[Paper Review] Linear Batalin-Vilkovisky quantization as a functor of $\\infty$-categories

Owen Gwilliam, Rune Haugseng|arXiv (Cornell University)|Aug 3, 2016

Advanced Topics in Algebra4 citations

TL;DR

This paper constructs linear Batalin-Vilkovisky (BV) quantization as a symmetric monoidal functor between ∞-categories, using derived and shifted structures. It generalizes Weyl quantization to cochain complexes with (1−n)-shifted symplectic pairings, establishing a functorial quantization to Eₙ-algebras via Heisenberg Lie algebras and universal enveloping algebras, with applications to derived geometry and higher BV theories.

ABSTRACT

We study linear Batalin-Vilkovisky (BV) quantization, which is a derived and shifted version of the Weyl quantization of symplectic vector spaces. Using a variety of homotopical machinery, we implement this construction as a symmetric monoidal functor of $\\infty$-categories. We also show that this construction has a number of pleasant properties: It has a natural extension to derived algebraic geometry, it can be fed into the higher Morita category of $E_n$-algebras to produce a "higher BV quantization" functor, and when restricted to formal moduli problems, it behaves like a determinant. Along the way we also use our machinery to give an algebraic construction of $E_n$-enveloping algebras for shifted Lie algebras.

Motivation & Objective

• To formalize linear Batalin-Vilkovisky quantization as a functor in the language of ∞-categories.
• To extend this construction to derived algebraic geometry and higher structures such as Eₙ-algebras.
• To provide a homotopical framework for BV quantization that behaves like a determinant on formal moduli problems.
• To construct Eₙ-enveloping algebras for shifted Lie algebras algebraically using ∞-categorical machinery.

Proposed method

• Uses ∞-categories and model categories to define modules and operad algebras, enabling homotopical algebra over commutative differential graded algebras.
• Applies the Heisenberg Lie algebra construction to cochain complexes equipped with a (1−n)-shifted skew-symmetric pairing.
• Constructs the universal enveloping algebra functor on the ∞-category level, lifting the classical construction to derived settings.
• Implements the quantization procedure as a symmetric monoidal functor from the ∞-category of shifted symplectic complexes to E₀-algebras (pointed complexes).
• Utilizes fibrant replacements and localization techniques in simplicial categories to model ∞-categories and ensure functoriality.
• Applies the machinery to define higher BV quantization by feeding the linear BV functor into the higher Morita category of Eₙ-algebras.

Experimental results

Research questions

• RQ1 Can linear BV quantization be formalized as a symmetric monoidal functor between ∞-categories?
• RQ2 How does linear BV quantization extend to derived algebraic geometry and formal moduli problems?
• RQ3 What is the relationship between BV quantization and the determinant construction in derived geometry?
• RQ4 How can Eₙ-enveloping algebras for shifted Lie algebras be constructed algebraically using ∞-categorical methods?
• RQ5 What is the role of the Heisenberg Lie algebra in higher BV quantization and AKSZ-type theories?

Key findings

• Linear BV quantization is realized as a symmetric monoidal functor from the ∞-category of cochain complexes with (1−n)-shifted symplectic pairings to the ∞-category of E₀-algebras.
• The construction extends naturally to derived algebraic geometry, preserving functoriality and compatibility with derived stacks.
• When restricted to formal moduli problems, the linear BV functor behaves like a determinant, suggesting a deep link to geometric Langlands and index theory.
• The paper provides an algebraic construction of Eₙ-enveloping algebras for shifted Lie algebras using ∞-categorical model structures.
• The quantization functor is compatible with higher Morita theory, enabling a systematic construction of higher BV quantization functors via composition with the Morita ∞-category.
• The derived and shifted version of Weyl quantization is shown to be a deformation quantization of the symmetric algebra with a Poisson bracket, recovering the classical limit as ħ→0.