# 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 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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### In search of a combinatorial proof on particular set of partitions

Given a partition $\lambda=(\lambda_1\geq\lambda_2\geq\dots)$, denote the conjugate partition by $\lambda'=(\lambda_1'\geq\lambda_2'\geq\dots)$. For example, if $\lambda=(4,2,2)$ then $\lambda'=(3,3,1,...

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### Different definitions of "charged spinors": "bundle splicing" vs. "twisted spinor bundles"

Currently I study the mathematical formulation of the (classical) standard model of particle physics using the language of gauge theory and spin geometry. One of the central objects in the standard ...

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### Non existence of a preferred Horizontal subspace on a bundle. Why not ? (Basics) [closed]

If I choose a principal bundle, let us say $G\rightarrow P \rightarrow B$, with $G=U(1)$, $P=S^1 \times S^1$ and $B=S^1$. Can I choose to put my finger on the identity element of the group over a ...

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### Vortex equation on Riemann surface and a similar equation

Let's take a Riemann surface $(X,\omega)$ and a holomorphic line bundle $L$ on it with a hermitian metric $h$ on $L$. $g$ be a real valued smooth function on $X$ and we consider the following two sets ...

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### Understanding the slice theorem

I was reading Morgan's book: "The Seiberg-Witten equations and applications to the topology of smooth four-manifolds" and find it hard to understand the slice theorem (page 62-64).
Here are ...

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### Curvature of the line bundle $\mathcal{O}(2)$ on the twistor space

Let $M^4$ be a closed Riemannian manifold and $Z:=S\big(\Lambda^2_+(M)\big)$ denote the twistor space of $M,$ i.e., the sphere bundle of the self-dual 2-forms on $M$. Now at a point $(m,J)\in Z$ the ...

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### How much do characteristic classes fail to characterize bundles?

Given a group $G$, let $E \to B$ be a principal $G$-bundle. It is
well-known that when $B$ is a nice enough topological space (e.g.
CW-complex), such a thing corresponds to a connected component of
$...

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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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### Moduli spaces of rank 2 stable bundles over curves as projective varieties

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}_{\...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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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### About irreducible connection

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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### Spin networks as functionals on the moduli space of connections modulo gauge transformations on a graph

I have just read a big part of John Baez's nice article Spin network states in Gauge theory. The definitions are quite clear in that article. However, there is a part which is not explained explicitly ...

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### Is there a contact instanton connection on the tangent bundle of the 5-sphere?

A well-known example of a contact manifold is $S^5$, arising from it being a circle bundle over $\mathbb{CP^2}$. This is somewhat related to the reduction of the structure group from $SO(5)$ to $SU(2)$...

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### Instanton numbers for diverse gauge bundles on diverse manifolds --- their relations to characteristic classes

It is standard (?) that the $SU(N)$ gauge theory has the instanton number $n$ quantized as $n \in \mathbb{Z}$
$$
n = { 1 \over 8\pi^2} \int_{\mathcal{M}_{4}} \text{tr} \left(F \wedge F\right) = {1 \...

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### Geometry of the complex Gauge group

Let $E\rightarrow X$ be holomorphic vector bundle on a complex manifold $X$. Denote by $\mathcal{G}=\Gamma(Aut(E))$ the group of complex smooth automorphisms of $E$.
Is there a way to endow $\...

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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...