• #PDF
• [[#Integrability and the meaning of \(U\) ---]]
• #Wilson loops and Wilson lines ---
• #Yang--Mills theory -- the equations of motion ---
• #The dual field strength and the Bianchi identity ---
• #Euclidean solutions of the YM action ---
• #The Bogomolny bound and (anti) self dual solutions ---
• #The topological term as a total derivative ---
• #Behaviour at Euclidean infinity ---
• #Homotopy classes and winding number ---

Prof. Sachindeo Vaidya (CHEP, IISc) | PDF

Previous: Lecture 18 | Next: Lecture 20

Given \(U\), the equation \(D_\mu\psi = 0\) has the solution

\[ \psi \;=\; U\psi_0 \ ,\qquad \psi_0 = \text{const. vector.} \]

\(U\) tells us how the frame rotates as we move from \(x\to y\) along \(C\).

\[ D_\mu U \;=\; 0 \qquad\Longleftrightarrow\qquad F_{\mu\nu} \;=\; 0 \]

(the condition of integrability, i.e. the answer does not depend on the path).

Indeed, if \(D_\mu U = 0\) then also \(D_\nu D_\mu U = 0\), so

\[ \left(D_\mu D_\nu - D_\nu D_\mu\right)U \;=\; 0 \qquad\Longrightarrow\qquad \left[D_\mu,D_\nu\right]U \;=\; 0 \qquad\Longrightarrow\qquad F_{\mu\nu}U \;=\; 0 \]

i.e. \(F_{\mu\nu} = 0\).

If \(F_{\mu\nu} = 0\), then we can write

\[ A_\mu \;=\; -\left(\partial_\mu U\right)U^{-1} \]

If \(F_{\mu\nu}\neq 0\), then for a small closed loop we have

\[ U \;=\; P\exp\oint_{C}A_\mu\,dx^\mu \;\simeq\; 1 - F_{\mu\nu}\,\sigma^{\mu\nu} \]

where \(\sigma^{\mu\nu}\) is the area enclosed by \(C\).

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\[ P\exp\oint_{C}A_\mu\,dx^\mu \;\equiv\; W(C) \qquad\longrightarrow\qquad \text{Wilson loop.} \]

\(W\) is a non-local object, as it is labelled by the curve \(C\).

Under gauge transformations,

\[ A \;\longrightarrow\; gAg^{-1} + \left(\partial g\right)g^{-1} \ ,\qquad W(C) \;\longrightarrow\; g\,W(C)\,g^{-1} \]

More generally,

\[ U\!\left(x,y;C,A^{g}\right) \;=\; g(x)\,U(x,y;C;A)\,g^{-1}(y) \qquad\longrightarrow\qquad \text{Wilson line.} \]

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\[ S_{YM} \;=\; \frac{-1}{2g^2}\int d^4x\ \mathrm{Tr}\left(F_{\mu\nu}F^{\mu\nu}\right) \]

EOM: variation of \(S\), \(\delta S = 0\), i.e.

\[ \delta\left[\frac{-1}{g^2}\int d^4x\ \mathrm{Tr}\left(F_{\mu\nu}F^{\mu\nu}\right)\right] \;=\; 0 \]

Using

\[ \delta F^{\mu\nu} \;=\; \partial^\mu\delta A^\nu + i\,\delta A^\mu A^\nu + i\,A^\mu\delta A^\nu - \left(\partial^\nu\delta A^\mu + i\,\delta A^\nu A^\mu + i\,A^\nu\delta A^\mu\right) \]

we get

\[ \delta S \;=\; \frac{-2}{g^2}\int d^4x\ \mathrm{Tr}\left( F_{\mu\nu}\left[\partial^\mu\delta A^\nu + i\,\delta A^\mu A^\nu + i A^\mu\delta A^\nu - \left(\mu\leftrightarrow\nu\right)\right]\right) \]

Integrate by parts and throw away the surface term:

\[ \delta S \;=\; \frac{2}{g^2}\int d^4x\ \mathrm{Tr}\left[ \left(\partial^\mu F_{\mu\nu} + i\left[A^\mu,F_{\mu\nu}\right]\right)\delta A^\nu\right] \]

so that

\[ \text{EOM:}\qquad D^\mu F_{\mu\nu} \;=\; 0 \qquad(\text{source free}). \]

This is \underline{non-linear} in \(A_\mu\) -- it is linear only in electrodynamics, where superposition holds.

Look at the Hodge dual of \(F_{\mu\nu}\):

\[ \tilde{F}_{\mu\nu} \;=\; \epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma} \]

Then

\[ D^\mu\tilde{F}_{\mu\nu} \;=\; 0 \qquad\longrightarrow\qquad \text{Bianchi identity.} \]

This is \underline{not} an equation of motion, but an identity; it follows from

\[ \left[D_\mu,\left[D_\rho,D_\sigma\right]\right] + \text{cyclic permutations} \;=\; 0 \]

Non-abelian gauge theories are harder because we do not have linear equations of motion: we
have lost superposition (a linear combination of solutions is not a solution), since
\(D^\mu F_{\mu\nu} = 0\) is non-linear in \(A_\mu\).

