# 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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### Langlands dual group in math vs. Goddard-Nyuts-Olive dual group in physics

Given a group $G$, there is a so-called Langlands dual group $G^{∨}$.
Given a group $G$, there is also a so-called Goddard-Nyuts-Olive dual group $G^{'}$ that relates to the magnetic charge.
Are ...

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### Spectral flow of Dirac operator twisted by instanton

Suppose $E$ is a $SU(2)$-bundle over a closed three manifold $M$ and $S$ is the spinor bundle over $M$. Also assume $D_{A(t)}:\Gamma(S\otimes_{\mathbb c} E)\to \Gamma(S\otimes_{\mathbb c} E)$ is a ...

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### Cobordism monopole Floer homology

From the famous book: Monopole and three manifold, Kronheimer and Mrowka(https://www.maths.ed.ac.uk/~v1ranick/papers/kronmrowka.pdf). It is known that:
Let $Y$ be a closed oriented $3$ manifold, ...

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### What can be an appropriate notion of principal bundle over a category (with an appropriate notion of local trivialisation)?

Motivation for my question:
It is a well-known fact that there exists a bijection between the set of isomorphism class of
principal $G$ bundles over a nice topological space $X$ and the set $[X,B'G]$...

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### Derivative of the Bott-Chern forms

The Bott-Chern forms are constructed formally in Bismut's "Analytic Torsion and Holomorphic Determinant Bundle I" (page 74). This construction can be found as well in "Lectures on Arakelov Geometry" ...

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### Canonical connection on $\mathcal{A}\times X$

Let $E\rightarrow X$ be a vector bundle and let $\mathcal{A}$ denote the space of connections on $E$. Pulling back $E$ by the second projection we obtain a vector bundle $\mathbb{E}=p_2^*E\rightarrow ...

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### Does holonomy determine parallel transport? [duplicate]

Let $p: P \longrightarrow M$ be a smooth $G$-principal bundle endowed with a connection that determines the holonomies:
$$\Phi_{\gamma}: P_{x} \overset{\cong}{\longrightarrow} P_{x}$$
for any fiber ...

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### Solving equations of motion of holomorphic BF theory - pure gauge in complex coordinates

In this paper by Bailieu and Tanzini, aspects of holomorphic BF theory are presented.
Holomorphic BF theory on a four dimensional Kahler manifold is discussed from page 5, and on page 8 the ...

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### Weak 2-groups and non-abelian gerbe over a manifold

In literature, a strict 2-group is defined as a group object in the category of categories or groupoids. It has many well known equivalent descriptions viz:
1. A strict monoidal category in which all ...

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### Poincaré connection encode torsion and curvature

I'm trying to understand something that is written in Baez & Wise paper "Teleparallel Gravity as a Higher Gauge Theory". In section 4, they discuss Poincaré connection and, first of all, split ...

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### Uniqueness of Witten-Dijkgraaf 2D TQFT at 0th dimension

If I understand correctly, Dijkgraaf-Witten TQFT in dimension 2 is the
following. Fix any finite group $G$, we define a field over a closed
2-manifold to be a principle $G$ bundle (it's automatically ...

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408 views

### 1-dimensional pure gauge theory

I am learning TQFT from compact Lie groups by Freed, Hopkins, Lurie,
and Teleman: https://arxiv.org/abs/0905.0731 , and got stuck very hard
even in the first section ($n = 1$), which was "trivial but ...

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264 views

### Electromagnetism as a $U(1)$-gauge theory

I would like to learn gauge theory, starting from the simplest case. I have heard that I should start with electromagnetism, which is just the $U(1)$-gauge theory. All the references I know are ...

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### Inverse semigroups and partial symmetries

I recently ran across the idea of inverse semi-groups in the context of partial symmetries, where the symmetry only acts on part of the system and not the entire system (e.g., in quasi-crystals).
My ...

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**1**answer

149 views

### Self-dual differential on $4$-manifold with boundary

Let $(M,g)$ be an oriented compact Riemannian $4$-manifold with boundary $\partial M$.
Let $a\in \Omega^1$ such that $*a\big|_{\partial M}=0$, i.e. $a(\nu)=0$ on $\partial M$, where $\nu$ denotes the ...

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163 views

### Does Dijkgraaf-Witten theory have a time-reversal symmetry?

By having a time-reversal symmetry I mean that there is a local anti-unitary symmetry (representing the non-trivial element of $Z_2$) of the state-sum construction (or, if you want, of the associated ...

