Chapter 6 of 10

§Commutants and cosets

The coset construction keeps the states whose fields commute with those of a chosen subalgebra. Its conformal vector will be the difference of the ambient and subalgebra conformal vectors, under the compatibility condition below.

§6.1 The commutant

Definition 6.1 (Commutant)

Let $U\subseteq V$ be a vertex subalgebra. Its commutant in $V$ is

\operatorname{Com}_V(U) =\{a\in V:Y(u,x)a\in V[[x]]\text{ for every }u\in U\}.

Thus $a$ belongs to the commutant precisely when $Y(u,x)a$ has no singular terms for every $u\in U$ $U$ .

Proposition 6.2 (Commuting fields)

For $a\in V$ $V$ , the following are equivalent:

\begin{aligned} a\in\operatorname{Com}_V(U) &\quad\Longleftrightarrow\quad [Y(u,x),Y(a,y)]=0\quad(u\in U)\\ &\quad\Longleftrightarrow\quad Y(a,x)u\in V[[x]]\quad(u\in U). \end{aligned}

Proof.

Taking the residue in the iterate variable in Proposition 2.10 gives

[Y(u,x),Y(a,y)] =\operatorname{Res}_t\,y^{-1}\delta\!\left(\frac{x-t}{y}\right) Y(Y(u,t)a,y).

The delta kernel is expanded in nonnegative powers of $t$ $t$ . If $Y(u,t)a$ is regular in $t$ $t$ , the integrand has no $t^{-1}$ term, so the commutator vanishes.

Conversely, apply the commuting-field identity to $\mathbf1$ and take the constant term in $y$ $y$ . On the left this gives $Y(u,x)a$ $Y (u, x) a$ . On the right, $Y(a,y)Y(u,x)\mathbf1=Y(a,y)e^{xT}u$ has no negative powers of $x$ $x$ , so $Y(u,x)a$ is regular. The last equivalence follows from skew-symmetry, $Y(a,x)u=e^{xT}Y(u,-x)a$ $e^{x T} Y (u, - x) a$ .

Proposition 6.3 (Vertex-algebra structure of the commutant)

The space $C=\operatorname{Com}_V(U)$ is a vertex subalgebra of $V$ $V$ , with vacuum $\mathbf1$ $1$ , translation $T|_C$ $T ∣_{C}$ , and fields $Y_C(a,x)=Y_V(a,x)|_C$ $Y_{V} (a, x) ∣_{C}$ .

Proof.

Creation gives $\mathbf1\in C$ $C$ . For $a,b\in C$ and $u\in U$ $U$ , Proposition 6.2 gives

Y(u,z)Y(a,x)b=Y(a,x)Y(u,z)b.

The right-hand side has no negative powers of $z$ $z$ , because $b\in C$ $C$ . Taking any coefficient in $x$ shows that every coefficient of $Y(a,x)b$ belongs to $C$ $C$ . Thus $Y(a,x)b\in C((x))$ $C ((x))$ , and Proposition 4.2 applies.

§6.2 The conformal vector of the coset

Now let $(U,\omega_U)$ be a VOA subalgebra of $(V,\omega_V)$ in the sense of Definition 4.3. The ambient field $Y_V(\omega_U,x)$ acts on all of $V$ $V$ , even though $\omega_U$ is the conformal vector of $U$ $U$ .

Theorem 6.4 (Coset VOA)

Let $U\subseteq V$ be a VOA subalgebra and set $C=\operatorname{Com}_V(U)$ $Com_{V} (U)$ . Assume

\omega_C:=\omega_V-\omega_U\in C.

Then $C$ is a VOA with conformal vector $\omega_C$ $ω_{C}$ , central charge and grading

c_C=c_V-c_U,\qquad C_h=C\cap V_h.

We call $(C,Y_C,\mathbf1,\omega_C)$ the coset VOA. If $V$ is of CFT type, so is $C$ $C$ .

Proof.

Write the conformal fields on $V$ as

\begin{aligned} Y_V(\omega_V,x)&=\sum_n L_n^Vx^{-n-2},\\ Y_V(\omega_U,x)&=\sum_n L_n^Ux^{-n-2},\\ L_n^C&=L_n^V-L_n^U. \end{aligned}

The operators $L_n^U$ satisfy the Virasoro relations on $V$ $V$ : this follows from Jacobi and the conformal-vector products in $U$ $U$ , just as in Proposition 3.3. Since $\omega_U\in U$ and $\omega_C\in C$ $C$ , their fields commute, so $[L_m^U,L_n^C]=0$ $0$ . Consequently,

\begin{aligned} [L_m^C,L_n^C] &=[L_m^V,L_n^V]-[L_m^U,L_n^U]\\ &=(m-n)L_{m+n}^C +\frac{c_V-c_U}{12}(m^3-m)\delta_{m+n,0}\operatorname{id}_V. \end{aligned}

For $a\in C$ $C$ , the series $Y_V(\omega_U,x)a$ is regular. Its $x^{-1}$ and $x^{-2}$ coefficients therefore vanish: $L_{-1}^Ua=L_0^Ua=0$ $0$ . Hence

L_{-1}^C|_C=T|_C,\qquad L_0^C|_C=L_0^V|_C.

The modes of $\omega_C\in C$ preserve $C$ $C$ , so these relations give translation compatibility and the inherited grading. Finite-dimensional weight spaces, the lower bound on weights, and CFT type follow from those of $V$ $V$ .

Remark 6.5 (Full subalgebras)

If $U$ is full, then $\omega_U=\omega_V$ $ω_{V}$ . For $a\in\operatorname{Com}_V(U)$ $Com_{V} (U)$ , regularity of $Y(\omega_V,x)a$ forces $L_0^Va=0$ $0$ . Thus, when $V$ is of CFT type,

\operatorname{Com}_V(U)=\mathbb C\mathbf1.

A nontrivial coset in this setting therefore requires a subalgebra with its own conformal vector.

§6.3 Elementary commutant relations

Proposition 6.6 (Inclusions and double commutants)

For vertex subalgebras $U_1\subseteq U_2\subseteq V$ $V$ ,

\operatorname{Com}_V(U_2)\subseteq\operatorname{Com}_V(U_1).

For every vertex subalgebra $U\subseteq V$ $V$ ,

U\subseteq\operatorname{Com}_V(\operatorname{Com}_V(U)).

Exercise. Use the definition for the first assertion and the symmetry of commuting fields for the second. Equality in the double-commutant inclusion is an additional property, not part of the coset construction.

§6.4 References

Hiroshi Yamauchi, A Theory of Simple Current Extensions of Vertex Operator Algebras and Applications to the Moonshine Vertex Operator Algebra, §3.8: commutants and cosets.
Yanjun Chu and Zongzhu Lin, The varieties of Heisenberg vertex operator algebras, §§1–2: semi-conformal subalgebras and complementary conformal vectors.

License · CC0 1.0 · Peerpedia collaboration terms