In this note, we determine the structure of the associative algebra generated by the differential operators
μ
¯
,
∂
¯
,
∂
\overline{\mu },\overline{\partial },\partial
, and
μ
\mu
that act on complex-valued differential forms of almost complex manifolds. This is done by showing that it is the universal enveloping algebra of the graded Lie algebra generated by these operators and determining the structure of the corresponding graded Lie algebra. We then determine the cohomology of this graded Lie algebra with respect to its canonical inner differential
[
d
,
−
]
\left[d,-]
, as well as its cohomology with respect to all its inner differentials.
Given a braided fusion category C, it is well known that the natural map F: C ⊠ Cbop → Z(C) from the square of C to the (Drinfeld) categorical center Z(C) is an equivalence if and only if C is modular. This provides a non-constructive structure theorem for Z(C) for the modular case. However, it is not clear how to construct the inverse. In this work, we provide an explicit construction using insights from a specific quantum field theory. In particular, we construct an adjoint functor for F that is its inverse precisely when C is modular. The witnessing natural transformations are also constructed as values at certain cobordism of a specific 4-dimensional extended topological quantum field theory, the Crane-Yetter model. Such construction provides a (partial) factorization of the structure of Z(C) even when C is not modular. It is useful for understanding the extended structure of the Crane-Yetter model (future work).
MSC: 18M20, 57K30, 57K40.
Given a braided fusion category C, it is well known that the natural map C C bop → Z(C) from the square of C to the (Drinfeld) categorical center Z(C) is an equivalence if and only if C is modular. However, it is not clear how to construct the inverse and the natural isomorphisms. In this work, we provide an explicit construction using insights from a specific quantum field theory, and explore how the equivalence fails for the degenerate cases.
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