Recent zbMATH articles in MSC 58Jhttps://www.zbmath.org/atom/cc/58J2021-04-16T16:22:00+00:00WerkzeugLie group methods for eigenvalue function.https://www.zbmath.org/1456.580232021-04-16T16:22:00+00:00"Nazarkandi, H. A."https://www.zbmath.org/authors/?q=ai:nazarkandi.hossain-alizadehSummary: By considering a \(C^\infty\) structure on the ordered non-increasing of elements of \(\mathbb R^n\), we show that it is a differentiable manifold. By using of Lie groups, we show that eigenvalue function is a submersion. This fact is used to prove some results. These results is applied to prove a few facts about spectral manifolds and spectral functions. Orthogonal matrices act on the real symmetric matrices as a Lie transformation group. This fact, also, is used to prove the results.Optimisation of the lowest Robin eigenvalue in the exterior of a compact set. II: Non-convex domains and higher dimensions.https://www.zbmath.org/1456.351472021-04-16T16:22:00+00:00"Krejčiřík, David"https://www.zbmath.org/authors/?q=ai:krejcirik.david"Lotoreichik, Vladimir"https://www.zbmath.org/authors/?q=ai:lotoreichik.vladimirIsoperimetric problems for the lowest eigenvalue of the Robin Laplacian on the exterior \(\Omega^{\mathrm{ext}}\) of a bounded, smooth open set \(\Omega \subset \mathbb{R}^d\), \(d \geq 2\), are studied. Denote by \(\lambda_1^\alpha (\Omega^{\mathrm{ext}})\) the minimum of the spectrum of the negative Laplacian in \(L^2 (\Omega^{\mathrm{ext}})\) subject to the boundary condition
\[
\frac{\partial u}{\partial n} = \alpha u \quad \text{on}~\partial \Omega,
\]
where the Robin parameter \(\alpha < 0\) is a constant and \(\frac{\partial}{\partial n}\) denotes the derivative with respect to the outer unit normal vector to \(\Omega\) (i.e.\ the normal pointing inside \(\Omega^{\mathrm{ext}}\)); in dimension \(d = 2\), \(\lambda_1^\alpha (\Omega^{\mathrm{ext}})\) is always a discrete, negative eigenvalue, while for \(d \geq 3\) this is true for all \(\alpha\) below a certain threshold.
In the first main result of this article, it is shown for \(d = 2\), fixed \(\alpha < 0\) and fixed \(c > 0\) that
\[
\max_{\frac{|\partial \Omega|}{N_\Omega} = c} \lambda_1^\alpha (\Omega^{\mathrm{ext}}) = \lambda_1^\alpha (B^{\mathrm{ext}}),
\]
where the maximum is taken over all smooth, bounded open sets \(\Omega\) consisting of a finite number of connected components (the latter number denoted by \(N_\Omega\)) such that \(\frac{|\partial \Omega|}{N_\Omega} = c\) and \(B\) is the disk with perimeter \(c\). This improves upon an earlier result by the same authors where only convex \(\Omega\) where allowed.
The second main result concerns the higher-dimensional case \(d \geq 3\); here convexity of \(\Omega\) is required. With the notation
\[
\mathcal{M} (\partial \Omega) := \frac{1}{|\partial \Omega|} \int_{\partial \Omega} \left( \frac{\kappa_1 + \dots + \kappa_{d - 1}}{d - 1} \right)^{d - 1},
\]
where \(\kappa_1, \dots, \kappa_{d - 1}\) denote the principle curvatures of \(\partial \Omega\), the authors prove that, for each \(\alpha < 0\) and \(c > 0\),
\[
\max_{\mathcal{M} (\partial \Omega) = c} \lambda_1^\alpha (\Omega^{\mathrm{ext}}) = \lambda_1^\alpha (B^{\mathrm{ext}}),
\]
where the maximum is taken over all convex, smooth, bounded open sets \(\Omega\) such that \(\mathcal{M} (\partial \Omega) = c\), and \(B\) is the ball with \(\mathcal{M} (\partial B) = c\).
For Part I, see [the authors, J. Convex Anal. 25, No. 1, 319--337 (2018; Zbl 1401.35223)].
