# 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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### Gauge invariance issues of YM theories in 2D Euclidean space

In order to be clear, I will write down every component explicitly. Also, I assume Euclidean metric in this post, so that spacetime indices are written as $i,j$ rather than $\mu, \nu$. Following Wiki, ...
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### Question about the index of two elliptic operators over a 4-dimensional Riemannian manifold

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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 ...
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### An interesting identity involving skew-Schur functions

Denote $\rho=(-\frac12,-\frac32,-\frac52,\dots)$. I was reading this interesting paper, where in particular, the authors claim that one can get the expression in (2.9) \begin{align*} \prod_{k\geq1}(1+...
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### Question from Taubes' SW$\Rightarrow$ Gr

I am trying to understand Taubes' paper on SW$\Rightarrow$ Gr. I don't understand how either of the equations 2.16 or 2.17 appears, I would be happy to understand how the curvature term $F_a$ appears ...
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### Taubes' SW$\Rightarrow$ Gr

I am reading Taubes' paper on SW$\Rightarrow$ Gr and lost in some analysis, can anyone help me to see how to get equation 2.19 from equation 2.18? Is this some version of Kato for the Laplacian?
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### 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 ...
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### 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-...
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### 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 ...
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### 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 ...
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### 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 ...
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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 $\omega$ ...
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### 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 ...
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### 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 ...
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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 9 votes 0 answers 200 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 ... • 5,038 8 votes 0 answers 279 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 ... • 17.3k 4 votes 1 answer 835 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 ... • 247 5 votes 1 answer 340 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 ... • 1,209 3 votes 0 answers 244 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 0 answers 167 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 ... • 975 3 votes 1 answer 446 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 ... • 247 2 votes 0 answers 288 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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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 $H_A$ ...
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### The notion of a "relatively" flat connection

Suppose that $X$ is a connected smooth manifold and $\Gamma$ is a group acting smoothly, freely, properly and discretely on $X$, so that $Y=X/\Gamma$ is another smooth manifold endowed with a covering ...
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### 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. ...
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### 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, ...
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