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In $SU(n)$-Yang-Mills theory an instanton is a field configuration with non-vanishing second Chern class that minimizes the Yang-Mills energy.
Let $(X,g)$ be a compact Riemannian manifold of dimension 4. Let $G$ be a compact Lie group.
A field configuration of $G$-Yang-Mills theory on $(X,g)$ is a $G$-principal bundle $P \to X$ with connection $\nabla$.
For $G = SU(n)$ the special unitary group, there is canonically an associated complex vector bundle $E = P \times_G \mathbb{C}^n$.
Write $F_\nabla \in \Omega^2(X,End(E))$ for the curvature 2-form of $\nabla$.
One says that $\nabla$ is an instanton configuration if $F_\nabla$ is Hodge-self dual
where $\star : \Omega^k(X) \to \Omega^{4-k}(X)$ is the Hodge star operator induced by the Riemannian metric $g$.
The second Chern class of $P$, which by the Chern-Weil homomorphism is given by
is called the instanton number or the instanton sector of $\nabla$.
Notice that therefore any connection, even if not self-dual, is in some instanton sector, as its underlying bundle has some second Chern class, meaning that it can be obtained from shifting a self-dual connection. The self-dual connections are a convenient choice of “base point” in each instanton sector.
We discuss how Yang-Mills instantons may be understood as trajectories of the gradient flow of the Chern-Simons theory action functional.
Let $(\Sigma,g_\Sigma)$ be a compact 3-dimensional Riemannian manifold .
Let the cartesian product
of $\Sigma$ with the real line be equipped with the product metric of $g$ with the canonical metric on $\mathbb{R}$.
Consider field configurations $\nabla$ of Yang-Mills theory over $\Sigma \times \mathbb{R}$ with finite Yang-Mills action
These must be such that there is $t_1 \lt t_2 \in \mathbb{R}$ such that $F_\nabla(t \lt t_1) = 0$ and $F_\nabla(T \gt t_2) = 0$, hence these must be solutions interpolating between two flat connections $\nabla_{t_1}$ and $\nabla_{t_2}$.
For $A \in \Omega^1(U\times \mathbb{R}, \mathfrak{g})$ the Lie algebra valued 1-form corresponding to $\nabla$, we can always find a gauge transformation such that $A_{\partial_t} = 0$ (“temporal gauge”). In this gauge we may hence equivalently think of $A$ as a 1-parameter family
of connections on $\Sigma$. Then the self-duality condition on a Yang-Mills instanton
reads equivalently
On the linear configuration space $\Omega^1(\Sigma, \mathfrak{g})$ of Lie algebra valued forms on $\Sigma$ define the Hodge inner product metric
where $\langle-,-\rangle : \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{R}$ is the Killing form invariant polynomial on the Lie algebra $\mathfrak{g}$.
The instanton equation
is the equation characterizing trajectories of the gradient flow of the Chern-Simons action functional
with respect to the Hodge inner product metric on $\Omega^1(\Sigma,\mathfrak{g})$.
The variation of the Chern-Simons action is
(see Chern-Simons theory for details).
In other words, we have the 1-form on $\Omega^1(\Sigma,\mathfrak{g})$:
The corresponding gradient vector field
is uniquely defined by the equation
With the formula (see Hodge star operator)
we find therefore
Hence the gradient flow equation
is indeed
Since flat connections are the critical loci of $S_{CS}$ this says that a finite-action Yang-Mills instanton on $\Sigma \times \mathbb{R}$ is a gradient flow trajectory between two Chern-Simons theory vacua .
Often this is interpreted as saying that “a Yang-Mills instanton describes the tunneling? between two Chern-Simons theory vacua”.
In $SU(2)$-YM theory: see BPTS instanton .
In $SU(3)$-YM theory, QCD/strong nuclear force: see instanton in QCD
contact instanton?
Introductions and surveys include
J. Zinn-Justin, The principles of instanton calculus, Les Houches (1984)
M.A. Shifman et al., ABC of instantons, Fortschr.Phys. 32,11 (1984) 585
For a fairly comprehensive list of literature see the bibliography of
The multi-instantons in $SU(2)$-Yang-Mills theory (BPTS instantons) were discovered in
A. A. Belavin, A.M. Polyakov, A.S. Schwartz, Yu.S. Tyupkin, Pseudoparticle solutions of the Yang-Mills equations, Phys. Lett. B 59 (1), 85-87 (1975) doi
A. A. Belavin, V.A. Fateev, A.S. Schwarz, Yu.S. Tyupkin, Quantum fluctuations of multi-instanton solutions, Phys. Lett. B 83 (3-4), 317-320 (1979) doi
See also
Michael Atiyah, Nigel Hitchin, J. M. Singer, Deformations of instantons, Proc. Nat. Acad. Sci. U.S. 74, 2662 (1977)
Edward Witten, Some comments on the recent twistor space constructions, Complex manifold techniques in theoretical physics (Proc. Workshop, Lawrence, Kan., 1978), pp. 207–218, Res. Notes in Math., 32, Pitman, Boston, Mass.-London, 1979.
Methods of algebraic geometry were introduced in
The more general ADHM construction in terms of linear algebra of vector bundles on projective varieties is proposed in
Monographs with the standard material include
Dan Freed, Karen Uhlenbeck, Instantons and four-manifolds, Springer-Verlag, (1991)
Robbert Dijkgraaf, Topological gauge theories and group cohomology (ps)
Nicholas Manton, Paul M. Sutcliffe, Topological solitons, Cambridge Monographs on Math. Physics, gBooks
Yang-Mills instantons on spaces other than just spheres are explicitly discussed in
based on
Chris Isham Gabor Kunstatter, Phys. Letts. v.102B, p.417, 1981. (doi)
Chris Isham Gabor Kunstatter, J. Math. Phys. v.23, p.1668, 1982. (doi)
In
is a discussion of Yang-Mills instantons on a 7-dimensional manifold with special holonomy.