Reviewer: Jonathan Rohleder (Stockholm)Concise notes on special holonomy with an emphasis on Calabi-Yau and \(G_2\)-manifolds.https://www.zbmath.org/1456.530062021-04-16T16:22:00+00:00"Oliveira, Gonçalo"https://www.zbmath.org/authors/?q=ai:oliveira.goncaloSummary: These are notes for a very short introduction to some selected topics on special Riemannian holonomy with a focus on Calabi-Yau and \(G_2\)-manifolds. No material in these notes is original and more on it can be found in the papers/books of Bryant, Hitchin, Joyce and Salamon referenced during the text.Note on the Green's function formalism and topological invariants.https://www.zbmath.org/1456.814982021-04-16T16:22:00+00:00"Zhou, Yehao"https://www.zbmath.org/authors/?q=ai:zhou.yehao"Liu, Junyu"https://www.zbmath.org/authors/?q=ai:liu.junyuWilson loop algebras and quantum K-theory for Grassmannians.https://www.zbmath.org/1456.814352021-04-16T16:22:00+00:00"Jockers, Hans"https://www.zbmath.org/authors/?q=ai:jockers.hans"Mayr, Peter"https://www.zbmath.org/authors/?q=ai:mayr.peter"Ninad, Urmi"https://www.zbmath.org/authors/?q=ai:ninad.urmi"Tabler, Alexander"https://www.zbmath.org/authors/?q=ai:tabler.alexanderSummary: We study the algebra of Wilson line operators in three-dimensional \(\mathcal{N} = 2\) supersymmetric \(\mathrm{U}(M)\) gauge theories with a Higgs phase related to a complex Grassmannian \(\mathrm{Gr}(M,N)\), and its connection to K-theoretic Gromov-Witten invariants for \(\mathrm{Gr}(M,N)\). For different Chern-Simons levels, the Wilson loop algebra realizes either the quantum cohomology of \(\mathrm{Gr}(M,N)\), isomorphic to the Verlinde algebra for \(\mathrm{U}(M)\), or the quantum K-theoretic ring of Schubert structure sheaves studied by mathematicians, or closely related algebras.Microstate geometries from gauged supergravity in three dimensions.https://www.zbmath.org/1456.831152021-04-16T16:22:00+00:00"Mayerson, Daniel R."https://www.zbmath.org/authors/?q=ai:mayerson.daniel-r"Walker, Robert A."https://www.zbmath.org/authors/?q=ai:walker.robert-a|walker.robert-a-ii"Warner, Nicholas P."https://www.zbmath.org/authors/?q=ai:warner.nicholas-pSummary: The most detailed constructions of microstate geometries, and particularly of superstrata, are done using \(\mathcal{N} = (1, 0)\) supergravity coupled to two anti-self-dual tensor multiplets in six dimensions. We show that an important sub-sector of this theory has a consistent truncation to a particular gauged supergravity in three dimensions. Our consistent truncation is closely related to those recently laid out by Samtleben and Sarıoğlu, which enables us to develop complete uplift formulae from the three-dimensional theory to six dimensions. We also find a new family of multi-mode superstrata, indexed by two arbitrary holomorphic functions of one complex variable, that live within our consistent truncation and use this family to provide extensive tests of our consistent truncation. We discuss some of the future applications of having an intrinsically three-dimensional formulation of a significant class of microstate geometries.Four-point functions in large \(N\) Chern-Simons fermionic theories.https://www.zbmath.org/1456.815002021-04-16T16:22:00+00:00"Kalloor, Rohit R."https://www.zbmath.org/authors/?q=ai:kalloor.rohit-rSummary: We compute all four-point functions involving the operators \(J_0\) and \(J_1\) in large-\(N\) Chern-Simons fermionic theories, in the regime where all external momenta lie along the \(z\)-axis. We find that our result for \(\langle J_0J_0J_0J_0 \rangle\) agrees with previous computations, and that the other correlators fall in line with expectations from bootstrap arguments.Polyakov-Alvarez type comparison formulas for determinants of Laplacians on Riemann surfaces with conical singularities.https://www.zbmath.org/1456.580242021-04-16T16:22:00+00:00"Kalvin, Victor"https://www.zbmath.org/authors/?q=ai:kalvin.victorSummary: We present and prove Polyakov-Alvarez type comparison formulas for the determinants of Friederichs extensions of Laplacians corresponding to conformally equivalent metrics on a compact Riemann surface with conical singularities. In particular, we find how the determinants depend on the orders of conical singularities. We also illustrate these general results with several examples: based on our Polyakov-Alvarez type formulas we recover known and obtain new explicit formulas for determinants of Laplacians on singular surfaces with and without boundary. In one of the examples we show that on the metrics of constant curvature on a sphere with two conical singularities and fixed area \(4 \pi\) the determinant of Friederichs Laplacian