On operad structures of moduli spaces and string theory

Abstract

Abstract. We construct a real compactification of the moduli space of punctured rational algebraic curves and show how its geometry yields operads governing homotopy Lie algebras, gravity algebras and Batalin-Vilkovisky algebras. These algebras appeared recently in the context of string theory, and we give a simple deduction of these algebraic structures from the formal axioms of conformal field theory and string theory. The physical importance of these structures is that they lead toward the classification of string theories at the tree level, because the structure constants of the algebras appear as all correlators of the theory. We suggest that an appropriate background for putting together those algebraic structures is the structure of an operad. On the one hand, as we point out, a conformal field theory at the tree level is equivalent to an algebra over the operad of Riemann spheres with punctures, cf. Huang and Lepowsky [22]. On the other hand, this one operad gives rise to several other operads creating these various algebraic structures. The relevance to physics is that theories such as conformal field theory or string-field theory provide a representation of the geometry of the moduli space of such punctured Riemann spheres in the category of differential graded vector spaces. This paper, one of a series, deals with a part of these algebraic structures, namely with the structure of a homotopy Lie algebra and the related structures of the gravity algebra and Batalin-Vilkovisky algebra. A richer structure, the moduli space of Riemann spheres, induces a homotopy version of a Gerstenhaber algebra, which contains

BV Quantization of Topological Open Membranes

“…The L ∞ structure of closed string field theory was demonstrated in [24]. It was discussed in the context of topological strings in [25,26]. It is indeed this L ∞ structure that is naturally deformed by bulk membrane couplings [23].…”

“…We will be interested in the effect of the bulk terms on the string theory living on the boundary. This topological closed string field theory has the structure of a L ∞ algebra [24,26,25], generating its closed string field theory. The bracket in this L ∞ algebra is defined as the current algebra bracket of the boundary string,…”

TOPOLOGICAL OPEN P-BRANES

“…Similar structures also appear as Fukaya's A ∞ -category [25] in the open string version of the so called A model of the topological sigma model [93]. The open-closed string field theory [103] is related to the algebra over Swiss-Cheese operads, while the closed string field theory is based on the L ∞ part of the homotopy 2-algebra [102,45,46]. It is also natural to expect that the homological mirror conjecture [49] can eventually be the physical equivalence of open-closed string field theory.…”

Cyclic operad formality for compactified moduli spaces of genus zero surfaces

“…The operad composition is obtained by gluing stable curves at marked points and tensoring the corresponding rays. This operad was introduced by Kimura-Stasheff-Voronov [14], and it is closely related to the modular operad of Deligne-Mumford compactified moduli spaces of curves whose formality is proved in [9]. Moreover f M is homotopy equivalent, as a cyclic operad, to the operad of genus zero Riemann surfaces with boundary, where the operad composition is defined by gluing along boundary components.…”