The Department of Mathematics and Statistics at the University of Massachusetts Amherst will host its fourth annual summer school on Physical Mathematics of Quantum Field Theory during the week of June 9 - 13, 2025. The aim of the workshop is to introduce young researchers on the boundary between mathematics and physics to recent ideas in classical and quantum field theory using notions from complex geometry and algebraic topology. This year there is a focus on 4d gauge theories and their relationship with representation theory, algebraic geometry, and topology. The target audience of the workshop is advanced graduate students and post-docs, from both mathematics and theoretical physics. The schedule will be designed to include a large amount of time for participants to interact. We hope especially to encourage conversations between mathematicians and physicists.
LECTURE SERIES (with references initially collapsed)
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
(2) Algebraic geometry and Vafa-Witten theory by Martijn Kool (Utrecht University) Lecture notes are here.
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.
(3) Topological Correlators of N=2 SYM and Four-Manifold Invariants by Jan Manschot (Trinity College Dublin)
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!
SCHEDULE: The plan is here. The lectures will take place at in room S131 of ILC (the Integrative Learning Center).
REGISTRATION: Registration is now closed.
PAST SUMMER SCHOOLS: 2024 2023 2022
If you have any questions, you can contact the organizers by e-mail at the following addresses:
Owen Gwilliam --- ogwilliam@umass.edu
Chris Elliott --- celliott@amherst.edu