Hochschild cohomology of the universal associative conformal envelope of the Virasoro Lie conformal algebra with coefficients in all finite modules

TL;DR

The paper computes the Hochschild cohomology groups $\mathrm{H}^n(U(3),M)$ of the universal associative conformal envelope $U(3)$ of the Virasoro Lie conformal algebra with locality bound $N=3$, using an Anick resolution of the coefficient algebra $\mathcal{A}_+(U(3))$ and algebraic discrete Morse theory. The method relies on expressing conformal cohomology through $\mathcal{A}_+(C)$ and constructing explicit Anick resolutions from a Gröbner–Shirshov basis, including an extension to differential algebras. The main results include precise dimensions: for conformal modules $M_{(\alpha,\Delta)}$, $\dim\mathrm{H}^1(U(3),M) = 2$ only in the special case $\Delta=1$, $\alpha=(0,0)$; for $\alpha\neq0$ the higher Hochschild cohomology vanishes, and $\mathrm{H}^2(U(3),M)$ and higher vanish in many parameter regimes, with nontrivial $\mathrm{H}^2$ arising for certain $M=(0,\Delta)$. Furthermore, $\mathrm{H}^k(U(3),M)=0$ for all $k\ge4$ for every finite module $M$, and the same vanishing propagates to quotients in finite Virasoro-type filtrations. Altogether, the work yields a detailed, parameter-sensitive map of the Hochschild cohomology landscape for $U(3)$ with finite coefficients, clarifying deformations and module structures in this conformal setting.

Abstract

In this paper, we find the Hochschild cohomology groups of the universal associative conformal envelope $U(3)$ of the Virasoro Lie conformal algebra with respect to associative locality $N=3$ on the generator with coefficients in all finite modules. In order to obtain this result, we construct the Anick resolution via the algebraic discrete Morse theory and Gröbner--Shirshov basis.

Theorems & Definitions (6)

• proof
• proof
• proof
• proof
• proof
• proof