First-order deformations of vertex algebras

Deformation theory of vertex algebras is an important yet difficult problem in both mathematics and physics. We approach this problem by computing the second cohomology $H^2_{1/2}(V, V)$ constructed by Yi-Zhi Huang. Joint with Vladimir Kovalchuk, we created an algorithm for classifying all the first-order deformations of freely generated vertex algebras. Using these results, we explicitly determine the first-order deformations of the universal Virasoro VOA $Vir_c$, universal affine VOA $V^l(\g)$, Heisenberg VOA $V^l(\h)$, and the universal Zamolodchikov VOA $W_3^c$. It has long been conjectured that rational vertex algebras are deformation rigid, i.e., they admit no first-order deformations. In the recent work joint with Andrew Linshaw, we proved this conjecture for simple affine VOA with positive integer levels.