Associated varieties of modules over Kac-Moody algebras and $C_2$-cofiniteness of W-algebras

TL;DR

The paper builds a comprehensive BRST-based bridge between modules over affine Kac–Moody algebras and affine $W$-algebras, establishing a precise identification $X_{H^{\frac{\infty}{2}+0}_{f}(M)} \cong X_M \cap \mathcal{S}$ for graded modules and proving the Feigin–Frenkel conjecture that the associated varieties of $G$-integrable admissible representations lie in the nilcone and are closures of specific nilpotent orbits. It develops jet-space and Slodowy-slice machinery, proves key vanishing theorems for BRST cohomology, and shows the BRST reduction preserves and reflects geometric data of associated varieties. These results yield a broad C2-cofiniteness proof for many simple $W$-algebras, including all exceptional ones, and provide explicit orbit–theoretic classifications of associated varieties for admissible representations. The work thus advances the chiralization program linking Premet’s finite $W$-algebras to the representation theory of affine algebras with deep implications for rationality and modular behavior of W-algebras.

Abstract

First, we establish the relation between the associated varieties of modules over Kac-Moody algebras \hat{g} and those over affine W-algebras. Second, we prove the Feigin-Frenkel conjecture on the singular supports of G-integrable admissible representations. In fact we show that the associated variates of G-integrable admissible representations are irreducible G-invariant subvarieties of the nullcone of g, by determining them explicitly. Third, we prove the C_2-cofiniteness of a large number of simple W-algebras, including all minimal series principal W-algebras and the exceptional W-algebras recently discovered by Kac-Wakimoto.

Theorems & Definitions (90)

• Lemma 2.1.1: Ara12
• Definition 2.2.1
• Lemma 2.2.2
• Lemma 2.2.3
• Lemma 2.6.1
• Lemma 3.1.1
• Lemma 3.2.1
• Proposition 3.4.1
• Corollary 3.4.2
• Proposition 3.5.1