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.

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What is the "gauge field" on the base space in a gauge field theory?

Suppose we have a principal $G$-bundle $P\xrightarrow{\pi} M$, and we want to consider a classical gauge field theory (with a field Lagrangian) on $M$ for this bundle, in the physics sense. In the ...
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Moduli Space of Flat connection over Homology 3-Shpere

I'm trying to understand the space of flat connections over the trivial $SU(2)$ -bundle of a closed, oriented homology three-sphere (for the purpose of understanding the instanton Floer homology of it)...
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Complex gauge group

We define the action of complex gauge group on a connection $d_A$ satisfies the condition of curvature: $F_A\in \Omega^{1,1}$ as follows: Suppose $d_A=\partial_A+\bar{\partial}_A$, for any $g \in \...
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Standard metric on adjoint bundle

Consider we have a principal $G$-bundle $P$ over a closed manifold $V$. Denote $\mathfrak{g}_P$ by the associated bundle $P\times_G \mathfrak{g}$ where $G$ acts by adjoint action. Denote $\mathscr{G}$ ...
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Symplectic Reduction of 3-D Chern Simons Theory

So, I'm new to gauge theories and symplectic reduction and was trying to analyze the Chern Simons theory in three dimensions. I have a few questions regarding the steps towards reduction. First off, ...
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What is / why the connection one-form from a physics point of view?

Take the Yang-Mills gauge theory for example. Gauge field $A$ is the pullback of the connection one-form to the base manifold. Other concepts of gauge theory also find their definition in fiber ...
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determinant line bundle of ASD moduli space

For a principal bundle $P$ and the space of irreducible connection $\mathcal{A}^*$. We consider the configuration space $\mathcal{B}^*=\mathcal{A}^*/\mathcal{G}$. The book Instantons and Four-...
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Defining Sobolev space of Connections on Determinant Line Bundle

Background: If we have a vector bundle $S^+(\tilde{P})$ over a Riemannian manifold $M$, then we have a metric $g$ and Levi-Civita connection $\nabla$ on $M$, and we can choose a metric $h$ and a ...
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Orientation of the Space of Self-Dual Forms

Let $X$ be a closed, orientable, smooth 4-manifold. Let us give an orientation to $X$ by setting $e^1, e^2, e^3, e^4$ to be an oriented local orthonormal basis for its cotangent bundle. Does this ...
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"Covariant" hodge star operator

I've some question about not(at)ions i've found on this article https://arxiv.org/pdf/hep-th/0403048.pdf Let $(M,g)$ be a $n$-dimensional ortientable compact riemannian manifold with boundary $\Gamma$....
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Does $(S^1\times S^3)\#(S^1\times S^3)\#(S^2\times S^2)$ admit a symplectic form?

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 $\omega$ is a symplectic form, then the real cohomology class $[\omega]$...
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Derivation of Yang-Mills functional

I was trying to calculate the critical points of Yang-Mills functional. And I failed to show that $F_{A+ta}=F_{A}+t\nabla_Aa+t^2a\wedge a$. Here is my attempt: Suppose all calculation is in a local ...
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Confusion regarding definition of gauge transformation

Let $E \to M$ be a principal $G$-bundle. The gauge group is the group of $G$-bundle automorphisms of $E$. A connection on $E$ can be thought of as a global $g$-valued 1-form on $E$ where $g$ = Lie$(G)$...
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Are the Cartan structure equations invariant under gauge transformations?

(I'm a trying to build a specific gauge connection for a physical theory, but it turns out I need to use a kind of spherical basis for the Lie algebra, and it's confusing me. I'm trained in general ...
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D-brane gauge theories

Reading Yang-Hui He's review on quiver gauge theories, I read that D-brane gauge theories manifest as a natural description of symplectic quotients and their resolutions in geometric invariant theory....
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What is the correspondence between gauge field terminology and bundle terminology in electromagnetism?

In electromagnetism, the electromagnetic field tensor can be expressed as $$F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu.$$ If we let $A= A_\mu dx^\mu$, since $F= \frac{1}{2} F_{\mu \nu} dx^\...
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Non existence of a preferred Horizontal subspace on a bundle. Why not ? (Basics)

If I choose a principal bundle, let us say $G\rightarrow P \rightarrow B$, with $G=U(1)$, $P=T^2$ (2-torus) and $B=S^1$. Can I choose to put my finger on the identity element of the group over a point ...
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Why is $F_{\mu \nu}$ fundamentally geometric in nature?

