[Paper Review] Heat Kernel Renormalization on Manifolds with Boundary

Benjamin I. Albert|arXiv (Cornell University)|Sep 7, 2016

Geometric Analysis and Curvature Flows9 references3 citations

TL;DR

This paper extends Costello's inductive position space renormalization procedure—based on heat kernel regularization—to manifolds with boundary, establishing a rigorous framework for local counterterms and effective field theory construction. The key contribution is a systematic inductive method for constructing local counterterms using heat kernel asymptotics and Taylor expansions near the boundary, valid even when the Riemannian metric is not smooth across the double of the manifold.

ABSTRACT

In the monograph Renormalization and Effective Field Theory, Costello made two major advances towards the mathematical formulation of quantum field theory. Firstly, he developed an inductive position space renormalization procedure for constructing effective field theories that is based on heat kernel regularization of the propagator. Secondly, he gave a rigorous formulation of quantum gauge theory within effective field theory that makes use of the BV formalism. In this work, we extend Costello's inductive renormalization procedure from manifolds without boundary to a class of manifolds with boundary. In addition, we reorganize the presentation of the preexisting material, filling in details and strengthening the results.

Motivation & Objective

• To generalize Costello's inductive renormalization procedure—originally formulated for manifolds without boundary—to a class of manifolds with boundary.
• To provide a rigorous construction of local counterterms in the presence of boundaries using heat kernel regularization.
• To reorganize and strengthen the foundational material from Costello's monograph, particularly regarding Feynman diagram expansions and Wick's theorem on bounded domains.
• To establish conditions under which the smooth double of a Riemannian manifold with boundary admits a smooth metric, enabling global counterterm construction.
• To develop a framework for effective field theory on curved manifolds with boundary through iterative renormalization and error bounding.

Proposed method

• Uses heat kernel regularization of the propagator as the core mechanism for defining Feynman weights and counterterms in position space.
• Applies Wick’s theorem on finite-dimensional subspaces of fields restricted to energy intervals, extending it to half-lines and bounded intervals.
• Employs spanning tree coordinates and Taylor expansions of field functionals to isolate divergent contributions near the boundary.
• Introduces a double of the manifold to extend local computations to global ones, analyzing smoothness of the extended metric via normal derivatives of the metric tensor.
• Implements an inductive construction of counterterms by bounding errors in asymptotic expansions of the heat kernel and field functionals.
• Uses the condition that the boundary is totally geodesic and odd-order normal derivatives of the curvature vanish to ensure smoothness of the double metric.

Experimental results

Research questions

• RQ1 How can Costello’s inductive renormalization procedure be extended from manifolds without boundary to manifolds with boundary?
• RQ2 What conditions ensure that the smooth double of a Riemannian manifold with boundary admits a smooth metric?
• RQ3 How can local counterterms be systematically constructed in the presence of a boundary using heat kernel asymptotics?
• RQ4 What role does the Taylor expansion of the field functional play in isolating divergent contributions near the boundary?
• RQ5 Under what geometric conditions can global counterterms be constructed on curved manifolds with boundary?

Key findings

• A systematic inductive construction of local counterterms is achieved on flat manifolds with boundary using heat kernel asymptotics and Taylor expansions of field functionals.
• The smooth double of a Riemannian manifold with boundary admits a smooth metric if and only if the boundary is totally geodesic and all odd-order normal derivatives of the curvature tensor vanish.
• The condition $ abla^{2k+1}_{ abla_{ ext{inward}}} ext{Ricci} = 0$ for $k o rac{1}{2}$ is equivalent to the vanishing of odd-order normal derivatives of the metric tensor components.
• The method allows for the construction of effective field theories on curved manifolds with boundary by replacing the heat kernel with its asymptotic series and bounding the error via Taylor expansion.
• The framework generalizes Costello’s original construction to include boundaries, providing a rigorous path to renormalization in quantum field theory on manifolds with boundary.
• The paper identifies that the assumption of parallel curvature tensor is sufficient but not necessary for doublability, and suggests a generalization via Taylor expansion of the geodesic distance squared.