Recent zbMATH articles in MSC 58Jhttps://zbmath.org/atom/cc/58J2021-06-15T18:09:00+00:00WerkzeugDerivation of asymptotics of partial differential equations in a neighborhood of irregular singular points.https://zbmath.org/1460.351632021-06-15T18:09:00+00:00"Korovina, M. V."https://zbmath.org/authors/?q=ai:korovina.mariya-victorovna"Smirnov, V. Yu."https://zbmath.org/authors/?q=ai:smirnov.v-yuSummary: This article is devoted to the study of irregular singular points of the linear partial differential equations with holomorphic coefficients. In this paper we consider two important cases of differential equations. In the first case we consider partial differential equation with a special condition on the main symbol of the differential operator, in the second case considered Laplace equation on a manifold with cuspidal singularities. For these cases we construct asymptotics of solution with the help of resurgent analysis.Renormalization of Feynman amplitudes on manifolds by spectral zeta regularization and blow-ups.https://zbmath.org/1460.810612021-06-15T18:09:00+00:00"Dang, Nguyen Viet"https://zbmath.org/authors/?q=ai:dang.nguyen-viet"Zhang, Bin"https://zbmath.org/authors/?q=ai:zhang.bin.2|zhang.bin.4|zhang.bin.1|zhang.bin.3Summary: Our goal in this paper is to present a generalization of the spectral zeta regularization for general Feynman amplitudes on Riemannian manifolds. Our method uses complex powers of elliptic operators but involves several complex parameters in the spirit of \textit{analytic renormalization} by Speer, to build mathematical foundations for the renormalization of perturbative interacting quantum field theories. Our main result shows that spectrally regularized Feynman amplitudes admit analytic continuation as meromorphic germs with linear poles in the sense of the works of Guo-Paycha and the second author. We also give an explicit determination of the affine hyperplanes supporting the poles. Our proof relies on suitable resolution of singularities of products of heat kernels to make them smooth.
As an application of the analytic continuation result, we use a universal projection from meromorphic germs with linear poles on holomorphic germs to construct renormalization maps which subtract singularities of Feynman amplitudes of Euclidean fields. Our renormalization maps are shown to satisfy consistency conditions previously introduced in the work of Nikolov-Todorov-Stora in the case of flat space-times.Global stability of some totally geodesic wave maps.https://zbmath.org/1460.352332021-06-15T18:09:00+00:00"Abbrescia, Leonardo Enrique"https://zbmath.org/authors/?q=ai:abbrescia.leonardo-enrique"Chen, Yuan"https://zbmath.org/authors/?q=ai:chen.yuanSummary: We prove that wave maps that factor as \(\mathbb{R}^{1 + d} \xrightarrow{\varphi_\mathrm{S}} \mathbb{R} \xrightarrow{\varphi_1} M\), subject to a sign condition, are globally nonlinear stable under small compactly supported perturbations when \(M\) is a spaceform. The main innovation is our assumption on \(\varphi_{\mathrm{S}} \), namely that it be a semi-Riemannian submersion. This implies that the background solution has infinite total energy, making this, to the best of our knowledge, the first stability result for factored wave maps with infinite energy backgrounds. We prove that the equations of motion for the perturbation decouple into a nonlinear wave-Klein-Gordon system. We prove global existence for this system and improve on the known regularity assumptions for equations of this type.Random band matrices in the delocalized phase. III: Averaging fluctuations.https://zbmath.org/1460.600092021-06-15T18:09:00+00:00"Yang, Fan"https://zbmath.org/authors/?q=ai:yang.fan.1|yang.fan.4|yang.fan.2|yang.fan.6"Yin, Jun"https://zbmath.org/authors/?q=ai:yin.jun.1|yin.junSummary: We consider a general class of symmetric or Hermitian random band matrices \(H=(h_{xy})_{x,y \in\lbrack\lbrack 1,N\rbrack\rbrack^d}\) in any dimension \(d\ge 1\), where the entries are independent, centered random variables with variances \(s_{xy}=\mathbb{E}|h_{xy}|^2\). We assume that \(s_{xy}\) vanishes if \(|x-y|\) exceeds the band width \(W\), and we are interested in the mesoscopic scale with \(1\ll W\ll N\). Define the generalized resolvent of \(H\) as \(G(H,Z):=(H-Z)^{-1}\), where \(Z\) is a deterministic diagonal matrix with entries \(Z_{xx}\in\mathbb{C}_+\) for all \(x\). Then we establish a precise high-probability bound on certain averages of polynomials of the resolvent entries. As an application of this fluctuation averaging result, we give a self-contained proof for the delocalization of random band matrices in dimensions \(d\ge 2\). More precisely, for any fixed \(d\ge 2\), we prove that the bulk eigenvectors of \(H\) are delocalized in certain averaged sense if \(N\le W^{1+\frac{d}{2}}\). This improves the corresponding results in [\textit{Y. He} and \textit{M. Marcozzi}, J. Stat. Phys. 177, No. 4, 666--716 (2019; Zbl 1448.60023)] that imposed the assumption \(N\ll W^{1+\frac{d}{d+1}}\), and the results in [\textit{L. Erdős} and \textit{A. Knowles}, Ann. Henri Poincaré 12, No. 7, 1227--1319 (2011; Zbl 1247.15033); Commun. Math. Phys. 303, No. 2, 509--554 (2011; Zbl 1226.15024)] that imposed the assumption \(N\ll W^{1+\frac{d}{6}}\). For 1D random band matrices, our fluctuation averaging result was used in Part II and Part I [\textit{P. Bourgade} et al., J. Stat. Phys. 174, No. 6, 1189--1221 (2019; Zbl 1447.60018); Commun. Pure Appl. Math. 73, No. 7, 1526--1596 (2020; Zbl 1446.60005)] to prove the delocalization conjecture and bulk universality for random