Landau-Ginzburg

On the geometry and (bi)category theory of isolated hypersurface singularities. Currently on hiatus.

The aim of the seminar is to provide an accessible and detailed construction of the bicategory of Landau-Ginzburg models (which has isolated hypersurface singularities as objects and matrix factorisations as 1-morphisms). Partly this is meant as background for work on semantics of linear logic involving this structure. You might the talking boards in LC001 useful.

• Organiser: Daniel Murfet.
• Venue: the Rising Sea.

Our principal reference is

• N. Carqueville, D. Murfet, “Adjoints and defects in Landau-Ginzburg models”, Adv. Math 2016.

But see below for (much) more.

Schedule

Dates are AEDT.

• 13-1-22 (Dan Murfet): Introduction Part 1 (notes, video).
• 20-1-22 (Dan Murfet): Introduction Part 2 (video).
• 27-1-22 (Rohan Hitchcock): Bicategories Part 1 (notes, video).
• 3-2-22 (Daniel Murfet): From dynamical systems to quadratic forms (notes, video).
• 10-2-22 (Daniel Murfet): From quadratic forms to bicategories (notes, video).
• 17-2-22 (Rohan Hitchcock): Matrix factorisations and geometry (notes, video).
• 24-2-22 (Rohan Hitchcock): The perturbation lemma I (notes, video).
• 3-3-22 (Rohan Hitchcock): The perturbation lemma II (notes, video).
• 14-7-22 (Rohan Hitchcock): Composition in LG (video).
• 18-7-22 (Daniel Murfet): Matrix factorisation and quantum error correcting codes (video).
• 4-8-22 (Rohan Hitchcock): The cut operation Part 1 (video).
• 11-8-22 (Rohan Hitchcock): The cut operation Part 2 (video).
• 25/8/22 (Rohan Hitchcock): Differentiation and Euclidean division (video).
• 20/10/22 (Rohan Hitchcock): Differentiation, division and the bicategory of Landau-Ginzburg models (MSc thesis presentation) (video, pocket).

Notes

• Rohan Hitchcock: idempotent completions.

References

• D. Eisenbud, “Homological algebra on a complete intersection, with an application to group representations”, Trans. Amer. Math. Soc. 260 (1980), 35–64.
• For the relation between Clifford algebras and matrix factorisations see R.-O. Buchweitz, D. Eisenbud, J. Herzog “Cohen-Macaulay modules on quadrics”.

• M. Khovanov and L. Rozansky, “Matrix factorizations and link homology”, Fund. Math. 199 (2008), 1–91.

• 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”).
• 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 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).
• 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.
• 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.