The study of quantum field theory has lead to myriad insights in pure mathematics. In the context of four-dimensional $\mathcal{N} = 2$ superconformal field theory (SCFT), one of the hallmark examples of this appears in work of Beem, Lemos, Liendo, Peelaers, Rastelli, and van Rees [BLLPRvR15]: to each such superconformal field theory there is an associated vertex operator algebra (VOA): \[
\textrm{SCFT}~\mathcal{T} \rightsquigarrow \textrm{VOA}~\mathbb{V}[\mathcal{T}].
\] and is often dubbed the SCFT / VOA correspondence. Although the whole four-dimensional SCFT $\mathcal{T}$ lacks a mathematically satisfactory definition, VOAs date back to Borcherds' work on the Moonshine conjecture [Bor86] and have been objects of great interest to representation theorists, geometers, and number theorists alike. Thus, the SCFT / VOA correspondence allows one to probe a certain "protected'' sector of $\mathcal{T}$ within the world of pure mathematics.
The SCFT / VOA correspondence is quite far from surjective: the physics of the SCFT $\mathcal{T}$ imposes many constraints on the VOA $\mathbb{V}[\mathcal{T}]$. Some of these constraints were already witnessed in the original work [BLLPRvR15]; for example, the central charge $c$ (a numerical invariant associated to any VOA coming from its natural Virasoro symmetry) is constrained by four-dimensional physics to be negative. Ardehalli, Beem, Lemos, and Rastelli recently introduced a much more rigid requirement on these VOAs in[ABLR25]: the vector space underlying the VOA $\mathbb{V}[\mathcal{T}]$ must be equipped with Hermitian inner product satisfying certain compatibility constraints with its VOA structure. They called such VOAs graded unitary.
In a recent paper [BG26] with fellow Oxford Mathematician Christopher Beem, I studied aspects of the relative semi-infinite cohomology of graded unitary VOAs; this cohomology can be thought of as an analogue of Marsden-Weinstein reduction for VOAs equipped with a Hamiltonian action of an affine Lie algebra (at twice-critical level) and arises in the SCFT / VOA correspondence when discussing superconformal gauge theories. One of the most striking observations that we made is that the complex underlying the relative semi-infinite cohomology of graded unitary VOAs (with suitable assumptions on the Hamiltonian $\widehat{\mathfrak{g}}$ action) bears a striking resemblance to the complex differential forms on a compact Kähler manifold equipped with the exterior differential. Namely, we find that the underlying VOA comes equipped with a positive-definite inner product (analogous to the Hodge inner product on differential forms), two pairs of adjoint differentials with matching Laplacians (analogous to $\partial$, $\overline{\partial}$ and their adjoints and the Hodge Laplacian), and the commensurate Hodge decomposition.
With this observation, many structural properties of differential forms on a compact Kähler manifold can be readily deduced for the relative semi-infinite complex of graded unitary VOAs mutatis mutandis. For example, the underlying VOA comes naturally equipped with an action of $\textrm{SU}(2)$ by unitary automorphisms and, although it does not preserve the differential, the action is inherited by the cohomology; in light of this analogy to Kähler manifolds, this is a version of the Lefschetz $\mathfrak{sl}(2)$ action on the homology of a compact Kähler manifold. One of the most famous such properties is due to Deligne, Griffiths, Morgan, and Sullivan [DGMS75] which asserts that compact Kähler manifolds are formal in the homotopy-theoretic sense, i.e. that the homotopy type of any compact Kähler manifold is uniquely determined by their homology. Similarly, although the homotopy theory of VOAs has yet to be developed, we are able to deduce that the vertex-algebraic homotopy type (whatever that is) of a relative semi-infinite complex of a graded unitary VOA is uniquely determined by its cohomology.
Niklas Garner is a Postdoctoral Research Associate in Oxford Mathematics
References
[ABLR25] A. A. Ardehali, C. Beem, M. Lemos, and L. Rastelli, Graded Unitarity in the SCFT/VOA Correspondence, 7 2025.
[BG26] C. Beem and N. Garner, On the semi-infinite cohomology of graded-unitary vertex algebras, J. Algebra 698 (2026), 172--223.
[BLLPRvR15] C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli, and B. C. van Rees, Infinite Chiral Symmetry in Four Dimensions, Commun. Math. Phys. 336 (2015), no. 3, 1359--1433.
[Bor86] R. Borcherds, Vertex algebras, kac-moody algebras, and the monster, Proc. Natl. Acad. Sci. 83 (1986), no. 10, 3068--3071.
[DGMS75] P. Deligne, P. Griffiths, J. Morgan, and D. Sullivan, Real homotopy theory of kähler manifolds, Invent. Math. 29 (1975), no. 3, 245--274.