[Paper Review] On the unitary structures of vertex operator superalgebras

Chunrui Ai, Xingjun Lin|arXiv (Cornell University)|Oct 29, 2015

Algebraic structures and combinatorial models18 references3 citations

TL;DR

This paper introduces the concept of unitary vertex operator superalgebras (VOSupAs), proving that those associated with unitary highest weight representations of the Neveu-Schwarz Lie superalgebra, Heisenberg superalgebras, and positive definite integral lattices are unitary. The key contribution is showing that any unitary VOSupA decomposes as a direct sum of strong CFT-type unitary simple VOSupAs, and classifying unitary VOSupAs generated by conformal weight ≤1 as tensor products of affine Kac-Moody and Heisenberg VOSupAs.

ABSTRACT

In this paper, the notion of unitary vertex operator superalgebra is introduced. It is proved that the vertex operator superalgebras associated to the unitary highest weight representations for the Neveu-Schwarz Lie superalgebra, Heisenberg superalgebras and to positive definite integral lattices are unitary vertex operator superalgebras. The unitary structures are then used to study the structures of vertex operator superalgebras, it is proved that any unitary vertex operator superalgebra is a direct sum of strong CFT type unitary simple vertex operator superalgebras. The classification of unitary vertex operator superalgebras generated by the subspaces with conformal weights less than or equal to $1$ is also considered.

Motivation & Objective

• To define and establish the notion of unitary vertex operator superalgebras as a generalization of unitary VOAs.
• To prove that VOSupAs from unitary highest weight representations of the Neveu-Schwarz Lie superalgebra, Heisenberg superalgebras, and positive definite integral lattices are unitary.
• To study the structural properties of unitary VOSupAs, particularly their decomposition into simple components.
• To classify unitary VOSupAs generated by subspaces of conformal weight ≤1, focusing on those generated by weight 1 components.

Proposed method

• Introduce a positive definite invariant Hermitian form on VOSupAs, defining unitary VOSupAs via anti-linear involutions.
• Use the unitary structure of highest weight modules for Lie superalgebras to induce unitary structures on associated VOSupAs.
• Apply the theory of contravariant forms and semisimplicity to show that unitary VOSupAs decompose into direct sums of strong CFT-type simple VOSupAs.
• Analyze the conformal weight 1 subspace and use the decomposition of the vertex operator algebra into Heisenberg and lattice VOSupA components.
• Leverage results from [DM1] and [X] on simple VOSupAs generated by weight 1/2 and weight 1 subspaces to classify the full structure.
• Prove that unitary VOSupAs generated by conformal weights ≤1 are isomorphic to tensor products of affine Kac-Moody VOSupAs and Heisenberg VOSupAs.

Experimental results

Research questions

• RQ1 What conditions ensure that a vertex operator superalgebra admits a unitary structure?
• RQ2 How do unitary structures on highest weight modules for Lie superalgebras induce unitary structures on the corresponding VOSupAs?
• RQ3 Can any unitary VOSupA be decomposed into a direct sum of simple, strong CFT-type unitary VOSupAs?
• RQ4 What is the structure of unitary VOSupAs generated by subspaces of conformal weight ≤1?
• RQ5 Which unitary VOSupAs generated by conformal weight 1 subspaces can be classified without assuming C2-cofiniteness or regularity?

Key findings

• The vertex operator superalgebras associated to unitary highest weight representations of the Neveu-Schwarz Lie superalgebra, Heisenberg superalgebras, and positive definite integral lattices are unitary VOSupAs.
• Any unitary VOSupA is isomorphic to a direct sum of strong CFT-type unitary simple VOSupAs.
• The unitary structure of VOSupAs associated to the Neveu-Schwarz and Heisenberg superalgebras is induced from the unitary structures of their respective highest weight modules.
• The conformal vector of a unitary VOSupA with abelian V1 is equal to the conformal vector of the Heisenberg subalgebra, implying V1 carries a positive definite even lattice structure.
• Unitary VOSupAs generated by conformal weights ≤1 are isomorphic to tensor products of VOSupAs associated to unitary highest weight representations of affine Kac-Moody algebras and Heisenberg algebras.
• A unitary simple VOSupA with V1 abelian and of dimension c (central charge) is isomorphic to VL for some positive definite even lattice L, under rationality or C2-cofiniteness conditions.