nLab Lagrangian cobordism

Idea

A Lagrangian cobordism is roughly a cobordism between Lagrangian submanifolds which is itself a Lagrangian submanifold in a suitable sense.

More in detail, given an ambient symplectic manifold $(X, \omega)$, a Lagrangian cobordism in $X$ is a cobordism with an embedding into $([0,1] \times \mathbb{R} \times X )$ as a Lagrangian submanifold (where $[0,1] \times \mathbb{R} \hookrightarrow \mathbb{R}^2$ is equipped with its canonical symplectic structure) and such that the boundary components are Lagrangian submanifolds of $X$. (for details see e.g. Biran-Cornea 11, 2.1.1).

Lagrangian cobordisms in $(X,\omega)$ arrange into a stable (infinity,1)-category which is supposed to be at least closely related to the Fukaya category of $(X,\omega)$ (Nadler-Tannaka 11).

Related concepts

• Fukaya category

References

The notion was introduced in

• Vladimir Arnold, Lagrange and Legendre cobordisms. I, II, Funkts. Anal. Prilozh. 14:3, 1–13 (1980) article; 14:4 (1980), 8–17 article

Further developments are in

• Paul Biran, Octav Cornea, Lagrangian cobordism I, (arxiv/1109.4984)

• Paul Biran, Octav Cornea, Lagrangian cobordism III(arxiv/1304.6032)

A stable $(\infty,1)$-category of Lagrangian cobordisms is discussed in

• David Nadler, Hiro Tannaka, A stable infinity-category of Lagrangian cobordisms, Adv. Math. 366 3 (2020) 107026 [arXiv:1109.4835, doi:10.1016/j.aim.2020.107026]

• Wenyuan Li, Lagrangian cobordism functor in microlocal sheaf theory I, arXiv:2108.10914