New moduli spaces of pointed curves and pencils of flat connections.

By A. Losev, Yuri I. Manin. 2000

214 citations recorded by the source corpus.

It is well known that formal solutions to the Associativity Equations are the same as cyclic algebras over the homology operad $(H_*(\bar{M}_{0,n+1}))$ of the moduli spaces of $n$--pointed stable curves of genus zero. In this paper we establish a similar relationship between the pencils of formal flat connections (or solutions to the Commutativity Equations) and homology of a new series $\bar{L}_n$ of pointed stable curves of genus zero. Whereas $\bar{M}_{0,n+1}$ parametrizes trees of $\bold{P}^1$'s with pairwise distinct nonsingular marked points, $\bar{L}_n$ parametrizes strings of $\bold{P}^1$'s stabilized by marked points of two types. The union of all $\bar{L}_n$'s forms a semigroup rather than operad, and the role of operadic algebras is taken over by the representations of the appropriately twisted homology algebra of this union.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9149415dbc72d751281db9f9024a6b38944bf0a4

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Citations in the corpus

MODULI SPACES OF RATIONAL WEIGHTED STABLE CURVES AND TROPICAL GEOMETRY (2014, background): “These spaces were introduced and studied in [19] and play a role, for instance, in the theory of relative stable maps, as a target for branch morphisms.”
Moduli spaces of weighted pointed stable curves (2002, methodology): “[ LM ] 7.5). The most important step is the following:”
Integrals of ψ-classes over double ramification cycles (2012, background): “We have This simple but very useful statement ﬁrst appeared in Ionel’s paper [3].”
Arc Operads and Arc Algebras (2002, background): “…6 December 2002 Seconded: Ralph Cohen, Vaughan Jones Accepted: 8 June 2003 c© Geometry & Topology Publications Introduction With the rise of string theory and its companions topological and conformal field theories, there has been a very fruitful interaction between physical ideas and topology.”