band matrices with \(N\ll W^{4/3}\).Is entanglement a probe of confinement?https://zbmath.org/1460.810072021-06-15T18:09:00+00:00"Jokela, Niko"https://zbmath.org/authors/?q=ai:jokela.niko"Subils, Javier G."https://zbmath.org/authors/?q=ai:subils.javier-gSummary: We study various entanglement measures in a one-parameter family of three-dimensional, strongly coupled Yang-Mills-Chern-Simons field theories by means of their dual supergravity descriptions. A generic field theory in this family possesses a mass gap but does not have a linear quark-antiquark potential. For the two limiting values of the parameter, the theories flow either to a fixed point or to a confining vacuum in the infrared. We show that entanglement measures are unable to discriminate confining theories from non-confining ones with a mass gap. This lends support on the idea that the phase transition of entanglement entropy at large-\(N\) can be caused just by the presence of a sizable scale in a theory. and just by itself should not be taken as a signal of confinement. We also examine flows passing close to a fixed point at intermediate energy scales and find that the holographic entanglement entropy, the mutual information, and the \(F\)-functions for strips and disks quantitatively match the conformal values for a range of energies.Non-relativistic symmetries in three space-time dimensions and the Nappi-Witten algebra.https://zbmath.org/1460.810362021-06-15T18:09:00+00:00"Peñafiel, Diego M."https://zbmath.org/authors/?q=ai:penafiel.diego-m"Salgado-Rebolledo, Patricio"https://zbmath.org/authors/?q=ai:salgado-rebolledo.patricioSummary: We show that the Extended Bargmann and Newton-Hooke algebras in \(2+1\) dimensions can be obtained as expansions of the Nappi-Witten algebra. The procedure can be generalized to obtain two infinite families of non-relativistic symmetries, which include the Maxwellian Exotic Bargmann symmetry, its generalized Newton-Hooke counterpart, and its Hietarinta dual. In each case, the invariant bilinear form on the Nappi-Witten algebra leads to the invariant tensor on the expanded algebra, allowing one to construct the corresponding Chern-Simons gravity theory.An isoperimetric inequality for the harmonic mean of the Steklov eigenvalues in hyperbolic space.https://zbmath.org/1460.352422021-06-15T18:09:00+00:00"Verma, Sheela"https://zbmath.org/authors/?q=ai:verma.sheelaFor any bounded and smooth domain \(\Omega\) of the standard hyperbolic space \(\mathbb H^n\), an inequality for harmonic mean of the first \(n\) positive eigenvalues \(\mu_1(\Omega),...,\mu_n(\Omega)\) of the Steklov problem on \(\Omega\) is proved, namely \[ \sum_{i=1}^n\frac{1}{\mu_i(\Omega)}\geq\sum_{i=1}^n\frac{1}{\mu_i(B_R)}, \] where \(B_R\) is the ball in \(\mathbb H^n\) with the same volume of \(\Omega\). Equality holds if and only if \(\Omega\) is a ball. The proof relies on the construction of a suitable set of test functions for the Rayleigh quotient in terms of normal coordinate functions.
Reviewer: Luigi Provenzano (Padova)Geometric and obstacle scattering at low energy.https://zbmath.org/1460.352452021-06-15T18:09:00+00:00"Strohmaier, Alexander"https://zbmath.org/authors/?q=ai:strohmaier.alexander"Waters, Alden"https://zbmath.org/authors/?q=ai:waters.aldenSummary: We consider scattering theory of the Laplace Beltrami operator on differential forms on a Riemannian manifold that is Euclidean at infinity. The manifold may have several boundary components caused by obstacles at which relative boundary conditions are imposed. Scattering takes place because of the presence of these obstacles and possible non-trivial topology and geometry. Unlike in the case of functions eigenvalues generally exist at the bottom of the continuous spectrum and the corresponding eigenforms represent cohomology classes. We show that these eigenforms appear in the expansion of the resolvent, the scattering matrix, and the spectral measure in terms of the spectral parameter \(\lambda\) near zero, and we determine the first terms in this expansion explicitly. In dimension two an additional cohomology class appears as a resonant state in the presence of an obstacle. In even dimensions the expansion is in terms of \(\lambda\) and \(\log \lambda \). The theory of Hahn holomorphic functions is used to describe these expansions effectively. We also give a Birman-Krein formula in this context. The case of one-forms with relative boundary conditions has direct applications in physics as it describes the scattering of electromagnetic waves.Exponential convergence of parabolic optimal transport on bounded domains.https://zbmath.org/1460.352172021-06-15T18:09:00+00:00"Abedin, Farhan"https://zbmath.org/authors/?q=ai:abedin.farhan"Kitagawa, Jun"https://zbmath.org/authors/?q=ai:kitagawa.junThe authors study the asymptotic behavior of solutions to the second boundary value problem for a parabolic PDE of Monge-Ampère type arising from optimal mass transport. They are able to prove an exponential rate of convergence for solutions of this evolution equation to the stationary solution of the optimal transport problem. They obtain this important exponential convergence and the control of the oscillation in time of solutions to the parabolic equation by deriving a differential Harnack inequality for a special class of functions that solve the linearized problem and by certain techniques specific to mass transport. Additionally, in the course of the proof, they discover an interesting connection with the pseudo-Riemannian framework introduced by Kim and McCann in the context of optimal transport.
