The main content of this paper is a carefully worked-through homotopical transfer formula for algebras over modular operads. Particular care is taken with various versions of the associative and \(A_\infty\) operad, with various twistings. The formulas, like similar formulas for operads, cyclic operads, and properads, are realized in terms of linear transfer data as sums over decorated graphs. Such formulas can generally be repackaged so that they are understood as solutions to a master equation; the specifics of which master equation depend on the algebraic structure in question. In the case of modular operads, the governing master equation is the Batalin-Vilkovisky master equation.
In order to generate the transfer formula in the most familiar context, one begins with a differential graded algebra \((A,\cdot)\) equipped with a compatible bilinear pairing. The linear transfer data consists of a choice of representatives for the homology \(H\) of \(A\) (understood as an inclusion operator), a projection operator from \(A\) to \(H\), and a chain homotopy between the composition of these two and the identity in the algebra. One generates operations on \(H\) by considering decorated trivalent ribbon graphs whose external edges are decorated with inclusion and projection operators, whose internal edges are decorated with the chain homotopy, and whose trivalent vertices are decorated with the product in \(A\). Using the bilinear pairing, such a decorated graph yields a well-defined operation among the tensor powers of \(H\). Summing appropriately over these yields a modular \(A_\infty\) structure on \(H\) equivalent to the original structure on \(A\).
In more generality, one could start with an algebra \(A\) over any modular operad \(\mathcal{P}\) and the same type of linear transfer data to its homology \(H\). In this context, vertices of valence \(n\) may be decorated with operations of arity \(n-1\) in \(\mathcal{P}\). The resulting structure on \(H\) is a strongly homotopy \(\mathcal{P}\)-algebra (or \(\mathcal{P}_\infty\) algebra).
The author further extends the construction, which is generally only taken in the differential-graded framework, to a physically-motivated case where the algebra \(A\) is equipped not with a differential but with a looser structure, a unary operator \(I\) compatible with the algebra which does not necessarily square to zero. In this case, one can still posit the existence of a linear space \(H\) equipped with an inclusion to \(A\) and a projection from \(A\) that are compatible with \(I\), and a “chain homotopy” which commutes with \(I^2\) and satisfies the usual relations of linear transfer data. If this data is given (contrary to an aside in the paper, a given algebra of this sort may only have trivial linear transfer data) then the same formulas still give a \(\mathcal{P}_\infty\) structure on \(H\). This extension is interesting and timely. There has been increased interest in curved algebraic structures in recent years and this extension seems to apply well to curved structures, among others.
The final section of the paper is an application of the formulas in a particular class of example, that of equivariant \(A_\infty\) matrix integrals. The input is a cyclic \(A_\infty\) algebra \(V\). Combining the transfer formula with his previous work, the author can construct an equivariantly closed differential form on a space closely related to \(V\).