arXiv subjects

Nguyen Viet Dang

Publications and source records attributed to Nguyen Viet Dang.

At least 19 recordsLinked to original sources

arXiv2026

The Yang-Mills measure on surfaces via Morse theory

We introduce a Morse theoretical approach to the construction of the Yang--Mills measure on the space of connections of a compact Riemannian surface. This provides a direct continuous version of this measure which was previously obtained through lattice approximations by Chevyrev in the case of the flat torus and by one of the authors and Nohra for general compact Riemannian surfaces. The starting point is the new notion of a Morse gauge together with the resolution of random cohomological equations associated to Morse--Smale vector fields. This is achieved by improving exponential convergence to equilibrium results for Morse--Smale gradient flows that were obtained by two of the authors in the context of the study of Ruelle spectra and by Jia, Stewart and Sverak in the context of simplified models from fluid mechanics. Combining these random solutions with the data given by the Morse complex, we introduce a free Yang-Mills measure on space of connections and, using classical tools from stochastic differential equations, we show how to make sense of holonomies for random connections along a large class of curves. Finally, by setting a proper conditioning of this free measure through these random holonomies, we define the Yang--Mills measure and we compute its partition function together with the law of random holonomies with respect to this measure, recovering the formulas from the works of Migdal, Witten and Lévy.

math.PR↗

arXiv2026

Semiclassical analysis for Yang--Mills random connections on compact surfaces

We introduce anisotropic Banach spaces of distributional 1-forms on compact surfaces designed to capture the fine regularity properties of Morse gauge-fixed Yang--Mills random connections. Using singular connections supported on unstable curves of a Morse gradient flow, we provide a new description of the moduli space of flat connections and of its canonical Atiyah--Bott--Goldman symplectic form. When the group is the special unitary group, we prove that in the zero-area limit, the Yang--Mills random connection converges, within these anisotropic spaces, to a random distributional 1-form whose law coincides with our Morse-theoretic representative of the Atiyah--Bott--Goldman measure. Our approach extends the works of Witten, Forman, Liu, and Sengupta by establishing the zero-area limit of the Yang--Mills measure at the level of random distributional connections, rather than only at the level of holonomies. This answers a question of Thierry Lévy on the semiclassical limits of Yang--Mills random connections.

math.PR↗

arXiv2026

The Yang--Mills measure on compact surfaces as a universal scaling limit of lattice gauge models

In this article, we study the 2 dimensional Yang--Mills measure on compact surfaces from a unified continuum and discrete perspective. We construct the Yang--Mills measure as a random distributional 1 form on surfaces of arbitrary genus equipped with an arbitrary smooth area form, using the analytic concept of pseudo-coordinates. Our approach yields a canonical noise-flat decomposition of the measure, reflecting the topology of the surface. We prove a universality theorem stating that the continuum Yang--Mills measure arises as the scaling limit of a wide class of lattice gauge theories -- including Wilson, Manton, and Villain actions -- on any compact surface. We study the convergence in natural spaces of distributions with anisotropic regularity. As further consequences, we obtain a new intrinsic construction of the Yang--Mills measure, independent of the previous constructions in the literature, and prove the convergence of correlation functions and Segal amplitudes on all compact surfaces.

math.PR↗

arXiv2026

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_ζ(Δ_{N})}{\text{Vol}(M_N)} $ when $N\rightarrow +\infty$ where $Δ_N$ 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

Poincar{é} series for analytic convex bodies

We study Poincar{é} series associated to strictly convex bodies in the Euclidean space. These series are Laplace transforms of the distribution of lengths (measured with the Finsler metric associated to one of the bodies) from one convex body to a lattice. Assuming that the convex bodies have analytic boundaries, we prove that the Poincar{é} series, originally defined in the right complex half-plane, continues holomorphically to a conical neighborhood of this set, removing a countable set of cuts and points. The latter correspond to the spectrum of a dual elliptic operator. We describe singularities of the Poincar{é} series at each of these branching points. One of the steps of the proof consists in showing analytic continuation of the resolvent of multiplication operators by a real-valued analytic Morse function on the sphere as a branched holomorphic function, a result of independent interest.

