Abstract
We present the geometric model of pure gauge theory, including the basics concerning the structure of the classical configuration space, and derive the Yang-Mills equation from the classical action functional. Next, we study the theory of instantons in a systematic way: we present the BPST-instanton family and the famous ADHM-construction, including a partial proof that this construction yields all instanton solutions. Moreover, we study the instanton moduli space. Finally, we present the classical stability analysis of the Yang-Mills equation as developed by Bourguignon and Lawson and discuss non-minimal solutions.
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Notes
- 1.
Note that such an inner product always exists if G is compact. In that case, it may be obtained from any auxiliary inner product by averaging over the group with respect to the Haar measure. In many applications, the gauge group G is compact and semisimple. Then, for \(\langle \cdot , \cdot \rangle _{\mathfrak g}\) one can choose the negative of the Killing form k. Compactness implies that \(-k\) is positive-definite, cf. Sect. 5.5 of Volume I.
- 2.
We must restrict ourselves to square integrable forms. In particular, we may consider forms with compact support.
- 3.
For basics of the theory of Sobolev spaces, we refer to Sect. 5.7.
- 4.
- 5.
This is proved by the same arguments as in the proof of Proposition I/6.3.4.
- 6.
We have used the notation of Volume I here.
- 7.
The first group of Maxwell equations is of purely geometric character. It says that the 2-form f is closed. In terms of connection theory, this equation clearly coincides with the Bianchi identity.
- 8.
The factor \(\frac{1}{2}\) is chosen according to the conventions used in physics.
- 9.
Since we use the language of quaternions, we consistently write \(\mathrm{Sp}(1)\). Recall that \(\mathrm{Sp}(1) \cong \mathrm{SU}(2)\) as real Lie groups.
- 10.
Since this expression is given in terms of the associative multiplication in \(\mathbb {H}\), in the sequel it will be worthwhile to work with the associative exterior calculus, cf. Remark 1.4.8 /1.
- 11.
Choosing, instead, the transition mapping \(\rho _{n, s}\) results in a change of sign of these mapping degrees.
- 12.
Then, under the identification \(\mathrm S^4 = \mathbb {R}^4 \cup \{\infty \}\), infinity corresponds to the south pole \(-{\mathbf {e}}_0\).
- 13.
See [130].
- 14.
- 15.
Cf. Example 1.3.19 for further details.
- 16.
That is, the transition functions may be chosen to be rational functions of the complex coordinates.
- 17.
This is the terminology of complex geometry. Instead, we could call \(\sigma \) a quaternionic structure in that case.
- 18.
Cf. Remark 1.1.21.
- 19.
Clearly, it may also be identified with the positive projective spinor bundle.
- 20.
If we adopt this point of view, Theorem 4.1 of [37] cited in Remark 5.5.8 guarantees the integrability of the almost complex structure constructed there. Clearly, given the homogeneous presentation (6.4.40) one can define the almost complex structure in terms of the corresponding Lie algebra decomposition. Then, checking the integrability is a purely algebraic task, see [218] for details.
- 21.
Cf. Definition I/7.2.2.
- 22.
\(\overline{\mathscr {L}}\) denotes the bundle conjugate to \(\mathscr {L}\). Note that \(\sigma ^*\overline{\mathscr {L}}\) is a holomorphic bundle, because \(\sigma \) is anti-holomorphic, and \(\sigma ^*\overline{\sigma ^*\overline{ {\mathscr {L}}} } = {\mathscr {L}}\).
- 23.
See also [42] for a semicontinuity argument. Alternatively, one may deduce \({{\mathsf {c}}}_1(\mathscr {L})=0\) from property 1 of \(\mathscr {L}\) by observing that any fibre of \(\pi \) represents a generator of \(H_{2}{(\mathbb {C}\mathrm P^3)}\).
- 24.
Remember that we may identify \(\sigma ^*\overline{\mathscr {L}} \) with \(\mathscr {L}\), cf. Remark 6.4.13.
- 25.
Cf. Sect. 5.7. Note that \(\mathrm{Ad}(P)\) is redundant here.
- 26.
