Categorical Bernstein operators and the Boson-Fermion correspondence

Abstract

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 ${\mathcal {H}}$.

Appendix A: Examples

1.1 A.1.

Let $a=2$ and $b=3$. By Theorem 6.6 we know that $\mathsf {B}_1 \otimes \mathsf {B}_3 \simeq \mathsf {B}_2 \otimes \mathsf {B}_2 [1]$. We show explicitly how such an equivalence could be true. In particular, $\mathsf {B}_2 \otimes \mathsf {B}_2$ is given by total complex of the bi-complex,

Applying Proposition 4.6 followed by Proposition 4.8 we obtain an isomorphism with the bi-complex,

Applying Lemma 3.7 along the isomorphisms denoted in red, we see that

$$\begin{aligned} \mathsf {B}_2 \otimes \mathsf {B}_2 \simeq \cdots \rightarrow \mathsf {P}^{(3,2)}\mathsf {Q}[1] \rightarrow \mathsf {P}^{(2,2)}[0]. \end{aligned}$$

Likewise, $\mathsf {B}_1 \otimes \mathsf {B}_3$ is given by the total complex of the bi-complex,

which after applications of Propositions 4.6 and 4.8 becomes,

Once again, cancelling summands along the isomorphisms in red and blue, by Lemma 3.7 we have

$$\begin{aligned} \mathsf {B}_1 \otimes \mathsf {B}_3 \simeq \cdots \rightarrow \mathsf {P}^{(3,2)}\mathsf {Q}[2] \rightarrow \mathsf {P}^{(2,2)}[1]. \end{aligned}$$

Continuing in this manner along all homological degrees it becomes apparent how one obtains the homotopy equivalence $\mathsf {B}_1 \otimes \mathsf {B}_3 \simeq \mathsf {B}_2 \otimes \mathsf {B}_2[1]$.

1.2 A.2

Now suppose $a=b=2$ and consider $\mathsf {B}_1 \otimes \mathsf {B}_2$. By Theorem 6.6 this product should be nullhomotopic. To see why, consider the following bi-complex,

Once again, by Propositions 4.6 and 4.8 there an isomorphism with the bi-complex below, with isomorphisms along the red and blue arrows.

After applying Lemma 3.7 along the red and blue isomorphisms, all chain groups in homological degrees zero and one can be canceled. Continuing in this manner one can eliminate the summands in all degrees, thus obtaining the desired nullhomotopy.

1.3 A.3

Lastly, let $a=1$ and $b=2$ and suppose $\mathsf {Q}^{\otimes 4} =0$. By Theorem 6.14 we know that $\mathsf {B}_1 \otimes \mathsf {B}_2^* \simeq \mathsf {B}_1^*\otimes \mathsf {B}_0$. In particular, we have that $\mathsf {B}_1 \otimes \mathsf {B}_2^*$ is given by the total complex of the bi-complex below,

Thus, after applying Lemma 3.7 along the isomorphism in red, we see that

$$\begin{aligned} \mathsf {B}_1 \otimes \mathsf {B}_2^* \simeq \mathsf {P}^{(2)}\mathsf {Q}^{(2,1)} [1] \rightarrow \mathsf {P}\mathsf {Q}^{(2)}[0] \rightarrow \mathsf {P}\mathsf {P}\mathsf {Q}^{(3)}[-1]. \end{aligned}$$

Now, if we consider $\mathsf {B}_1^* \otimes \mathsf {B}_0$ we see it is the total complex of the bi-complex

which after applying Propositions 4.6 and 4.7 is homotopic to,

But $\mathsf {Q}^{(2,1)}=\mathsf {Q}^{(2,1)^t}$, thus after Gaussian elimination along the isomorphisms in red we obtain

$$\begin{aligned} \mathsf {B}_1^*\otimes \mathsf {B}_0 \simeq \mathsf {P}^{(2)}\mathsf {Q}^{(2,1)^t} [2] \rightarrow \mathsf {P}\mathsf {Q}^{(2)}[1] \rightarrow \mathsf {P}\mathsf {P}\mathsf {Q}^{(3)}[0] \simeq \mathsf {B}_1\otimes \mathsf {B}_2^*[1]. \end{aligned}$$