[Paper Review] Factorization Algebras for Bulk-Boundary Systems

Eugene Rabinovich|arXiv (Cornell University)|Nov 2, 2021

Algebraic structures and combinatorial models44 references4 citations

TL;DR

This paper extends Costello and Gwilliam's factorization algebra framework for perturbative quantum gauge theories to manifolds with boundary, constructing a factorization algebra of quantum observables for bulk-boundary systems. It establishes a quasi-isomorphism between classical observable models using Atiyah-Bott-type duality, proving equivalence of two formulations of observables and enabling quantization of boundary-conditioned field theories via renormalization and the quantum master equation.

ABSTRACT

Costello and Gwilliam have given both 1) a general definition of perturbative quantum gauge theory on a manifold M and 2) a construction of a factorization algebra of quantum observables assigned to every quantum gauge theory. In this dissertation, we extend these constructions to a certain general class of field theories on manifolds with boundary.

Motivation & Objective

• To generalize Costello and Gwilliam's perturbative quantum gauge theory framework to field theories on manifolds with boundary.
• To define a factorization algebra of quantum observables for bulk-boundary systems, extending the existing framework for closed manifolds.
• To establish the equivalence between two models of classical observables—via compactly supported fields and via dual spaces—using duality theorems.
• To develop a quantization procedure for free and interacting bulk-boundary systems using renormalization and the quantum master equation.
• To provide a rigorous mathematical foundation for quantum field theories with boundaries, including applications to BF theory and topological mechanics.

Proposed method

• Adapts the factorization algebra framework to manifolds with boundary by introducing boundary conditions via a Lagrangian subsheaf $\mathscr{L}$, defining $\mathscr{L}$-conditioned fields.
• Uses $D$-module theory to describe local action functionals and their pushforwards/pullbacks, particularly in the context of boundary conditions.
• Applies deformation retractions to relate compactly supported cochains $\mathscr{E}_{\mathscr{L},c}[1](U)$ to dual spaces $\mathscr{E}_{\mathscr{L}}^\vee(U)$, especially near the boundary.
• Employs the Atiyah-Bott lemma in the form of a quasi-isomorphism between $\mathscr{L}^\perp_c(V) \to \mathscr{E}_{\mathscr{L},c}[1](U^\prime)$ and $\mathscr{E}_{\mathscr{L}}^\vee(U^\prime) \to \mathscr{L}^\vee(V)$ to prove duality.
• Constructs the quantum factorization algebra via renormalization group flow and the quantum master equation, using counterterms to handle divergences.
• Applies the doubling trick and regularized heat kernels to define propagators and parametrices in the presence of boundary conditions.

Experimental results

Research questions

• RQ1 How can the factorization algebra framework of Costello and Gwilliam be extended to field theories on manifolds with boundary?
• RQ2 What is the correct mathematical formulation of classical and quantum observables for bulk-boundary systems, particularly under boundary conditions?
• RQ3 Are the two models of classical observables—based on compactly supported fields and on dual spaces—quasi-isomorphic, and if so, under what conditions?
• RQ4 How can the quantum master equation be formulated and solved for bulk-boundary systems, especially in the presence of boundary conditions?
• RQ5 What role do renormalization and counterterms play in constructing a consistent quantum factorization algebra for such systems?

Key findings

• The natural map $ (\mathscr{E}_{\mathscr{L},c}(U)[1]^{{\widehat{\otimes}}_{\beta}k})_{S_k} \to \underline{CVS}((\mathscr{E}_{\mathscr{L}}(U))^{{\widehat{\otimes}}_{\beta}k},\mathbb{R})_{S_k} $ is a quasi-isomorphism of differentiable vector spaces, proving equivalence of two classical observable models.
• The map $ \mathscr{E}_{\mathscr{L},c}[1](U^\prime) \to \mathscr{E}_{\mathscr{L}}^\vee(U^\prime) $ is a quasi-isomorphism for sufficiently small neighborhoods $ U^\prime $ near the boundary, established via deformation retractions and duality.
• For $ U^\prime = V \times [0,T^\prime) $, the composite map $ \mathscr{L}^\perp_c(V) \to \mathscr{E}_{\mathscr{L},c}[1](U^\prime) \to \mathscr{E}_{\mathscr{L}}^\vee(U^\prime) \to \mathscr{L}^\vee(V) $ realizes the Atiyah-Bott quasi-isomorphism.
• The factorization algebra of quantum observables is constructed via renormalization and the quantum master equation, with counterterms ensuring consistency.
• In one-dimensional BF theory on $ \mathbb{R}_{\geq 0} $, the quantization procedure is explicitly verified, and the quantum observables are computed via parametrices and Feynman diagrams.
• The framework successfully quantizes topological mechanics and the free Poisson sigma model on manifolds with boundary, demonstrating the robustness of the construction.