We prove a conjecture of Cautis and Sussan providing a categorification of the Boson-Fermion correspondence as formulated by Frenkel and Kac. We lift the Bernstein operators to infinite chain complexes in Khovanov’s Heisenberg category \({\mathcal {H}}\) and from them construct categorical analogues of the Kac-Frenkel fermionic vertex operators. These fermionic functors are then shown to satisfy categorical Clifford algebra relations, solving a conjecture of Cautis and Sussan. We also prove another conjecture of Cautis and Sussan demonstrating that the categorical Fock space representation of \({\mathcal {H}}\) is a direct summand of the regular representation by showing that certain infinite chain complexes are categorical Fock space idempotents. In the process, we enhance the graphical calculus of \({\mathcal {H}}\) by lifting various Littlewood-Richardson branching isomorphisms to the Karoubian envelope of \({\mathcal {H}}\).

中文翻译：

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分类伯恩斯坦算子和玻色子-费米翁对应

我们证明了Cautis和Sussan的猜想提供了Frenkel和Kac提出的玻色子-费米翁对应关系的分类。我们将Bernstein算子提升到Khovanov's Heisenberg类别\（{\ mathcal {H}} \）中的无限链复合体，并从中构造Kac-Frenkel铁氧体顶点算子的分类类似物。然后证明这些费米仿函数满足分类Clifford代数关系，从而解决了Cautis和Sussan的猜想。我们还证明了Cautis和Sussan的另一个猜想，证明了\（{\ mathcal {H}} \）的Fock空间分类表示通过证明某些无限链复合物是Fock空间绝对幂等范畴，是正则表示的直接加法式。在此过程中，我们通过将各种Littlewood-Richardson分支同构提升到\（{\ mathcal {H}} \}的Karoubian包络中，来增强\（{\ mathcal {H}} \）的图形演算。