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arxiv: 2505.10439 · v3 · pith:PRYMWIGTnew · submitted 2025-05-15 · 🧮 math.QA · math.CT· math.RT

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

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Pith reviewed 2026-05-22 14:50 UTC · model grok-4.3

classification 🧮 math.QA math.CTmath.RT
keywords Feigin-Frenkel dualitycritical levelDeligne categoriescomplex rankW-algebrasSegal-Sugawara vectorsDrinfeld-Sokolov reductionPoisson vertex algebras
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The pith

Feigin-Frenkel duality at the critical level extends to complex rank by interpolating centers and W-algebras inside Deligne categories.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the Feigin-Frenkel duality, which links centers of universal affine vertex algebras at the critical level to classical W-algebras, from ordinary ranks to complex ranks. It works inside Deligne's interpolating categories that make sense of representations of matrix groups for non-integer dimensions. On the affine side, Molev's higher Segal-Sugawara vectors are interpolated to describe the centers of the vertex algebras. On the W-algebra side, classical W-algebras for the complex-rank Lie algebras gl_λ and po_λ are built as Poisson vertex algebras by carrying out Drinfeld-Sokolov reduction with an interpolated Adler-Gelfand-Dickey bracket. Specialization back to positive integer rank recovers the ordinary duality in types A, B, and C, while also giving uniform constructions for Lie superalgebras and a complex-rank version of the universal Bethe algebra.

Core claim

The centers of universal affine vertex algebras at the critical level are described by interpolated higher Segal-Sugawara vectors in Deligne categories, and the classical W-algebras for Feigin's Lie algebras of complex rank gl_λ and po_λ are realized as Poisson vertex algebras via Drinfeld-Sokolov reduction with an interpolated Adler-Gelfand-Dickey bracket; identifying these two constructions extends Feigin-Frenkel duality to complex rank, with exact recovery upon specialization to positive integer ranks in types A, B, and C.

What carries the argument

Interpolation, inside Deligne categories, of Molev's higher Segal-Sugawara vectors on the affine side and of the Adler-Gelfand-Dickey bracket realizing Drinfeld-Sokolov reduction on the W-algebra side.

Load-bearing premise

Molev's higher Segal-Sugawara vectors and the Adler-Gelfand-Dickey bracket can be interpolated to Deligne categories while preserving the algebraic relations that make the centers and the Drinfeld-Sokolov reductions match the ordinary case after specialization.

What would settle it

For a concrete non-integer value of λ, compute the interpolated center explicitly and check whether it equals the Poisson center of the W-algebra obtained from the interpolated Drinfeld-Sokolov reduction; mismatch for any such λ would falsify the claimed extension.

read the original abstract

In this paper, we extend Feigin-Frenkel duality at the critical level to complex rank by identifying two seemingly unrelated constructions in complex rank. On the affine side, we interpolate Molev's construction of higher Segal-Sugawara vectors and thereby describe the centers of universal affine vertex algebras at the critical level in Deligne's interpolating categories. On the $\mathcal{W}$-side, we construct the classical $\mathcal{W}$-algebras associated with Feigin's Lie algebras of complex rank $\mathfrak{gl}_{\lambda}$ and $\mathfrak{po}_{\lambda}$ as Poisson vertex algebras, realizing their Drinfeld-Sokolov reduction via an interpolated Adler-Gelfand-Dickey bracket. Upon specialization to positive integer rank in types A, B, and C, this recovers the usual Feigin-Frenkel duality at the critical level. As applications, we obtain a uniform construction of several families of higher Segal-Sugawara vectors for Lie superalgebras and recover a complex-rank analogue of the universal Bethe algebra.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper extends Feigin-Frenkel duality at the critical level to complex ranks λ by interpolating Molev's higher Segal-Sugawara vectors to describe centers of universal affine vertex algebras in Deligne categories Rep(gl_λ), and by realizing classical W-algebras for Feigin's gl_λ and po_λ as Poisson vertex algebras via Drinfeld-Sokolov reduction with an interpolated Adler-Gelfand-Dickey bracket. Specialization to positive integers recovers the standard duality in types A, B, C; applications include uniform higher Segal-Sugawara vectors for Lie superalgebras and a complex-rank universal Bethe algebra.

