Abstract Hodge Decomposition and Minimal Models for Cyclic Algebras

Joseph Chuang,Andrey Lazarev

doi:10.1007/s11005-009-0314-7

Joseph Chuang, Andrey Lazarev

Open Access

https://doi.org/10.1007/s11005-009-0314-7

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Abstract

We show that an algebra over a cyclic operad supplied with an additional linear algebra datum called Hodge decomposition admits a minimal model whose structure maps are given in terms of summation over trees. This minimal model is unique up to homotopy.

Highlights

Operadic algebras were originally introduced for the needs of homotopy theory but figure prominently in algebraic geometry and certain parts of theoretical physics, in those aspects which concern mirror symmetry
In [5] we introduced a new approach to the construction of explicit minimal models, giving a conceptual explanation for the Merkulov tree sum formulas
We worked in an operadic setting, so our results apply to homotopy algebras interpreted in the broadest sense, including as special cases C∞-algebras and L∞-algebras

Summary

Introduction

Operadic algebras (such as A∞- , C∞- and L∞-algebras) were originally introduced for the needs of homotopy theory but figure prominently in algebraic geometry and certain parts of theoretical physics, in those aspects which concern mirror symmetry. As a special case we recovered the theory of minimal models for homotopy algebras equipped with non-degenerate bilinear forms. This is important because the most interesting examples such as de Rham and Dolbeault algebras are infinite dimensional and cannot support a non-degenerate inner product Their minimal models tend to be finite dimensional and so it is natural to ask whether their inner products could be made compatible with the higher multiplications. Our main result states that an algebra V (not necessarily finite dimensional) over a cyclic operad admits an explicit minimal model whose structure maps are given by a Merkulovtype formula provided the underlying complex of V possesses a Hodge decomposition. Our results extend to algebras over arbitrary cyclic operads

Cyclic operads and their algebras

Main theorem: an explicit form of a minimal model of cyclic algebras