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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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348 views

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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1answer
127 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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166 views

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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62 views

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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94 views

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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1answer
125 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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164 views

$U(1)$ 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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34 views

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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2answers
196 views

$spin_{\mathbb{C}}$ 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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1answer
94 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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180 views

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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1answer
347 views

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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3answers
1k views

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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349 views

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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226 views

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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97 views

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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1answer
178 views

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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1answer
226 views

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 ...
7
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1answer
202 views

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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147 views

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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1answer
203 views

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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90 views

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 ...
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1answer
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Todd genus of symplectic $4$-manifolds a smooth invariant?

Suppose that $(M_{1},\omega_{1})$ and $(M_{2},\omega_{2})$ are compact symplectic $4$-manifolds, that are (oriented) diffeomorphic. Is it true that the Todd genus ($\frac{1}{12} (c_{1}^{2} + c_{2})(M_{...
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2answers
416 views

Elliptic operator becomes Fredholm

Let $X$ be a Riemannian $n$-manifold with tubular end $\mathbb R^+\times Y$, where $Y$ is a closed $n-1$-manifold. Suppose $L:L^{p,w}_2(X)\to L^{p,w}(X)$ is the Laplacian operator which is ...
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3answers
1k views

About Simon Donaldson's book on four dimensional manifold

Recently I'm reading Donaldson's Geometry of four manifolds. It seems to me that the book requires a lot for background. Additionally, the proof in the book is too sketchy without too much detail. I ...
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0answers
274 views

A Question about Hermitian Yang-Mills Equations

Let E be a holomorphic bundle over algebra surface X, let $H$ be a Hermitian metric of $E$, recall the Hermitian-Yang-mills equation is $\wedge F_H=\lambda.1$. Let $H_t$ be Hermitian metrics over $E$ ...
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2answers
422 views

Is the space of connections modulo gauge equivalence paracompact?

I find this question interesting, but need to get it out of my system: is the space of connections (modulo gauge) on a compact four-manifold paracompact, in the Sobolev topology? If so, I believe it ...
2
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0answers
97 views

On nowhere zero self-dual 2-forms

Let $(X, g_x)$ be a smooth, oriented, Riemannian 4-diemnsional manifold. Let $\Lambda^2$ denote the bundle of 2-forms over $X$. Then the Hodge-star decomposes $\Lambda^2$ into the space of self-dual ...
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166 views

Connected components of gauge groups of principal bundles over generalized flag manifolds

Let $G$ be a compact connected Lie group and $P$ a principal $G$-bundle over a finite CW complex $X$. The gauge group $\mathcal{G}(P)$ is defined to be the group of principal bundle automorphisms of $...
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1answer
202 views

Killing fields for Yang-Mills

Physicists frequently talk about symmetries of a theory, and them being generated by Killing vectors. While this is clear to me in the context of gravity, where a Killing field $\xi$ is defined by $\...
10
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1answer
376 views

Monopole Floer Homology vs. Heegaard-Floer theory

I have a (possibly very naive) question: what is the relation between Monopole Floer Homology and Heegaard-Floer theory? (both known and conjectured) Is there some version of Atiyah-Floer conjecture ...
10
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1answer
499 views

Every 4-manifold has a $Spin^c$ Structure

I'm having trouble understanding the proof given in Morgan's The Seiberg-Witten Equations that every 4-manifold $X$ admits a $Spin^c$ structure (Lemma 3.1.2). One can easily see from the exact ...
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2answers
295 views

Homology Sphere Embedding into $\mathbb R^4$

Let $Y$ be an oriented closed $3$-manifold, with trivial homology group, i.e. integer homological sphere. Q: If $Y$ can be embedded into $\mathbb R^4$, is there any example, that such a $Y$ admits a ...
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1answer
468 views

Gottsche Nakajima Yoshioka define a weird slant product

In their article Instanton counting and Donaldson invariants the authors define the slant product for $\beta \in H_i(X)$ (where $X$ is a manifold) as following. Let $P \to X$ and SO(3) bundle and $M(...
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1answer
206 views

Computing the Cartan1-form for $Sp(2)$

Context: I must find the Cartan 1-form for $Sp(2)$ before I start dealing with the natural connection of the Hopf fibration $S^3 \hookrightarrow S^7 \overset{\mathcal P}\to S^4$. To do so, the idea ...
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0answers
160 views

Analog of Gauss-Bonnet formula for principal bundles over manifolds with boundary

The Gauss-Bonnet formula gives a topological invariant as an integral over a local density on the given manifold. In particular, when there is a boundary, GB formula has to be supplemented by a ...
4
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1answer
177 views

partitions into odd parts vs hooks and symplectic contents

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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1answer
269 views

Is a closed basic 2-form on a principal $S^1$ bundle the curvature of a connection?