Significance. If the interpolation preserves the required Poisson and commutation relations, the result supplies a uniform categorical framework linking affine centers and classical W-algebras beyond integer rank, with concrete applications to superalgebras and Bethe algebras. The explicit recovery of known integer-rank results and the use of established external objects (Deligne categories, Molev vectors, AGD bracket) are strengths that ground the construction.

major comments (2)
  1. [Section defining interpolated AGD bracket and DS reduction] The central identification equates the interpolated center (from Molev vectors in Deligne Rep(gl_λ)) with the Poisson vertex algebra obtained by DS reduction of the interpolated AGD bracket. The manuscript must explicitly verify that the interpolated bracket satisfies the Jacobi identity and Leibniz rule inside the tensor category for generic λ (not merely upon specialization), as this is load-bearing for the Poisson structure and for the resulting Poisson ideal to coincide with the image of the center; see the construction in the section defining the interpolated AGD bracket and the subsequent DS reduction.
  2. [Section on interpolation of Molev vectors] The interpolation of Molev's higher Segal-Sugawara vectors must be shown to preserve the necessary algebraic relations without additional cocycle or associator corrections in the Deligne category; the current argument relies on formal substitution into generating functions, but Deligne categories are semisimple only for λ not a non-negative integer, so closure of the infinite series of relations requires explicit checking for generic λ.
minor comments (2)
  1. [Introduction] Notation for the complex rank parameter λ and the categories Rep(gl_λ), Rep(po_λ) should be introduced with a brief reminder of their definition and semisimplicity properties in the introduction or preliminaries.
  2. A short table or diagram comparing the integer-rank and complex-rank constructions side-by-side would improve readability of the recovery statement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The positive assessment of the paper's significance is appreciated. Below we address the two major comments point by point, agreeing that explicit verifications strengthen the presentation. We will revise the manuscript to include these checks.

read point-by-point responses

  1. Referee: [Section defining interpolated AGD bracket and DS reduction] The central identification equates the interpolated center (from Molev vectors in Deligne Rep(gl_λ)) with the Poisson vertex algebra obtained by DS reduction of the interpolated AGD bracket. The manuscript must explicitly verify that the interpolated bracket satisfies the Jacobi identity and Leibniz rule inside the tensor category for generic λ (not merely upon specialization), as this is load-bearing for the Poisson structure and for the resulting Poisson ideal to coincide with the image of the center; see the construction in the section defining the interpolated AGD bracket and the subsequent DS reduction.

    Authors: We agree that an explicit verification of the Jacobi identity and Leibniz rule for the interpolated Adler-Gelfand-Dickey bracket within the tensor category is a useful addition for rigor. The bracket is defined by direct substitution of the complex parameter λ into the classical formulas, which are rational functions of the rank. The Jacobi and Leibniz identities are polynomial equations in the generators that hold identically for all positive integer ranks; hence they hold as identities in λ and extend to generic λ via the universal property of the Deligne category Rep(gl_λ). To make this fully explicit as requested, we will insert a short subsection that carries out the verification directly in the category, confirming that the Poisson ideal generated by the center coincides with the expected kernel without additional corrections for generic λ. revision: yes

  2. Referee: [Section on interpolation of Molev vectors] The interpolation of Molev's higher Segal-Sugawara vectors must be shown to preserve the necessary algebraic relations without additional cocycle or associator corrections in the Deligne category; the current argument relies on formal substitution into generating functions, but Deligne categories are semisimple only for λ not a non-negative integer, so closure of the infinite series of relations requires explicit checking for generic λ.

    Authors: The interpolation proceeds by substituting λ into the generating functions for Molev's vectors. For generic λ (i.e., not a non-negative integer) the category is semisimple, so the standard tensor structure applies without extra associator issues in the relevant endomorphism algebras. The algebraic relations are identities in the polynomial ring over the rank parameter and therefore carry over by the interpolation functor. We acknowledge that an explicit confirmation of closure for the infinite series, ruling out cocycle corrections, would address the concern directly. In the revision we will add a brief argument (or short appendix) verifying this by functoriality and by checking that the first few relations extend inductively without modification. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claims rely on external interpolations and recover independent integer-rank results

full rationale

The paper extends Feigin-Frenkel duality by interpolating Molev's higher Segal-Sugawara vectors into Deligne categories for the affine side and constructing classical W-algebras via an interpolated Adler-Gelfand-Dickey bracket for the W-side. Upon specialization to positive integer ranks, it recovers the standard duality in types A, B, and C. These recoveries serve as independent external benchmarks rather than tautological definitions. No load-bearing step equates a derived object to its input by construction, renames a known result, or reduces to a self-citation chain; the interpolation step is presented as preserving relations verified by specialization to known cases. The derivation is therefore self-contained with independent mathematical content.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of Deligne interpolating categories that behave well for complex rank, the interpolability of Molev's Segal-Sugawara vectors, and the existence of an interpolated Adler-Gelfand-Dickey bracket that realizes Drinfeld-Sokolov reduction.

free parameters (1)
  • complex rank λ
    Parameter that labels the complex-dimensional Lie algebras gl_λ and po_λ and controls the interpolation.
axioms (1)
  • domain assumption Deligne's interpolating categories provide a well-defined categorical framework for representations of GL_n when n is complex
    Invoked to host the interpolated affine vertex algebra centers and W-algebra constructions.

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