Locus LC001 - Landau-Ginzburg

For the study of isolated hypersurface singularities, matrix factorisations, Landau-Ginzburg models (a particular kind of topological quantum field theory) and associated algebra, geometry and higher category theory.

• Links: Roblox, Discord voice channel LC001.
• Office hours: for current office hours, see the main page.

LC001.01 - General introduction

• Video: YouTube.
• D. Eisenbud, “Homological algebra on a complete intersection, with an application to group representations”, Trans. Amer. Math. Soc. 260 (1980), 35–64.
• P. Dirac “The quantum theory of the electron”, Proc. R. Soc. Lond. A117 (1928) 610–624.
• M. Khovanov and L. Rozansky, “Matrix factorizations and link homology”, Fund. Math. 199 (2008), 1–91.

LC001.02 - Exterior algebras

• Video: YouTube.
• The main reference is my notes on Tensor, Exterior, Symmetric algebras.
• If you want to know more about universal properties, functors and categories you could see my old course but there are many fine references for category theory (my usual reference is Borceux’s “Handbook of categorical algebra”).

LC001.03 - Clifford algebras

• Video: YouTube.
• Beyond the Wikipedia entry the main reference for Clifford algebras is T. Friedrich “Dirac operators in Riemannian geometry”, Graduate Studies in Mathematics Vol. 25, AMS.
• For the relation between Clifford algebras and matrix factorisations see R.-O. Buchweitz, D. Eisenbud, J. Herzog “Cohen-Macaulay modules on quadrics”.
• For the identity defect see N. Carqueville, D. Murfet “Adjoints and Defects in Landau-Ginzburg models”, Advances in Mathematics, 2016.
• Notes from two of my talks on the relation between Clifford algebras and matrix factorisations “Monoidal bicategories of critical points” (2019) and “From critical points to extended TQFTs” (2020).

LC001.04 - Exterior algebra as a Hilbert space

• Video: YouTube.
• For the pairing on the exterior algebra see TES.
• See the postulates of quantum mechanics. Another physics reference I recommend is A. L. Fetter and J. D. Walecka “Quantum theory of many-particle systems”, McGraw-Hill.
• See Wikipedia for the basics on fermionic Fock states.

LC001.05 - Entanglement

• Video: YouTube.
• The standard survey reference for entanglement is R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, “Quantum entanglement”.
• I also recommend the textbook M. A. Nielsen and I. L. Chuang, “Quantum Computation and Quantum Information” 10th Anniversary Edition (available freely online PDF).
• See also Preskill’s notes specifically Preskill Ch.4 and Preskill Ch.7.