is unbounded from above and attains its local maximum on the metric of standard round sphere. In another example we deduce the famous Aurell-Salomon formula for the determinant of Friederichs Laplacian on polyhedra with spherical topology, thus providing the formula with mathematically rigorous proof.Exact results and Schur expansions in quiver Chern-Simons-matter theories.https://www.zbmath.org/1456.814422021-04-16T16:22:00+00:00"Santilli, Leonardo"https://www.zbmath.org/authors/?q=ai:santilli.leonardo"Tierz, Miguel"https://www.zbmath.org/authors/?q=ai:tierz.miguelSummary: We study several quiver Chern-Simons-matter theories on the three-sphere, combining the matrix model formulation with a systematic use of Mordell's integral, computing partition functions and checking dualities. We also consider Wilson loops in ABJ(M) theories, distinguishing between typical (long) and atypical (short) representations and focusing on the former. Using the Berele-Regev factorization of supersymmetric Schur polynomials, we express the expectation value of the Wilson loops in terms of sums of observables of two factorized copies of \(\mathrm{U}(N\)) pure Chern-Simons theory on the sphere. Then, we use the Cauchy identity to study the partition functions of a number of quiver Chern-Simons-matter models and the result is interpreted as a perturbative expansion in the parameters \(t_j = - e^{2 \pi m_j }\), where \(m_j\) are the masses. Through the paper, we incorporate different generalizations, such as deformations by real masses and/or Fayet-Iliopoulos parameters, the consideration of a Romans mass in the gravity dual, and adjoint matter.Vertex algebras and 4-manifold invariants.https://www.zbmath.org/1456.580182021-04-16T16:22:00+00:00"Dedushenko, Mykola"https://www.zbmath.org/authors/?q=ai:dedushenko.mykola"Gukov, Sergei"https://www.zbmath.org/authors/?q=ai:gukov.sergei"Putrov, Pavel"https://www.zbmath.org/authors/?q=ai:putrov.pavelFor the entire collection see [Zbl 1408.14005].Thin-shell wormholes in \(\mathrm{AdS}_5\) and string dioptrics.https://www.zbmath.org/1456.830922021-04-16T16:22:00+00:00"Chernicoff, Mariano"https://www.zbmath.org/authors/?q=ai:chernicoff.mariano"García, Edel"https://www.zbmath.org/authors/?q=ai:garcia.edel-b"Giribet, Gaston"https://www.zbmath.org/authors/?q=ai:giribet.gaston-e"de Celis, Emilio Rubín"https://www.zbmath.org/authors/?q=ai:de-celis.emilio-rubinSummary: We consider string probes in a traversable wormhole geometry that connects two locally \(\mathrm{ AdS}_5\) asymptotic regions. Holographically, this describes two interacting copies of a 4-dimensional gauge theory. We consider string configurations whose endpoints are located either in the same boundary or in the two different boundaries of the wormhole. A string with both endpoints in the same boundary is dual to a quark-antiquark pair charged under the same gauge field, while a string extending through the wormhole describes a pair of colored particles charged under two different gauge fields. When one considers a quark-antiquark pair in each boundary, the system undergoes a phase transition: while for small separation each pair of charges exhibits Coulomb interaction, for large separation the charges in different field theories pair up. This behavior had previously been observed in other geometric realizations such as locally \(\mathrm{AdS}_5\) wormhole solutions with hyperbolic throats. The geometries we consider here, in contrast, are stable thin-shell wormholes with flat codimension-one hypersurfaces at fixed radial coordinate. They appear as electrovacuum solutions of higher-curvature gravity theories coupled to abelian gauge fields. The presence of the thin-shells produces a refraction of the string configurations in the bulk, leading to the presence of cusps in the phase space diagram. We discuss these and other features of the phase diagram, including the analogies and difference with other wormhole solutions considered in related contexts.Elliptic \(G\)-operators on manifolds with isolated singularities.https://www.zbmath.org/1456.580172021-04-16T16:22:00+00:00"Savin, A. Yu."https://www.zbmath.org/authors/?q=ai:savin.anton-yu"Sternin, B. Yu."https://www.zbmath.org/authors/?q=ai:sternin.boris-yuSummary: In the present work we study elliptic operators on manifolds with singularities in the situation where the manifold is endowed with an action of a discrete group \(G\). As usual in elliptic theory, the Fredholm property of an operator is governed by the properties of its principal symbol. We show that the principal symbol in our situation is a pair consisting of the symbol on the main stratum (interior symbol) and