The First Modular Zhu Algebra of $L(\frac12,0)_F$ in Characteristic Seven

Abstract

Let $F$ be an algebraically closed field of characteristic $7$, and let $V=L(\frac12,0)_F$. For the degree-four singular vector $q=(L(-2)^2-2L(-4))\mathbf1$ in the universal vacuum vertex operator algebra $\bar V$, set $V_q=\bar V/I(q)$. We prove that the natural quotient map induces an isomorphism $A_1(V_q)\cong A_1(V)\cong F[T]/(T(T-4)(T-5))\cong F^3$. Explicit $O_1(V_q)$-relations and four families of triangular PBW straightening relations give the upper bound, while three one-dimensional homogeneous subspaces give the matching lower bound.