Algebraic topology for theoretical physicists / 数理物理学 (2024)

Overview

The slot with the title Mathematical Physics / 数理物理学 in the Graudate School of Physics was dormant for about twenty years. I decided to revive it this year to have an introductory course on algebraic topology meant for theoretical physicists. The rationale is as follows.

In the last ten years, theoretical physicsits realized that algebraic topology is useful and necessary to have a systematic understanding of symmetry-protected topological phases of matter and of anomalies in particle physics. Before that, only a very rudimentary part of algebraic topology was used in theoretical physics. As such, the techniques of algebraic topology required to follow recent advances are not adequately covered in the standard courses of mathematics for physicisists.

This set of lectures is an effort to fill this gap. I plan to use various physics topics to illustrate how concepts from algebraic topology are used.

Topics covered

• On the math side, I plan to cover
  • Homotopy groups
  • Homology and cohomology groups
  • Fiber bundles
  • Classifying spaces
  • Characteristic classes
  • K-theory and its periodicity
  • Bordism groups
• I will plan to use physics topics such as
  • topological solitons
  • integer quantum Hall effect
  • topological insulators and superconductors
  • symmetry-protected topological phases
  • anomalies of massless fermions
  to illustrate mathematical ideas introduced.

Misc info

Please also see the course page provided by the university.

To get credits

You'll need to submit the term-end report through the link above.

Date and time

2nd slot of Thursdays, from 10:25 to 12:10.

email: yuji.tachikawa_at_ipmu.jp