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### Construction of $G$-invariant map between manifolds

Let $M,N$ be two closed differential manifolds and let $G$ be a compact Lie group. Assume that $G$ acts on both manifolds $M,N$ nicely (i.e. free and proper so that $M/G$ and $N/G$ have the structure ...

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161 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 ...

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**1**answer

161 views

### Orientability of moduli space and determinant bundle of ASD operator

Setting
In instanton gauge theory, given a $G$-principal bundle $P\to X^4$, the orientability of the moduli space of ASD connections
$$\mathcal{M}_k = \{A \in L^{2}_{k}(X, \Lambda^1 \otimes\mathrm{...

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285 views

### When is the action of the gauge group on the space of connections free?

Let $G$ be a compact Lie group. Let $\mathcal{A}$ be the space of connections on the principal trivial $G$-bundle $G\times \mathbb{R}^4$ possibly with some growth condition (to specify it is a part of ...

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159 views

### Gauge structure of teleparallel gravity

I am interested in references that treat teleparallel gravity in a mathematically rigorous manner, especially in regards to it being a "gauge theory of the translation group".
The standard reference ...

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### Airy stress, Beltrami stress and gauge fields

The following problem comes from the theory of elasticity, but reduces to a pure geometric problem. Consider a $d$-dimensional Riemannian manifold $(M,g)$ with boundary representing the intrinsic ...

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276 views

### Gauge invariance of Chern-Simons functional integral for a 3-manifold with boundary

I am trying to understand how the functional integral for Chern-Simons theory for a possibly non-compact 3-manifold with boundary is made gauge invariant.
For a compact 3-manifold, $M$, without ...

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**1**answer

282 views

### Reference request: Gauge theory [closed]

What are some good introductory texts to gauge theory? I have some basic differential geometry knowledge, but I don’t know any algebraic geometry.
Also, as a side question, what intuitively is a ...

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274 views

### On the classification of $\mathrm{SU}(mn)/\mathbb{Z}_n$ principal bundles over 4-complexes

In
The Classification of Principal PU(n)-bundles Over a 4-complex, J. London Math. Soc. 2nd ser. 25 (1982) 513–524, doi:10.1112/jlms/s2-25.3.513
Woodward proposed a classification of $\mathrm{PU}...

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332 views

### Curvature of tautological connections over the space of connections

My question is about computing the curvature of a quotient connection, specifically for the case of the quotient of the tautological connection of a universal bundle on the moduli space of connections....

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### 3d Chern-Simons TQFT of gauge group (E8)$_1$ = SO(16)$_1 \otimes$ a trivial spin TQFT = Cartan E$_8$ matrix

In this post, we like to relate the following 3 bosonic TQFTs that can be defined on generic non-spin manifold $M^3$.
Given a non-abelian Chern-Simons (CS) TQFT of a gauge group $G$ and the $k$ named ...

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### Gauge theory on schemes

Gauge theories are traditionally formulated in space-time, superspace, manifolds, or supermanifolds.
Are there formulations of gauge theory in the context of algebraic geometry (e.g. defining fields ...

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186 views

### Is conjugation by gauge transformation of $G$-bundle contained in $\mathfrak{g}$?

Before painstakingly defining all these terms, let me ask my question in plain english: given a $G$-bundle, is every conjugate of a vector field by a gauge transformation an element of the Lie algebra?...

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### Reference Request: A “Chevalley-Eilenberg”-style formulation of the $L_\infty$ algebra minimal model theorem?

The nicest definition of $L_\infty$-algebras ---which I will call a "Chevalley-Eilenberg" style definition after the obvious analogy with the Chevalley-Eilenberg differential of Lie algebras--- is the ...

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### Infinite-dimensional manifold $ΩG$ of loops and the complex projective $n$-space

In a review seminar today, I heard the speaker takes the below for granted, but I have no enough background
"Yang-Mills (YM) instantons in 4D can be naturally identified with (i.e. have the same ...

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### Does $\Lambda^2_{+}$ generate a differential ideal for a self-dual $4$-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-...

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### Determine all possible magnetic monopole of gauge theories

In Wikipedia, it states about the magnetic monopole of the gauge theory is determined by the fact:
This argument for monopoles is a restatement of the lasso argument for a pure U(1) theory. It ...

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184 views

### compact manifold as a hyperkahler quotient of an infinite-dimensional affine space

Is it possible to obtain K3 (or any other compact
hyperkahler manifold) with its hyperkahler structure
as a hyperkahler quotient of an infinite-dimensional affine
quaternionic vector space with an ...