I am studying gauge theory, and we derive the Lagrangian for electrodynamics by wanting the Lagrangian to be gauge invariant under U(1) symmetry group. That is, invariant under the phase rotation, $$\...
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Associate bundles, equivariant sections and tangent elements

Consider a principal $G$-bundle over a manifold $X$, and consider yet another manifold $B$ endowed with a $G$-action. Everything is assumed to be smooth. The associated bundle $$ \mathcal{B}=P\times_G ...
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calculate the dimension of moduli space

I'm learning gauge theoretic topics about 4-manifolds, and I get stuck when I try to calculate the dimension of ASD moduli space. For an oriented closed 4-manifold $M$, we first fix a riemannian ...
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local model of gauge theoretic moduli space

I'm a beginner of gauge theory and I find that most materials state the theorem(without proof) that: Given a principal bundle $P$, for a connection $A$ over $P$, a isotopy group $G_A$ of $A$ consist ...
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Principal connections on $P$ and covariant derivatives on associated vector bundle $E=P\times_\rho V$

I would like to have a concrete proof or reference to the following fact: Let $P\rightarrow M$ be a principal $G$-bundle over an $n$-dimensional manifold $M$, and let $E=P\times_\rho V$ be an ...
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What does it mean for a gauge field to have no curvature?

The electromagnetic gauge field is $A + d\theta$, where $\theta \colon \mathbb{R}^n \to \mathbb{R}$ comes from a gauge function, $e^{i\theta(\vec x)} $. Let's set $A=0$. The curvature form is $0$ ...
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Moduli space of connection on line bundle

I want to show that for a $U(1)$ bundle $P$ over a connected smooth 4-manifold $X$, the moduli space of Yang-Mills connection over $P$ is the torus $H^1(X,\mathbb{R})/H^1(X,\mathbb{Z})$. Now I reduce ...
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stabilizer of connection on SU(2) bundle

Suppose $P$ a principal bundle over connected manifold $B$ with correspondent Lie group $G=SU(2)$, and $A$ a connection on $P$. We say a map $\sigma \in Aut(P)$ a stabilizer of $A$ if $\sigma^*A=A$....
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differentiate gauge transformation

Suppose $A_0$ a connection on principal bundle $P$, pick $\xi$ an elements in $\Omega^1(adP)$ and $e^{t\xi} \in \Omega^0(AdP)$ a one-parameter group generated by $\xi$. (edited:Suppose $G$ the ...
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Understanding gauge transformations and relation to Lie Groups?

I am starting to study gauge theory. I have a background in basic group theory, multivariable calculus, and the idea of symmetry in relation to group theory. I am trying to understand why the ...
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On the definition of the transformation of the connection $1-$form.

Introduction I'm studying principal fiber bundles to deal with the gauge sector of standard model lagrangian. Now, references $[1]$ and $[2]$, gives us a definition on how the connection $1$-form ...
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$k=1$ instantons on $S^4$

In the book Instantons and four-manifolds written by Uhlenbeck and Freedman, they say that: We identify $\mathbb{R}^8$ as $\mathbb{H}^2$ with the standard inner product of $\mathbb{R}^8$, suppose $(...
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Question about gauge transformation

I'm studying something fundamental about gauge theory and I find that many materials state(without proof) that: for a principal bundle $P$ with correspondent connection $\omega$ and correspondent Lie ...
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Problem about multiplication of functions in Sobolev spaces and regularity related to gauge theory

Let $X$ be a closed Riemannian 4-manifold. Let $f\in L^2_k(X)$ (Sobolev space of maps $X\to \mathbb{C}$ of regularity $k$), with $k>2$. Suppose also that $f|_{\mathcal O} \not\equiv 0$ for any ...
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From Monad construction of Instantons to ADHM data

In reference to Atiyah's book - "Geometry of Yang-Mills Fields" (1979). In chapter 5, section 3, he describes how the monad construction for $Sp(n)$ potentials can be interpreted in terms of ...
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Is the Poincaré gauge theory a real gauge theory in the mathematical sense?

First, I want to say that I posted this in the physics forum but no one seems to be interested to respond so because I really believe that my question can be figured out by mathematicians Am here. ...
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2 votes
1 answer
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Transition of Z2 gauge theory in four dimensions

Are there rigorous results (apart from the self-duality of the system and its self-dual transition point) concerning the character of the transition of the Ising lattice gauge theory in four ...
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2 votes
1 answer
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What does covariantly constant mean in the context of $G$-equivariant fiber bundle diffeomorphism?

