From the instanton moduli spaces to the Vafa-Witten invariants
This course begins with the study of the geometry and topology of moduli spaces arising from four-dimensional gauge theory, starting with anti-self-dual (ASD) instantons and their moduli spaces. We examine the structure of these moduli spaces and discuss their compactifications. These considerations lead to the definition of Donaldson invariants, which provide powerful diffeomorphic invariants of the underlying smooth four-manifolds. We also introduce the Seiberg–Witten invariants and explore their relationship with the Donaldson invariants.
We then turn to the algebro-geometric perspective via the Donaldson–Uhlenbeck–Yau correspondence, as well as algebro-geometric compactifications using stability conditions in the sense of Mumford–Takemoto and Gieseker, and move on to the study of Vafa–Witten invariants on projective surfaces, incorporating perfect obstruction theory and related developments in algebraic geometry, and highlighting the deep interplay among differential geometry, algebraic geometry, and representation theory.
We then turn to the algebro-geometric perspective via the Donaldson–Uhlenbeck–Yau correspondence, as well as algebro-geometric compactifications using stability conditions in the sense of Mumford–Takemoto and Gieseker, and move on to the study of Vafa–Witten invariants on projective surfaces, incorporating perfect obstruction theory and related developments in algebraic geometry, and highlighting the deep interplay among differential geometry, algebraic geometry, and representation theory.
Lecturer
Date
18th March ~ 17th June, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Wednesday | 13:30 - 16:55 | A14-202 | ZOOM 04 | 482 240 1589 | BIMSA |
Prerequisite
manifolds, vector bundles, principal bundles, and the characteristic classes
Syllabus
1. Course overview
2. Anti-self dual instantons, gauge transformations, and the moduli spaces
3. Sobolev spaces and elliptic theory
4. Atiyah–Hitchin–Singer complex and the Kuranishi model
5. The generic smoothness theorem by Freed–Uhlenbeck
6. Uhlenbeck compactness and the Uhlenbeck compactification
7. Stable range and fundamental class of the moduli space
8. Donaldson invariants
9. Seiberg-Witten invariants and Witten’s conjecture
10. Hermitian–Einstein connections
11. Slope (Mumford–Takemoto) stability
12. Vanishing theorem
13. Existence of Hermitian–Einstein metrics
14. Gieseker stability and compactification
15. Algebro-Geometric Donaldson invariants and Li–Morgan’s Map
16. Donaldson–Mochizuki invariants on projective surfaces
17. Betti numbers of the moduli spaces of bundles/sheaves
18. Euler characteristics of the instanton moduli spaces
19. Donaldson’s theory and N=2 super Yang–Mills theory
20. N=4 super Yang–Mills theory and S-duality conjecture
21. Donaldson-Uhlenbeck-Yau theorem for the Vafa-Witten theory
22. Perfect obstruction theory and virtual enumerative invariants
23. Vafa-Witten invariants on projective surfaces
24. Generating series of the Vafa-Witten invariants and modular forms
2. Anti-self dual instantons, gauge transformations, and the moduli spaces
3. Sobolev spaces and elliptic theory
4. Atiyah–Hitchin–Singer complex and the Kuranishi model
5. The generic smoothness theorem by Freed–Uhlenbeck
6. Uhlenbeck compactness and the Uhlenbeck compactification
7. Stable range and fundamental class of the moduli space
8. Donaldson invariants
9. Seiberg-Witten invariants and Witten’s conjecture
10. Hermitian–Einstein connections
11. Slope (Mumford–Takemoto) stability
12. Vanishing theorem
13. Existence of Hermitian–Einstein metrics
14. Gieseker stability and compactification
15. Algebro-Geometric Donaldson invariants and Li–Morgan’s Map
16. Donaldson–Mochizuki invariants on projective surfaces
17. Betti numbers of the moduli spaces of bundles/sheaves
18. Euler characteristics of the instanton moduli spaces
19. Donaldson’s theory and N=2 super Yang–Mills theory
20. N=4 super Yang–Mills theory and S-duality conjecture
21. Donaldson-Uhlenbeck-Yau theorem for the Vafa-Witten theory
22. Perfect obstruction theory and virtual enumerative invariants
23. Vafa-Witten invariants on projective surfaces
24. Generating series of the Vafa-Witten invariants and modular forms
Reference
M. F. Atiyah, Geometry of Yang-Mills fields, Scuola Normale Superiore Pisa, 1979.
S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford University Press, 1990.
