Questions tagged [gauge-theory]
Gauge theory in physics and mathematics refers to a field theory whose fields include principal bundles with connection.
157
questions
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63 views
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 ...
6
votes
0answers
100 views
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 ...
6
votes
1answer
136 views
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, ...
3
votes
1answer
167 views
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 $[X,B'G]$...
5
votes
0answers
170 views
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" ...
2
votes
1answer
117 views
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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vote
0answers
78 views
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 ...
2
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0answers
32 views
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 ...
4
votes
0answers
148 views
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 ...
1
vote
0answers
56 views
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 ...
5
votes
0answers
72 views
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 ...
6
votes
2answers
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 ...
4
votes
3answers
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 ...
3
votes
0answers
101 views
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 ...
4
votes
1answer
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 ...
4
votes
0answers
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 ...
2
votes
1answer
124 views
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 ...
6
votes
1answer
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 ...
4
votes
1answer
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{...
3
votes
2answers
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 ...
3
votes
0answers
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 ...
4
votes
0answers
80 views
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 ...
5
votes
1answer
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 ...
1
vote
1answer
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 ...
8
votes
1answer
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 $\mathrm{PU}...
9
votes
2answers
332 views
Curvature of tautological connections over the space of connections
My question is about computing the curvature of a quotient connection, specifically for the case of the quotient of the tautological connection of a universal bundle on the moduli space of connections....
4
votes
0answers
114 views
3d Chern-Simons TQFT of gauge group (E8)$_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 ...
10
votes
1answer
534 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 ...
1
vote
1answer
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?...
2
votes
0answers
111 views
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 ...
6
votes
0answers
182 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 ...
5
votes
0answers
68 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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vote
0answers
98 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 ...
1
vote
1answer
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 ...
7
votes
0answers
212 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
...
1
vote
0answers
35 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 $...
3
votes
2answers
454 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}}...
2
votes
1answer
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 ...
6
votes
0answers
201 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 $\...
12
votes
1answer
441 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 ...
13
votes
3answers
2k 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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0answers
424 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 ...
13
votes
0answers
274 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 ...
4
votes
0answers
101 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 ...
8
votes
1answer
209 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 $\...
4
votes
1answer
286 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
votes
1answer
224 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 ...
6
votes
0answers
153 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 $P : X \to \mathrm{B}G$, ...
12
votes
1answer
220 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^+...
4
votes
0answers
111 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 ...