Reviewer: Vincenzo Vespri (Firenze)Comparison of partition functions in a space-time random environment.https://zbmath.org/1460.601132021-06-15T18:09:00+00:00"Junk, Stefan"https://zbmath.org/authors/?q=ai:junk.stefanSummary: Let \(Z^1\) and \(Z^2\) be partition functions in the random polymer model in the same environment but driven by different underlying random walks. We give a comparison in concave stochastic order between \(Z^1\) and \(Z^2\) if one of the random walks has ``more randomness'' than the other. We also treat some related models: The parabolic Anderson model with space-time Lévy noise; Brownian motion among space-time obstacles; and branching random walks in space-time random environments. We also obtain a necessary and sufficient criterion for \(Z^1 \preceq_{cv} Z^2\) if the lattice is replaced by a regular tree.Analysis and geometry on graphs and manifolds. Selected papers of the conference, University of Potsdam, Potsdam, Germany, July 31 -- August 4, 2017.https://zbmath.org/1460.580022021-06-15T18:09:00+00:00"Keller, Matthias (ed.)"https://zbmath.org/authors/?q=ai:keller.matthias"Lenz, Daniel (ed.)"https://zbmath.org/authors/?q=ai:lenz.daniel-h"Wojciechowski, Radoslaw K. (ed.)"https://zbmath.org/authors/?q=ai:wojciechowski.radoslaw-krzysztofPublisher's description: The interplay of geometry, spectral theory and stochastics has a long and fruitful history, and is the driving force behind many developments in modern mathematics. Bringing together contributions from a 2017 conference at the University of Potsdam, this volume focuses on global effects of local properties. Exploring the similarities and differences between the discrete and the continuous settings is of great interest to both researchers and graduate students in geometric analysis. The range of survey articles presented in this volume give an expository overview of various topics, including curvature, the effects of geometry on the spectrum, geometric group theory, and spectral theory of Laplacian and Schrödinger operators. Also included are shorter articles focusing on specific techniques and problems, allowing the reader to get to the heart of several key topics.
The articles of this volume will be reviewed individually.Dispersionless multi-dimensional integrable systems and related conformal structure generating equations of mathematical physics.https://zbmath.org/1460.170402021-06-15T18:09:00+00:00"Hentosh, Oksana Ye."https://zbmath.org/authors/?q=ai:hentosh.oksana-ye"Prykarpatsky, Yarema A."https://zbmath.org/authors/?q=ai:prykarpatsky.yarema-anatoliyovych"Blackmore, Denis"https://zbmath.org/authors/?q=ai:blackmore.denis-l"Prykarpatski, Anatolij K."https://zbmath.org/authors/?q=ai:prykarpatsky.anatoliy-karolevychSummary: Using diffeomorphism group vector fields on \(\mathbb{C}\)-multiplied tori and the related Lie-algebraic structures, we study multi-dimensional dispersionless integrable systems that describe conformal structure generating equations of mathematical physics. An interesting modification of the devised Lie-algebraic approach subject to spatial-dimensional invariance and meromorphicity of the related differential-geometric structures is described and applied in proving complete integrability of some conformal structure generating equations. As examples, we analyze the Einstein-Weyl metric equation, the modified Einstein-Weyl metric equation, the Dunajski heavenly equation system, the first and second conformal structure generating equations and the inverse first Shabat reduction heavenly equation. We also analyze the modified Plebański heavenly equations, the Husain heavenly equation and the general Monge equation along with their multi-dimensional generalizations. In addition, we construct superconformal analogs of the Whitham heavenly equation.Density of resonances for covers of Schottky surfaces.https://zbmath.org/1460.580172021-06-15T18:09:00+00:00"Pohl, Anke"https://zbmath.org/authors/?q=ai:pohl.anke-d"Soares, Louis"https://zbmath.org/authors/?q=ai:soares.louisSummary: We investigate how bounds of resonance counting functions for Schottky surfaces behave under transitions to covering surfaces of finite degree. We consider the classical resonance counting function asking for the number of resonances in large (and growing) disks centered at the origin of \(\mathbb{C}\), as well as the (fractal) resonance counting function asking for the number of resonances in boxes near the axis of the critical exponent. For the former counting function we provide a transfer-operator-based proof that bounding constants can be chosen such that the transformation behavior under transition to covers is as for the Weyl law in the case of surfaces of finite area. For the latter counting function we deduce a bound in terms of the covering degree and the minimal length of a periodic geodesic on the covering surface. This yields an improved fractal Weyl upper bound. In the setting of Schottky surfaces, these estimates refine previous results due to \textit{L. Guillopé} and \textit{M. Zworski} [J. Funct. Anal. 129, No. 2, 364--389 (1995; Zbl 0841.58063); Ann. Math. (2) 145, No. 3, 597--660 (1997; Zbl 0898.58054)] and \textit{L. Guillopé} et al. [Commun. Math. Phys. 245, No. 1, 149--176 (2004; Zbl 1075.11059)]. When applied to principal congruence covers, these results yield new estimates for the resonance counting functions in the level aspect, which have recently been investigated by \textit{D. Jakobson} and \textit{F. Naud} [Isr. J. Math. 213, 443--473 (2016; Zbl 1391.11076)]. The techniques used in this article are based on the thermodynamic formalism for L-functions (twisted Selberg zeta functions), and twisted transfer operators.On the quotient quantum graph with respect to the regular representation.https://zbmath.org/1460.580192021-06-15T18:09:00+00:00"Mutlu, Gökhan"https://zbmath.org/authors/?q=ai:mutlu.gokhanSummary: Given a quantum graph \(\Gamma\), a finite symmetry group \(G\) acting on it and a representation \(R\) of \(G\), the quotient quantum graph \(\Gamma/R\) is described and constructed in the literature [1,2,18]. In particular, it was shown that the quotient graph \(\Gamma/\mathbb{C}G\) is isospectral to \(\Gamma\) by using representation theory where \(\mathbb{C}G\) denotes the regular representation of \(G\) [18]. Further, it was conjectured