Holomorphic modularity of the Vk(g) trace series at admissible levels

Determine whether, for the simple affine vertex algebra V_k(g) associated to a simple Lie algebra g, the series tr_{V_k(g)} e^{2πi T (L_0 − 24 C_k)} is a holomorphic modular function on the upper half-plane Im T > 0 if and only if the level k is either a principal admissible level or a subprincipal admissible level. Here C_k = k · dim g / (k + h∨), and principal/subprincipal admissible levels are the rational levels specified by k = −h∨ + p/u with the respective arithmetic conditions on u and bounds on p.

Background

In the non-twisted case associated to a simple Lie algebra g, the irreducible highest weight module L(kΛ_0) carries the structure of the simple affine vertex algebra V_k(g) at level k. Admissible levels are rational and split into principal and subprincipal types, distinguished by number-theoretic conditions on the denominator u and lower bounds on the numerator p.

The paper proves modular invariance of normalized characters for subprincipal admissible levels (Theorem 5.1) and provides explicit modular transformation formulas. Conjecture 0.1 asks for a complete characterization of levels k for which the V_k(g) trace series yields a holomorphic modular function, asserting that this occurs if and only if k is principal or subprincipal admissible.