A note on the bicategory of Landau-Ginzburg models (ℒ𝒢K)

Abstract

The bicategory of Landau-Ginzburg models denoted by ℒ𝒢K possesses adjoints and this helps in explaining a certain duality that exists in the setting of Landau-Ginzburg models in terms of some specified relations. The construction of ℒ𝒢K is reminiscent of, but more complex than, the construction of the bicategory of associative algebras and bimodules. In this paper, we review this complex but very inspiring construction in order to expose it more to pure mathematicians. In particular, we spend some time explaining the intricate construction of unit morphisms in this bicategory from a new vantage point. Besides, we briefly discuss how this bicategory could be constructed in more than one way using the variants of the Yoshino tensor product. Furthermore, without resorting to Atiyah classes, we prove that the left and right unitors in this bicategory have direct right inverses but do not have direct left inverses.

Keywords

• bicategory of landau-ginzburg models
• matrix factorizations
• tensor product
• polynomials