that \(\Gamma\) can be obtained as a quotient \(\Gamma/\mathbb{C}G\) [18]. However, proving this by construction of the quotient quantum graphs has remained as an open problem. In this paper, we solve this problem by proving by construction that for a quantum graph \(\Gamma\) and a finite symmetry group \(G\) acting on it, the quotient quantum graph \(\Gamma/\mathbb{C}G\) is not only isospectral but rather identical to \(\Gamma\) for a particular choice of a basis for \(\mathbb{C}G\). Furthermore, we prove that, this result holds for an arbitrary permutation representation of \(G\) with degree \(|G|\), whereas it doesn't hold for a permutation representation of \(G\) with degree greater than \(|G|\).The effective action of superrotation modes.https://zbmath.org/1460.830082021-06-15T18:09:00+00:00"Nguyen, Kévin"https://zbmath.org/authors/?q=ai:nguyen.kevin"Salzer, Jakob"https://zbmath.org/authors/?q=ai:salzer.jakobSummary: Starting from an analysis of four-dimensional asymptotically flat gravity in first order formulation, we show that superrotation reparametrization modes are governed by an Alekseev-Shatashvili action on the celestial sphere. This two-dimensional conformal theory describes spontaneous symmetry breaking of Virasoro superrotations together with the explicit symmetry breaking of more general \( \mathrm{Diff} ({\mathcal{S}}^2)\) superrotations. We arrive at this result by first reformulating the asymptotic field equations and symmetries of the radiative vacuum sector in terms of a Chern-Simons theory at null infinity, and subsequently performing a Hamiltonian reduction of this theory onto the celestial sphere.\(L^p\)-estimates for the heat semigroup on differential forms, and related problems.https://zbmath.org/1460.580152021-06-15T18:09:00+00:00"Magniez, Jocelyn"https://zbmath.org/authors/?q=ai:magniez.jocelyn"Ouhabaz, El Maati"https://zbmath.org/authors/?q=ai:ouhabaz.el-maatiFor a \(M^n\) complete Riemannian manifold and \(\Delta\) the (non-negative) Laplace-Beltrami operator, consider \((e^{-t\Delta})_{t\geq 0}\) the associated heat semigroup acting as a contraction semigroup on \(L^p(M)\) for all \(1\leq p \leq \infty\), and the semigroup is strongly continuous on \(L^p(M)\) for \(1\leq p < \infty\). Now instead of \(\Delta\), one may consider the Hodge-de Rham Laplacian \(\overrightarrow{\Delta}_k=d_k^*d_k+d_{k-1}d_{k-1}^*\) and the associated contraction semigroup \((e^{-t\Delta_k})_{t\geq 0}\) on \(L^2(\Lambda^kT^*M)\). Note that \(\overrightarrow{\Delta}_k\) is non-negative.
The article under review studies \(L^p\)-estimates of the semigroup \((e^{-t\Delta_k})_{t\geq 0}\). Apparently the precise estimate of the \(L^p\)-norm \(\|(e^{-t\Delta_k})_{t\geq 0}\|_{p-p}\) is not easy, and the article proves in Theorem 1.2 (i) such an estimate. To be more specific, consider the Bochner's formula \(\overrightarrow{\Delta}_k=\Delta^*\Delta+R_k\) where \(\Delta\) the Levi-Civita connection and \(R_k\) a symmetric section of \(\text{End}(\Lambda^k T^*M)\). Denote by \(R^\pm_k\) the positive and negative part of \(R_k\). The article, under an assumption of a volume-doubling property (Compare Equation (D) in p.3003), the Gaussian upper bound condition (Compare Equation (G) in p.3004), \(R_k^-\) in the enlarged Kato class \(\hat{K}\) (compare Definition 1.1), shows that \(\|(e^{-t\Delta_k})\|_{p-p}\leq C(t\log t)^{\frac{D}{4}(1-\frac{2}{p})}\) for large \(t\) where \(D\) is a homogeneous dimension appearing in the volume doubling property.
In addition, using an \(\epsilon\)-subcritical condition for \(R_k^-\) for \(\epsilon\in[0,1)\), the authors prove in Theorem 1.4 better estimates than that of Theorem 1.2. The article also provides in Proposition 1.5 a side result that under an additional assumption of uniform boundedness of \((e^{-t\Delta})_{t\geq 0}\) on \(L^p(\Lambda^1 T^*M)\) for some fixed \(p\), the Riesz transformation \(d\Delta^{-1/2}\) is bounded on \(L^r(M)\) for all \(r\in (1,\text{max}(p,p'))\).
Reviewer: Byungdo Park (Cheongju)Five-dimensional gauge theories on spheres with negative couplings.https://zbmath.org/1460.811022021-06-15T18:09:00+00:00"Minahan, Joseph A."https://zbmath.org/authors/?q=ai:minahan.joseph-a"Nedelin, Anton"https://zbmath.org/authors/?q=ai:nedelin.antonSummary: We consider supersymmetric gauge theories on \(S^5\) with a negative Yang-Mills coupling in their large \(N\) limits. Using localization we compute the partition functions and show that the pure \( \mathrm{SU} (N)\) gauge theory descends to an \(\mathrm{SU} (N/2)_{+N/2} \times \mathrm{SU} (N/2)_{-N /2} \times \mathrm{SU}(2)\) Chern-Simons gauge theory as the inverse 't Hooft coupling is taken to negative infinity for \(N\) even. The Yang-Mills coupling of the \( \mathrm{SU} (N/2)_{\pm N /2}\) is positive and infinite, while that on the SU(2) goes to zero. We also show that the odd \(N\) case has somewhat different behavior. We then study the \(\mathrm{SU} (N/2)_{N/2}\) pure Chern-Simons theory. While the eigenvalue density is only found numerically, we show that its width equals 1 in units of the inverse sphere radius, which allows us to find the leading correction to the free energy when turning on the Yang-Mills term. We then consider \( \mathrm{USp} (2N)\) theories with an antisymmetric hypermultiplet and \(N_f< 8\) fundamental hypermultiplets and carry out a similar analysis. Along the way we show that the one-instanton contribution to the partition function remains exponentially suppressed at negative coupling for the \(\mathrm{SU} (N)\) theories in the large \(N\) limit.Almost global existence for the Klein-Gordon equation with the Kirchhoff-type nonlinearity.https://zbmath.org/1460.352362021-06-15T18:09:00+00:00"Han, Zheng"https://zbmath.org/authors/?q=ai:han.zheng"Fang, Daoyuan"https://zbmath.org/authors/?q=ai:fang.daoyuanSummary: We prove an almost global existence result for the Klein-Gordon equation with the Kirchhoff-type nonlinearity on \(\mathbb{T}^d\) with Cauchy data of small amplitude \(\epsilon\). We show a lower bound \(\epsilon^{-2N-2}\) for the existence time with any natural number \(N\). The proof relies on the method of normal forms and induction. The structure of the nonlinearity is good enough that proceeds normal forms up to any order.Non-relativistic three-dimensional supergravity theories and semigroup