Introduction to Vertex Operator Algebras

--- Base version+++ Proposed version@@ -0,0 +1,142 @@+# 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.++## The commutant++::: definition [Commutant] {@def: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 $u\in U$.+:::++::: proposition [Commuting fields] {@prop:commutant-fields}+For $a\in V$, 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 @prop:jacobi-identity 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 $t$. If $Y(u,t)a$ is regular in $t$, 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 $y$. On the left this gives $Y(u,x)a$. On the right, $Y(a,y)Y(u,x)\mathbf1=Y(a,y)e^{xT}u$ has no negative powers of $x$, so $Y(u,x)a$ is regular. The last equivalence follows from skew-symmetry, $Y(a,x)u=e^{xT}Y(u,-x)a$.+:::++::: proposition [Vertex-algebra structure of the commutant] {@prop:commutant-va}+The space $C=\operatorname{Com}_V(U)$ is a vertex subalgebra of $V$, with vacuum $\mathbf1$, translation $T|_C$, and fields $Y_C(a,x)=Y_V(a,x)|_C$.+:::++::: proof+Creation gives $\mathbf1\in C$. For $a,b\in C$ and $u\in U$, @prop:commutant-fields gives++$$+Y(u,z)Y(a,x)b=Y(a,x)Y(u,z)b.+$$++The right-hand side has no negative powers of $z$, because $b\in C$. Taking any coefficient in $x$ shows that every coefficient of $Y(a,x)b$ belongs to $C$. Thus $Y(a,x)b\in C((x))$, and @prop:inherited-va applies.+:::++## The conformal vector of the coset++Now let $(U,\omega_U)$ be a VOA subalgebra of $(V,\omega_V)$ in the sense of @def:voa-subalgebra. The ambient field $Y_V(\omega_U,x)$ acts on all of $V$, even though $\omega_U$ is the conformal vector of $U$.++::: theorem [Coset VOA] {@thm:coset-voa}+Let $U\subseteq V$ be a VOA subalgebra and set $C=\operatorname{Com}_V(U)$. Assume++$$+\omega_C:=\omega_V-\omega_U\in C.+$$++Then $C$ is a VOA with conformal vector $\omega_C$, 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 $C$.+:::++::: 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 $V$: this follows from Jacobi and the conformal-vector products in $U$, just as in @prop:virasoro-covariance. Since $\omega_U\in U$ and $\omega_C\in C$, their fields commute, so $[L_m^U,L_n^C]=0$. 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 $a\in C$, the series $Y_V(\omega_U,x)a$ is regular. Its $x^{-1}$ and $x^{-2}$ coefficients therefore vanish: $L_{-1}^Ua=L_0^Ua=0$. Hence++$$+L_{-1}^C|_C=T|_C,\qquad L_0^C|_C=L_0^V|_C.+$$++The modes of $\omega_C\in C$ preserve $C$, 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 $V$.+:::++::: remark [Full subalgebras] {@rem:full-subalgebra-coset}+If $U$ is full, then $\omega_U=\omega_V$. For $a\in\operatorname{Com}_V(U)$, regularity of $Y(\omega_V,x)a$ forces $L_0^Va=0$. 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.+:::++## Elementary commutant relations++::: proposition [Inclusions and double commutants] {@prop:double-commutant}+For vertex subalgebras $U_1\subseteq U_2\subseteq V$,++$$+\operatorname{Com}_V(U_2)\subseteq\operatorname{Com}_V(U_1).+$$++For every vertex subalgebra $U\subseteq V$,++$$+U\subseteq\operatorname{Com}_V(\operatorname{Com}_V(U)).+$$+:::++**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.++## References++- Hiroshi Yamauchi, [A Theory of Simple Current Extensions of Vertex Operator Algebras and Applications to the Moonshine Vertex Operator Algebra](https://www.math.twcu.ac.jp/~yamauchi/math/phd/myphd.pdf), §3.8: commutants and cosets.+- Yanjun Chu and Zongzhu Lin, [The varieties of Heisenberg vertex operator algebras](https://arxiv.org/abs/1508.02963), §§1–2: semi-conformal subalgebras and complementary conformal vectors.