Suppose one has an $S^1$ principal bundle $p: P\rightarrow M$, and a closed 2-form $F$ on $M$. Then the pullback form $p^*F$ is closed, vanishes on vertical vectors, and is invariant under the action ...
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1answer
164 views

Transformation between two conventions of Hitchin equation

Recall that for a given Riemann surface $\Sigma$ Hitchin's self-duality equation consists of a complex rank $r$ vector bundle $E$ (with degree 0 for simplicity), a connection $d_A: \Omega^k(\Sigma, E) ...
3
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1answer
241 views

About the Weitzenböck Formula for $SL(2,\mathbb{C})$ connection

Suppose $M$ is a compact four manifold and $P$ is an $SU(2)$ bundle, let $\mathfrak{g}$ be the adjoint bundle of $P$, given a connection $A$ on this bundle. Given $\phi\in \Omega^1(\mathbb{g})$, we ...
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0answers
866 views

Linking formulas by Euler, Pólya, Nekrasov-Okounkov

Consider the formal product $$F(t,x,z):=\prod_{j=0}^{\infty}(1-tx^j)^{z-1}.$$ (a) If $z=2$ then on the one hand we get Euler's $$F(t,x,2)=\sum_{n\geq0}\frac{(-1)^nx^{\binom{n}2}}{(x;x)_n}t^n,$$ on the ...
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208 views

Exact triangle for monopole Floer homology with $\mathbb{Z}$-coefficient

Let $Y$ be oriented 3 manifold with torus boundary and let $\gamma_{j}$ (j=0,1,2) be three curves on its boundary with $\#(\gamma_{j}\cap \gamma_{j+1})=-1$. We denote by $Y_{j}$ the manifold obtained ...
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1answer
196 views

Elementary question: Curvature change under Complexified Gauge Transformation

Forgive me for this elementary question. Let $E$ be a holomorphic vector bundle over a Riemann surface $M$ equipped with a Hermitian metric. Let $\nabla$ be the compatible connection on $E$ amd $g$ ...
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1answer
236 views

Geometric Construct for Integrating Symmetric Tensors?

I'm interested in finding the appropriate geometric construct for the integration of symmetric tensors, analogous to the way differential forms can be integrated over manifolds. The motivation comes ...
8
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2answers
401 views

Flat connections on 3-manifold with boundary

Suppose $Y$ is a 3-manifold and the boundary $\Sigma:=\partial Y$ is non-empty. Let $G$ be a Lie group with trivial center. Let $\overline {\mathcal A}_{flat}(\Sigma)$ and $\overline {\mathcal A}_{...
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1answer
202 views

Homotopy Type of the Based Mapping Space $Map_*^{(k,l)}(\mathbb{C}P^2,BU(2))$

Path components of the based mapping space $Map_*(\mathbb{C}P^2,BU(2))$ are indexed by a pair of integers $(k,l)$ determined by the values of the first two Chern classes that a map $f:\mathbb{C}P^2\...
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230 views

Deformation of the covariant Laplacian

Let $M$ be a Riemann surface and $P \to M$ a principal $G$-bundle (with compact structure group $G$). Fix a connection $A$ in $P$ and consider a nearby connection $B$, which is in Coulomb gauge ...
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233 views

Is central extension of a group equivalent to a bundle with gauge field?

Let $\tilde G$ be a central extension of a group $G$ by $U(1)$. One common elegant definition is that there should be a short exact sequence of groups: $0 \to U(1) \to \tilde G \to G \to 0$ However,...
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1answer
100 views

A question on 2-bundles

In this paper, the authors +John Baez and +Urs Schreiber defined (page 15) "transition functions" for a special kind of 2-bundles (those whose the base space is a ordinary smooth space augmented to a ...