Foliation Matter Phase and Godbillon-Vey Invariant

It has been a main topic in today's physics to classify matter phases. Especially, topologically ordered phases are attracting much attension from broad perspective. However, most of mathematical structures other than the topology are not investigated yet in physics. In this talk, we present a physical model which is strongly connected to the foliation structure of the space manifold, and its field theoretical description. In such a foliation field theory, we can see the structure is highly connected to the mathematical invariant of foliation structures called Godbillon-Vey invariant. This work would be a fiest step toward shining a light on mathematical structures used in physics. This work is based on arXiv:2408.05048 with Hiromi Ebisu, Masazumi Honda, and Soichiro Shimamori.