Chapter 5 of 10

§G-orbifolds

We construct VOAs by taking fixed points of automorphism groups, using the subalgebra and ideal properties from the preceding chapter.

§5.1 The fixed-point VOA

Definition 5.1 (Fixed-point subalgebra and orbifold)

Let $G\leq\operatorname{Aut}(V)$ $Aut (V)$ . The fixed-point subalgebra is

V^G=\{a\in V:ga=a\text{ for every }g\in G\}, \qquad Y_{V^G}(a,x)=Y_V(a,x)|_{V^G}.

When $G$ is finite, we call $V^G$ the $G$ $G$ -orbifold of $V$ $V$ .

Proposition 5.2 (Inherited conformal structure)

For any $G\leq\operatorname{Aut}(V)$ $Aut (V)$ , $V^G$ is a full VOA subalgebra of $V$ $V$ , with

\mathbf1_{V^G}=\mathbf1_V,\qquad \omega_{V^G}=\omega_V,\qquad (V^G)_h=(V_h)^G,\qquad c_{V^G}=c_V.

CFT type is inherited. In particular, $V$ is an extension of $V^G$ $V^{G}$ , proper whenever $G\ne\{1\}$ ${1}$ .

Exercise. Prove Proposition 5.2 using Proposition 4.2 and the identity

g\bigl(Y(a,x)b\bigr)=Y(ga,x)gb=Y(a,x)b \quad(a,b\in V^G).

Thus taking fixed points restricts the states while retaining the conformal field; it does not divide the central charge by $|G|$ $∣ G ∣$ .

§5.2 Averaging

For the rest of this chapter, assume $G$ is finite.

Proposition 5.3 (Averaging projection)

The averaging operator

R_G=\frac1{|G|}\sum_{g\in G}g

preserves weights and satisfies

R_G^2=R_G,\qquad \operatorname{im}R_G=V^G, \qquad V=V^G\oplus\ker R_G.

For $u\in V^G$ and $v\in V$ $V$ ,

R_G\bigl(Y(u,x)v\bigr)=Y(u,x)R_Gv, \qquad R_G\bigl(Y(v,x)u\bigr)=Y(R_Gv,x)u.

Consequently, both summands are preserved by the fields of $V^G$ $V^{G}$ .

Exercise. Prove Proposition 5.3 by averaging the automorphism identity. The projection $R_G$ is generally not a VOA homomorphism: the identities above require one state to be invariant.

§5.3 Simplicity

Theorem 5.4 (Simplicity of finite orbifolds)

If $V$ is simple and $G\leq\operatorname{Aut}(V)$ is finite, then $V^G$ is simple.

Proof.

We use the cyclicity result that, for any nonzero $u$ in a simple VOA $V$ $V$ , the coefficients of $Y(a,x)u$ $Y (a, x) u$ , as $a$ ranges over $V$ $V$ , span $V$ $V$ ; see Dong–Li–Mason, Proposition 2.1.

Let $0\ne I\subseteq V^G$ be an ideal and choose $0\ne u\in I$ $I$ . By cyclicity, there are finitely many $a_i\in V$ and Laurent polynomials $f_i(x)$ such that

\mathbf1=\sum_i\operatorname{Res}_x f_i(x)Y(a_i,x)u.

Apply $R_G$ and use Proposition 5.3:

\mathbf1=\sum_i\operatorname{Res}_x f_i(x)Y(R_Ga_i,x)u\in I.

Since $R_Ga_i\in V^G$ $V^{G}$ , each residue lies in $I$ $I$ . Creation now gives $a=\left.Y(a,x)\mathbf1\right|_{x=0}\in I$ for every $a\in V^G$ $V^{G}$ . Thus $I=V^G$ $V^{G}$ .

§5.4 Fixed points in stages

Proposition 5.5 (Subgroups and successive orbifolds)

If $H\leq G$ $G$ , then

V^G\subseteq V^H\subseteq V

are full VOA inclusions. If $H\trianglelefteq G$ $G$ , the quotient $G/H$ acts on $V^H$ by VOA automorphisms through $(gH)a=ga$ $g a$ , and

(V^H)^{G/H}=V^G.

Exercise. Check that normality makes the quotient action well-defined and that being fixed by $G/H$ inside $V^H$ is equivalent to being fixed by $G$ $G$ . The quotient action need not be faithful for a general $V$ $V$ ; the displayed equality uses its fixed vectors.

The construction above requires no rationality assumption. Questions about modules, rationality, and extensions involving twisted sectors will be treated after module theory is introduced.

§5.5 References

Chongying Dong, Haisheng Li, Geoffrey Mason, Compact automorphism groups of vertex operator algebras, Proposition 2.1 and Theorem 2.4: cyclicity and simplicity of fixed-point VOAs.

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