Introductory class on instanton gauge theory
With the assistance of Felix Eberhart
We will start with some foundational material (connections, curvature, Sobolev spaces...) and will then discuss the ASD equation, the Chern-Simons function, and we will consider instantons on the 4-sphere. We will study the linearisation of the equation and index theory, and we will discuss Uhlenbeck compactification and transversality. We will then aim to introduce instanton Floer homology, and we aiming for a proof of the exact triangle sequence. If there is enough time we will also discuss the sutured version of instanton gauge theory. Prerequisits will be basic algebraic and geometric topology, and some familiarity with analysis on manifolds (differential forms, bundles...)
The schedule is:
Class: 10:15 -- 12:00 (with 15min break) on Thursdays, starting 15 April (No class on 13.5. and 3.6. because of holidays)Exercise class: 14:00 -- 15:45 (with 15min break) on Tuesdays, starting 20 April (note the time shift by 15 minutes we had agreed on in class)
If you are interested in attending then please send me an email or just come.
Zoom information is:
Meeting-ID: 629 7257 0398 (for the class)Meeting-ID: 682 5786 5957 (for the exercise class)
In both cases, the password is the name of the german mathematician who has introduced Floer homology in the instanton setting.
| Date and Time | Topics | Literature | Handwritten notes | Recording |
|---|---|---|---|---|
| Thursday 15.4. | Class: Motivation, principal and associated bundles, connections | [B, Ch. 2 and 3] | 1st and 2nd lecture | Recording |
| Tuesday 20.4. | Class: Exterior differential, covariant derivative | [B, Ch. 2 and 3] | 1st and 2nd lecture | Partial recording |
| Tuesday 20.4. | Exercise class: Hopf bundles, tautological bundle | [B, Ch. 2 and 3] | Exercise notes | Recording |
| Thursday 22.4. | Class: Parallel transport, curvature | [B, Ch. 2 and 3] | up to 3rd lecture | Recording |
| Tuesday 27.4. | Exercise class: Bundles, connections | [B, Ch. 2 and 3] | Exercice notes | Recording |
| Thursday 29.4. | Class: Chern-Weil theory I, reduction and extension of structure group | [B, Ch. 4 and 5] | Lecture 4 | Recording |
| Tuesday 4.5. | Exercise class: Two notions of parallel transport, curvature in principal bundle vs curvature of the associated covariant derivative in the associated bundle | [B, Ch. 2 and 3] | Exercise notes | Recording |
| Thursday 6.5. | Class: Push-forward and pull-back of connections, Chern-Weil theory II, Chern- and Pontryagin classes | [B, Ch. 4 and 5] | Class 4 and 5 notes | Recording |
| Tuesday 11.5. | Exercise class: Exercises 2,3, and 7 | [B, Ch. 4 and 5] | Exercise notes | Recording |
| Thursday 13.5. | No class: National holiday (Christi Himmelfahrt) | |||
| Tuesday 18.5. | Exercise class: Classification of bundles | [Milnor-Stasheff: Characteristic classes, Steenrod: Fibre bundles] | Exercise notes | Recording |
| Thursday 20.5. | Class: Hodge star, Yang-Mills functional, Yang-Mills equation, instantons, energy bounds | [B, Ch. 7], [Atiyah, Ch.2], [DK, Ch. 3.4] | Class notes | Recording |
| Tuesday 25.5. | Exercise class: Charge of the Hopf bundle S^7 -> S^4 via Chern-Weil theory | [B, Ch. 7], [Atiyah, Ch.2], [DK, Ch. 3.4] | Exercise notes | Recording |
| Thursday 27.5. | Class: Conformal transformations on S^4 and charge=1 instantons | [B, Ch. 7], [Atiyah, Ch.2], [DK, Ch. 3.4] | Class notes | Recording |
| Tuesday 1.6. | Exercise class: Ambrose-Singer theorem, p_1(su(E)) via Chern-Weil theory | [B, Ch. 4 and 6] | Exercise notes | Recording |
| Tuesday 8.6 | Exercise class: Discussion of exercise sheets | [B, Ch. 4 and 6], [Milnor-Stasheff: Characterstic classes] | Exercise notes | Recording |
| Thursday 10.6. | Class: Reducibles, Sobolev norms | [DK, Ch. 4], [M] | Class notes | Recording |
| Thursday 17.6. | Class: Sobolev embedding theorems, Sobolev multiplication theorems | [DK, Ch. 4], [M] | Class notes | Recording |
| Tuesday 22.6. | Class: Sobolev completions of space of connections and gauge transformations, the configuration space is Hausdorff | [DK, Ch. 4], [M] | Class notes | Recording |
| Thursday 24.6. | Class: Elliptic operators, fundamental inequality, Hodge decomposition | [Warner: | Class notes | Recording |
| Tuesday 29.6. | Class: Construction of slices | [DK, Ch. 4], [Mr2] | Class notes | Recording (missing after the break) |
| Thursday 1.7. | Class: Uhlenbeck's fundamental lemma | [DK, Ch. 2.3], [Mr2] | Class notes | Recording |
| Tuesday 6.7. | Uhlenbeck compactification, continued (Felix' replacement) | [DK, Ch. 4], [Mr2] | Class notes | Recording |
| Thursday 8.7. | A sketch of proof of Donaldson's diagonalisation theorem | [DK, Ch. 2.3], [Mr2] | Class notes | Recording |
| Tuesday 13.7. | A few concluding remarks about compactness and transversality on closed manifolds, Chern-Simons function, extended Hessian | [DK, Ch. 2.3], [Mr2] | Class notes | Recording |
| Thursday 15.7. | A (very rough) sketch of the construction of instanton Floer homology | [D, Ch. 5], [Mr2] | Class notes | Recording |
We will mainly be using the following Literature:
- [A] M. Atiyah, Geometry on Yang-Mills fields, Scuola Normale Superiore Pisa, Pisa, 1979.
- [B] H. Baum, Eichfeldtheorie, Springer Masterclass. See also the free notes on her webpage .
- [D] S. Donaldson, Floer homology groups in Yang-Mills theory, Cambridge Tractats in Mathematics
- [DK] S. Donaldson, P. Kronheimer, Geometry of four-manifolds, Oxford Mathematical Monographs
- [Mr1] T. Mrowka, Instanton gauge theory class notes 2011, available for instance on Danny Ruberman's webpage:
- [Mr2] T. Mrowka, Instanton gauge theory class notes 2019, notes by Donghao Wang, available on Donghao's webpage:
- [W] F. Warner, Foundations of Differentiable Manifolds and Lie groups, Springer GTM
Course notes
Piotr Suwara is typesetting the lecture notes that you can download here.