Vertex Operator Algebras

Sometimes, in order to define a consistent Conformal Field Theory it is nice to set the details of the fields as particular types of operator valued distributions with certain properties, and instead describe an algebra with these properties axiomatically present instead.

This can help us abstract a lot of the formalism when it comes to interpreting things like boundary conditions as other types of mathematical objects, and more!

Formal Distributions

The building blocks of vertex algebras are formal distributions. These are the field operator interpretation on the language we are trying to build.

Building Blocks

Definition: Let $Z$ be a finite set of $n\in \mathbb{N}$ elements and $R$ be an algebra over some field $\mathbb{K}$. The vector space of formal distributions in $Z$ over $R$ defined as the set

$$R[[Z]]:=R^{⟨Z⟩}=\{f:⟨Z⟩\to R\},$$

where $⟨Z⟩$ is the free group on $Z$, together with the following operations.

1. Elementwise addition: For any two $a=(a_1,a_2,\cdots ),b=(b_1,b_2,\cdots )\in R[[Z]]$ their sum is the sequence

   $$a+b=(a_1+b_1,a_2+b_2,\cdots )\in R[[z]].$$

2. Scalar multiplication: For any $a=(a_1,a_2,\cdots )\in R[[Z]]$ and $c\in \mathbb{K}$ we have that

   $$c\cdot a=ca=(c{a}_1,c{a}_2,\cdots )\in R[[z]].$$

Corollary: The vector space of formal distributions is a topological vector space under the product topology under the isomorphism $R^{⟨Z⟩}\cong R^{{\mathbb{Z}}^n}$ by treating $R$ with the discrete topology.

Corollary: $R[[Z]]$ is a closed metrizable space.

Remark: The best we can do here is a vector space. The reason is that since we have infinitely many terms in both directions, multiplication with arbitrary elements might be hard to define. However, here is the next best thing.

Definition: Let $Z$ be a finite set of $n\in \mathbb{N}$ elements and $R$ be a ring. The ring of formal power series in $Z$ over $R$ is the set

$$R^{\prime }[[Z]]:=R^{Z_{\ast }}=\{f:Z_{\ast }\to R\},$$

where $Z_{\ast }$ is the set of all words generated by $Z$, with elementwise addition and Cauchy multiplication, i.e. by Interpreting $a,b\in R[[Z]]$ as maps $a,b:⟨Z⟩\to R$ their product is such that for any $z\in Z_{\ast }$

$$(a\times b)(z)=\sum _{w\in ⟨Z⟩}a(w)b(w^{-1}z).$$

Remark: The ring of formal power series is a ring. Also since $Z_{\ast }\subset ⟨Z⟩$ the ring of formal power series is a subset of the vector space of formal distributions.

We often view any element of the formal distribution ring as a sum of terms that belong in the free group. Since we can index $⟨Z⟩\cong {\mathbb{Z}}^{|Z|}$ we can write $a\in R[[z_1,z_2,\cdots ,z_n]]$ as

$$a=\sum _{j\in {\mathbb{Z}}^n}{a}_j{z}^j=\sum _{j\in {\mathbb{Z}}^n}{a}_{j_1{j}_2\cdots j_n}{z}_1^{j_1}{z}_2^{j_2}\cdots z_n^{j_n},$$

where $a_{j_2{j}_2\cdots j_n}\in R$, which looks like a polynomial in the variables $z_i$. This is a nice interpretation that will help us define the algebraic tools needed in the following sections.

The next object of interest is a subset of elements in $R[[Z]]$ that can be multiplied with everything else. We call them the formal Laurent polynomials.

Definition: The ring of formal Laurent polynomials in $Z$ over an algebra $R$ is the subset

$$R[Z]:=\{a\in R[[Z]]\mid \mathrm{\exists }N\in \mathbb{N}\text{ where }|z|>N⟹a(z)=0\}$$

In other words it is the polynomials that have a finite degree on both ends.

Remark: For any Laurent polynomial $a\in R[Z]$ there exist $n\le m\in {\mathbb{Z}}^{|Z|}$ such that

$$a=\sum _{j=n}^m{a}_j{z}^j,$$

where we have used that $n\le m⟺n_i\le m_i\mathrm{\forall }i=1,\cdots ,|Z|$.

The cool thing about formal Laurent polynomials is that we can multiply them with everything.

Proposition: The Cauchy product of a formal Laurent polynomial with any formal distribution is well defined.

Definition: The ring of formal Laurent series in $Z$ over $R$ is the localization of $R^{\prime }[[Z]]$ by the multiplicatively closed set $S$ of positive formal distributions in $Z$, i.e.

$$R((Z)):=S^{-1}{R}^{\prime }[[Z]].$$

where

$$S:=\{z^j=z_1^{j_1}{z}_2^{j_2}\cdots z_n^{j_n}\in R^{\prime }[[Z]]|j\in {\mathbb{N}}^n\}=Z_{\ast }.$$

Example: Defining Laurent series as elements of the localization might be a bit confusing at first, so we can untangle it as follows. By definition of the localization we know that for any $a\in R^{\prime }[[Z]]$ and $s\in S$ there exists $b\in R((Z))$ such that

$$a=sb.$$

For example, consider the series $a=1+z+z^2+\cdots \in R^{\prime }[[Z]]$ for some $z\in Z$. By picking $s=z\in S$ we can find the Laurent series

$$b=z^{-1}+1+z+z^2+\cdots \in R[[Z]].$$

The reason for this definition of the laurent series as the Localization is that we can think of them as fractions in $R[[Z]]$ which is going to be really helpful later.

Operations

Here are some very useful operations we are going to play with all the time.

Defintion: The residue of a distribution $f\in R[[Z]]$ at a variable $z\in Z$ is defined as

$${\mathrm{Res}}_{z}f=f(z^{-1})\in R.$$

The formal derivative at a variable $z\in Z$ is the linear map

$${\partial }_z:R[[Z]]\to R[[Z]].$$

such that ${\partial }_{z}z=1$, ${\partial }_{z}1=0$ and for any reduced word $w=a_1{a}_2\cdots \in R[[Z]]$ we get

$${\partial }_{z}w=({\partial }_z{a}_1)a_2\cdots +a_1({\partial }_z{a}_2)\cdots +\cdots$$

Example: Given a formal distribution $f=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_k{z}^k$ the derivative is given by

$${\partial }_{z}f=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}(k+1)f_{k+1}{z}^k.$$