First-order deformations of freely generated vertex algebras

We solve the problem of how to classify the first-order vertex-algebraic deformations for any grading-restricted vertex algebra $V$ that is freely generated by homogeneous elements of positive weights. We approach by computing the second cohomology $H^2_{1/2}(V, V)$ constructed by Yi-Zhi Huang. We start with the cocycle on two generators and show that its cohomology class is completely determined by its singular part. To extend the cocycle to any pair of elements in $V$, we take a generating function approach, formulate the cocycle equation, and show that all the complementary solutions are coboundaries. Then we use a very general procedure to construct a particular solution. The procedure applies to vertex algebras that are not freely generated. As a by-product, we show that $H^2_{1/2}(V, V) = H^2_\infty(V, V)$. Using these results, we explicitly determine the first-order deformations of the universal Virasoro VOA $Vir_c$, universal affine VOA $V^l(\mathfrak{g})$, Heisenberg VOA $V^l(\mathfrak{h})$, and the universal Zamolodchikov VOA $W_3^c$.