Graded-unitary Vertex Algebras Demonstrate Kähler Manifold Structure in Semi-Infinite Cohomology

The study of vertex algebras, mathematical structures underpinning much of theoretical physics, recently gained new momentum with the introduction of ‘graded unitarity’, a concept extending traditional unitarity to a broader class of algebras. Christopher Beem and Niklas Garner investigate the semi-infinite cohomology of these graded-unitary vertex algebras, specifically those linked to affine current algebras, and reveal a surprising connection to the geometry of compact Kähler manifolds. Their work demonstrates that this semi-infinite cohomology possesses a graded-unitary structure, establishing unitarity for a significant range of vertex operator algebras originating from supersymmetric field theories, and further uncovers a previously unknown symmetry mirroring the Lefschetz action in Kähler geometry. By establishing a deep analogy between vertex algebras and classical geometric structures, this research provides powerful new tools for understanding the mathematical foundations of theoretical physics and opens avenues for exploring related areas such as Poisson vertex algebras.

The research investigates the relative semi-infinite cohomology of graded-unitary vertex algebras possessing a chiral quantum moment map for an affine current algebra at twice the critical level. The study demonstrates that the relative semi-infinite chain complex for such a graded-unitary vertex algebra exhibits a structure analogous to that of differential forms on a compact Kähler manifold, thereby generalizing a strong form of the classic construction developed by Banks, Peskin and Frenkel, Garland, Zuckerman. Consequently, the research establishes that the relative semi-infinite cohomology is itself graded-unitary, which confirms graded unitarity for a large class of vertex operator algebras originating from three- and four-dimensional supersymmetric quantum field theories.

Semi-Infinite Cohomology and Vertex Algebra Structures

This research explores the deep connections between Vertex Operator Algebras, Conformal Field Theory, and geometric structures like Higgs branches and symplectic varieties. It forms part of a broader program seeking to establish a precise mathematical equivalence between certain physical theories and abstract algebraic structures. Scientists are investigating how to use semi-infinite cohomology to understand and classify VOAs, particularly those that arise from geometric data. Vertex Operator Algebras are powerful algebraic structures that generalize Lie algebras, appearing in conformal field theory, string theory, and representation theory.

They encode information about symmetries and fields in physical systems. Conformal Field Theory is a quantum field theory invariant under conformal transformations, used to describe critical phenomena and closely related to VOAs. Semi-Infinite Cohomology is a sophisticated tool from representation theory used to study VOAs, providing information about their structure and classifying their modules. Graded-Unitary VOAs possess additional structure, a grading and a unitary invariant bilinear form, crucial for ensuring a well-behaved and rich representation theory. Higgs Branches are geometric spaces parameterizing the vacuum states of supersymmetric gauge theories, closely related to the representation theory of VOAs.

Symplectic Varieties are geometric spaces equipped with a symplectic form, appearing in classical mechanics and quantum field theory. The central conjecture of this work, the SCFT/VOA Correspondence, states that there is a precise mathematical equivalence between certain Superconformal Field Theories and Vertex Operator Algebras. The research classifies a specific class of graded-unitary VOAs using semi-infinite cohomology, providing a deeper understanding of their structure and representation theory. Scientists explore how to realize VOAs geometrically, using Higgs branches and symplectic varieties, connecting abstract algebra and geometry.

The work provides evidence for the SCFT/VOA correspondence, demonstrating that the classified VOAs arise from geometric data associated with SCFTs. Researchers investigate constructing VOAs using free fields, fundamental objects in quantum field theory, providing a concrete way to build and understand these algebras. The technique of Poisson reduction is used to study the structure of VOAs and their associated geometric spaces, simplifying analysis and extracting key information. This research provides a rigorous mathematical framework for understanding the connections between VOAs, CFTs, and geometry.

It uses sophisticated tools from representation theory and algebraic geometry to classify VOAs and understand their properties. The work provides evidence for the SCFT/VOA correspondence, a major open problem in mathematical physics. This paper is a significant contribution to the field of vertex operator algebras with implications for both mathematics and physics.

Graded Unitarity in Non-Unitary Vertex Algebras

Scientists have established a novel structure termed graded unitarity for a specific class of vertex algebras, extending the concept of unitarity beyond traditional definitions. This work stems from investigations into four-dimensional superconformal field theories and their connection to vertex algebras realized through collections of local observables called Schur operators. Researchers discovered that while these vertex algebras are not unitary in the conventional sense, often possessing negative Virasoro central charges, they inherit a form of unitarity from their four-dimensional counterparts through a subtle, graded structure. The team defined graded unitarity by leveraging a Hermitian form derived from two-point functions within the four-dimensional theory, accounting for kinematical phases and pairing operators with their conjugates.

This approach assigns a positive norm to operators, establishing a consistent inner product within the vertex algebra. Investigations revealed constraints on the allowed forms of certain vertex algebras, specifically the M(p, q) type, limiting them to the form M(2, 2n + 3). Similarly, the allowed levels, k, for affine current algebras Lk(sl(2)) and Lk(sl(3)) were found to be precisely the boundary admissible levels. Further analysis demonstrated that the relative semi-infinite chain complex for these graded-unitary vertex algebras exhibits a structure analogous to differential forms on a compact Kähler manifold, generalizing a construction previously established by Banks-Peskin and Frenkel-Garland-Zuckerman.

Crucially, the team proved that the relative semi-infinite cohomology itself is graded-unitary, confirming the existence of this structure for vertex operator algebras arising from three- and four-dimensional supersymmetric field theories. Measurements also revealed an outer USp(2) action on the semi-infinite cohomology, mirroring the Lefschetz action in Kähler geometry, and demonstrated that the semi-infinite chain complex is quasi-isomorphic to its cohomology, analogous to the formality result for de Rham cohomology. These findings establish a powerful connection between vertex algebras, supersymmetric field theories, and geometric structures, opening new avenues for research in mathematical physics.

Vertex Algebras Mirror Kähler Manifold Geometry

This work establishes a deep connection between graded-unitary vertex algebras and the geometry of Kähler manifolds, revealing a surprising level of mathematical structure within these algebraic systems. Researchers demonstrated that the relative semi-infinite cohomology of graded-unitary vertex algebras, under specific conditions involving chiral moment maps, mirrors the properties of differential forms on compact Kähler manifolds. This analogy extends to the existence of a graded-unitary structure on the semi-infinite cohomology itself, establishing graded unitarity for a broad class of vertex operator algebras originating from supersymmetric field theories. Furthermore, the team identified an outer USp action on this cohomology, analogous to the Lefschetz action in Kähler geometry, and proved that the semi-infinite chain complex is quasi-isomorphic to its cohomology, a result comparable to the formality theorem for Kähler manifolds.

These findings significantly advance the understanding of vertex algebras and their relationship to geometric structures, offering new tools for studying both algebraic and geometric systems. The researchers acknowledge that their results rely on specific conditions regarding the existence of chiral moment maps and the properties of the associated affine current algebras, representing a limitation for broader applicability. Future research directions include exploring the implications of these findings for Poisson vertex algebras and derived Poisson reductions.