[Paper Review] Towards a categorical boson-fermion correspondence

Tian Yin|arXiv (Cornell University)|Oct 31, 2017

Algebraic structures and combinatorial models8 references3 citations

TL;DR

This paper constructs a categorical boson-fermion correspondence by introducing a modified Heisenberg category that serves as the Heisenberg counterpart to Khovanov's categorification of the Heisenberg algebra. It lifts vertex operators from the Clifford algebra to endofunctors on the Fock space categorification, establishing a derived equivalence between the Heisenberg and Clifford categories via a chain of Morita equivalences, thereby categorifying the classical boson-fermion correspondence at the level of categories and functors.

ABSTRACT

We construct the Heisenberg counterpart of a Clifford categorification. It is a modification of Khovanov's Heisenberg categorification. We express generators of the Heisenberg category as a complex of generators of the Clifford category. Certain vertex operators associated to the Clifford algebra are lifted to endofunctors of the Fock space categorification.

Motivation & Objective

• To construct a categorical boson-fermion correspondence by defining a Heisenberg counterpart to the Clifford categorification in [16].
• To generalize the DG algebra construction from characteristic 2 to characteristic zero, enabling a representation-theoretic interpretation of the geometric structure underlying the Clifford categorification.
• To lift vertex operators from the Clifford algebra to endofunctors on the Fock space categorification using a derived equivalence.
• To establish a derived Morita equivalence between the Heisenberg algebra B and the Clifford algebra F via a chain of quasi-isomorphisms and derived categories.

Proposed method

• Construct a $$\mathbf{k}$-algebra $B$ containing $\bigoplus_{n=0}^\infty \mathbf{k}[S(n)]$ as a subalgebra, whose homotopy category $\mathcal{B} = \operatorname{Kom}(B)$ categorifies the bosonic Fock space.
• Define the Heisenberg category $\mathcal{DH}$ as a full triangulated monoidal subcategory of $D(B^e)$, generated by $B$, $P$, and $Q$, where $P$ and $Q$ are bimodules corresponding to induction and restriction functors.
• Generalize the DG algebra $R$ from $\mathbb{F}_2$ to $\mathbf{k}$ of characteristic zero, with $R_0$ formal and quasi-isomorphic to its cohomology, and define the Clifford category $\mathcal{CL}$ as a full triangulated monoidal subcategory of $D(R^e)$ generated by $R$ and $T(i)$ bimodules.
• Establish a chain of Morita equivalences: $R_0 \leftrightarrow H(R_0) \leftrightarrow \widetilde{H}(R_0) \cong F \leftrightarrow B$, showing that $B$ and $F$ are Morita equivalent.
• Construct two objects $\overline{Q}, \overline{P}$ in $D(R_0^e)$ that lift the expressions $g(q)$ and $g(p)$ in terms of Clifford generators $t_i$, and prove via derived equivalence $\mathcal{G}: D(B^e) \to D(R_0^e)$ that $\mathcal{G}(Q) \cong \overline{Q}$ and $\mathcal{G}(P) \cong \overline{P}$.

Experimental results

Research questions

• RQ1 How can the classical boson-fermion correspondence be lifted to a categorical level via categorified Heisenberg and Clifford algebras?
• RQ2 What modifications to Khovanov’s Heisenberg categorification are necessary to establish a symmetric counterpart to the Clifford categorification?
• RQ3 How can vertex operators from the Clifford algebra be lifted to endofunctors on the Fock space categorification in the derived category?
• RQ4 What is the role of the derived equivalence $\mathcal{G}$ in relating the Heisenberg and Clifford categories, and how does it preserve the structure of the generators?

Key findings

• The Heisenberg category $\mathcal{DH}$ is constructed as a full triangulated monoidal subcategory of $D(B^e)$, generated by $B$, $P$, and $Q$, and admits an infinite chain of adjoint pairs, extending beyond Khovanov’s original construction.
• The Clifford category $\mathcal{CL}$ is defined as a full triangulated monoidal subcategory of $D(R^e)$, generated by $R$ and $T(i)$, with classes $t_i = [T(i)]$ in the Grothendieck group satisfying the Clifford algebra relation $t_i t_j + t_j t_i = \delta_{|i-j|,1} 1$.
• The derived equivalence $\mathcal{G}: D(B^e) \to D(R_0^e)$ satisfies $\mathcal{G}(Q) \cong \overline{Q}$ and $\mathcal{G}(P) \cong \overline{P}$, where $\overline{Q}, \overline{P}$ are lifts of the vertex operator expressions $g(q), g(p)$ in terms of $t_i$, proving Theorem 5.3.
• The algebras $B$ and $F = \widetilde{H}(R_0)$ are Morita equivalent, and $F$ is isomorphic to a quiver algebra, establishing a derived Morita equivalence between the Heisenberg and Clifford categorifications.
• The generating series $\overline{t}(z) = \sum_{i \in \mathbb{Z}} t_{2i+1} z^i$ and $t(z) = \sum_{i \in \mathbb{Z}} t_{2i} z^{-i}$ are associated to the Clifford algebra, and the action of $\mathcal{G}$ on $P$ and $Q$ corresponds to the action of these series on the Fock space.
• The map $\mathbf{x}: K_0(F_n) \to V_F^{(0)}$ satisfies $\mathbf{x}([\mathcal{S}_n(P(\lambda))]) = \tau \circ \sum_{t=0}^n (-1)^{s_t} t_{2s_t}(\mathbf{x}(\lambda))$, and this matches the symmetric function action $S_n(\mathbf{x}(\lambda))$, confirming the compatibility of the categorified action with the classical boson-fermion correspondence.