expansion method.https://zbmath.org/1460.831112021-06-15T18:09:00+00:00"Concha, Patrick"https://zbmath.org/authors/?q=ai:concha.patrick"Ipinza, Marcelo"https://zbmath.org/authors/?q=ai:ipinza.marcelo"Ravera, Lucrezia"https://zbmath.org/authors/?q=ai:ravera.lucrezia"Rodríguez, Evelyn"https://zbmath.org/authors/?q=ai:rodriguez.evelynSummary: In this work we present an alternative method to construct diverse non-relativistic Chern-Simons supergravity theories in three spacetime dimensions. To this end, we apply the Lie algebra expansion method based on semigroups to a supersymmetric extension of the Nappi-Witten algebra. Two different families of non-relativistic superalgebras are obtained, corresponding to generalizations of the extended Bargmann superalgebra and extended Newton-Hooke superalgebra, respectively. The expansion method considered here allows to obtain known and new non-relativistic supergravity models in a systematic way. In particular, it immediately provides an invariant tensor for the expanded superalgebra, which is essential to construct the corresponding Chern-Simons supergravity action. We show that the extended Bargmann supergravity and its Maxwellian generalization appear as particular subcases of a generalized extended Bargmann supergravity theory. In addition, we demonstrate that the generalized extended Bargmann and generalized extended Newton-Hooke supergravity families are related through a contraction process.Integrable lattice models and holography.https://zbmath.org/1460.830732021-06-15T18:09:00+00:00"Ashwinkumar, Meer"https://zbmath.org/authors/?q=ai:ashwinkumar.meerSummary: We study four-dimensional Chern-Simons theory on \(D \times \mathbb{C}\) (where \(D\) is a disk), which is understood to describe rational solutions of the Yang-Baxter equation from the work of \textit{K. Costello} et al. [ICCM Not. 6, No. 1, 120--146 (2018; Zbl 1405.81044)]. We find that the theory is dual to a boundary theory, that is a three-dimensional analogue of the two-dimensional chiral WZW model. This boundary theory gives rise to a current algebra that turns out to be an ``analytically-continued'' toroidal Lie algebra. In addition, we show how certain bulk correlation functions of two and three Wilson lines can be captured by boundary correlation functions of local operators in the three-dimensional WZW model. In particular, we reproduce the leading and subleading nontrivial contributions to the rational \(R\)-matrix purely from the boundary theory.BPS invariants for 3-manifolds at rational level \(K\).https://zbmath.org/1460.810902021-06-15T18:09:00+00:00"Chung, Hee-Joong"https://zbmath.org/authors/?q=ai:chung.hee-joongSummary: We consider the Witten-Reshetikhin-Turaev invariants or Chern-Simons partition functions at or around roots of unity \(q={e}^{\frac{2\pi i}{K}}\) with a rational level \(K = \frac{r}{s}\) where \(r\) and \(s\) are coprime integers. From the exact expression for the \(G = \mathrm{SU}(2)\) Witten-Reshetikhin-Turaev invariants of the Seifert manifolds at a rational level obtained by \textit{R. Lawrence} and \textit{L. Rozansky} [Commun. Math. Phys. 205, No. 2, 287--314 (1999; Zbl 0966.57017)], we provide an expected form of the structure of the Witten-Reshetikhin-Turaev invariants in terms of the homological blocks at a rational level. Also, we discuss the asymptotic expansion of knot invariants around roots of unity where we take a limit different from the limit in the standard volume conjecture.Tube structures of co-rank 1 with forms defined on compact surfaces.https://zbmath.org/1460.580142021-06-15T18:09:00+00:00"Hounie, J."https://zbmath.org/authors/?q=ai:hounie.j"Zugliani, G."https://zbmath.org/authors/?q=ai:zugliani.giuliano-angeloSummary: We study the global solvability of a locally integrable structure of tube type and co-rank 1 by considering a linear partial differential operator \(\mathbb{L}\) associated to a general complex smooth closed 1-form \(c\) defined on a smooth closed \(n\)-manifold. The main result characterizes the global solvability of \(\mathbb{L}\) when \(n=2\) in terms of geometric properties of a primitive of a convenient exact pullback of the form \(\mathfrak{Im}(c)\) as well as in terms of homological properties of \(\mathfrak{Re}(c)\) related to small divisors phenomena. Although the full characterization is restricted to orientable surfaces, some partial results hold true for compact manifolds of any dimension, in particular, the necessity of the conditions, and the equivalence when \(\mathfrak{Im}(c)\) is exact. We also obtain informations on the global hypoellipticity of \(\mathbb{L}\) and the global solvability of \(\mathbb{L}^{n-1}\) -- the last non-trivial operator of the complex when \(M\) is orientable.Lévy Laplacians and instantons on manifolds.https://zbmath.org/1460.580052021-06-15T18:09:00+00:00"Volkov, Boris O."https://zbmath.org/authors/?q=ai:volkov.boris-oThe main result of the paper under review is that a connection in a complex vector bundle over a 4-dimensional orientable Riemannian manifold \(M\) satisfies the anti-self-duality Yang-Mills equations if and only if its corresponding parallel transport satisfies a system of Laplace equations corresponding to three modified Lévy Laplacians. In some more detail, let \(E\to M\) be a complex vector bundle. Let \(\Omega\) be the set of all \(H^1\)-curves \(\gamma\colon[0,1]\to M\). The set \(\Omega\) has the natural structure of a manifold modeled on an separable infinite-dimensional real Hilbert space. Moreover, one has a natural vector bundle \(\mathcal{E}\to\Omega\), whose fiber over \(\gamma\in\Omega\) is the set of all linear maps from the fiber of \(E\) over \(\gamma(0)\in M\) to the fiber of \(E\) over \(\gamma(1)\in M\). For any \(m\in M\) we also denote \(\Omega_m:=\{\gamma\in\Omega\mid\gamma(0)=m\}\) and let \(\mathcal{E}_m:=\mathcal{E}\vert_{\Omega_m}\). The parallel transport associated to a connection in the vector bundle \(E\) can be regarded as a section of the vector bundle \(\mathcal{E}\). The aforementioned modified Lévy Laplacians are second-order differential operators that act in section spaces of bundles \(\mathcal{E}_m\) as above, and their action is defined via so-called modified Lévy traces which are associated with certain \(C^1\)-curves in the special orthogonal group \(\mathrm{SO}(4)\).
