Geometry, integrability and field theories

Lecture course "Geometry, integrability and field theories"

Sylvain Lavau (EIMI)

The idea of the course is to provide notions of topology and geometry to mathematical physi-

cists, as well as present concrete applications of such notions to pure algebraists and geometers.

The idea would be not to fall in a too abstract presentation, and to anchor it into examples taken
from physics. I propose to start from the basics and grow in complexity to reach higher grounds
which are much more intricate. Here is a tentative plan of what i would like to talk about. I
think most items need more than one seance and maybe not everything could be addressed.
1. Basics on differential forms: exterior algebra, Hodge star operator, differential forms on
$\mathbb R^n$, de Rham cohomology and div/grad/curl, codifferential operator and Laplace-Beltrami
operator, Poincaré Lemma, Aharanov-Bohm effect.

2. Integration of differential forms: manifolds, tangent vectors, covectors and differen-
tial forms on a manifold, volume form, integration of differential forms, manifolds with

boundaries, Stokes theorem, Maxwell equations.
3. Poisson geometry and analytical mechanics: polyvector fields, Schouten-Nuijenhuis
bracket, Poisson manifolds, Hamiltonian mechanics, tautological one form and Legendre
transform, Liouville theorem, constraints, coisotropic reduction, Dirac bracket.

4. Yang-Baxter Equations: Lie groups, Lie algebras, Lie bi-algebras, the R-matrix, Yang-
Baxter equations, Poisson Lie groups.

If you have ideas that would be beneficial to the program please do not hesitate to share it
with me! What lies below are more advanced topics that I would really like to reach eventually
(in a few months):
5. Gauge theories: Vector bundles, principal bundles, connections, curvatures, Bianchi
identities, Yang-Mills theory
6. Topological field theories: Chern-Simons Theory, BF theory, sigma models, graded
geometry, AKSZ formalism.
7. Quantization of gauge systems: ghosts/antighosts, BRST & Fradkin-Batalin-Vilkoviski
formalism, gerstenhaber & BV algebras, classical and quantum master equations.
8. Scattering processes: S-matrix, amplituhedron, L∞-algebras.
I propose to advance at a steady pace following a physically informed mathematical path.
I would not focus on the logico-deductive process of mathematical proofs but rather on the
physical ideas that led to the invention of these notions. I intend to write as many lecture notes
as possible, and would hopefully put them on my website every week.