moduli ...
7 votes
1 answer
597 views

Is there a program to solve The Yang–Mills Existence and Mass Gap problem similar to the Hamilton's program to solve Poincaré Conjecture?

According to Wikipedia: "Hamilton's program was started in his 1982 paper in which he introduced the Ricci flow on a manifold and showed how to use it to prove some special cases of the Poincaré ...
6 votes
1 answer
246 views

Different definitions of "charged spinors": "bundle splicing" vs. "twisted spinor bundles"

Currently I study the mathematical formulation of the (classical) standard model of particle physics using the language of gauge theory and spin geometry. One of the central objects in the standard ...
6 votes
1 answer
514 views

Questions on R. Bryant's paper "Calibrated embeddings in the special Lagrangian and coassociative cases"

I am reading the paper "Calibrated embeddings in the special Lagrangian and coassociative cases" by R. Bryant (here the link: http://arxiv.org/abs/math/9912246) and there are certain things ...
1 vote
0 answers
87 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}(...
2 votes
0 answers
78 views

Weitzenbock- Anti-selfdual

In "The Theory of Gauge Fields in Four Manifolds", B.Lawson proves the Bochner-Weitzenbock, for an anti-self-dual field $\Psi \in \Omega^2_-(\mathfrak{G}_E)$,where $\mathfrak{G}_E$ is the ...
0 votes
0 answers
219 views

Finding a unitary operator on L^{2}(\mathbb{R}^{2},dxdy)

I have a one parameter (r) family of self-adjoint representations of the universal enveloping algebra of some nilpotent Lie group on $L^{2}(\mathbb{R}^{2},dxdy)$ as follows: $$\hat{X}^{r}=\hat{x}-i(r-...
11 votes
4 answers
1k views

Literature for gauge field theory on the lattice in geometrical formulation

I have found an article by Huebschmann, Rudolph and Schmidt about "A Gauge Model for Quantum Mechanics on a Stratified Space" and I am very interested in this subject, but I don't have any ...
0 votes
1 answer
138 views

Non existence of preferred Horizontal subspace on a bundle [closed]

If I choose a principal bundle, let us say $G\rightarrow P \rightarrow B$, with $G=U(1)$, $P=S^1 \times S^1$ and $B=S^1$. Can I follow the identity element of the group over a curve at the base. How ...
1 vote
0 answers
74 views

Uhlenbeck's compactness for abelian gauge

I am looking for a simpler proof (if possible) for the Uhlenbeck's compactness result (bound on connection up-to gauge from bound on curvature) on an open ball. I know that a proof exists using Hodge-...
2 votes
0 answers
134 views

Gauge invariance of a QFT path integral

If we consider the usual formal construction of a path integral over fields with gauge symmetries e.g as in Weinbergs "The Quantum Theory of Fields - Volume 2" the notion of gauge invariance ...
3 votes
1 answer
232 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 ...
12 votes
1 answer
461 views

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 $\omega$ ...
5 votes
1 answer
296 views

In search of a combinatorial proof on particular set of partitions

Given a partition $\lambda=(\lambda_1\geq\lambda_2\geq\dots)$, denote the conjugate partition by $\lambda'=(\lambda_1'\geq\lambda_2'\geq\dots)$. For example, if $\lambda=(4,2,2)$ then $\lambda'=(3,3,1,...
5 votes
1 answer
372 views

Stabilizer groups of Yang-Mills connections

Let $G$ be a compact Lie group with complexification $G^c$, and consider a principal $G^c$-bundle $P^c \to M$ together with a reduction $P \subseteq P^c$ to $G$. Assume that $M$ is a Riemann surface. ...
2 votes
0 answers
94 views

Vortex equation on Riemann surface and a similar equation

Let's take a Riemann surface $(X,\omega)$ and a holomorphic line bundle $L$ on it with a hermitian metric $h$ on $L$. $g$ be a real valued smooth function on $X$ and we consider the following two sets ...
6 votes
0 answers
471 views

Yang–Mills existence and mass gap official statement on Euclidean $\mathbb{R}^4$, why not Minkowski $ \mathbb{R}^{3,1}$?