Reviewer: Daniel Beltiţă (Bucureşti)Most general theory of 3d gravity: covariant phase space, dual diffeomorphisms, and more.https://zbmath.org/1460.830662021-06-15T18:09:00+00:00"Geiller, Marc"https://zbmath.org/authors/?q=ai:geiller.marc"Goeller, Christophe"https://zbmath.org/authors/?q=ai:goeller.christophe"Merino, Nelson"https://zbmath.org/authors/?q=ai:merino.nelsonSummary: We show that the phase space of three-dimensional gravity contains two layers of dualities: between diffeomorphisms and a notion of ``dual diffeomorphisms'' on the one hand, and between first order curvature and torsion on the other hand. This is most elegantly revealed and understood when studying the most general Lorentz-invariant first order theory in connection and triad variables, described by the so-called Mielke-Baekler Lagrangian. By analyzing the quasi-local symmetries of this theory in the covariant phase space formalism, we show that in each sector of the torsion/curvature duality there exists a well-defined notion of dual diffeomorphism, which furthermore follows uniquely from the Sugawara construction. Together with the usual diffeomorphisms, these duals form at finite distance, without any boundary conditions, and for any sign of the cosmological constant, a centreless double Virasoro algebra which in the flat case reduces to the \( \mathrm{BMS}_3\) algebra. These algebras can then be centrally-extended via the twisted Sugawara construction. This shows that the celebrated results about asymptotic symmetry algebras are actually generic features of three-dimensional gravity at any finite distance. They are however only revealed when working in first order connection and triad variables, and a priori inaccessible from Chern-Simons theory. As a bonus, we study the second order equations of motion of the Mielke-Baekler model, as well as the on-shell Lagrangian. This reveals the duality between Riemannian metric and teleparallel gravity, and a new candidate theory for three-dimensional massive gravity which we call teleparallel topologically massive gravity.Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional.https://zbmath.org/1460.353372021-06-15T18:09:00+00:00"Chen, Bo"https://zbmath.org/authors/?q=ai:chen.bo.4|chen.bo.3|chen.bo.2|chen.bo.1"Wang, Youde"https://zbmath.org/authors/?q=ai:wang.youdeSummary: We follow the idea of \textit{Y.-D. Wang} [J. Math. Phys. 39, No. 1, 363--371 (1998; Zbl 0938.58017)] to show the existence of global weak solutions to the Cauchy problems of Landau-Lifshtiz type equations and related heat flows from a \(n\)-dimensional Euclidean domain \(\Omega\) or a \(n\)-dimensional closed Riemannian manifold \(M\) into a 2-dimensional unit sphere \(\mathbb{S}^2\). Our conclusions extend a series of related results obtained in the previous literature.Inverse spectral theory for perturbed torus.https://zbmath.org/1460.580162021-06-15T18:09:00+00:00"Isozaki, Hiroshi"https://zbmath.org/authors/?q=ai:isozaki.hiroshi"Korotyaev, Evgeny L."https://zbmath.org/authors/?q=ai:korotyaev.evgeny-lThe authors focus on studying inverse problems for Laplacians on a suitable manifold. Therefore, among the results, they state the existence of an analytic isomorphism map between an explicit real separable Hilbert space of real functions and the direct sum of the real Hilbert space of square-summable sequences. The proof is based on employing nonlinear functional analysis tools and on a result of explicit perturbed Riccati mappings.
Reviewer: Mohammed El Aïdi (Bogotá)Small time equivalents for the density of a planar quadratric diffusion.https://zbmath.org/1460.580202021-06-15T18:09:00+00:00"Franchi, Jacques"https://zbmath.org/authors/?q=ai:franchi.jacquesThis paper presents asymptotic estimates for the joint probability density function \(p_\varepsilon (w,y)\) of the random vector \(\left( B_\varepsilon , \int_0^\varepsilon B^2_s ds \right)\) as \(\varepsilon\) tends to zero, where \((B_s)_{s\in {\mathbb R}_+}\) is a standard Brownian motion. Exact equivalents are given for \(p_\varepsilon (0, y)\) and \(p_\varepsilon (w, y)\), and for \(p_\varepsilon (0,\varepsilon y)\), \(p_\varepsilon (w,\varepsilon y)\) in the scaled regime \((w,\varepsilon y)\), \(w \in {\mathbb R}^*\), \(y > 0\). The proofs rely on expressions of the joint Fourier transform of quadratic Langevin-type diffusion processes, and on their inversion via a careful analysis of contour and oscillatory integrals by saddle-point methods.
Reviewer: Nicolas Privault (Singapore)An index formula for the intersection Euler characteristic of an infinite cone.https://zbmath.org/1460.580182021-06-15T18:09:00+00:00"Ludwig, Ursula"https://zbmath.org/authors/?q=ai:ludwig.ursula-beateThis article is part of the author's program to extend the Cheeger-Müller theorem to singular spaces with conical singularities. The main idea is to generalize the strategy of \textit{J.-M. Bismut} and \textit{W. Zhang} [An extension of a theorem by Cheeger and Müller. With an appendix by François Laudenbach. Paris: Société Mathématique de France (1992; Zbl 0781.58039)]. This program was concluded in [\textit{U. Ludwig}, Duke Math. J. 169, No. 13, 2501--2570 (2020; Zbl 1458.58016)].
