# 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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29 views

### Relation between Poisson equation and Wilson lattice gauge invariance theory

I've recently started writing a library of numerical solvers for elliptic partial differential equations, with particular focus on the Poisson equation. If one considers typical Poisson equation in ...
23 views

### The map $C^\infty(M,U(A_F))\to C^\infty(M,U(A_F)/\mathfrak{H}(F))$ is an homomorphism.

I am reading "Noncommutative Geometry and Particle Physics" by van Suijlekom. I have problems to identify one map as homomorphism. Let $M\times F$ be an almost-commutative manifold. The ...
26 views

### The Grassmann connection is a connection

Let $\mathcal{A}$ be a *-algebra and $p\in M_N(\mathcal{A})$ an orthogonal projection. I need to show that $\nabla=p\circ d$ defines a connection on $\mathcal{E}=p\mathcal{A}^N$, where $d$ is acting ...
237 views

### Intuition behind connection 1-forms and Ehresmann connections

I am learning about mathematical gauge theory and so far I have been able to develop an intuition behind all the objects I've read about such as principal bundles, associated bundles, and vertical/...
85 views

### Why does the adjoint representation appear in the definition of a connection 1-form?

When defining the a connection 1-form $A$ on a principal $G$-bundle we require that $$r^*_gA = Ad_{g^{-1}} \circ A, \quad \forall g \in G.$$ Here $r_g^*$ is the pullback of right multiplication by $g$,...
70 views

### Invariant differential forms?

In the first page of the paper "On the spaces of maps inducing isomorphic connections" by T.R. Ramadas, one can read that the automorphisms of a connection $\nabla$ on a principal $G$-bundle ...
69 views

### Trivial principal bundles and curvature.

Let $\mathcal{M}$ be a smooth manifold, $G$ a Lie group with Lie algebra $\mathfrak{g}$ and $\mathcal{P}\xrightarrow{\pi}\mathcal{M}$ a principal bundle. If $A\in\Omega^{1}(\mathcal{P},\mathfrak{g})$ ...
1 vote
67 views

### The set of all Yang-Mills connections is infinite dimensional

I am reading Jost's Riemannian Geometry and Geometric Analysis (7-Ed) and having a question about the Yang-Mills functional (page 182). Let $M$ be a compact manifold and $E$ be a metric bundle where ...
81 views

### Is gauge group infinite dimensional Lie group?

Let $P$ be a smooth principal bundle on a manifold $M$ with a structure Lie group $G$. Then we define the gauge group $\mathcal{G}$ by the automorphism group of $P$, that is, the group of ...
1 vote
73 views

### How do gauge transformations reflect on the associated bundles?

The way we define an associated bundle to a principle bundle is by constructing the trivial bundle over the principle bundle with the desired fibre and then quotienting out by the structure group (...
1 vote
61 views

### Lagrangian field theories define a canonical $\mathbb R$-torsor/$\mathbb R$-bundle

Why is it possible to define an $\mathbb R$-torsor/$\mathbb R$-bundle out of the Lagrangian and to construct a projective limit? (I'm guessing this has to do with time evolution acting on the space of ...
27 views

### Residual gauge freedom and complete residual gauge fixing in lorenz gauge

I posted this question on physics stack exchange but nobody answers it so hope to get an answer on math stack exchange, What I understand after reading all answers from physics stack exchange related ...
61 views

### Cohomology of mixed degree differential forms

$\newcommand{\d}{\mathrm{d}}\newcommand{\H}{\mathrm{H}}$Consider the space, $\Omega^n(M)$, of differential $n$-forms on a smooth, torsion-free manifold, $M$, without boundary, of dimension $D$, and ...
26 views

### When does the Cartan formalism have a gauge that makes the tetrad constant?

I hope I am using the terms correctly: by Cartan formalism I am thinking of a general (orthonormal) tetrad, defined with respect to a coordinate system $\mathbb{R}^n$, along with the spin connection. ...
1 vote
56 views

