[Paper Review] Homotopy Batalin-Vilkovisky algebras

Imma Gálvez-Carrillo, Andrew Tonks+1 more|arXiv (Cornell University)|Jul 13, 2009

Advanced Topics in Algebra45 references17 citations

TL;DR

This paper constructs an explicit cofibrant resolution of the operad governing Batalin–Vilkovisky (BV) algebras using an extended Koszul duality theory for operads with quadratic and linear relations. The resolution provides a homotopy BV-algebra structure that lifts BV-algebra structures on homology, enabling deformation theory, obstruction theory, and verification of the cyclic Deligne conjecture.

ABSTRACT

This paper provides an explicit cofibrant resolution of the operad encoding Batalin-Vilkovisky algebras. Thus it defines the notion of homotopy Batalin-Vilkovisky algebras with the required homotopy properties. To define this resolution we extend the theory of Koszul duality to operads and properads that are defind by quadratic and linear relations. The operad encoding Batalin-Vilkovisky algebras is shown to be Koszul in this sense. This allows us to prove a Poincare-Birkhoff-Witt Theorem for such an operad and to give an explicit small quasi-free resolution for it. This particular resolution enables us to describe the deformation theory and homotopy theory of BV-algebras and of homotopy BV-algebras. We show that any topological conformal field theory carries a homotopy BV-algebra structure which lifts the BV-algebra structure on homology. The same result is proved for the singular chain complex of the double loop space of a topological space endowed with an action of the circle. We also prove the cyclic Deligne conjecture with this cofibrant resolution of the operad BV. We develop the general obstruction theory for algebras over the Koszul resolution of a properad and apply it to extend a conjecture of Lian-Zuckerman, showing that certain vertex algebras have an explicit homotopy BV-algebra structure.

Motivation & Objective

• To develop a homotopy theory for Batalin–Vilkovisky algebras, which are fundamental in mathematical physics and topology but lack cofibrant resolutions.
• To extend Koszul duality theory to operads defined by quadratic and linear relations, enabling resolution of the BV operad.
• To construct an explicit small quasi-free resolution of the BV operad, defining homotopy BV-algebras with proper homotopy properties.
• To prove that topological conformal field theories and double loop spaces with circle actions carry homotopy BV-algebra structures lifting their homology-level BV structures.
• To establish deformation and obstruction theories for homotopy BV-algebras using the resolution, and verify the cyclic Deligne conjecture.

Proposed method

• Extend inhomogeneous Koszul duality theory to operads with quadratic and linear relations, proving the BV operad is Koszul in this generalized sense.
• Construct the Koszul dual coproperad $\mathscr{P}^{\scriptstyle\text{\rm!`}}$ as a graded coproperad with a relative grading $[n]$ that tracks the number of $sV_1$ generators.
• Define the homotopy BV-algebra via the cofibrant resolution $\mathscr{BV}^\text{!`}$, giving four equivalent definitions of the structure.
• Use the convolution dg Lie algebra $\mathfrak{g} = \mathrm{Hom}_{\mathbb{S}}(\mathscr{BV}^{\scriptstyle\text{\rm!`}}, \mathrm{End}_A)$ with a relative grading to describe deformation and obstruction theory.
• Apply the Maurer–Cartan equation in $\mathfrak{g}$ with decomposition $\gamma = \gamma_0 + \gamma_1 + \cdots$, where $\gamma_0$ is a homotopy $\mathscr{P}_0$-algebra structure.
• Leverage formality of the framed little discs operad and the cofibrancy of the resolution to prove that any topological conformal field theory or double loop space with $S^1$-action carries a homotopy BV-algebra structure.

Experimental results

Research questions

• RQ1 Can a cofibrant resolution be explicitly constructed for the operad encoding Batalin–Vilkovisky algebras?
• RQ2 Does the extended Koszul duality theory for operads with quadratic and linear relations allow for a Poincaré–Birkhoff–Witt theorem and small resolution?
• RQ3 Can homotopy BV-algebra structures be lifted from homology to chain complexes in topological conformal field theories and double loop spaces?
• RQ4 Is the cyclic Deligne conjecture valid when using the cofibrant resolution of the BV operad?
• RQ5 What is the deformation and obstruction theory for homotopy BV-algebras, and how can it be described via the convolution Lie algebra?

Key findings

• The operad $\mathscr{BV}$ encoding Batalin–Vilkovisky algebras is shown to be Koszul in the extended sense of quadratic-linear relations, enabling a cofibrant resolution.
• An explicit small quasi-free resolution of $\mathscr{BV}$ is constructed, providing a conceptual definition of homotopy BV-algebras with full homotopy properties.
• The Poincaré–Birkhoff–Witt theorem holds for the $\mathscr{BV}$ operad, confirming a structural property of its universal enveloping algebra.
• Any topological conformal field theory carries a homotopy BV-algebra structure that lifts the BV-algebra structure on its homology.
• The double loop space of a topological space with an $S^1$-action carries a homotopy BV-algebra structure lifting the BV-algebra structure on homology.
• The cyclic Deligne conjecture is proven using the cofibrant resolution, showing that the Hochschild cochain complex of a Frobenius algebra admits a homotopy BV-algebra structure.