Constructive Quantum Field Theory on Curved Surfaces and Related Topics

TL;DR

This thesis constructs a rigorous $P()_2$ quantum field theory on curved surfaces, extending Segal's axioms, and explores entanglement entropy and zeta determinant asymptotics with geometric and analytic methods.

Contribution

It develops a local regularization for QFT on curved surfaces satisfying Segal's axioms and extends the formalism to include entanglement entropy and large genus asymptotics.

Findings

Constructed $P()_2$ QFT on curved surfaces satisfying Segal's axioms.

Extended Segal's formalism to surfaces with slits for entanglement entropy.

Provided geometric and heat kernel proofs for zeta determinant asymptotics.

Abstract

This is the Ph.D. thesis of the author. In this thesis, we construct the $ P(\phi)_2 $ Quantum Field Theory (QFT) model on curved surfaces and show that it satisfies Segal's axioms (arXiv:2403.12804). An important ingredient in this construction is the use of a local regularization procedure to define the interaction as a random variable with respect to the Gaussian Free Field (GFF). We provide a counterexample demonstrating that spectral truncation regularization violates locality (arXiv:2312.15511). We then explain how Segal's formalism can be extended to the gluing of surfaces with slits, which offers a geometric interpretation of the entanglement entropy. Using this interpretation, we exploit the Polyakov anomaly formula in Conformal Field Theory (CFT) and apply a simple renormalization procedure to define a quantity corresponding to entanglement entropy within this geometric interpretation. We then show that this quantity behaves like a CFT correlation function. This allows us to rigorously derive an entropy calculation of Cardy and Calabrese (arXiv:2501.19014). Finally, Segal's formalism is also related to the asymptotics of zeta determinants on surfaces of large genus where the genus tends to infinity (arXiv:2505.01586). We provide a geometric proof--independent of Segal's axioms--of the corresponding result using heat kernels, in addition to another proof based on Segal's axioms. Both proofs are presented in the thesis.