These have interesting properties and have deep implications for the non-perturbative properties
of the quantum theory. Go from Minkowski to Euclidean space by

\[ t \;=\; -i\tau \]

Identity:

\[ \mathrm{Tr}\left[\left(F_{\mu\nu}\pm\tilde{F}_{\mu\nu}\right)\left(F_{\mu\nu}\pm\tilde{F}_{\mu\nu}\right)\right] \;\geq\; 0 \] \[ \mathrm{Tr}\left[F_{\mu\nu}F_{\mu\nu} \pm F_{\mu\nu}\tilde{F}_{\mu\nu} \pm \tilde{F}_{\mu\nu}F_{\mu\nu} + \tilde{F}_{\mu\nu}\tilde{F}_{\mu\nu}\right] \;\geq\; 0 \] \[ \mathrm{Tr}\left[F_{\mu\nu}F_{\mu\nu} + \tilde{F}_{\mu\nu}\tilde{F}_{\mu\nu}\right] \;\geq\; \mp\,2\,\mathrm{Tr}\left[F_{\mu\nu}\tilde{F}_{\mu\nu}\right] \]

Using

\[ \mathrm{Tr}\left(\tilde{F}_{\mu\nu}\tilde{F}_{\mu\nu}\right) \;=\; \mathrm{Tr}\left(\tfrac{1}{2}\epsilon_{\mu\nu\rho_1\sigma_1}F^{\rho_1\sigma_1} \cdot\tfrac{1}{2}\epsilon_{\mu\nu\rho_2\sigma_2}F^{\rho_2\sigma_2}\right) \;=\; \mathrm{Tr}\left(F_{\mu\nu}F_{\mu\nu}\right) \]

we obtain

\[ \mathrm{Tr}\left(F_{\mu\nu}F_{\mu\nu}\right) \;\geq\; \mp\,\mathrm{Tr}\left(F_{\mu\nu}\tilde{F}_{\mu\nu}\right) \]

The identity is \underline{saturated} when \(F_{\mu\nu} = \pm\tilde{F}_{\mu\nu}\), i.e.

• \(F_{\mu\nu} = +\tilde{F}_{\mu\nu}\) --- self dual solution
• \(F_{\mu\nu} = -\tilde{F}_{\mu\nu}\) --- anti self dual solution

\(\int d^4x\,\mathrm{Tr}\left(F_{\mu\nu}\tilde{F}_{\mu\nu}\right)\) is a total derivative:

\[ \epsilon^{\mu\nu\rho\sigma}\,\mathrm{Tr}\left(F_{\mu\nu}F_{\rho\sigma}\right) \;=\; 4\,\partial_\varsigma W^{\varsigma} \]

where

\[ W^{\varsigma} \;=\; \epsilon^{\varsigma\rho\mu\nu}\,\mathrm{Tr} \left(A_\rho\partial_\mu A_\nu + \frac{2i}{3}A_\rho A_\mu A_\nu\right) \]

So

\[ \int d^4x\ \mathrm{Tr}\left(F_{\mu\nu}\tilde{F}_{\mu\nu}\right) \;=\; 2\int d^4x\ \partial_\mu W_\mu \]

and therefore

\[ S^{YM}_{E} \;=\; \frac{1}{2g^2}\int d^4x\ \mathrm{Tr}\left(F_{\mu\nu}F_{\mu\nu}\right) \;\geq\; \frac{2}{g^2}\left|\oint_{S^3}d^3\sigma_\mu\,W_\mu\right| \]

The minimum value of the action depends on the properties of \(W_\mu\) (the gauge field) at \(S^3\)
(at \(\infty\)) --- Euclidean infinity.

Say \(F_{\mu\nu}\to 0\) as \(|x|\to\infty\). Then

\[ A_\mu \;\longrightarrow\; i\,U\partial_\mu U^{\dagger} \qquad(\text{pure gauge}) \quad\text{for } |x|\to\infty \]

\(i\partial_\mu U\,U^{\dagger}\) is gauge related to zero. Look at the \(W_\mu\)'s for such \(A_\mu\)'s:

\[ W_\mu \;=\; \frac{1}{3}\epsilon_{\mu\nu\rho\sigma}\,\mathrm{Tr} \left(\left(U\partial_\nu U^{\dagger}\right)\left(U\partial_\rho U^{\dagger}\right)\left(U\partial_\sigma U^{\dagger}\right)\right) \] \[ S^{YM}_{E} \;\geq\; \frac{2}{3g^2}\oint_{S^{3}_{\infty}} \epsilon_{\mu\nu\rho\sigma}\,\mathrm{Tr} \left(U\partial_\nu U^{\dagger}\,U\partial_\rho U^{\dagger}\,U\partial_\sigma U^{\dagger}\right) \]

Case of \(SU(2)\): \(U\) depends upon 3 ``angles'' \(\varphi_1,\varphi_2,\varphi_3\), and

\[ U = U(x) \qquad\Longleftrightarrow\qquad \varphi_i = \varphi_i(x) \]

\(U(x)\) (for \(|x|\to\infty\)) is a map from \(S^{3}_{\infty}\to SU(2)\).

Such mappings are characterised by equivalence classes, called \underline{homotopy classes}.
Roughly, the idea of the classes is that they correspond to the ``wrapping'' number of
\(S^{3}_{\infty}\) onto \(G\ (= SU(2))\). Note that

\[ U(1)\sim\text{circle} \ ,\qquad SU(2)\sim S^3 \ ,\qquad SU(3)\sim S^5\otimes S^3 \]

Homotopy: ``class 1'' means that \(S^{3}_{\infty}\) is mapped exactly once onto the sphere \(S^3\),
for maps \(\left(S^{3}_{\infty}\to SU(2)\right)\).

Let us look at (based) maps \(S^{1}_{\infty}\to U(1)\), with \(U(1) = \left\{e^{i\alpha}\right\}\).
Say \(f: S^1 \to U(1)\).

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\[ f:\ \varphi \;\longrightarrow\; \alpha \ ,\qquad\text{i.e.}\quad \alpha = \varphi \]

This is called a \underline{winding number 1} map.

If \(\alpha = 2\varphi\), then one rotation in \(S^1\) is two rotations in \(\alpha\), so such a map
is called winding number 2. Similarly \(\alpha = -\varphi\) is a winding number \(-1\) map.

All those maps which do not wind fully make up the class with winding number \(0\).