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Abstract Using the invariant theory of arc spaces, we find minimal strong generating sets for certain cosets of affine vertex algebras inside free field algebras that are related to classical Howe duality. These results have several applications. First, for any vertex algebra $${{\mathcal {V}}}$$, we have a surjective homomorphism of differential algebras $$\mathbb {C}[J_{\infty }(X_{{{\mathcal {V}}}})] \rightarrow \text {gr}^{F}({{\mathcal {V}}})$$; equivalently, the singular support of $${{\mathcal {V}}}$$ is a closed subscheme of the arc space of the associated scheme $$X_{{{\mathcal {V}}}}$$. We give many new examples of classically free vertex algebras (i.e., this map is an isomorphism), including $$L_{k}({{\mathfrak {s}}}{{\mathfrak {p}}}_{2n})$$ for all positive integers $$n$$ and $$k$$. We also give new examples where the kernel of this map is nontrivial but is finitely generated as a differential ideal. Next, we prove a coset realization of the subregular $${{\mathcal {W}}}$$-algebra of $${{\mathfrak {s}}}{{\mathfrak {l}}}_{n}$$ at a critical level that was previously conjectured by Creutzig, Gao, and the 1st author. Finally, we give some new level-rank dualities involving affine vertex superalgebras.
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This content will become publicly available on June 1, 2026
Conformal Blocks on Smoothings via Mode Transition Algebras
Here we introduce a series of associative algebras attached to a vertex operator algebra V of CFT type, called mode transition algebras, and show they reflect both algebraic properties of V and geometric constructions on moduli of curves. Pointed and coordinatized curves, labeled by admissible V-modules, give rise to sheaves of coinvariants. We show that if the mode transition algebras admit multiplicative identities satisfying certain natural properties (called strong identity elements), these sheaves deform as wanted on families of curves with nodes. This provides new contexts in which coherent sheaves of coinvariants form vector bundles. We also show that mode transition algebras carry information about higher level Zhu algebras and generalized Verma modules. To illustrate, we explicitly describe the higher level Zhu algebras of the Heisenberg vertex operator algebra, proving a conjecture of Addabbo–Barron.
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- Award ID(s):
- 2401420
- PAR ID:
- 10616315
- Publisher / Repository:
- Springer Nature
- Date Published:
- Journal Name:
- Communications in Mathematical Physics
- Volume:
- 406
- Issue:
- 6
- ISSN:
- 0010-3616
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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