[Paper Review] Quantum $L_\infty$ Algebras and the Homological Perturbation Lemma

M. Doubek, Branislav Jurčo+1 more|arXiv (Cornell University)|Dec 7, 2017

Black Holes and Theoretical Physics16 references3 citations

TL;DR

This paper establishes a homological perturbation lemma (HPL) framework for quantum $L_\infty$ algebras, showing that the effective action in the finite-dimensional Batalin–Vilkovisky formalism arises as a Feynman diagram expansion. The key contribution is a rigorous construction of a minimal model on cohomology and a homotopy between the original and effective actions via HPL, generalizing perturbative path integral methods to quantum $L_\infty$ structures.

ABSTRACT

Quantum $L_\infty$ algebras are a generalization of $L_\infty$ algebras with a scalar product and with operations corresponding to higher genus graphs. We construct a minimal model of a given quantum $L_\infty$ algebra via the homological perturbation lemma and show that it's given by a Feynman diagram expansion, computing the effective action in the finite-dimensional Batalin-Vilkovisky formalism. We also construct a homotopy between the original and this effective quantum $L_\infty$ algebra.

Motivation & Objective

• To generalize the homological perturbation lemma to quantum $L_\infty$ algebras with a scalar product and higher genus operations.
• To construct a minimal model of a quantum $L_\infty$ algebra on the cohomology of its differential using the HPL.
• To show that the effective action obtained via integration over the complement of the cohomology is itself a solution to the quantum master equation.
• To establish a homotopy between the original action and the effective action, proving their equivalence in the context of quantum $L_\infty$ structures.

Proposed method

• Use a deformation retract between the original graded vector space $V$ and its cohomology $H$, induced by projection $p$, inclusion $i$, and homotopy $s$.
• Lift the deformation retract to the space of functionals $F(V)$ and $F(H)$ using the Batalin–Vilkovisky formalism.
• Apply the HPL to perturb the differential $\lambda_1^0$ by $\hbar\Delta$, where $\Delta$ is the BV Laplacian.
• Derive explicit formulas for the perturbed projection $P_1: F(V) \to F(H)$, showing it corresponds to a path integral.
• Construct the effective action $W$ via $e^{W/\hbar} = \int_{H^c} e^{S/\hbar}$, interpreted as a Feynman diagram expansion.
• Prove that the resulting $W$ satisfies the quantum master equation on $F(H)$, thus defining a quantum $L_\infty$ algebra.

Experimental results

Research questions

• RQ1 How can the homological perturbation lemma be extended to quantum $L_\infty$ algebras with higher genus operations and a scalar product?
• RQ2 What is the precise diagrammatic structure of the effective action obtained via the HPL in the BV formalism?
• RQ3 How is the homotopy between the original action $S$ and the effective action $W$ constructed, and what does it imply for the equivalence of quantum $L_\infty$ structures?
• RQ4 Can the path integral interpretation of the HPL be rigorously linked to the Feynman diagram expansion of the effective action?
• RQ5 What is the role of the BV Laplacian $\Delta$ in the perturbation and how does it ensure the quantum master equation is preserved?

Key findings

• The HPL construction yields a minimal quantum $L_\infty$ algebra on the cohomology $H$, with operations encoded in a Feynman diagram expansion.
• The effective action $W$ is obtained as a path integral over the complement of $H$ in $V$, and it satisfies the quantum master equation on $F(H)$, ensuring it defines a valid quantum $L_\infty$ algebra.
• The projection map $P_1: F(V) \to F(H)$ is explicitly shown to be equivalent to the path integral, providing a diagrammatic interpretation of the HPL in this context.
• A homotopy between the original action $S$ and the effective action $W$ is constructed, proving they are homotopic solutions to the quantum master equation.
• The framework generalizes and unifies prior constructions by Costello, Gwilliam, Braun & Maunder, and Barannikov, showing their path integral and diagrammatic methods are instances of the HPL in the quantum $L_\infty$ setting.
• The propagator in the Feynman expansion is identified as the inverse of the symplectic form $\sigma = \langle -, d- \rangle$, matching known results in the literature.