Yang–Mills existence and mass gap problem is officially stated by Clay Mathematics Institute: Yang–Mills Existence and Mass Gap.'' Prove that for any compact simple gauge group G, a non-trivial ...
5 votes
0 answers
166 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 ...
2 votes
1 answer
630 views

Understanding the slice theorem

I was reading Morgan's book: "The Seiberg-Witten equations and applications to the topology of smooth four-manifolds" and find it hard to understand the slice theorem (page 62-64). Here are ...
13 votes
3 answers
2k views

The "miracle" of Heegard Floer.

Taking tori in symmetric products and "miraculously" proving that the Floer homology is independent of choices always seemed, well, miraculous. Some time ago Max Lipyanski explained to me the origins ...
4 votes
2 answers
515 views

How much do characteristic classes fail to characterize bundles?

Given a group $G$, let $E \to B$ be a principal $G$-bundle. It is well-known that when $B$ is a nice enough topological space (e.g. CW-complex), such a thing corresponds to a connected component of $...
9 votes
1 answer
1k views

Casson's invariant and the trivial connection contribution to witten's 3-manifold invariant

This question might turn out not to make any sense, but here it is: Witten's (and Reshetikhin and Turaev's) 3-manifold invariant can be "defined" as an integral over the space of connections on the ...
2 votes
0 answers
94 views

Chern-Simons Functional for $S^1$-bundles over Riemann Surfaces

If $M$ is a closed, orientable 3-manifold $M$ and also happens to be an $S^1$ bundle over some Riemann surface $\Sigma$, then it seems like a natural Heegaard splitting of $M$ could come from ...
4 votes
0 answers
294 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}_{\...
9 votes
0 answers
197 views

Donaldson invariants for piecewise-linear $4$-manifolds

It is well known that in dimension $4$, the notion of piecewise linear manifolds and the notion of smooth manifolds are the same [1][2]. On the other hand, the computations of Donaldson invariants ...
4 votes
1 answer
808 views

The Yang-Mills Higgs Lagrangian

Let's say we have a principal bundle $(P,B,\pi;G)$ and associated bundle $E=P \times_{(G,\rho)}V$and $Ad(P)=P\times_{(G,Ad)} \mathfrak{g}$ the adjoint bundle. The Yang-Mills-Higgs action (without ...
8 votes
0 answers
276 views

Infinitely many nonempty Seiberg-Witten moduli spaces

The classic "finiteness" statement in Seiberg-Witten (SW) theory is that, for any smooth closed connected 4-manifold, there are only finitely many spin-c structures with nontrivial SW ...
3 votes
1 answer
412 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 ...
13 votes
1 answer
4k views

Curvature as infinitesimal holonomy

Let $P \to M$ be a principal $G$-bundle, assume as much regularity as you want (compact $G$ or compact base manifold, ect). Via parallel transport, a connection $A$ on $P$ gives rise to the holonomy ...
5 votes
1 answer
328 views

Spin connection in the tetradic Palatini-formalism of general relativity

$\DeclareMathOperator\SO{SO}$I am trying to understand the tetradic Palatini-formalism of general relativity from a mathematical point of view. I am graduate student and quite new to mathematical ...
4 votes
0 answers
166 views

Compactly supported geometric representatives for Seiberg-Witten invariant

The question is introduced at the end of the second paragraph. Readers familiar with Seiberg-Witten theory may well skip the first paragraph. The first paragraph is meant to set up some notation which ...
3 votes
0 answers
234 views

Tensor product of associated vector bundles

Let $(P, X, \pi, G)$ and $(P', X, \pi', G')$ be two principal bundles (with Lie groups $G$, $G'$ respectively). Given a vector space $V$ and representations $\rho, \rho'$ of the Lie groups in this ...
4 votes
1 answer
1k views

Uhlenbeck's theorem novelty

This link provides a short introduction to the contributions of Uhlenbeck about regular gauge fixing. However, I feel quite puzzled about it and I do not understand the real novelty apported by this ...
2 votes
0 answers
281 views

Maxwell $U(1)$ gauge theory's electric and magnetic sources turned on simultaneously in the classical differential geometry

Question: How do we couple $U(1)$ electric (E) and magnetic (M) sources simultaneously in the classical differential geometry language, in a $U(1)$ gauge theory based on $U(1)$ gauge bundle and its $U(...

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