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### $U(1)$ v.s. $SU(N)$ v.s. $SO(N)$ instantons

I am interested in knowing the details of the comparison between $U(1)$, $SU(N)$ and $SO(N)$ instantons for their gauge theories in 4 spacetime dimensions., in terms of:
Chern class (1st, 2nd), and
...

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### Fourier Lapalacian over periodic end

This is a technical question on Taubes' paper: Gauge theory over periodic end. on Page 378.
Recall that:
Let $Y$ be a closed manifold, with $b_1=1$, and $\tilde Y$ be the $\mathbb Z$-covering of $...

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### $spin_{\mathbb{C}}$ Connection and Charge Parity

From the paper "Gapped Boundary Phases of Topological Insulators via Weak Coupling" on page 11,
https://arxiv.org/abs/1602.04251
the authors states that on a curved manifold with a $spin_{\mathbb{C}}...

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111 views

### Extend a gauge transformation

Suppose $M$ is a smooth manifold and $P$ is a principal bundle on $M$. Let $U^\prime\Subset U\Subset M$ be strictly contained precompact open subsets. Let $g\in C^\infty(U, \hbox{Ad}P|_U)$ be a ...

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### Correspondence between $O(n)$-instantons on $S^4$ and holomorphic bundles over $\mathbb{P}^3$

Theorem 2.9 on page 49 of Atiyah's Geometry of Yang-Mills Fields states that there is a natural correspondence between $U(n)$-instantons on $S^4$ and holomorphic vector bundles of rank $n$ over $\...

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### SO(3) monopole Floer homology

From what I understand about work on the Witten conjecture relating Donaldson and Seiberg-Witten invariants, the main strategy has been to relate them with the use of the "SO(3) monopole" theory ...

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### What is an “Instanton” in classical gauge theory? (to a mathematician)

There's already a question about the same topic but I think its aim is different.
Classical (non-quantum) gauge theory is a completely rigorous mathematical theory. It can be phrased in completely ...

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### Hartshorne's Conjectures about Algebraic Bundles?

In 1978 Hartshorne published a list of 26 open problems about algebraic bundles on projective spaces [Hart], proceeding from an Oxford conference organized by Atiyah.
I understand that many of these ...

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### Finite dimensional approximation of Donaldson theory

In addition to the Seiberg-Witten invariant there has been further success with "finite dimensional approximations" of the Seiberg-Witten theory: Bauer-Furuta's stable (co)homotopy invariants, and ...

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### The topology of subgroups of gauge groups

I am reading Atiyah and Bott's classic paper "Yang-Mills Equations over Riemann surfaces" and struggling with proposition 2.16 (p. 542)
Let $P$ be a principal $U(n)$-bundle over a compact Riemann ...

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### Deformation-Obstruction Theory of YM Instantons

In Donaldson-Kronhiemer Section 4.2.5. (local models of the moduli space of YM instantons) they first get local models of the moduli space $M$ inside the space of all connections modulo gauge $\...

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### Gauge group of tangent bundle and diffeomorphism group

I'm not exactly a differential geometer, so I hope this isn't too elementary a question.
From a naive point of view, it seems as if there are two natural group actions on the space of connections on ...

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### Extension problem for Seiberg-Witten solutions

Let $X$ be a compact $4$-manifold, possibly with boundary.
Theorem 17.1.2 of Kronheimer-Mrowka's book "Monopoles and Three-Manifolds" states
Let $X' \subset X$ be a codimension-zero submanifold ...

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### Are 2d gauge anomalies determined by genus-one data?

Let $G$ be a (finite, say) group and $\alpha \in \mathrm{H}^3(\mathrm{B}G; \mathrm{U}(1))$ a 3-cohomology class. For each oriented 3-manifold $X^3$ equipped with a $G$-bundle $P : X \to \mathrm{B}G$, ...

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### Example of ''annihilation'' of Seiberg-Witten Equation solutions

The proof that the Seiberg-Witten invariants of a 4-manifold $X$ with fixed Spin$^c$ structure really are invariant wrt the metric used to define them goes roughly as follows (for simplicity let $b_2^+...

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### tertiary characteristic class: integration of the Chern-Simons form

Let $P \to M$ be a trivial principal circle bundle with connection $A$ over a closed 3-manifold $M$. The Chern-Simons 3-form of the connection is defined by $\mathrm{CS}(A) = A \wedge dA$. Suppose ...