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Interpolating Feigin-Frenkel Duality at the Critical Level to Matrices of Complex Size

Andrew Riesen

TL;DR

The work develops a comprehensive framework to interpolate Feigin–Frenkel duality to complex rank. It constructs two families of objects in Deligne’s interpolating categories: centers of universal affine vertex algebras at the critical level in complex rank, and the classical Adler–Gelfand–Dickey/W–algebras $ rak{W}( rak{g}_ u)$ for Lie algebras of complex rank, realized as Poisson vertex algebras. It then proves interpolated FF isomorphisms $ rak{f}_ u$ (and $ rak{f}_T^B, rak{f}_T^C$) that map the interpolated centers to the interpolated $ rak{W}$–algebras, compatible with parity dualities and recovering the classical FF duality when the rank is an integer. The results unify and extend the FF correspondence to a complex-rank setting, providing explicit interpolated Segal–Sugawara vectors and Adler-map descriptions, and establishing a robust algebraic framework for complex-rank representation theory and its applications in geometric Langlands-type contexts. The construction uses polynomial structures in $T$ and $ f(T)$-valued morphisms, ensuring well-behaved evaluations at fixed ranks and preserving the Poisson-vertex-algebra structure through the interpolation process.

Abstract

In this paper, we extend Feigin-Frenkel duality at the critical level to the setting of complex rank. This is accomplished by considering the center of a vertex algebra in Deligne's interpolating categories, along with Feigin's Lie algebras of complex rank, $\mathfrak{gl}_λ$ and $\mathfrak{po}_λ$. More precisely, we define the universal affine vertex algebras associated with Lie algebras in $\underline{\mathrm{Re}}\mathrm{p}(\mathrm{GL}_α,\mathbb{F})$, $\underline{\mathrm{Re}}\mathrm{p}(\mathrm{O}_α,\mathbb{F})$ and $\underline{\mathrm{Re}}\mathrm{p}(\mathrm{Sp}_α,\mathbb{F})$, and describe their centers at the critical level explicitly by interpolating Molev's construction of Segal-Sugawara vectors. Using the formalism of Poisson vertex algebras, we identify a natural set of generators for the Drinfeld-Sokolov reduction of $\mathfrak{gl}_λ$ and $\mathfrak{po}_λ$, denoted by $\mathcal{W}(\mathfrak{gl}_λ)$ and $\mathcal{W}(\mathfrak{po}_λ)$, respectively. Finally, we show that the interpolated Feigin-Frenkel isomorphism maps the interpolated Segal-Sugawara vectors to these generators.

Interpolating Feigin-Frenkel Duality at the Critical Level to Matrices of Complex Size

TL;DR

The work develops a comprehensive framework to interpolate Feigin–Frenkel duality to complex rank. It constructs two families of objects in Deligne’s interpolating categories: centers of universal affine vertex algebras at the critical level in complex rank, and the classical Adler–Gelfand–Dickey/W–algebras for Lie algebras of complex rank, realized as Poisson vertex algebras. It then proves interpolated FF isomorphisms (and ) that map the interpolated centers to the interpolated –algebras, compatible with parity dualities and recovering the classical FF duality when the rank is an integer. The results unify and extend the FF correspondence to a complex-rank setting, providing explicit interpolated Segal–Sugawara vectors and Adler-map descriptions, and establishing a robust algebraic framework for complex-rank representation theory and its applications in geometric Langlands-type contexts. The construction uses polynomial structures in and -valued morphisms, ensuring well-behaved evaluations at fixed ranks and preserving the Poisson-vertex-algebra structure through the interpolation process.

Abstract

In this paper, we extend Feigin-Frenkel duality at the critical level to the setting of complex rank. This is accomplished by considering the center of a vertex algebra in Deligne's interpolating categories, along with Feigin's Lie algebras of complex rank, and . More precisely, we define the universal affine vertex algebras associated with Lie algebras in , and , and describe their centers at the critical level explicitly by interpolating Molev's construction of Segal-Sugawara vectors. Using the formalism of Poisson vertex algebras, we identify a natural set of generators for the Drinfeld-Sokolov reduction of and , denoted by and , respectively. Finally, we show that the interpolated Feigin-Frenkel isomorphism maps the interpolated Segal-Sugawara vectors to these generators.
Paper Structure (36 sections, 57 theorems, 276 equations)

This paper contains 36 sections, 57 theorems, 276 equations.

Table of Contents

  1. Introduction
  2. The Center of $V^k(\mathfrak{g})$ at Critical Level
  3. Categorical Background
  4. Tensor Category Basics
  5. Deligne's Interpolating Categories $\underline{\mathrm{Re}}\mathrm{p}(\mathrm{GL}_{\alpha},\mathbb{F}), \underline{\mathrm{Re}}\mathrm{p}(\mathrm{O}_{\alpha},\mathbb{F}), \underline{\mathrm{Re}}\mathrm{p}(\mathrm{Sp}_{\alpha},\mathbb{F})$
  6. Interpolating Type $\mathrm{GL}_n$
  7. Interpolating $\mathrm{O}_n,\mathrm{Sp}_{m}$
  8. Polynomial Structures in Interpolating Categories
  9. Evaluating Polynomial Structures
  10. Categorical Vertex Algebras
  11. Vertex Algebras and Basic Results
  12. Polynomial Vertex Algebras in $\mathcal{C}_{\alpha,\mathbb{F}}$ and $V^k(\underline{\mathfrak{g}}_{{\raisebox{1.5pt}{$\alpha$}}})$
  13. Interpolating the Center at Critical Level
  14. The Center at Critical Level and its Poisson Vertex Algebra Structure
  15. Segal-Sugawara Vectors of Classical Lie Algebras
  16. ...and 21 more sections

Key Result

Proposition 2.4

St_Interpolating_Deligne

Theorems & Definitions (132)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • proof
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 122 more