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Using the Zhu algebra for a certain category of C-graded vertex algebras V, we prove that if V is finitely Ω-generated and satisfies suitable grading conditions, then V is rational, i.e., it has semi-simple representation theory, with a one-dimensional level zero Zhu algebra. Here, Ω denotes the vectors in V that are annihilated by lowering the real part of the grading. We apply our result to the family of rank one Weyl vertex algebras with conformal element ωμ parameterized by μ∈C and prove that for certain non-integer values of μ, these vertex algebras, which are non-integer graded, are rational, with a one-dimensional level zero Zhu algebra. In addition, we generalize this result to appropriate C-graded Weyl vertex algebras of arbitrary ranks.
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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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- Free Publicly Accessible Full Text
-
Accepted Manuscript
- Journal Article:
- https://doi.org/10.1007/s00220-025-05237-1