As a consequence of the regularity of solutions to elliptic equations, these spaces remain unchanged after completing \(\varOmega ^p(M, \mathrm{Ad}(P))\) with respect to any Sobolev norm.
- 27.
- 28.
For a detailed presentation of the Sobolev-type arguments involved, we refer to Part IV in [83].
- 29.
Of course, from the previous sections, we know already that self-dual connections exist.
- 30.
This will follow from \(H_0 = 0\), that is, in particular, the assumption that \(\omega \) be irreducible is essential here.
- 31.
The adjoint representation induces a natural representation \(T = \mathrm{Ad}\otimes \mathrm{Ad}^*: G \rightarrow {{\mathrm{Aut}}}({{\mathrm{End}}}(\mathfrak g))\) via \(T(g)(\eta ) := \mathrm{Ad}(g) \circ \eta \circ \mathrm{Ad}(g^{-1})\).
- 32.
Recall that \(\mathrm{Spin}_{r, s} = \mathrm{Spin}_{s, r}\), that is, we could also take \(\mathrm{SO}_+(5, 1)\) below.
- 33.
Clearly, this is the Poincaré model of the hyperbolic 5-space.
- 34.
These morphisms are induced from the group homomorphism \(\mathrm{SO}(4) \rightarrow \mathrm{SO}(4)/\mathbb {Z}_2 = \mathrm{SO}(3) \times \mathrm{SO}(3)\) combined with the canonical projections onto the first and the second \(\mathrm{SO}(3)\)-component, respectively.
- 35.
See Definition 2.3.12.
- 36.
Cf. Definition 5.7.56.
- 37.
Clearly, we have \([{\mathscr {F}} + \mathrm {d}\alpha ]= u\).
- 38.
If \({\mathsf {s}}_M\) is expressed as a matrix with integer entries, then \(\det ({{\mathsf {s}}}_M) = \pm 1\).
- 39.
If M is simply connected, vanishing of the second Stiefel–Whitney class is equivalent to the signature being of type II.
- 40.
The only additional input we need is elementary knowledge of cobordism theory. For our purposes, the information contained in Appendix B of [213] is sufficient. For a more detailed presentation, see e.g. [104], Sect. 16 of Chap. II.
- 41.
We present the theorem in its original formulation. The assumption of being simply connected may be dropped, see also the Remark after Theorem 6.5.14.
- 42.
In such a situation, we say that p is resolvable. By a general theorem of Quinn [527], for any compact topological 4-manifold M whose Kirby-Siebenmann invariant is zero, the following holds: M has a smooth structure defined outside a finite set of singular points such that each of these points is resolvable. The manifold considered in the example fulfils the assumptions of this theorem.
- 43.
In the ordinary \(\mathbb {R}^4\), any compact set can be enclosed by a smoothly embedded 3-sphere.
- 44.
The authors of [95] assign this result to J. Simons.
- 45.
In more abstract terms, this fact is an immediate consequence of the isomorphism (2.8.11).
- 46.
See [170], \(\, \mathrm{Sect. II.5, n^o 17}\).
- 47.
See Remark 1.9.14/2.
- 48.
For the notation, see Remark 1.2.3.
- 49.
Note that, for a symmetric space, \(\sum _i [e_i, [e_i, \cdot ] ] \) coincides with the second Casimir operator of \(\mathrm{ad}{(\mathfrak h)}_{\upharpoonright { \mathfrak m}}\).
- 50.
Cf. Sect. 6.3 for details.
- 51.
Cf. the proof of Theorem 6.7.7.
- 52.
In sharp contrast, the induced connections \(\omega ^\pm \) given by (6.3.8) are (anti-)self-dual.
- 53.
The cohomogeneity of a G-action is the dimension of the orbit space.
- 54.
This is the irreducible representation of spin 2.
- 55.
For good reasons, these bundles are called quadrupole bundles, see [44] for an explanation.
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Rudolph, G., Schmidt, M. (2017). The Yang–Mills Equation. In: Differential Geometry and Mathematical Physics. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-0959-8_6
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DOI: https://doi.org/10.1007/978-94-024-0959-8_6
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