In this article, the author establishes an index formula for the intersection Euler characteristic of a cone. To do that, it is studied the spectral properties of the model Witten Laplacian on the infinite cone. An explicit formula for the zeta function of the model Witten Laplacian is presented (Theorem I). Latter, the author express the absolute and relative intersection Euler characteristic of the cone as a sum of two terms, a local term and the Cheeger invariant (Theorem II). This local term is build over the Chern-Simons class of \((0,\infty)\times L\) with the Levi-Civita connections of the metrics \(dr^2+r^2g\) and \(dr^2+g\), where \(g\) is the metric on \(L\).
Reviewer: Luiz Hartmann (São Carlos)Stability in the inverse Steklov problem on warped product Riemannian manifolds.https://zbmath.org/1460.811082021-06-15T18:09:00+00:00"Daudé, Thierry"https://zbmath.org/authors/?q=ai:daude.thierry"Kamran, Niky"https://zbmath.org/authors/?q=ai:kamran.niky"Nicoleau, François"https://zbmath.org/authors/?q=ai:nicoleau.francoisSummary: In this paper, we study the amount of information contained in the Steklov spectrum of some compact manifolds with connected boundary equipped with a warped product metric. Examples of such manifolds can be thought of as deformed balls in \(\mathbb{R}^d\). We first prove that the Steklov spectrum determines uniquely the warping function of the metric. We show in fact that the approximate knowledge (in a given precise sense) of the Steklov spectrum is enough to determine uniquely the warping function in a neighbourhood of the boundary. Second, we provide stability estimates of log-type on the warping function from the Steklov spectrum. The key element of these stability results relies on a formula that, roughly speaking, connects the inverse data (the Steklov spectrum) to the \textit{Laplace transform} of the difference of the two warping factors.Regularity of fully non-linear elliptic equations on Kähler cones.https://zbmath.org/1460.351482021-06-15T18:09:00+00:00"Yuan, Rirong"https://zbmath.org/authors/?q=ai:yuan.rirongSummary: We derive quantitative boundary estimates, and then solve the Dirichlet problem for a general class of fully non-linear elliptic equations on annuli of Kähler cones over closed Sasakian manifolds. This extends extensively a result concerning the geodesic equations in the space of Sasakian metrics due to \textit{P. Guan} and \textit{X. Zhang} [Adv. Math. 230, No. 1, 321--371 (2012; Zbl 1245.58016)]. Our results show that the solvability is deeply affected by the transverse Kähler structures of Sasakian manifolds. We also discuss possible extensions of the results to equations with right-hand side depending on unknown solutions.On the hidden mechanism behind non-uniqueness for the anisotropic Calderón problem with data on disjoint sets.https://zbmath.org/1460.353872021-06-15T18:09:00+00:00"Daudé, Thierry"https://zbmath.org/authors/?q=ai:daude.thierry"Kamran, Niky"https://zbmath.org/authors/?q=ai:kamran.niky"Nicoleau, François"https://zbmath.org/authors/?q=ai:nicoleau.francoisSummary: We show that there is generically non-uniqueness for the anisotropic Calderón problem at fixed frequency when the Dirichlet and Neumann data are measured on disjoint sets of the boundary of a given domain. More precisely, we first show that given a smooth compact connected Riemannian manifold with boundary \((M, g)\) of dimension \(n\geq 3\), there exist in the conformal class of \(g\) an infinite number of Riemannian metrics \(\tilde{g}\) such that their corresponding DN maps at a fixed frequency coincide when the Dirichlet data \(\Gamma _D\) and Neumann data \(\Gamma _N\) are measured on disjoint sets and satisfy \(\overline{\Gamma _D \cup \Gamma _N} \neq \partial M\). The conformal factors that lead to these non-uniqueness results for the anisotropic Calderón problem satisfy a nonlinear elliptic PDE of Yamabe type on the original manifold \((M, g)\) and are associated with a natural but subtle gauge invariance of the anisotropic Calderón problem with data on disjoint sets. We then construct a large class of counterexamples to uniqueness in dimension \(n\geq 3\) to the anisotropic Calderón problem at fixed frequency with data on disjoint sets and \textit{modulo this gauge invariance}. This class consists in cylindrical Riemannian manifolds with boundary having two ends (meaning that the boundary has two connected components), equipped with a suitably chosen warped product metric.Feature matching and heat flow in centro-affine geometry.https://zbmath.org/1460.530102021-06-15T18:09:00+00:00"Olver, Peter J."https://zbmath.org/authors/?q=ai:olver.peter-j"Qu, Changzheng"https://zbmath.org/authors/?q=ai:qu.changzheng"Yang, Yun"https://zbmath.org/authors/?q=ai:yang.yunSummary: In this paper, we study the differential invariants and the invariant heat flow in centro-affine geometry, proving that the latter is equivalent to the inviscid Burgers' equation. Furthermore, we apply the centro-affine invariants to develop an invariant algorithm to match features of objects appearing in images. We show that the resulting algorithm compares favorably with the widely applied scale-invariant feature transform (SIFT), speeded up robust features (SURF), and affine-SIFT (ASIFT) methods.Chern characters in equivariant basic cohomology.https://zbmath.org/1460.530292021-06-15T18:09:00+00:00"Liu, Wenran"https://zbmath.org/authors/?q=ai:liu.wenranSummary: The purpose of this Note is to establish a geometric realization of the cohomological isomorphism in the case of a transversely oriented Killing foliation on a compact smooth manifold through equivariant basic Chern characters.Self-adjoint local boundary problems on compact surfaces. I: Spectral