nLab
Yang-Mills instanton

Idea

In $\mathrm{SU}(n)$-Yang-Mills theory an instanton is a field configuration with non-vanishing second Chern class that minimizes the Yang-Mills energy.

Definition

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 $G=\mathrm{SU}(n)$ the special unitary group, there is canonically an associated complex vector bundle $E=P{\times }_G{ℂ}^n$.

Write $F_{\nabla }\in {\Omega }^2(X,\mathrm{End}(E))$ for the curvature 2-form of $\nabla$.

One says that $\nabla$ is an instanton configuration if $F_{\nabla }$ is Hodge-self dual

$$\star F_{\nabla }=-F_{\nabla },$$\star F_\nabla = - F_\nabla \,,

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

$$c_2(E)={\int }_X\mathrm{Tr}(F_{\nabla }\wedge F_{\nabla })=k\in H^4(X,\mathbb{Z})$$c_2(E) = \int_X Tr(F_\nabla \wedge F_\nabla) = k \in H^4(X, \mathbb{Z})

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.

Properties

As gradient flows between flat connections.

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

$$X=\Sigma \times ℝ$$X = \Sigma \times \mathbb{R}

of $\Sigma$ with the real line be equipped with the product metric of $g$ with the canonical metric on $ℝ$.

Consider field configurations $\nabla$ of Yang-Mills theory over $\Sigma \times ℝ$ with finite Yang-Mills action

$$S_{\mathrm{YM}}(\nabla )={\int }_{\Sigma \times ℝ}{F}_{\nabla }\wedge \star F_{\nabla }<\mathrm{\infty }.$$S_{YM}(\nabla) = \int_{\Sigma \times \mathbb{R}} F_\nabla \wedge \star F_\nabla \,\,\lt \infty \,.

These must be such that there is $t_1<t_2\in ℝ$ such that $F_{\nabla }(t<t_1)=0$ and $F_{\nabla }(T>t_2)=0$, hence these must be solutions interpolating between two flat connections ${\nabla }_{t_1}$ and ${\nabla }_{t_2}$.

For $A\in {\Omega }^1(U\times ℝ,𝔤)$ 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

$$t\mapsto A(t)\in {\Omega }^1(\Sigma ,𝔤)$$t \mapsto A(t) \in \Omega^1(\Sigma, \mathfrak{g})

of connections on $\Sigma$. Then the self-duality condition on a Yang-Mills instanton

$$F_{\nabla }=-\star F_{\nabla }$$F_\nabla = - \star F_\nabla

reads equivalently

$$\frac{d}{dt}A=-{\star }_g{F}_A\in {\Omega }^1(\Sigma ,𝔤).$$\frac{d}{d t} A = -\star_{g} F_A \,\,\, \in \Omega^1(\Sigma, \mathfrak{g}) \,.

Definition

On the linear configuration space ${\Omega }^1(\Sigma ,𝔤)$ of Lie algebra valued forms on $\Sigma$ define the Hodge inner product metric

$$G(\alpha ,\beta ):={\int }_{\Sigma }⟨\alpha \wedge {\star }_g\beta ⟩,$$G(\alpha, \beta) := \int_{\Sigma} \langle \alpha \wedge \star_g \beta \rangle \,,

where $⟨-,-⟩:𝔤\otimes 𝔤\to ℝ$ is the Killing form invariant polynomial on the Lie algebra $𝔤$.

Proposition

The instanton equation

$$\frac{d}{dt}A=-{\star }_g{F}_A$$\frac{d}{d t} A = -\star_{g} F_A

is the equation characterizing trajectories of the gradient flow of the Chern-Simons action functional

$$S_{\mathrm{CS}}:{\Omega }^1(\Sigma ,𝔤)\to ℝ$$S_{CS} : \Omega^1(\Sigma, \mathfrak{g}) \to \mathbb{R}

$$A\mapsto {\int }_{\Sigma }\mathrm{CS}(A)$$A \mapsto \int_\Sigma CS(A)

with respect to the Hodge inner product metric on ${\Omega }^1(\Sigma ,𝔤)$.

Proof

The variation of the Chern-Simons action is

$$\delta S_{\mathrm{CS}}(A)={\int }_{\Sigma }⟨\delta A\wedge F_A⟩$$\delta S_{CS}(A) = \int_\Sigma \langle \delta A \wedge F_A\rangle

(see Chern-Simons theory for details).

In other words, we have the 1-form on ${\Omega }^1(\Sigma ,𝔤)$:

$$\delta S_{\mathrm{CS}}(-{)}_A={\int }_{\Sigma }⟨-\wedge F_A⟩.$$\delta S_{CS}(-)_A = \int_\Sigma \langle - \wedge F_A \rangle \,.

The corresponding gradient vector field

$$\nabla S_{\mathrm{CS}}:=G^{-1}\delta S_{\mathrm{CS}}$$\nabla S_{CS} := G^{-1} \delta S_{CS}

is uniquely defined by the equation

$$\begin{array}{rl}\delta S_{\mathrm{CS}}(-)& =G(-,\nabla S_{\mathrm{CS}})\\ {\int }_{\Sigma }⟨-,\star \nabla S_{\mathrm{CS}}⟩\end{array}.$$\begin{aligned} \delta S_{CS}(-) & = G(-,\nabla S_{CS}) \\ \int_\Sigma \langle - , \star \nabla S_{CS}\rangle \end{aligned} \,.

With the formula (see Hodge star operator)

$$\star \star A=(-1{)}^{1(3+1)}A=A$$\star \star A = (-1)^{1(3+1)} A = A

we find therefore

$$\nabla S_{\mathrm{CS}}={\star }_g{F}_A.$$\nabla S_{CS} = \star_g F_A \,.

Hence the gradient flow equation

$$\frac{d}{dt}A+\nabla S_{\mathrm{CS}}{}_A=0$$\frac{d}{d t} A + \nabla S_{CS}_A = 0

is indeed

$$\frac{d}{dt}A=-{\star }_g{F}_A.$$\frac{d}{d t} A = - \star_g F_A \,.

Since flat connections are the critical loci of $S_{\mathrm{CS}}$ this says that a finite-action Yang-Mills instanton on $\Sigma \times ℝ$ 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”.

Examples

• In $\mathrm{SU}(2)$-YM theory: see BPTS instanton .

• In $\mathrm{SU}(3)$-YM theory, QCD/strong nuclear force: see instanton in QCD

Related concepts

• instanton Floer homology

• self-dual higher gauge field

References

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

• Marcus Hutter, Instantons in QCD: Theory and Application of the Instanton Liquid Model (arXiv:hep-ph/0107098)

The multi-instantons in $\mathrm{SU}(2)$-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

• M. F. Atiyah, R. S. Ward, Instantons and algebraic geometry, Comm. Math. Phys. 55, n. 2 (1977), 117-124, MR0494098, euclid

The more general ADHM construction in terms of linear algebra of vector bundles on projective varieties is proposed in

• M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld, Yu.I. Manin, Construction of instantons, Physics Letters 65 A, 3, 185–187 (1978) pdf

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

In

• Henrique N. Sá Earp, Instantons on $G_2$−manifolds PhD thesis (2009) (pdf)

is a discussion of Yang-Mills instantons on a 7-dimensional manifold with special holonomy.

• Michael Atiyah, R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523–615, MR85k:14006, doi.