Tree-level S-matrix of superstring field theory with homotopy algebra structure

We show that the tree-level S-matrices of the superstring field theories based on the homotopy-algebra structure agree with those obtained in the first-quantized formulation. The proof is given in detail for the heterotic string field theory. The extensions to the type II and open superstring field theories are straightforward.

On the strong homotopy associative algebra of a foliation

An involutive distribution C on a smooth manifold M is a Lie-algebroid acting on sections of the normal bundle TM/C. It is known that the Chevalley–Eilenberg complex associated to this representation of C possesses the structure 𝕏 of a strong homotopy Lie–Rinehart algebra. It is natural to interpret 𝕏 as the (derived) Lie–Rinehart algebra of vector fields on the space P of integral manifolds of C. In this paper, we show that 𝕏 is embedded in an A∞-algebra 𝔻 of (normal) differential operators. It is natural to interpret 𝔻 as the (derived) associative algebra of differential operators on P. Finally, we speculate about the interpretation of 𝔻 as the universal enveloping strong homotopy algebra of 𝕏.

HOMOTOPY RELATIONS FOR TOPOLOGICAL VOA

We consider a parameter-dependent version of the homotopy associative part of the Lian–Zuckerman homotopy algebra and provide an interpretation of the multilinear operations of this algebra in terms of integrals over certain polytopes. We explicitly prove the pentagon relation up to homotopy and propose a construction of the higher operations.

A twisted tale of cochains and connections

Early in the history of higher homotopy algebra [Stasheff, Trans. Am. Math. Soc. 108: 293–312, 1963], it was realized that Massey products are homotopy invariants in a special sense, but it was the work of Tornike Kadeishvili that showed they were but a shadow of an 𝐴∞-structure on the homology of a differential graded algebra. Here we relate his work to that of Victor Gugenheim [J. Pure Appl. Algebra 25: 197–205, 1982] and K. T. (Chester) Chen [Ann. of Math. (2) 97: 217–246, 1973]. This paper is a personal tribute to Tornike and the Georgian school of homotopy theory as well as to Gugenheim and Chen, who unfortunately are not with us to appreciate this convergence.