• Introductory aspects of twisted QFT: 

[1] N. Garner, N. Paquette, TASI Lectures on the Mathematics of String Dualities, in Theoretical Advanced Study Institute in Elementary Particle Physics: Black Holes, Quantum Information, and Dualities (TASI 2021), 4, 2022; https://arxiv.org/abs/2204.01914

• Origin of descent in TQFT:

[1] E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. Vol. 117, p. 353, 1988.

• Algebraic structures from descent:

[1] C. Beem, D. Ben-Zvi, M. Bullimore, T. Dimofte, A. Neitzke, Secondary products in supersymmetric field theory, Ann. Henri Poincaré 21 (2020), no. 4 1235-1310; https://arxiv.org/abs/1809.00009

[2] K. Costello, O. Gwilliam, Factorization Algebras in QFT. Vol. 1, vol. 31 of New Mathematical Monographs. Cambridge University Press, 2016. https://websites.umass.edu/ogwilliam/

[3] D. Gaiotto, J. Kulp, J. Wu, Higher Operations in Perturbation Theory; https://arxiv.org/abs/2403.13049

• Survey of twists of 4d SYM:

[1] C. Elliott, P. Safronov, B. R. Williams, A Taxonomy of Twists of Supersymmetric Yang--Mills Theory, Selecta Math. 28 (2022), no. 4 Paper no. 73; https://arxiv.org/abs/2002.10517

[2] C. Elliott, O. Gwilliam, B. R. Williams, Higher Deformation Quantization for Kapustin-Witten Theories; https://arxiv.org/abs/2108.13392

• Symmetries in minimally twisted 4d theories:

[1] I. Saberi, B. R. Williams, Superconformal algebras and holomorphic field theories, Ann. Henri Poincaré 24 (2023), no. 2 541-604; https://arxiv.org/abs/1910.04120

[2] K. Budzik, D. Gaiotto, J. Kulp, B. R. Williams, J. Wu, Semi-chiral operators in 4d N=1 gauge theories, JHEP 05 (2024) 245; https://arxiv.org/abs/2108.13392

• Background in algebraic geometry: 

Review the notions of scheme, coherent sheaf, and sheaf cohomology (i.e., some Hartshorne) and consider watching these lectures

Richard Thomas ``Introduction to coherent sheaves 1" at https://youtu.be/8A17VWf7NHc

Richard Thomas ``Introduction to coherent sheaves 2" at https://youtu.be/99-SgxjOb-Y

• Books:

[1] D.~Huybrechts and M.~Lehn, The geometry of moduli spaces of sheaves, Cambridge University Press (2010).

[2] M.~Lehn, Lectures on Hilbert schemes, Algebraic structures and moduli spaces, CRM proceedings and lecture notes v.~38 AMS Providence, R.I., 2004.

[3] T.~Mochizuki, Donaldson type invariants for algebraic surfaces, Lecture Notes in Math.~1972, Springer-Verlag, Berlin (2009).

• Papers:

[1] B.~Fantechi and L.~G\"ottsche, Riemann-Roch theorems and elliptic genus for virtually smooth schemes, Geom.~Topol.~14 (2010) 83--115.

[2] A.~Gholampour and R.~P.~Thomas, Degeneracy loci, virtual cycles and nested Hilbert schemes II, Compos.~Math.~156 (2020) 1623--1663.

[3] L.~Göttsche, M.~Kool, and T. Laarakker, SU(r) Vafa-Witten invariants, Ramanujan's continued fractions, and cosmic strings, Mich.~Math.~Jour.~75 (2025) 3--63.

[4] A.~A.~Klyachko, Vector bundles and torsion free sheaves on the projective plane, preprint Max Planck Institut f\"ur Mathematik (1991).

[5] T.~Laarakker, Monopole contributions to refined Vafa-Witten invariants, Geom.~Topol.~24 (2020) 2781--2828.

[6] Y.~Tanaka and R.~P.~Thomas, Vafa-Witten invariants for projective surfaces I: stable case, Jour.~Alg.~Geom.~29 (2020) 603--668.

• Books:

Labastida & Marino “Topological Quantum Field Theory and Four-Manifolds” 

Donaldson & Kronheimer “The Geometry of Four-Manifolds”

• Classic papers:

Vafa, Witten “A Strong Coupling Test of S-duality" https://arxiv.org/abs/hep-th/9408074

Witten “Monopoles and Four-Manifolds” https://arxiv.org/abs/hep-th/9411102

Moore, Witten “Integrating over the u-plane in Donaldson Theory” https://arxiv.org/abs/hep-th/9709193

Losev, Nekrasov, Shatashvili “Issues in Topological Gauge Theory” https://arxiv.org/abs/hep-th/9711108

• Recent papers:

JM, Moore “Topological Correlators of SU(2), N=2* SYM on Four-Manifolds” https://arxiv.org/abs/2104.06492

JM “Four-Manifold Invariants and Donaldson-Witten Theory” https://arxiv.org/abs/2312.14709

• For the Donaldson invariants:

[1] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds.

[2] D. S. Freed and K. K. Uhlenbeck, Instantons and four-manifolds.

[3] M. F. Atiyah, Geometry of Yang-Mills fields.

• For the Hitchin-Kobayashi correspondence and the moduli spaces of semistable sheaves:

[4] S. Kobayashi, Differential geometry of complex vector bundles.

[5] M. Luebke and A. Teleman, The Kobayashi-Hitchin correspondence.

[6] D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves, Second edition. 

• For the Vafa-Witten theory in Algebraic Geometry:

[7] Y. Tanaka, Stable sheaves with twisted sections and the Vafa-Witten equations on smooth projective surfaces, https://arxiv.org/abs/1312.2673.

[8] Y. Tanaka and R. P. Thomas, Vafa-Witten invariants for projective surfaces I: stable case, https://arxiv.org/abs/1702.08487.

[9] Y. Tanaka and R. P. Thomas, Vafa-Witten invariants for projective surfaces II: semistable case, https://arxiv.org/abs/1702.08488

[10] C. Vafa and E. Witten, A strong coupling test of S-duality, https://arxiv.org/abs/hep-th/9408074.

Please have a look at also the references in Martijn and Jan's lecture series!