math.DG↗

arXiv2025

Combinatorial zeta functions counting triangles

In this paper, we compute special values of certain combinatorial zeta functions counting geodesic paths in the (n-1)-skeleton of a triangulation of a n-dimensional manifold. We show that they carry a topological meaning. As such, we recover the first Betti number and L2-Betti number of compact manifolds, and the linking number of pairs of null-homologous knots in a 3-manifold. The tool to relate the two sides (counting geodesics/topological invariants) are random walks on higher dimensional skeleta of the triangulation.

math.GT↗

arXiv2024

Dirac operators and local invariants on perturbations of Minkowski space

For small perturbations of Minkowski space, we show that the square of the Lorentzian Dirac operator $P= -D^2$ has real spectrum apart from possible poles in a horizontal strip. Furthermore, for $\varepsilon>0$ we relate the poles of the spectral zeta function density of $P-i\varepsilon$ to local invariants, in particular to the Lorentzian scalar curvature. The proof involves microlocal propagation and radial estimates in a resolved scattering calculus as well as high energy estimates in a further resolved classical-semiclassical calculus.

math.AP↗

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↗

arXiv2023

Length orthospectrum of convex bodies on flat tori

In analogy with the study of Pollicott-Ruelle resonances on negatively curved manifolds, we define anisotropic Sobolev spaces that are well-adapted to the analysis of the geodesic vector field associated with any translation invariant Finsler metric on the torus $\mathbb{T}^d$. Among several applications of this functional point of view, we study properties of geodesics that are orthogonal to two convex subsets of $\mathbb{T}^d$ (i.e. projection of the boundaries of strictly convex bodies of $\mathbb{R}^d$). Associated with the set of lengths of such orthogeodesics, we define a geometric Epstein function and prove its meromorphic continuation. We compute its residues in terms of intrinsic volumes of the convex sets. We also prove Poisson-type summation formulae relating the set of lengths of orthogeodesics and the spectrum of magnetic Laplacians.

math.AP↗

arXiv2023

Length orthospectrum and the correlation function on flat tori

This note presents some of the results obtained in arXiv:2207.05410 and it has beenthe object of a talk of the second author during the Journées "Équations auxDérivées Partielles" (Obernai, june 2022). We study properties of geodesics that are orthogonal to two convex subsets of the flat torus. We discuss meromorphic properties of a geometric Epstein zeta function associated to the set of lengths of such orthogeodesics. We also define the associated length distribution and discuss singularities of its Fourier transform. Our analysis relies on a fine study of the dynamical correlation function of the geodesic flow on the torus and the definition of anisotropic Sobolev spaces that are well-adapted to this integrable dynamics.

math.AP↗

arXiv2023

Poincar{é} series and linking of Legendrian knots

On a negatively curved surface, we show that the Poincar{é} series counting geodesic arcs orthogonal to some pair of closed geodesic curves has a meromorphic continuation to the whole complex plane. When both curves are homologically trivial, we prove that the Poincar{é} series has an explicit rational value at 0 interpreting it in terms of linking number of Legendrian knots. In particular, for any pair of points on the surface, the lengths of all geodesic arcs connecting the two points determine its genus, and, for any pair of homologically trivial closed geodesics, the lengths of all geodesic arcs orthogonal to both geodesics determine the linking number of the two geodesics.

math.DG↗

arXiv2022

Complex powers of the wave operator and the spectral action on Lorentzian scattering spaces

We consider perturbations of Minkowski space as well as more general spacetimes on which the wave operator $\square_g$ is known to be essentially self-adjoint. We define complex powers $(\square_g-i\varepsilon)^{-α}$ by functional calculus, and show that the trace density exists as a meromorphic function of $α$. We relate its poles to geometric quantities, in particular to the scalar curvature. The results allow us to formulate a spectral action principle which serves as a simple Lorentzian model for the bosonic part of the Chamseddine-Connes action. Our proof combines microlocal resolvent estimates, including radial propagation estimates, with uniform estimates for the Hadamard parametrix. The arguments operate in Lorentzian signature directly and do not rely on a transition from the Euclidean setting. The results hold also true in the case of ultrastatic spacetimes.

math.AP↗

arXiv2022

Dynamical residues of Lorentzian spectral zeta functions

We define a dynamical residue which generalizes the Guillemin-Wodzicki residue density of pseudo-differential operators. More precisely, given a Schwartz kernel, the definition refers to Pollicott-Ruelle resonances for the dynamics of scaling towards the diagonal. We apply this formalism to complex powers of the wave operator and we prove that residues of Lorentzian spectral zeta functions are dynamical residues. The residues are shown to have local geometric content as expected from formal analogies with the Riemannian case.

