SUSTech Topology

About

This is an undergraduate student seminar on geometry and topology. Here is a course description for you to get an idea of how it runs and particularly for those of you who wish to register for credit. If you would like to propose a topic in geometry and topology for now or the future, write to Yifei Zhu.

Our theme for Fall 2024 is the geometry and physics of Higgs bundles. References include Laura Schaposnik's lecture notes from a summer school on quantum field theory and manifold invariants (this volume also contains other topics of interest) and her expository article for the Notices of the AMS. See here for more motivations. Background materials can and should be supplemented according to participant needs. If you're thinking about getting involved, let us know.

We meet on Thursdays 4:20–6:10 pm in Lecture Hall 3-309. The first, organizational meeting is tentatively scheduled on September 19. To facilitate communication between participants, there is a QQ group: 811299006.

Talks

Sep 19, '24, Yifei Zhu, Organizational meeting / introduction: A partial, but motivated view of Higgs bundles

Why do we care about them?
- Vector bundles, parametrized objects, and moduli spaces: tangent bundle, line bundles over $S^1$, moduli of elliptic curves
- Motivation from physics (condensed matter and quantum materials): sensing and absorbing devices (invisibility cloak, anti-objects), optical devices (holography, laser), materials with exceptional properties not found in nature at the macroscopic level realized by engineering and fine-tuning against degeneracy at the microscopic/quantum level
- Hamiltonian $\stackrel{\rm mathematically}{=}$ square matrix with prescribed symmetry (Hermitian, reason), eigenvalue (energy) and eigenvector (state), evolution of eigenstates (or rotation of eigenframe $\leadsto$ eigenbundle), energy bands
- Examples: Hermitian 2-band system (orthogonal half Möbius bands, classified by $\pi_1(S^1)\cong{\mathbb Z}$) and 3-band system (classified by $\pi_1\Big({\rm SO}(3)\big/\big({\rm O}(1)\times{\rm O}(1)\big)\Big)\cong Q=\{\pm1,\pm i,\pm j,\pm k\}$, see Wu et al., Science 2019), non-Hermitian 2-band system (non-Hermitian = Hermitian with respect to a non-Euclidean Riemannian metric form, reason(s); stratified space, kissing half Möbius bands)
- Higgs bundle $\approx$ family of matrices: Peter Higgs, Nigel Hitchin, Carlos Simpson
- Riemann–Hilbert correspondence (de Rham's theorem, the Poincaré lemma), Simpson correspondence
Some specific goals (or rather questions to keep in mind):
- Classification of eigenbundles, up to homotopy: intersection homology and homotopy
- Metrics on the base space of a Higgs bundle and non-Hermiticity: hyperbolic metric (Kollár et al., Nature 2019; Gothen, Geometry on surfaces and Higgs bundles)
- Bulk–edge correspondence and (geometric) Langlands correspondence: roles played by Higgs bundles, explicit examples and calculations (Kienzle and Rayan, Hyperbolic band theory through Higgs bundles)
Background materials and a plan: course description for MA423
Exercise: Compute the eigenbundle for the real matrices $$\begin{bmatrix} f_1 & f_2 \\ f_2 & -f_1 \end{bmatrix}$$ [a solution] and find its relationship to the Hopf bundle $S^0\hookrightarrow S^1\to S^1$. What about the complex and quaternionic (and octonionic) cases? Hint: See here.