Suppose $M$ be a base (smooth) manifold and $P$ be a Principal $G$-bundle (with $\rho$ as a right $G$-action on it.). Let $\varphi : P \to P$ be a fiber bundle diffeomorphism which is $G$-equivariant, ...
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Relating the Chern classes of vector bundles with structure groups $G_1$, $G_2$, and $G_1\times G_2$

Let $E$ be a complex vector bundle whose structure group $G$ is a product: $G=G_1\times G_2$. If the fiber of $E$ is $n$-dimensional, then are the Chern classes of $E$ related to the Chern classes of ...
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2 answers
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Gauge covariant derivative of an adjoint action: $\psi(x) \to g \psi(x) g^{-1}$, instead of a left action $\psi(x)\to e^{iq\theta(x)} \psi(x)$

In the case where the transformation on $\psi$ is applied from the left: $$ \psi(x)\to e^{-iq\theta(x)}\psi(x). $$ The gauge covariant derivative is $$ D_\mu = \partial_\mu - iqA_\mu \tag{1} $$ and ...
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Definition of connection 1-forms - when we write $A(X^\#)=X$, is $A(X^\#)$ meant to be constant?

Let $A$ be a connection $1$-form on a principal $G$-bundle and $X\in\mathfrak{g}$. By the definition of connection $1$-forms, $A(X^\#)=X$. However, $A(X^\#)\in C^\infty(P,\mathfrak{g})$, so the ...
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Definition of pull-back in gauge transformation

In an online lecture given by a famous physicist Youshi Wu, he says: Consider a map $g:\quad M\rightarrow G:x\rightarrow g(x)\in G$ ($G$ is a Lie Group, $M$ can be physical space or spacetime), we ...
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Problem about the notation $\text{Spin}^c(V)\cong \text{Spin}(V)\times _{\{\pm1\}} S^1$

Lemma 2.6.1 of Morgan's book on Seiberg-Witten equations states that the group $\text{Spin}^c(V)$ is isomorphic to the group $\text{Spin}(V)\times _{\{\pm1\}} S^1$. The proof actually shows that $\...
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Gauging the GL(4,R) group

I am told from numerous sources that general relativity can be understood as the product of gauging the GL(4,R) group. However, no one seems able to provide me with a source, or a derivation. It is ...
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5 votes
2 answers
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Local coordinate expression for the equations of motion in gauge theory

Let's assume $P$ is a principal bundle, $F^A \in \Omega^2(M,Ad(P))$ the curvature 2-form, $Ad(P)$ the adjoint bundle. $d_A$ the covariant differential. For sections in the associated bundle $E=P \...
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Resources for understanding connection between gauge theory and differential cohomology?

As a physicist, I first learned about gauge theories as just theories with some local symmetry. In the interest of furthering my understanding, I've read sections from Nakahara's "Geometry, ...
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2 votes
2 answers
120 views

Textbook recommendations for the differential geometry of Yang-Mills fields

I was wondering if anyone could recommend text books or papers that could help me really understand the math behind Yang-Mills fields? Thanks!
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1 answer
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Why is the group of gauge transformations $\mathcal{G}$ on the frame bundle isomorphic to $\text{Diff}(M)$?

Let $LM \to M$ be the frame bundle on pseudo-Riemannian manifold $M$ and suppose that there exists a Lorentzian metric tensor $g \in \Gamma \big(T^0_2(M) \big)$ on $M$. Since the frame bundle is a ...
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$SU(2)$ bundles over a flat Euclidean $\mathbb{R}^d$ or Minkowski $\mathbb{R}^{d,1}$ space? [duplicate]

I know that we can have a $U(1)$ bundle over the base manifold like $S^2$. For example, we can choose the fibre $U(1)=S^1$ over $S^2$ as the $S^3$ Hopf fibration. I know that we can have a $SU(2)$ ...
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2 votes
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$U(1)$ bundles over a flat Euclidean $\mathbb{R}^d$ or Minkowski $\mathbb{R}^{d,1}$ space [duplicate]

I know that we can have a $U(1)$ bundle over the base manifold like $S^2$. For example, we can choose the fibre $U(1)=S^1$ over $S^2$ as the $S^3$ Hopf fibration. Now, can we have any nontrivial $U(1)$...
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1 vote
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$\operatorname{spin}^c$ structure through the Pauli matrices

Let $M$ be a n-dimensional compact oriented smooth manifold. As far as I know, a $\operatorname{spin}^c$ structure on $M$ (or $TM$) is either a principal $\operatorname{spin}^c(n)$-bundle $P$ such ...
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What do double angle brackets mean? E.g., $\langle \langle \exp (-G(A))\rangle \rangle$.

I came across this in an academic paper on the mass gap, and it had no explanation of the double angle bracket notation: $$\langle \langle \exp (-G(A))\rangle \rangle = (2\pi/e)^{-|\Lambda|}\int_C \...
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1 answer
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Gauge theory and well-posedness of a PDE

I am trying to understand the relationship between gauge theories (for example General Relativity) and the well-posedness of the underlying theory. In General Relativity, it is known that you must be ...
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