math.AP↗

arXiv2022

Lorentzian spectral zeta functions on asymptotically Minkowski spacetimes

In this note, we consider perturbations of Minkowski space as well as more general spacetimes on which the wave operator $\square_g$ is essentially self-adjoint. We review a recent result which gives the meromorphic continuation of the Lorentzian spectral zeta function density, i.e. of the trace density of complex powers $α\mapsto (\square_g-i \varepsilon)^{-α}$. In even dimension $n\geq 4$, the residue at $\frac{n}{2}-1$ is shown to be a multiple of the scalar curvature in the limit $\varepsilon\to 0^+$. This yields a spectral action for gravity in Lorentzian signature.

math.AP↗

arXiv2020

Dynamical torsion for contact Anosov flows

We introduce a new object, the dynamical torsion, which extends the potentially ill-defined value at $0$ of the Ruelle zeta function of a contact Anosov flow twisted by an acyclic representation of the fundamental group. We show important properties of the dynamical torsion: it is invariant under deformations among contact Anosov flows, it is holomorphic in the representation and it has the same logarithmic derivative as some refined combinatorial torsion of Turaev. This shows that the ratio between this torsion and the Turaev torsion is locally constant on the space of acyclic representations. In particular, for contact Anosov flows path connected to the geodesic flow of some hyperbolic manifold among contact Anosov flows, we relate the leading term of the Laurent expansion of $ζ$ at the origin, the Reidemeister torsion and the torsions of the finite dimensional complexes of the generalized resonant states of both flows for the resonance $0$. This extends previous work of~\cite{dang2018fried} on the Fried conjecture near geodesic flows of hyperbolic $3$--manifolds, to hyperbolic manifolds of any odd dimension.

math.DS↗

arXiv2020

Renormalization of determinant lines in Quantum Field Theory

On a compact manifold $M$, we consider the affine space $A$ of non self-adjoint perturbations of some invertible elliptic operator acting on sections of some Hermitian bundle, by some differential operator of lower order. We construct and classify all complex analytic functions on the Fréchet space $A$ vanishing exactly over non invertible elements, having minimal order and which are obtained by local renormalizations, a concept coming from quantum field theory, called renormalized determinants. The additive group of local polynomial functionals of finite degrees acts freely and transitively on the space of renormalized determinants. We provide different representations of the renormalized determinants in terms of spectral zeta determinants, Gaussian Free Fields, infinite product and renormalized Feynman amplitudes in perturbation theory in position space à la Epstein-Glaser. Specializing to the case of Dirac operators coupled to vector potentials and reformulating our results in terms of determinant line bundles, we prove our renormalized determinants define some complex analytic trivializations of some holomorphic line bundle over $A$ relating our results to a conjectural picture from some unpublished notes by Quillen [52] from April 1989.

math-ph↗

arXiv2020

Wick squares of the Gaussian Free Field and Riemannian rigidity

In the present paper, we show that on a compact Riemannian manifold $(M,g)$ of dimension $d\leqslant 4$ whose metric has negative curvature, the renormalized partition function $Z_g(λ)$ of a massive Gaussian Free Field determines the length spectrum of $(M,g)$ and imposes some strong geometric constraints on the Riemannian structure of $(M,g)$. In any finite dimensional family of Riemannian metrics of negative sectional curvature bounded from below and above and whose isometry group is trivial, there is only a \textbf{finite number of isometry classes} of metrics with given partition function $Z_g(λ)$. When $d<4$, the same result holds true if the random variable $\int_M:ϕ^2:dv$ has given probability distribution and without the lower bound on the sectional curvatures.

math-ph↗

arXiv2019

Fried conjecture in small dimensions

We study the twisted Ruelle zeta function $ζ_X(s)$ for smooth Anosov vector fields $X$ acting on flat vector bundles over smooth compact manifolds. In dimension $3$, we prove Fried conjecture, relating Reidemeister torsion and $ζ_X(0)$. In higher dimensions, we show more generally that $ζ_X(0)$ is locally constant with respect to the vector field $X$ under a spectral condition. As a consequence, we also show Fried conjecture for Anosov flows near the geodesic flow on the unit tangent bundle of hyperbolic $3$-manifolds. This gives the first examples of non-analytic Anosov flows and geodesic flows in variable negative curvature where Fried conjecture holds true.

math.DG↗