Categorical Bernstein operators and the Boson-Fermion correspondence
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Categorical Bernstein operators and the Boson-Fermion correspondence Nicolle S. González1
© Springer Nature Switzerland AG 2020
Abstract We prove a conjecture of Cautis and Sussan providing a categorification of the BosonFermion correspondence as formulated by Frenkel and Kac. We lift the Bernstein operators to infinite chain complexes in Khovanov’s Heisenberg category 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 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 H by lifting various Littlewood-Richardson branching isomorphisms to the Karoubian envelope of H. Mathematics Subject Classification 05E05 · 17B68 · 17B69 · 17B65 · 18G35 · 22E65
1 Introduction Building upon his ideas to categorify quantum groups and associated TQFTs, Igor Frenkel conjectured that conformal field theory could be categorified as well. As a starting point, he suggested that the Boson-Fermion correspondence should admit a categorification. One of the first breakthroughs came when Khovanov categorified the simplest Heisenberg algebra (see Sect. 4, [1,25]).1 Shortly after, Cautis and Licata [2] defined Heisenberg categories associated to every affine Dynkin diagram. Using this framework, they defined categorical vertex operators lifting the homogeneous realization of the basic representation of quantum affine algebras within the Frenkel1 In his original paper, Khovanov only proved injectivity h → K (H) and conjectured surjectivity. The 0
full bijection was not proved until much later in a recent paper by Brundan, Savage and Webster [1].
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Nicolle S. González [email protected] Department of Mathematics, UCLA, 520 Portola Plaza, Los Angeles, CA 90095-1555, USA 0123456789().: V,-vol
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Kac-Segal construction [11,33]. Specifically, they lifted vertex operators to infinite chain complexes in a Heisenberg category, inducing 2-representations of quantum affine algebras. The Boson-Fermion correspondence as formulated by Frenkel [13] and Kac [23] is analogous to the Frenkel-Kac-Segal construction in that it also defines vertex operators from a Heisenberg algebra. However, in the Boson-Fermion correspondence these vertex operators induce an action of the Clifford algebra instead of the quantum affine algebra. Motivated by this, Cautis and Sussan [6] conjectured a categorical formulation of the Boson-Fermion correspondence within the framework of Khovanov’s Heisenberg category H [25]. One of our main results is the proof of this conjecture (see Theorem 5.10). Our proof relies on a family of chain complexes in the homotopy category of H which
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