Center of the affine $\mathfrak{gl}_{n|1}$ at the critical level and pseudo-differential operators

TL;DR

The paper establishes that the center of the affine superalgebra $\\widehat{\\mathfrak{gl}}_{n|1}$ at the critical level is generated by the coefficients of the pseudo-differential operator $$(\partial_z-u_1(z))(\partial_z-u_2(z))\cdots (\partial_z-u_n(z))(\partial_z+u_{n+1}(z))^{-1}$$ with values in the Cartan subalgebra, presenting an affine Harish-Chandra isomorphism for the super case. It identifies this center with the Heisenberg coset of the regular $\\mathcal{W}$-superalgebra $\\mathcal{W}^{\\kappa_c}(\\mathfrak{gl}_{n|1})$, and shows its associated graded algebra is the algebra of affine supersymmetric polynomials, $\\operatorname{gr}\\mathfrak{z}(V^{\\kappa_c}(\\mathfrak{gl}_{n|1})) \\simeq \\Lambda^{n|1}_{\\text{aff}}$. A closed-form character formula is derived, matching the generating function for plane partitions with a pit condition, thereby confirming conjectures on the strong generation by higher Segal-Sugawara vectors. The work further develops a deformation to generic levels via the Heisenberg coset and pseudo-differential operators, and discusses a broader duality between regular and subregular $\\mathcal{W}$-algebras, Wakimoto realizations, and implications for the quantum geometric Langlands program in the super setting.

Abstract

We prove that the center of the affine Lie algebra $\widehat{\mathfrak{gl}}_{n|1}$ at the critical level is generated by the coefficients in the expansion of the pseudo-differential operator $(\partial_z-u_1(z))\cdots (\partial_z-u_n(z))(\partial_z+u_{n+1}(z))^{-1}$ taking values in the Cartan subalgebra. This is an affine analogue of the Harish-Chandra isomorphism in the finite case. The key ingredient of the proof is the identification of the center with the Heisenberg coset of the regular W-superalgebra of $\mathfrak{gl}_{n|1}$ at the critical level, whose associated graded algebra is realized as the affine supersymmetric polynomials. Based on this, we derive a character formula for the center, which coincides with the generating function of plane partitions with a pit condition. We also prove that the Heisenberg coset at generic levels has a similar interpretation in terms of pseudo-differential operators that deform the one at the critical level.

Figures (1)

• Figure :

Theorems & Definitions (43)

• Theorem A: Theorem \ref{['them: Detecting the FF center']}/\ref{['THM: FF center for gln1 vis pD op']}/\ref{['thm: character formula of the center']}
• Theorem B: Theorem \ref{['prop: coset for Wsalg']}
• Theorem 3.1: CGN21
• Theorem 3.2
• proof
• Proposition 3.3
• proof
• Theorem 3.4
• proof
• Remark 3.5