Graded Unitarity in Vertex Algebras

• Graded unitarity is defined as the compatibility between the graded structure and a positive-definite Hermitian form, reflecting physical constraints from 4D SCFTs.
• Filtration constraints, including the R-filtration and conformal grading, ensure that vertex operators preserve the refined energy and symmetry levels of the state space.
• Classification implications restrict physical VOAs to specific families, such as (2,p) minimal models and boundary-admissible affine Kac–Moody cases, consistent with 4D SCFT correspondence.

Graded unitarity for vertex algebras is a refinement of the standard unitarity condition, requiring compatibility between the graded structure of the vertex algebra and the Hermitian form introduced on its state space. The concept is particularly motivated by the properties of vertex algebras constructed from four-dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs), where the resulting vertex algebra typically fails to be unitary in the ordinary sense, yet retains a physically meaningful "graded" positive-definite structure reflective of the parent theory (Ardehali et al., 31 Jul 2025).

1. Motivation and Definition of Graded Unitarity in the SCFT/VOA Correspondence

The impetus for introducing graded unitarity arises from the observation that VOAs realized via the SCFT/VOA correspondence inherit a wealth of algebraic structure and representation-theoretic constraints from their four-dimensional origins. In this framework, a “graded-unitary” vertex algebra $(\mathcal{V},(-,-),\mathfrak{R}_\bullet,\rho)$ comprises:

• a vertex algebra $\mathcal{V}$ with a conformal weight grading (usually half-integral) and an additional $\mathfrak{R}$-grading inherited from $SU(2)_R$ symmetry,
• an invariant (bilinear or sesquilinear) form $(-,-)$,
• an $\mathfrak{R}$-filtration such that $\mathcal{V} = \bigoplus_{h,R} \mathcal{V}_{h,R}$,
• and an anti-linear conjugation $\rho$.

The structure is arranged so that after a refinement and standard Gram–Schmidt procedure, a sesquilinear form $\langle x | y \rangle = ((\sigma\circ\rho)(x), y)$ (with $\sigma$ a grading-refinement operator, e.g., $\sigma|_{\mathcal{V}_{R}} = i^{2R}$) is positive-definite on the whole state space. Thus, even when the parent VOA is not unitary (e.g., negative central charge, indecomposable representations), graded unitarity provides a robust Hilbert space structure compatible with the gradings inherited from the SCFT.

The physical motivation is that unitarity in four dimensions is always preserved, so the two-dimensional chiral algebra—obtained by topological or chiral reduction—must encode this in a nontrivial graded way, even when ordinary unitarity fails. The graded structure thus reflects the preserved $SU(2)_R$-charges and refined conformal weights that arise in the reduction.

2. Structural Properties and Filtration Constraints

Vertex algebras arising in these circumstances typically possess:

• a Virasoro subalgebra with a central charge $c$ fixed by the 4D anomaly,
• (possibly) affine Kac–Moody subalgebras at prescribed levels, reflecting flavor symmetries,
• a precisely structured $\mathfrak{R}$-filtration compatible with the field content and OPEs.

This $\mathfrak{R}$-filtration, or more generally an $\mathcal{F}$-filtration, organizes the states by “refined” energy levels, i.e., via conformal weight and internal symmetries (often $R$-charge). Compatibility with graded unitarity requires both:

• the positive-definiteness of the inner product on each graded piece,
• and preservation under the vertex operator map: $$Y(a,z)\mathcal{F}^k \subset \sum_{l\geq k-\ast} \mathcal{F}^l [[z,z^{-1}]]$$, ensuring OPEs respect the filtration.

These conditions restrict the possible degeneracies, and—in the presence of the $\mathfrak{R}$-filtration—admit only a subset of possible VOA representations that are “physical” from the perspective of the original four-dimensional theory (Ardehali et al., 31 Jul 2025).

3. Classification Implications for Virasoro and Affine VOA Families

The compatibility of a vertex algebra with graded unitarity imposes powerful constraints on the possible operator content and central charges:

• Virasoro VOAs: The only central charges for which a graded-unitary structure is possible correspond to the $(2,p)$ minimal models. These are precisely the cases that arise as chiral algebras of certain four-dimensional SCFTs—other central charges generally violate the positivity of the inner product or the grading compatibility.
• Affine Kac–Moody VOAs: For affine algebras (notably the cases of $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$), graded unitarity restricts the admissible levels to “boundary admissible” values, coinciding with the levels known to realize the SCFT/VOA correspondence.

This is summarized in the table below:

Underlying Subalgebra	Grading Constraint	Graded-Unitary Cases
Virasoro	$\mathfrak{R}$-filtration, positive-definite	$(2,p)$ minimal models
Affine Kac–Moody ($\mathfrak{sl}_2$ or $\mathfrak{sl}_3$)	Filtration, level admissibility	Boundary admissible levels

Note: “Boundary admissible” corresponds to levels that fit W-algebra quantum Hamiltonian reductions or to levels that match 4D flavor symmetry anomalies.

These constraints explain why only specific families of Virasoro and affine Kac–Moody vertex algebras appear in physically meaningful SCFT/VOA correspondences.

4. Filtration Formalism and Compatibility Conditions

Mathematical precision is critical in the construction:

• The VOA state space is bigraded, $V = \bigoplus_{h,R} V_{h,R}$, with $h$ (conformal weight) and $R$ (e.g., $SU(2)_R$ charge).
• The filtration is bounded from below, i.e., there is a lowest $R$-weight for each $h$.
• The action of the vertex operator $Y(a,z)$ is filtration-preserving (or, for non-chiral VOAs, preserves a total degree).
• The Hermitian form $\langle -, -\rangle$ is positive-definite and orthogonal on different graded components.

Explicitly, in examples with Virasoro symmetry: $$[L_m, L_n] = (m-n)L_{m+n} + \frac{c}{12}(m^3-m)\delta_{m+n,0}$$ is realized so that $L_0$ and the $\mathfrak{R}$-grading operators measure orthogonal degrees, and positivity in the graded sense is enforced.

5. Implications and Further Developments

Enforcing graded unitarity cuts down the set of admissible VOAs and modules to those compatible with physical expectations in four dimensions. In practice, this rules out many mathematically allowable—but physically unphysical—vertex algebras.

Key consequences include:

• The classification program for SCFT/VOA vertex algebras amounting to the determination of all graded-unitary VOAs with a given subalgebra structure and compatibility with prescribed filtrations.
• The possibility of transferring further four-dimensional structures, such as extra symmetries or dualities, to constraints or additional symmetry in the two-dimensional context.

Graded unitarity is thus an essential organizing principle in aligning vertex algebra representation theory with the physical data emerging from higher-dimensional SCFTs and underpins a wide array of recent developments relating chiral algebras, 4D SCFT dynamics, and the geometry of moduli spaces (Ardehali et al., 31 Jul 2025).

6. Summary

Graded unitarity encapsulates the structural legacy of four-dimensional unitarity in the language of vertex algebras, requiring a positive-definite Hermitian form compatible with both the conformal grading and the refined $\mathfrak{R}$-filtration or other symmetry gradings. Only those vertex algebras whose operator content, OPE, and representation theory are compatible with this structure reflect the full spectrum of chiral observables in physically admissible 4D $\mathcal{N}=2$ SCFTs. Central charges, levels, and module decomposition are accordingly tightly constrained, underlining the fundamental importance of graded unitarity in the modern landscape of mathematical physics and SCFT/VOA dualities.