### Existence and uniqueness of a twisted one form which satisfies some inner product property

Let $P\rightarrow M$ be a principal bundle over a (pseudo)-Riemannian manifold $(M,g)$, with compact structure group $G$ ($G$ is assumed to be a real Lie group), and a connection one form $A$. Let $W$ ...
1 vote
102 views

### Deriving Contracted Bianchi Identity from Einstein-Hilbert Action

It is well known and often argued that the contracted Bianchi identity (and vanishing divergence of the stress-energy tensor) of General Relativity can be seen as a consequence of the theory's (and ...
106 views

1 vote
55 views

### A determinant invariant with respect to a change of metric (covariant determinant?)

I use a multi-vector of G(2,R): $$\mathbf{u} = a+x\sigma_x+y \sigma_y + b \sigma_x\sigma_y$$ its matrix representation is $$\mathbf{u} \cong \pmatrix{a +x & y-b \\y+b & a-x}$$ and the ...
61 views

### Einstein field equation from covariant derivative

I am trying to derive the Einstein field equations from this approach: Let $\psi(t) = e^{-tM}$, where $M$ is a $4\times4$ matrix. Then $$\frac{d}{dt}\psi(t) = -M\psi(t)$$ Let us now suppose that I ...
63 views

### Existence of symplectic vortices

Consider for a closed Riemann surface $(\Sigma,j_{\Sigma})$, a compact Lie group $G$ and $P$ a principal $G$ bundle over $\Sigma$ and a connection 1-form $A$ on the principal bundle $P$ and $u:P \to M$...
71 views

### Constant $SU(2)$ ASD connections on $\mathbb R^4$ are Flat

Let $A \in \mathfrak{su}(2) \otimes \mathbb R^4$, so $A$ is a collection of $4$ elements of $\mathfrak {su}(2), (A_0, \dots, A_3)$. We can consider the system of equations  [A_0, A_1] + [A_2, A_3] = ...
194 views

### Torsion Free Spin Connection

Ok I am not exactly sure how much of this common notation/terminology, and how much is unique to the book I'm reading, so bear with me for a moment here. First we have a vector bundle $E$ associated ...
In Seiberg-Witten theory, the group of gauge transformations is $Map(M,S^1)$. For a configuration $(A,\psi)$, where $A$ is a unitary connection on the determinant line bundle, $\psi$ is a spinor, and $... 1 vote 0 answers 35 views ### Topological equivalent vs Holomorphic equivalent of line bundle. I know there are indeed some examples show that two line bundles over some projective manifold is topological equivalent but not holomorphic equivalent, but I find I give a "proof" shows ... 4 votes 0 answers 117 views ### How to set up Chern-Simons theory, compute its "topological gauge shift" term, and compute its quantum path integrals with Wilson lines? Suppose we want to consider Chern-Simons theory on an (odd-dimensional) compact boundaryless smooth manifold$X$for a Lie group (the "structure/gauge group")$G$. Is it possible to do so ... 1 vote 1 answer 82 views ### Gauge equivalence of Lie-valued forms on the base space of a principal bundle Given a principal$G$-bundle$P\xrightarrow{\pi} M$: Assuming the bundle is globally trivial, we define two Lie$G$-valued 1-forms$A_1,A_2$on$M$to be gauge-equivalent if there is a principal bundle ... 2 votes 0 answers 107 views ### What is the "gauge field" on the base space in a gauge field theory? Suppose we have a principal$G$-bundle$P\xrightarrow{\pi} M$, and we want to consider a classical gauge field theory (with a field Lagrangian) on$M$for this bundle, in the physics sense. In the ... 0 votes 0 answers 61 views ### Moduli Space of Flat connection over Homology 3-Shpere I'm trying to understand the space of flat connections over the trivial$SU(2)$-bundle of a closed, oriented homology three-sphere (for the purpose of understanding the instanton Floer homology of it)... 1 vote 0 answers 63 views ### Standard metric on adjoint bundle Consider we have a principal$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}\$ ... 