VOA Extensions, Orbifolds, Cosets, and Commutants

Extensions, orbifolds, and commutants are three different operations on a VOA. An extension adjoins modules and must pass a locality and integral-grading test; an orbifold takes fixed points and acquires twisted sectors; a commutant selects fields mutually nonsingular with a subalgebra. Their representation theories interact, but no sector count alone identifies the resulting VOAs or proves rationality.

Required background. Modules, Intertwining Operators, and Tensor Categories supplies fusion, braiding, and twists. C₂-Cofiniteness, Rationality, and Regularity supplies the finiteness distinctions. Cosets and Orbifolds supplies the physical constructions.

Helpful background. Four-Dimensional Chiral-Algebra Sectors and Lost Information illustrates why a protected chiral algebra does not retain every datum of its parent theory.

Local extensions and fixed points

Let $V$ be a simple VOA whose module category has a vertex tensor structure, and let

$$U\cong\bigoplus_i m_iM_i$$

as a $V$-module. To make $U$ a VOA extension with the same conformal vector, one needs an intertwining-operator multiplication $\mu:U\boxtimes U\to U$, a unit $V\to U$, associativity, braided commutativity, and trivial twist. Under semisimplicity, positivity of weights, and the other hypotheses stated by Huang, Kirillov, and Lepowsky, such extensions are equivalent to haploid commutative algebra objects $U$ with $\theta_U=1$ Huang, Kirillov, and Lepowsky 2015, Theorem 3.2, PDF pp. 10–15. Trivial twist means integral conformal spin on every summand used in an ordinary bosonic extension. It cannot be inferred merely from closure of fusion rules.

A simple current $J$ is an invertible simple module. A direct sum $\bigoplus_{g\in G}J_g$ can define a group-like extension only when its associativity obstruction is trivial and its monodromies permit a commutative multiplication. For one order-two current, nonintegral conformal weight gives $\theta_J=e^{2\pi ih_J}\neq1$ and blocks an ordinary local VOA extension. Depending on the twist, a super or more general graded extension may exist, but that is a different object.

For a finite automorphism group $G$, the orbifold $V^G$ is the fixed-point subVOA. The state-space restriction alone is not the complete orbifold theory: $g$-twisted $V$-modules contribute sectors of $V^G$. Their construction uses fractional modes and a twisted Jacobi identity. Rationality of $V$ does not, without further hypotheses, justify every desired rationality or completeness statement for $V^G$.

Lattice involution as a controlled example

The first application is the lattice and orbifold setting of Cosets and Orbifolds. Let $L$ be a positive-definite even unimodular lattice and $V_L$ its lattice VOA. Lift the isometry $\alpha\mapsto-\alpha$ to an involution $\theta$. Then

$$V_L=V_L^+\oplus V_L^-,\qquad V_L^+=(V_L)^{\langle\theta\rangle}.$$

Because $L=L^*$, $V_L$ has one untwisted irreducible module. There is also one irreducible $\theta$-twisted $V_L$-module $V_L^T$, which splits into its two $\theta$ eigenspaces. Abe and Dong’s classification therefore gives exactly four inequivalent irreducible $V_L^+$-modules,

$$V_L^+,\qquad V_L^-,\qquad (V_L^T)^+,\qquad (V_L^T)^-,$$

for this unimodular case Abe and Dong 2002, Theorem 7.7 and Remark 7.9, PDF pp. 27–28. This enumerates untwisted and twisted sectors; it does not derive their fusion or assert the same four-sector answer for a nonunimodular lattice, where cosets in $L^*/L$ add modules.

An elementary fixture checks the fixed-point decomposition. For $\alpha\in L$, the combinations $e^\alpha\pm e^{-\alpha}$ have $\theta$ parity $\pm$, while a single $e^\alpha$ is not invariant unless $\alpha=0$. Oscillator monomials acquire parity from the number of sign-reversed Heisenberg generators. This directly checks the untwisted $V_L^\pm$ grading.

Commutants and cosets

For a conformal subVOA $A\subset V$, the commutant is

$$\operatorname{Com}_V(A)= \{v\in V:Y(a,z_1)Y(v,z_2)\text{ has no singular part for every }a\in A\}.$$

Equivalently, $a_nv=0$ for every $a\in A$ and $n\geq0$, with the standard compatibility conditions. When the conformal vectors split, the coset stress tensor is $\omega_V-\omega_A$ and its central charge is $c_V-c_A$. A branching decomposition

$$V\cong\bigoplus_i A_i\otimes C_i$$

suggests $C_i$ as commutant modules, but equality with a proposed coset VOA and a double-commutant identity require proofs. In the lattice orbifold, the four-sector classification of $V_L^+$ does not determine a commutant inside it; one must compute which modes annihilate each candidate and then verify generation.

Adversarial extension test

Take a module $J$ whose fusion square contains the vacuum but whose lowest weight is $h_J\notin\mathbb Z$. The vector space $V\oplus J$ may close under candidate fusion coefficients, yet its twist on $J$ is not the identity. A field from $J$ acquires a nontrivial phase on exchange, so the proposed multiplication is not braided-commutative as an ordinary VOA. The strongest surviving conclusion is a graded vector-space candidate, possibly for a super or generalized extension after a separate analysis. Neither its orbifold nor its commutant data repairs the missing locality.

Exercises

1. Show why integral weight is necessary for a summand of an ordinary extension with trivial twist.

Solution

On a homogeneous module of lowest weight $h$, the categorical twist is $e^{2\pi iL_0}$ and acts on the lowest space by $e^{2\pi ih}$. Trivial twist therefore requires $h\in\mathbb Z$; descendants differ from $h$ by integers.

2. For $L=L_1\oplus L_2$, verify $V_L\cong V_{L_1}\otimes V_{L_2}$ and identify the commutant of $V_{L_1}$.

Solution

Oscillator and group-algebra factors split over the orthogonal sum, giving the tensor product. Modes from $V_{L_1}$ act trivially on the second factor, while any nonvacuum first-factor state has a singular action with some field of $V_{L_1}$. Hence $\operatorname{Com}_{V_L}(V_{L_1})=V_{L_2}$.

References

• Abe, Toshiyuki, and Chongying Dong. “Classification of Irreducible Modules for the Vertex Operator Algebra $V_L^+$: General Case.” Journal of Algebra 273 (2004), 657–685. Open PDF.
• Huang, Yi-Zhi, Alexander Kirillov Jr., and James Lepowsky. “Braided Tensor Categories and Extensions of Vertex Operator Algebras.” Communications in Mathematical Physics 337 (2015), 1143–1159. DOI. Open PDF.