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BIMSA-清华量子对称性讨论班
BIMSA-清华量子对称性讨论班
Fully exact and fully dualizable module categories: towards non-semisimple fully extended TQFTs
Fully exact and fully dualizable module categories: towards non-semisimple fully extended TQFTs
演讲者
Azat Gainutdinov
时间
2026年06月05日 10:30 至 12:00
地点
A3-3-301
线上
Zoom 242 742 6089
(BIMSA)
摘要
Let C be a finite braided tensor category over any field and C-mod denotes the monoidal 2-category of finite C-module categories. We define fully exact C-module categories, a subclass of exact C-module categories that is stable under the relative Deligne product. The monoidal 2-category of fully exact module categories strictly contains those of invertible and separable module categories. We show that each internal algebra of a fully exact module category is projectively separable, a generalization of separable algebras involving projective objects of C. In the semisimple C case, a C-module category is fully exact if and only if it is separable. In general, fully exact module categories are not dualizable inside their class, but if they are, they are fully dualizable objects in C-mod. We call such module categories perfect. Our main result is that perfect module categories form a rigid monoidal 2-subcategory C-perf containing all fully dualizable objects of C-mod. For symmetric braiding, we show that a module category is fully exact if and only if it is perfect.
As a detailed example, we classify perfect module categories over the symmetric tensor category Cpx(2) of 2-periodic chain complexes and compute their relative Deligne products, and the categories of 1-morphisms. In this case, C-perf has a continuum of isoclasses of indecomposable objects, with non-semisimple finite tensor categories of 1-endomorphisms, and only finitely many isoclasses of separable module categories. This work is motivated by fully extended TQFT constructions in the context of cobordism hypothesis. In particular, we get a continuum family of (framed) fully extended 2d TQFTs with values in the symmetric 2-category Cpx(2)-mod. This is the joint work arXiv:2601.22017 with R. Laugwitz.
As a detailed example, we classify perfect module categories over the symmetric tensor category Cpx(2) of 2-periodic chain complexes and compute their relative Deligne products, and the categories of 1-morphisms. In this case, C-perf has a continuum of isoclasses of indecomposable objects, with non-semisimple finite tensor categories of 1-endomorphisms, and only finitely many isoclasses of separable module categories. This work is motivated by fully extended TQFT constructions in the context of cobordism hypothesis. In particular, we get a continuum family of (framed) fully extended 2d TQFTs with values in the symmetric 2-category Cpx(2)-mod. This is the joint work arXiv:2601.22017 with R. Laugwitz.