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Ordinary VOA character

Definition

Let $V$ be a conformal vertex algebra of CFT type with central charge $c$ , and let $M$ be an ordinary $V$ -module. Write $L_{M} (0)$ for the conformal grading operator and $M_{h} = {m \in M : L_{M} (0) m = h m}$ for its eigenspaces. The formal ordinary character is

$χ_{M} (τ) = {Tr}_{M} (q^{L_{M} (0) - c / 24}) = \sum_{h \in Spec (L_{M} (0))} \dim_{C} (M_{h}) q^{h - c / 24}, q = e^{2 π i τ}, τ \in H,$

where $H = {τ \in C : Im τ > 0}$ , and the sum is a formal trace indexed by the actual $L_{M} (0)$ -eigenvalues. Every coefficient is finite. If the spectrum is contained in one coset $h_{0} + Z_{\geq 0}$ , the formal expression is $q^{h_{0} - c / 24} \sum_{n \geq 0} \dim (M_{h_{0} + n}) q^{n}$ ; the exponent $h_{0} - c / 24$ need not be integral. The convention $q^{α} = e^{2 π i τ α}$ fixes the branch when $τ$ is in the upper half-plane. An analytic character is the holomorphic function defined when this series converges locally uniformly on $H$ . Convergence is not part of the formal trace definition.

Definition references

Ordinary module

Virasoro algebra in the $L_{m}$ convention

Facts & Assumptions

Given: $M$ is ordinary, with finite-dimensional $L_{M} (0)$ -eigenspaces and lower-bounded real parts of its weights. The Virasoro central element acts on $M$ as $c {id}_{M}$ .

[F1]

Ordinary module: $M$ has its $V$ -module action, and $L_{M} (0)$ acts semisimply with finite-dimensional eigenspaces whose occurring weights have real parts bounded below.

[F2]

Virasoro algebra in the $L_{m}$ convention: the modes use the bracket $[L_{m}, L_{n}] = (m - n) L_{m + n} + \frac{1}{12} (m^{3} - m) δ_{m + n, 0} C$ ; a representation has central charge $c$ when $C$ acts as $c id$ .

Proof

1.1givenF1F2algebra

The ordinary-module assumption makes $M$ a $V$ -module, so its module Jacobi identity gives the Virasoro mode commutator formula for $Y_{M} (ω, z)$ . Comparing the products of $ω$ in the stated conformal-vertex-algebra structure and [F2] gives the Virasoro relations on $M$ with the same central scalar $c$ : the central term is a multiple of $1$ , and the module vacuum axiom gives $Y_{M} (1, z) = {id}_{M}$ . Thus the shift in the exponent is the scalar $c / 24$ . Since $L_{M} (0)$ is semisimple by [F1], $M$ is the algebraic direct sum of its eigenspaces $M_{h}$ . On each finite-dimensional $M_{h}$ , $q^{L_{M} (0) - c / 24}$ is multiplication by the formal monomial $q^{h - c / 24}$ , so its trace is $\dim (M_{h}) q^{h - c / 24}$ .

2.1givenF1step 1.1algebra∎

Summing these finite-dimensional traces over the direct-sum eigenspaces gives the displayed formal character. On a single support coset $h_{0} + Z_{\geq 0}$ , grouping exponents by $n$ gives the stated fractional leading power times a formal power series in $q$ . This construction is coefficientwise and asserts no convergence on $H$ .

Remarks

This is the ordinary trace of the grading operator with no intertwining-operator insertions. The separate genus-one pseudo-trace construction uses a projective generalized module and geometrically modified logarithmic intertwining operators; it is not a synonym for $χ_{M}$ .

Proof status

The formal trace and its fractional-power convention are established above. Analytic convergence and modular covariance remain theorem obligations of the consuming items.