Questions tagged [gauge-theory]

For questions about gauge theory in mathematical physics and differential geometry. Typical questions pertain to bundles, connections, spinors, and moduli spaces. Questions about the physics of gauge fields should be directed to physics.stackexchange.

Gauge theory is a branch of mathematical physics, differential geometry, and differential topology. The aim is to study the geometry and topology of a space by examining an appropriate moduli space of connections (and possibly spinors) which satisfy certain PDE. These PDE frequently have their origins in physics.

Topics of gauge theory include connections on principal bundles, gauge groups, Yang-Mills connections, (anti-)self-dual connections, stability of vector bundles, Donaldson invariants, the Seiberg-Witten equations and invariants, the Bogomolnyi (monopole) equation, Chern-Simons invariant, Donaldson-Thomas theory, relations to Gromov-Witten theory, applications to low-dimensional topology.

233 questions
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Intuition for Exotic $\mathbb R^4$'s

Today one of my professors told me that $\mathbb R^4$ admits uncountably many non-diffeomorphic differential structures. When I asked him whether there's an intuitive reason to expect a result like that he said it's just hard $4$-dimensional…
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Texts on Principal Bundles, Characteristic Classes, Intro to 4-manifolds / Gauge Theory

I am looking for a textbook that might serve as an introduction to principal bundles, curvature forms and characteristic classes, and perhaps towards 4-manifolds and gauge theory. Currently, the only books I know of in this regard are: "From…
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2 answers

Non-ellipticity of Yang-Mills equations

Let $D=\text{d}+A$ be a metric connection on a vector bundle with curvature $F=F_D$. How does one prove that the Yang-Mills equations $$ \frac{\partial}{\partial x^i}F_{ij}+[A_i,F_{ij}]=0 $$ from classical Yang-Mills theory are not elliptic? In…
14
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Good textbook or lecture notes on Seiberg-Witten theory.

I am looking for a good introductory book for Seiberg-Witten theory. The only textbook I have now is Morgan's "The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds". This book is self-contained and concise and…
M. K.
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Elliptic Bootstrapping for Gauge Transformations

I'm reading John Morgan's book "Seiberg-Witten equations and applications to the topology of smooth manifolds". I'm stuck in the proof of Lemma 4.5.3, which says the following. Suppose $(a_n,\psi_n)$ and $(b_n,\mu_n)$ are sequences of $W^{2,2}$…
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What does the symbol $\operatorname{Tr}$ in the Yang-Mills action mean?

I find that many authors write the Yang-Mills action as follows: $$\mathcal{J}= \int \operatorname{Tr}(F \wedge \star F).$$ I have yet to find a formal description of the symbol $\operatorname{Tr}$ and am especially interested in its relation to…
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Group actions and associated bundles

Let $P$ be a principal $G$-bundle over $B$, and let $G$ act on some space $F$ (feel free to work in your favorite category of spaces, if this helps). Then $\text{Aut}{P}$ (aka the group of gauge transformations of $P$) acts on the space of sections…
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Computing curvature of the quotient of the tautological connection

I am trying to understand a certain passage in the book "Geometry of Four-Manifolds" by Donaldson and Kronheimer (specifically, a computation in section 5.2.3). I am confused on the proof of Proposition 5.2.17, which states: Proposition (5.2.17)…
Andrew
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10
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What is the mathematical understanding behind what physicists call a gauge fixing?

I'm learning fiber bundle from my poor physicist point of view. I understand that a gauge transformation (physicist language) corresponds to the transformation of the connections built from an overlapping patch of the base space of the bundle. Said…
9
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Trace of tensor product vs Tensor contraction

I have come across various sources that talk about traces of tensors. How does that work? In particular, there seem to be such an equality: $$ \text{Tr}(T_1\otimes T_2)=\text{Tr}(T_1)\text{Tr}(T_2)\;\;\;...(1) $$ as found in this Stackexchange…
Argyll
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9
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Curvature of canonical connection on 4-manifold with self-dual harmonic 2 form

Let $X$ be an compact oriented Riemannian 4-manifold with $b_2^+\geq1$. Let $\omega$ be a self-dual harmonic two form vanishing transversely. On $X\setminus \omega^{-1}(0)$, the spinor bundle $S_+$ can be written as $E\oplus K^{-1}E$, where $E$ is…
9
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4 answers

Gentle introduction to fibre bundles and gauge connections

To better understand papers like this for example, which makes heavy use of fibre bundles and gauge connections to represent gauge fields, I am looking for a nice introduction to this topic. The only thing I have read so far is the corresponding…
8
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The classifying space of a gauge group

Let $G$ be a Lie group and $P \to M$ a principal $G$-bundle over a closed Riemann surface. The gauge group $\mathcal{G}$ is defined by $$\mathcal{G}=\lbrace f : P \to G \mid f(p \cdot g) = g^{-1}f(p)g\ (\forall g \in G, p \in P) \rbrace.$$ I want to…
7
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Holonomy and Differential Characters

This question is going to be rather vague, but I'm just trying to see if there are obvious connections between these two concepts. So the holonomy of a vector bundle with Lie group $G$ is $$h(A)=\mathcal{P}\exp\left(\int_\gamma A\right)$$ where…
levitopher
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7
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Formal Definition of Yang Mills Lagrangian

I have a question regarding the Lagrangian in non abelian gauge theory. Say, $G$ is the gauge group and $\mathfrak g$ the associated Lie algebra. The Lagrangian is often written as $$ \mathcal L=-\frac {1}{4} \text{tr} (F_{\mu \nu} F^{\mu \nu}) +…
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