# Classes on Compactifications of the Moduli Space of Curves Through Solutions to the Quantum Master Equation

@article{Hamilton2009ClassesOC, title={Classes on Compactifications of the Moduli Space of Curves Through Solutions to the Quantum Master Equation}, author={Alastair Hamilton}, journal={Letters in Mathematical Physics}, year={2009}, volume={89}, pages={115-130} }

In this paper we describe a construction which produces classes in compactifications of the moduli space of curves. This construction extends a construction of Kontsevich which produces classes in the open moduli space from the initial data of a cyclic A∞-algebra. The initial data for our construction are what we call a ‘quantum A∞-algebra’, which arises as a type of deformation of a cyclic A∞-algebra. The deformation theory for these structures is described explicitly. We construct a family of…

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## References

SHOWING 1-10 OF 25 REFERENCES

Noncommutative geometry and compactifications of the moduli space of curves

- Mathematics
- 2007

In this paper we show that the homology of a certain natural compactification of the moduli space, introduced by Kontsevich in his study of Witten's conjectures, can be described completely…

Cohomology theories for homotopy algebras and noncommutative geometry

- Mathematics
- 2009

This paper builds a general framework in which to study cohomology theories of strongly homotopy algebras, namely A∞–, C∞– and L∞–algebras. This framework is based on noncommutative geometry as…

Quantum A-infinity Structures for Open-Closed Topological Strings

- Physics, Mathematics
- 2006

We study factorizations of topological string amplitudes on higher genus Riemann surfaces with multiple boundary components and find quantum A-infinity relations, which are the higher genus analog of…

Modular operads and Batalin-Vilkovisky geometry

- Mathematics
- 2017

This is a copy of the article published in IMRN (2007). I describe the noncommutative Batalin-Vilkovisky geometry associated naturally with arbitrary modular operad. The classical limit of this…

Closed String Operators in Topology Leading to Lie Bialgebras and Higher String Algebra

- Mathematics
- 2004

Imagine a collection of closed oriented curves depending on parameters in a smooth d-manifold M. Along a certain locus of configurations strands of the curves may intersect at certain sites in M. At…

A Hopf algebra quantizing a necklace Lie algebra canonically associated to a quiver

- Mathematics
- 2004

V. Ginzburg, and independently R. Bocklandt and L. Le Bruyn, defined an infinite-dimensional "necklace" Lie algebra canonically associated to any quiver. Following suggestions of V. Turaev, P.…

The decorated Teichmüller space of punctured surfaces

- Mathematics
- 1987

A principal ℝ+5-bundle over the usual Teichmüller space of ans times punctured surface is introduced. The bundle is mapping class group equivariant and admits an invariant foliation. Several…

Loop Homotopy Algebras in Closed String Field Theory

- Mathematics
- 2001

Abstract: Barton Zwiebach constructed [20] “string products” on the Hilbert space of a combined conformal field theory of matter and ghosts, satisfying the “main identity”. It has been well known…

Symplectic C∞-algebras

- Mathematics
- 2008

In this paper we show that a strongly homotopy commutative (or C∞-) algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic C∞-algebra (an ∞-generalisation…

The virtual cohomological dimension of the mapping class group of an orientable surface

- Mathematics
- 1986

Let F = F ~ r be the mapping class group of a surface F of genus g with s punctures and r boundary components. The purpose of this paper is to establish cohomology properties of F parallel to those…