Quantizing deformation theory. II (Q2295839)

The central metatheorem in \textit{classical} deformation theory is that every reasonable deformation problem is controlled by a differential graded Lie algebra, or equivalently, by an \(L_\infty\)-algebra. Non-classical, i.e. \textit{quantum}, deformation problems arise in the context of modular operads, Frobenius algebras and maybe Gromov-Witten theory. In the paper under review, a generalization of the metatheorem to quantum deformation problems is proposed: Every reasonable quantum deformation problem should be controlled by a \(BV_\infty\)-algebra. A \(BV_\infty\)-algebra is a generalization of a differential graded Batalin-Vilkovisky algebra: It is a graded commutative algebra \(V\) together with a sequence \(\Delta_n\) of higher order Batalin-Vilkovisky operators and a conilpotent graded cocommutative coalgebra structure on \(V\). Every differential graded Batalin-Vilkovisky algebra \((V,d,\Delta)\) is considered a \(BV_\infty\)-algebra with \(\Delta_1 = d\), \(\Delta_2 = \Delta\) and \(\Delta_n = 0\) for \(n \geq 3\). The \textit{quantum deformation functor} \[QM_V: \mathbf{CLAlg} \to \mathbf{Set},\] which is defined on the category of complete local Noetherian algebras \(\mathbf{CLAlg}\), is an analogue of the Maurer-Cartan functor of an \(L_\infty\)-algebra (and not of its deformation functor). In the special case of a differential graded Batalin-Vilkovisky algebra, its value \(QM_V(R)\) at \(R \in \mathbf{CLAlg}\) is the set of solutions \(S \in V[[\hbar]]^2 \hat\otimes \mathfrak{m}_R\) of the quantum master equation \[dS + \hbar\Delta S + \frac{1}{2}\{S,S\} = 0.\] The article is not a sequel to a hypothetical first part by the author, but the title \textit{Quantizing Deformation Theory II} refers to \textit{J. Terilla}'s work \textit{Quantizing Deformation Theory} [in: Deformation spaces. Perspectives on algebro-geometric moduli. Including papers from the workshops held at the Max-Planck-Institut für Mathematik, Bonn, Germany, July 2007 and August 2008. Wiesbaden: Vieweg+Teubner. 135--141 (2010; Zbl 1206.14013)]. Therein, Terilla proposes a quantized deformation theory where \(\mathbf{CLAlg}\) is replaced by Frobenius algebras, and \(L_\infty\)-algebras are replaced by so-called \(IBL_\infty\)-algebras.