Vertex Operator Algebras and Representation Theory

Vertex operator algebras (VOAs) provide a rigorous algebraic framework for the operator product expansions of two-dimensional conformal field theories. At their core lies a graded vector space equipped with a vertex operation that encodes local fields, a vacuum vector and a translation operator, satisfying axioms of locality and associativity. Representation theory of VOAs investigates modules—generalised highest-weight representations—seeking to classify irreducible modules, understand their fusion products, and determine the behaviour of characters under modular transformations. In rational cases, finiteness conditions such as C₂-cofiniteness guarantee only finitely many simple modules and ensure that fusion rules define a semisimple tensor category. Beyond the rational realm, logarithmic theories admit indecomposable but non-semisimple modules, leading to richer tensor structures and novel modular phenomena. VOAs and their representations have deep connections to monstrous Moonshine, the theory of subfactors, quantum groups and knot invariants, as well as applications in statistical mechanics and string theory. Ongoing research continues to extend the scope of VOAs to non-cofinite examples, superconformal and fractional-level models, and to explore links with number theory, algebraic geometry and category theory.

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Technical terms

Vertex operator algebra (VOA): A graded vector space with a state–field correspondence satisfying axioms of locality, vacuum existence and translation covariance, modelling chiral algebras of conformal field theories.

Module: A representation space on which the VOA’s vertex operators act, generalising highest-weight modules and allowing for indecomposable structures in logarithmic cases.

Fusion rules: Algebraic prescriptions for decomposing the tensor or fusion product of two modules into a direct sum of irreducible or indecomposable modules.

Modular invariance: The property that the graded characters of modules transform among each other under the action of the modular group on the torus parameter, essential for consistency of conformal field theories.

C₂-cofiniteness: A finiteness condition requiring that the quotient of a VOA by the subspace spanned by elements of the form v₋₂w is finite-dimensional, often implying semisimplicity and rationality.

References

1. Bosonic Ghostbusting: The Bosonic Ghost Vertex Algebra Admits a Logarithmic Module Category with Rigid Fusion. Communications in Mathematical Physics (2022).
2. Unitary and non-unitary N = 2 minimal models. Journal of High Energy Physics (2019).
3. Cosets, characters and fusion for admissible-level osp ( 1 | 2 ) minimal models. Nuclear Physics B (2019).