flow.https://zbmath.org/1460.351252021-06-15T18:09:00+00:00"Prokhorova, Marina"https://zbmath.org/authors/?q=ai:prokhorova.marina-fSummary: The paper deals with first order self-adjoint elliptic differential operators on a compact oriented surface with non-empty boundary. We consider such operators with self-adjoint local boundary conditions. The paper is focused on paths in the space of such operators connecting two operators conjugated by a unitary automorphism. We compute the spectral flow for such paths in terms of the topological data over the boundary. The second result is the universality of the spectral flow: we show that the spectral flow is a universal additive invariant for such paths, if the vanishing on paths of invertible operators is required.Resurgent analysis of SU(2) Chern-Simons partition function on Brieskorn spheres \(\Sigma (2, 3, 6n + 5)\).https://zbmath.org/1460.810602021-06-15T18:09:00+00:00"Wu, David H."https://zbmath.org/authors/?q=ai:wu.david-hSummary: \( \hat{Z} \)-invariants, which can reconstruct the analytic continuation of the SU(2) Chern-Simons partition functions via Borel resummation, were discovered by GPV and have been conjectured to be a new homological invariant of 3-manifolds which can shed light onto the superconformal and topologically twisted index of 3d \(\mathcal{N} = 2\) theories proposed by GPPV. In particular, the resurgent analysis of \(\hat{Z}\) has been fruitful in discovering analytic properties of the WRT invariants. The resurgent analysis of these \(\hat{Z} \)-invariants has been performed for the cases of \(\Sigma (2, 3, 5) \), \( \Sigma (2, 3, 7)\) by GMP, \( \Sigma (2, 5, 7)\) by Chun, and, more recently, some additional Seifert manifolds by Chung and Kucharski, independently. In this paper, we extend and generalize the resurgent analysis of \(\hat{Z}\) on a family of Brieskorn homology spheres \(\Sigma (2, 3, 6n + 5)\) where \(n \in \mathbb{Z}_+\) and \(6n + 5\) is a prime. By deriving \(\hat{Z}\) for \(\Sigma (2, 3, 6n + 5)\) according to GPPV and Hikami, we provide a formula where one can quickly compute the non-perturbative contributions to the full analytic continuation of SU(2) Chern-Simons partition function.Multipeak solutions for the Yamabe equation.https://zbmath.org/1460.351442021-06-15T18:09:00+00:00"Rey, Carolina A."https://zbmath.org/authors/?q=ai:rey.carolina-a"Ruiz, Juan Miguel"https://zbmath.org/authors/?q=ai:ruiz.juan-miguelSummary: Let \((M,g)\) be a closed Riemannian manifold of dimension \(n\geq 3\) and \(x_0\in M\) be an isolated local minimum of the scalar curvature \(s_g\) of \(g\). For any positive integer \(k\) we prove that for \(\epsilon>0\) small enough the subcritical Yamabe equation \(-\epsilon^2\Delta u+(1+c_N\epsilon^2 s_g)u=u^{\text{q}}\) has a positive \(k\)-peaks solution which concentrate around \(x_0\), assuming that a constant \(\beta\) is non-zero. In the equation \(c_N=\frac{N-2}{4(N-1)}\) for an integer \(N>n\) and \(q=\frac{N+2}{N-2}\). The constant \(\beta\) depends on \(n\) and \(N\), and can be easily computed numerically, being negative in all cases considered. This provides solutions to the Yamabe equation on Riemannian products \((M\times X,g+\epsilon^2 h)\), where \((X, h)\) is a Riemannian manifold with constant positive scalar curvature. We also prove that solutions with small energy only have one local maximum.Large singular solutions for conformal \(Q\)-curvature equations on \(\mathbb{S}^n\).https://zbmath.org/1460.351362021-06-15T18:09:00+00:00"Du, Xusheng"https://zbmath.org/authors/?q=ai:du.xusheng"Yang, Hui"https://zbmath.org/authors/?q=ai:yang.hui.1Summary: In this paper, we study the existence of positive functions \(K \in C^1 (\mathbb{S}^n)\) such that the conformal \(Q\)-curvature equation
\[
P_m(v) = K v^{\frac{n + 2m}{n - 2m}} \quad \text{on } \mathbb{S}^n
\]
has a singular positive solution \(v\) whose singular set is a single point, where \(m\) is an integer satisfying \(1 \leq m < n / 2\) and \(P_m\) is the intertwining operator of order \(2m\). More specifically, we show that when \(n \geq 2m + 4\), every positive function in \(C^1 (\mathbb{S}^n)\) can be approximated in the \(C^1 (\mathbb{S}^n)\) norm by a positive function \(K \in C^1 (\mathbb{S}^n)\) such that (1) has a singular positive solution whose singular set is a single point. Moreover, such a solution can be constructed to be arbitrarily large near its singularity. This is in contrast to the well-known results of \textit{C.-S. Lin} [Comment. Math. Helv. 73, No. 2, 206--231 (1998; Zbl 0933.35057)] and \textit{J. Wei} and \textit{X. Xu} [Math. Ann. 313, No. 2, 207--228 (1999; Zbl 0940.35082)] which show that Eq. (1), with \(K\) identically a positive constant on \(\mathbb{S}^n, n > 2m\), does not exist a singular positive solution whose singular set is a single point.Sharp Adams-Moser-Trudinger type inequalities in the hyperbolic space.https://zbmath.org/1460.351152021-06-15T18:09:00+00:00"Ngô, Quốc Anh"https://zbmath.org/authors/?q=ai:ngo.quoc-anh"Nguyen, van Hoang"https://zbmath.org/authors/?q=ai:nguyen-van-hoang.1Summary: The purpose of this paper is to establish some Adams-Moser-Trudinger inequalities, which are the borderline cases of the Sobolev embedding, in the hyperbolic space \(\mathbb{H}^n\). First, we prove a sharp Adams' inequality of order two with the exact growth condition in \(\mathbb{H}^n\). Then we use it to derive a sharp Adams-type inequality and an Adachi-Tanakat-ype inequality. We also prove a sharp Adams-type inequality with Navier boundary condition on any bounded domain of \(\mathbb{H}^n\), which generalizes the result of Tarsi to the setting of hyperbolic spaces. Finally, we establish a Lions-type lemma and an improved Adams-type inequality in the spirit of Lions in \(\mathbb{H}^n\). Our proofs rely on the symmetrization method extended to hyperbolic spaces.