D. S. Freed and K. K. Uhlenbeck, Instantons and four-manifolds, Second edition, Mathematical Sciences Research Institute Publications, 1991.
R. Friedman and J. W. Morgan, Smooth four-manifolds and complex surfaces, Springer-Verlag, 1994.
R. Friedman and J. W. Morgan (eds.), Gauge theory and the topology of four-manifolds, IAS/Park City Mathematics Series, Volume 4, American Mathematical Society, 1998.
D. D. Joyce, Riemannian holonomy groups and calibrated geometry, Oxford Graduate Texts in Mathematics, 12, Oxford University Press, 2007.
S. Kobayashi, Differential geometry of complex vector bundles, Princeton University Press, 1987.
M. Luebke and A. Teleman, The Kobayashi-Hitchin correspondence, World Scientific, 1995.
T. Mochizuki, Donaldson type invariants for algebraic surfaces, Lecture Notes in Math. 1972, Springer, Berlin, 2009.
J. W. Morgan, The Seiberg–Witten equations and applications to the topology of smooth
of smooth four-manifolds, Mathematical Notes, 44 Princeton University Press NJ, 1996.
D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves, Second edition. Cambridge Mathematical Library, Cambridge University Press, 2010.
C. H. Taubes, Metrics, Connections and gluing theorems. CBMS Regional Conference Series in Mathematics, 89, American Mathematical Society, 1996.
C. H. Taubes, Differential Geometry. Bundles, connections, metrics and curvature, Oxford Graduate Texts in Mathematics, 23, Oxford University Press, 2011.
S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford University Press, 1990.
D. S. Freed and K. K. Uhlenbeck, Instantons and four-manifolds, Second edition, Mathematical Sciences Research Institute Publications, 1991.
R. Friedman and J. W. Morgan, Smooth four-manifolds and complex surfaces, Springer-Verlag, 1994.
R. Friedman and J. W. Morgan (eds.), Gauge theory and the topology of four-manifolds, IAS/Park City Mathematics Series, Volume 4, American Mathematical Society, 1998.
D. D. Joyce, Riemannian holonomy groups and calibrated geometry, Oxford Graduate Texts in Mathematics, 12, Oxford University Press, 2007.
S. Kobayashi, Differential geometry of complex vector bundles, Princeton University Press, 1987.
M. Luebke and A. Teleman, The Kobayashi-Hitchin correspondence, World Scientific, 1995.
T. Mochizuki, Donaldson type invariants for algebraic surfaces, Lecture Notes in Math. 1972, Springer, Berlin, 2009.
J. W. Morgan, The Seiberg–Witten equations and applications to the topology of smooth
of smooth four-manifolds, Mathematical Notes, 44 Princeton University Press NJ, 1996.
D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves, Second edition. Cambridge Mathematical Library, Cambridge University Press, 2010.
C. H. Taubes, Metrics, Connections and gluing theorems. CBMS Regional Conference Series in Mathematics, 89, American Mathematical Society, 1996.
C. H. Taubes, Differential Geometry. Bundles, connections, metrics and curvature, Oxford Graduate Texts in Mathematics, 23, Oxford University Press, 2011.
Audience
Advanced Undergraduate
, Graduate
, Postdoc
, Researcher
Video Public
No
Notes Public
No
Language
English
Lecturer Intro
My research interests are primarily centred on Gauge theory within mathematics. Recently, my focus has been on semistable Higgs sheaves on complex projective surfaces and associated gauge-theoretic invariants, employing algebro-geometric methods. However, I also have a strong interest in working within the analytic category.