Combinatorial Quantization of Chern-Simons theory I
This course will cover the background notions involved in the combinatorial quantization framework for 3-dimensional Chern-Simons theory, as developed by Alekseev-Grosse-Schomerus in ‘94-‘95. We will focus primarily on the Lie (bi)algebras, (Poisson-)Lie groups and the compact quantum groups that arise from the space of flat connections, graph holonomies and their canonical quantization. The content of this course is compatible with prof. Schwarz’s course “ Quantum mechanics and quantum field theory from algebraic and geometric viewpoints”.
At the end, if time allows, I will also briefly discuss how this combinatorial quantization framework is related to the Witten-Reshetikhin-Turaev TQFT.
At the end, if time allows, I will also briefly discuss how this combinatorial quantization framework is related to the Witten-Reshetikhin-Turaev TQFT.
Lecturer
Date
30th March ~ 24th June, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Wednesday | 09:50 - 11:25 | A3-3-201 | Zoom 15 | 204 323 0165 | BIMSA |
Prerequisite
Some familiarity with Lagrangian/Hamiltonian mechanics, field theory and the calculus of differential forms. Basic algebra (groups, rings, fields) and functional analysis (Banach/Hilbert spaces, norms, measures) is ideal.
Syllabus
The timeline is roughly as follows (may be subject to change):
1st-2nd week: Introduction — goals and expected covered materials. Lie algebra/Lie groups review; gauge theory: principal bundles, holonomies and gauge transformations.
3rd-4th week: Hamiltonian mechanics review and Dirac canonical quantization. T^*R example: from the Heisenberg/Weyl algebra to the Moyal-star (cf. prof. Schwarz’s lectures).
5th-7th week: Lie bialgebras, Poisson-Lie groups and the classical Yang-Baxter equations. Classical symmetries of the Chern-Simons action and the Fock-Rosly Poisson structure on flat connections.
8th-10th week: C*-algebras and Hopf algberas review. Compact quantum matrix groups of Woronowicz and the deformation quantization of Poisson-Lie groups. Quantum holonomies on a circle.
11th-12th week: The geometry of (ciliated) graphs on a Riemann surface; the quantum graph holonomies and their Hopf algebra structures.
1st-2nd week: Introduction — goals and expected covered materials. Lie algebra/Lie groups review; gauge theory: principal bundles, holonomies and gauge transformations.
3rd-4th week: Hamiltonian mechanics review and Dirac canonical quantization. T^*R example: from the Heisenberg/Weyl algebra to the Moyal-star (cf. prof. Schwarz’s lectures).
5th-7th week: Lie bialgebras, Poisson-Lie groups and the classical Yang-Baxter equations. Classical symmetries of the Chern-Simons action and the Fock-Rosly Poisson structure on flat connections.
8th-10th week: C*-algebras and Hopf algberas review. Compact quantum matrix groups of Woronowicz and the deformation quantization of Poisson-Lie groups. Quantum holonomies on a circle.
11th-12th week: The geometry of (ciliated) graphs on a Riemann surface; the quantum graph holonomies and their Hopf algebra structures.
Reference
References for the context of this course:
Drinfel’d (1983), “Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations,”
Alekseev-Grosse-Schomerus (1994-1995), “ Combinatorial Quantization of the Hamiltonian Chern-Simons Theory I & II”,
Fock-Rosly (1998), “Poisson structure on moduli of flat connections on Riemann surfaces and R-matrix”.
The course will mainly contain materials from:
Meusberger (2012), “Poisson-Lie groups and gauge theory”,
Schwarz (1993), “Quantum field theory and topology”,
Majid (2000), “Foundations of Quantum Group Theory”,
Drinfel’d (1983), “Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations,”
Alekseev-Grosse-Schomerus (1994-1995), “ Combinatorial Quantization of the Hamiltonian Chern-Simons Theory I & II”,
Fock-Rosly (1998), “Poisson structure on moduli of flat connections on Riemann surfaces and R-matrix”.
The course will mainly contain materials from:
Meusberger (2012), “Poisson-Lie groups and gauge theory”,
Schwarz (1993), “Quantum field theory and topology”,
Majid (2000), “Foundations of Quantum Group Theory”,
Audience
Advanced Undergraduate
, Graduate
, Postdoc
, Researcher
Video Public
Yes
Notes Public
Yes
Language
English