arXiv subjects

Jiasheng Lin

Publications and source records attributed to Jiasheng Lin.

5 recordsLinked to original sources

arXiv2025

Constructive Quantum Field Theory on Curved Surfaces and Related Topics

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.

quant-ph↗

arXiv2025

Asymptotics of zeta determinants of Laplacians on large degree abelian covers

Let $(M,g)$ be some smooth, closed, compact Riemannian manifold and $(M_N\mapsto M)_N$ be an increasing sequence of large degree cyclic covers of $M$ that converges when $N\rightarrow +\infty$, in a suitable sense, to some limit $\mathbb{Z}^p$ cover $M_\infty$ over $M$. Motivated by recent works on zeta determinants on random surfaces and some natural questions in Euclidean quantum field theory, we show the convergence of the sequence $ \frac{\log\det_\zeta(\Delta_{N})}{\text{Vol}(M_N)} $ when $N\rightarrow +\infty$ where $\Delta_N$ is the Laplace-Beltrami operator on $M_N$. We also generalize our results to the case of twisted Laplacians coming from certain flat unitary vector bundles over $M$.

math-ph↗

arXiv2025

Entanglement Entropy and Cauchy-Hadamard Renormalization

This note presents a purely geometric construction of the so-called twist-field correlation functions in Conformal Field Theory (CFT), derived from conical singularities. This approach provides a purely mathematical interpretation of the seminal results in physics by Cardy and Calabrese on the entanglement entropy of quantum systems. Specifically, we begin by defining CFT partition functions on surfaces with conical singularities, using a ``Cauchy-Hadamard renormalization'' of the Polyakov anomaly integral. Next, we demonstrate that for a branched cover $f:\Sigma_d\to \Sigma$ with $d$ sheets, where the cover inherits the pullback of a smooth metric from the base, a specific ratio of partition functions on the cover to the base transforms under conformal changes of the base metric in the same way as a correlation function of CFT primary fields with specific conformal weights. We also provide a discussion of the physical background and motivation for entanglement entropy, focusing on path integrals and the replica trick, which serves as an introduction to these ideas for a mathematical audience.

hep-th↗

arXiv2024

The Bayes Principle and Segal Axioms for $P(ϕ)_2$, with application to Periodic Covers

We construct a $P(ϕ)_2$ Gibbs state on infinite volume periodic surfaces (namely, with discrete ``time translations'') by analogy with 1-dimensional spin chains and establish the mass gap for our Gibbs state, there are no phase transitions. We also derive asymptotic properties of the $P(ϕ)_2$ partition function on certain towers of cyclic covers of large degrees that converge to the periodic surface in some appropriate sense. This gives the first construction of an interacting Quantum Field Theory on surfaces of infinite genus with a mass gap. The main ingredient in our approach is to reconcile the so-called $P(ϕ)_2$ model from classical constructive quantum field theory (CQFT) with Riemannian version of the axioms proposed by G. Segal in the 90's. We show the $P(ϕ)_2$ model satisfies these axioms, appropriately adjusted. One key ingredient in our proof is to use what we call ``the Bayes principle'' of conditional probabilities in the infinite dimensional setting. We also give a precise statement and full proof of the locality of the $P(ϕ)_2$ interaction.

math-ph↗

arXiv2023

Spectrally cut-off GFF, regularized $Φ^4$ measure, and reflection positivity

We argue that the spectrally cut-off Gaussian free field $Φ_Λ$ on a compact Riemannian manifold or on $\mathbb{R}^n$ cannot satisfy the spatial Markov property. Moreover, when the manifold is reflection positive, we show that $Φ_Λ$ fails to be reflection positive. We explain the difficulties one encounters when trying to deduce the reflection positivity property of the measure exp$(-\|ρΦ_Λ\|_{L^4}^4) μ_{\text{GFF}}(dΦ)$ from the reflection positivity property of the Gaussian free field measure $μ_{\text{GFF}}$ in a naive way. These issues are probably well-known to experts of constructive quantum field theory but to our knowledge, no detailed account can be found in the litterature. Our pedagogical note aims to fill this small gap.

math.PR↗