A vertex operator algebra is a tuple $(V, Y, \mathbf{1}, \omega)$ where:

1. $V$ is a graded vector space $V = \bigoplus_{n \in \mathbb{Z}} V_n$, where $V_n$ is the space of states of degree $n$, and $V_0$ typically contains the vacuum state.

2. $Y$ is a vertex operator map $Y: V \times V \to \text{End}(V)$, which satisfies the following:

3. For any $a \in V$, there is a map $Y(a, z)$ called the vertex operator, which is a formal power series in $z$: $$ Y(a, z) = \sum_{n \in \mathbb{Z}} a_n z^{-n-1}, $$ where $a_n \in \text{End}(V)$ are operators acting on the vector space $V$.

4. Vacuum vector $\mathbf{1}$: There exists a special element $\mathbf{1} \in V$ called the vacuum vector, which acts as a "neutral element" under the vertex operator algebra structure. This element satisfies:

5. $Y(\mathbf{1}, z) = \text{id}_V$ (the identity operator) when acting on any $v \in V$.

6. The vacuum vector is the unique element such that $Y(\mathbf{1}, z) v = v$ for all $v \in V$.

7. Conformal vector $\omega$: There is a special element $\omega \in V_2$ (often called the conformal vector), which governs the action of the Virasoro algebra. The Virasoro operators $L_n$ are defined in terms of the vertex operators: $$ Y(\omega, z) = \sum_{n \in \mathbb{Z}} L_n z^{-n-2}. $$ The conformal vector $\omega$ satisfies the vacuum condition:

8. $L_n \mathbf{1} = 0$ for all $n \geq 1$ (the vacuum is annihilated by positive modes of the Virasoro algebra).

9. The Virasoro generators $L_n$ satisfy the Virasoro algebra commutation relations:

   $$ [L_m, L_n] = (m-n) L_{m+n} + \frac{c}{12} (m^3 - m) \delta_{m+n, 0}, $$ where $c$ is the central charge.

satisfying the following properties:

• Grading: The vector space $V$ is graded by $\mathbb{Z}$ and contains a distinguished vacuum vector.
• Operator product expansion (OPE): The vertex operators have a well-defined operator product expansion, providing a way to multiply vertex operators.
• Commutativity and locality: The vertex operators satisfy certain commutation relations that encode locality in the quantum field theory context.
• Conformal symmetry: The Virasoro algebra, generated by the conformal vector $\omega$, encodes the symmetries of the theory, particularly the infinitesimal conformal transformations in two dimensions.