Topics in Mathematical Physics 2022



30 Dec, 2pm-4.30pm.

At 1.45pm, please join the Tencent Meeting: 995-182-781, password 301222.

You are required to keep one video camera switched on during the exam, for the purpose of ensuring that you are alone.

After the examination, you have an extra 15 minutes to submit a clear scanned/photographed copy of your solutions to my email address.

The examination will count for 60% of the final grade.

If you have applied to be assessed on a Pass/Fail basis, you must still take the exam because the assignments only count for 40% of the overall grade. But you only need to obtain enough marks in the exam to get an overall pass grade.

On the exam day (30 Dec), if you cannot sit for the examination due to medical reasons, please contact me in the morning and explain your circumstances.

(Exam Cover Page)



Graduates and advanced undergraduates from both mathematics and physics are welcome.

We will cover these topics (tentative):

Smooth manifolds: Generalities, Lie groups

Gauge theory constructions: Principal fibre bundles, connections, curvature, characteristic classes

Spin geometry: Clifford algebras, spinors, Dirac operators

Operators on Hilbert space: Spectral theory, Fredholm theory, unbounded operators, self-adjoint extensions

K-theory and Index theory: basic ideas

Quantum Hall and Aharonov-Bohm effects: real world manifestation of above ideas


Lecture notes

(Week 1) Informal introduction to gauge theory and quantum theory

(Week 2) Quantum mechanics and Dirac operator on Euclidean line, circle

(Week 3) Aharonov-Bohm effect, Self-adjointness issues, supersymmetric QM. Smooth manifolds, tangent/cotangent vectors

(Week 4) **Expanded 25/09** Derivatives, Immersions, submersions, embeddings. (Co)tangent bundles, general vector bundles

(Week 5) **Expanded 06/10** Principal bundles, gauge transformations. Preview of spin geometry

(Week 6) Differential forms, exterior calculus

(Week 7) **Update 26/10: See notation remark above Definition 35.** Integral curves, flows, Lie derivative. Lie groups, Lie algebras, fundamental vector fields, Maurer-Cartan form

(Week 8) Connections and curvature

(Week 9) **Typos corrected 02/11.** General and associated fibre bundles, reduction of structure group, parallel transport, covariant derivative, metric connections

(Week 10) **Update 16/11** Clifford algebras.

(Week 11) (18 November Summary) Spin groups and representations, spin manifolds

(Week 12) **Update 30/11, improved Lemmas 16.4, 16.5** Spinor bundle geometry

(Week 13) **Update 02/12, typo on page 183 corrected** Twisted Dirac operators, connection and spinor Laplacians, Lichnerowicz identity, Landau Hamiltonians and spectral geometry of quantum Hall effect.

(Week 14) Unbounded operators, Differential operators.

(Week 15) Self-adjoint extensions, Sobolev spaces, index theory idea

(Week 17) Exam






The above topics (except last one) are covered in many classical sources, but the presentation in the lectures will be rather different. It will be helpful to do some general reading of any of the following textbooks (alphabetical order), which develop mathematics alongside physics motivations


Baez, J., Muniain, J.P.: Gauge Fields, Knots, and Gravity, World Scientific, 1994

Bleecker, D., Booss-Bavnbek, B.: Index theory with applications to mathematics and physics. International Press, 2013

Choquet-Bruhat, Y., Dewitt-Morette, C.: Analysis, manifolds, and physics. Part I: Basics. North Holland, 1982

Eguchi, T., Gilkey, P.B., Hanson, A.J.: Gravitation, gauge theories and differential geometry. Physics Reports 66(6) 213--393, 1980

Hamilton, M.: Mathematical Gauge Theory. With Applications to the Standard Model of Particle Physics. Springer, 2017

Lawson, H.B., Michelsohn, M.-L.: Spin Geometry. Princeton Univ. Press, 1989

Naber, G.: Topology, Geometry, and Gauge Fields. Foundations. Texts in Appl. Math., Springer, 2011

Nash, C.: Differential Topology and Quantum Field Theory. Acad. Press, 1992

Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol. 1. Functional Analysis. and Vol. II. Fourier analysis, Self-adjointness, Acad. Press, 1980, 1975


If you don't know much about smooth manifolds and topology, some basic references are:

Guillemin, V., Pollack, A.: Differential Topology. Prentice-Hall, 1974

Milnor, J.: Topology from the differentiable viewpoint. Univ. Press Virginia, 1965


The following references are more Definition-Proof style, minimal physics:

Lee, J.M.: Introduction to smooth manifolds. Graduate Texts in Math. Vol 218, Springer 2013 (Detailed)

Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. Vol 1. Wiley, 1963

Steenrod, N.: The Topology of Fibre Bundles. Princeton Math. Series, 1960



Linear algebra, real analysis, basic metric and topological spaces

Basic abstract algebra definitions (especially groups)

Basic idea of differentiable manifolds

Some exposure to Hilbert spaces (at least semi-rigorously)


Exposure to theoretical physics is helpful, but not compulsory.