the symbol at the conical point (conormal symbol). The Fredholm property of elliptic elements is obtained.Bootstrapping conformal four-point correlators with slightly broken higher spin symmetry and \(3D\) bosonization.https://www.zbmath.org/1456.830832021-04-16T16:22:00+00:00"Li, Zhijin"https://www.zbmath.org/authors/?q=ai:li.zhijinSummary: Three-dimensional conformal field theories (CFTs) with slightly broken higher spin symmetry provide an interesting laboratory to study general properties of CFTs and their roles in the AdS/CFT correspondence. In this work we compute the planar four-point functions at arbitrary 't Hooft coupling \(\lambda\) in the CFTs with slightly broken higher spin symmetry. We use a bootstrap approach based on the approximate higher spin Ward identity. We show that the bootstrap equation is separated into two parts with opposite parity charges, and it leads to a recursion relation for the \(\lambda\) expansions of the correlation functions. The \(\lambda\) expansions terminate at order \(\lambda^2\) and the solutions are exact in \(\lambda\). Our work generalizes the approach proposed by Maldacena and Zhiboedov to four-point correlators, and it amounts to an on-shell study for the \(3D\) Chern-Simons vector models and their holographic duals in \(\mathrm{AdS}_4\). Besides, we show that the same results can also be obtained rather simply from bosonization duality of \(3D\) Chern-Simons vector models. The odd term at order \(O( \lambda)\) in the spinning four-point function relates to the free boson correlator through a Legendre transformation. This provides new evidence on the \(3D\) bosonization duality at the spinning four-point function level. We expect this work can be generalized to a complete classification of general four-point functions of single trace currents.A nonabelian M5 brane Lagrangian in a supergravity background.https://www.zbmath.org/1456.831142021-04-16T16:22:00+00:00"Gustavsson, Andreas"https://www.zbmath.org/authors/?q=ai:gustavsson.andreasSummary: We present a nonabelian Lagrangian that appears to have \((2, 0)\) superconformal symmetry and that can be coupled to a supergravity background. But for our construction to work, we have to break this superconformal symmetry by imposing as a constraint on top of the Lagrangian that the fields have vanishing Lie derivatives along a Killing direction.Gravitational dual of averaged free CFT's over the Narain lattice.https://www.zbmath.org/1456.830302021-04-16T16:22:00+00:00"Pérez, Alfredo"https://www.zbmath.org/authors/?q=ai:perez.alfredo"Troncoso, Ricardo"https://www.zbmath.org/authors/?q=ai:troncoso.ricardoSummary: It has been recently argued that the averaging of free CFT's over the Narain lattice can be holographically described through a Chern-Simons theory for \( \mathrm{U} (1)^D \times \mathrm{U}(1)^D\) with a precise prescription to sum over three-dimensional handlebodies. We show that a gravitational dual of these averaged CFT's would be provided by Einstein gravity on \( \mathrm{AdS}_3\) with \( \mathrm{U} (1)^{ D - 1}\times \mathrm{ U} (1)^{ D- 1}\) gauge fields, endowed with a precise set of boundary conditions closely related to the ``soft hairy'' ones. Gravitational excitations then go along diagonal \( \mathrm{SL} (2, \mathbb{R})\) generators, so that the asymptotic symmetries are spanned by \( \mathrm{U} (1)^D \times \mathrm{U} (1)^D\) currents. The stress-energy tensor can then be geometrically seen as composite of these currents through a twisted Sugawara construction. Our boundary conditions are such that for the reduced phase space, there is a one-to-one map between the configurations in the gravitational and the purely abelian theories. The partition function in the bulk could then also be performed either from a non-abelian Chern-Simons theory for two copies of \( \mathrm{SL} (2, \mathbb{R}) \times \mathrm{U} (1)^{ D- 1}\) generators, or formally through a path integral along the family of allowed configurations for the metric. The new boundary conditions naturally accommodate BTZ black holes, and the microscopic number of states then appears to be manifestly positive and suitably accounted for from the partition function in the bulk. The inclusion of higher spin currents through an extended twisted Sugawara construction in the context of higher spin gravity is also briefly addressed.Existence results for a super-Liouville equation on compact surfaces.https://www.zbmath.org/1456.580162021-04-16T16:22:00+00:00"Jevnikar, Aleks"https://www.zbmath.org/authors/?q=ai:jevnikar.aleks"Malchiodi, Andrea"https://www.zbmath.org/authors/?q=ai:malchiodi.andrea"Wu, Ruijun"https://www.zbmath.org/authors/?q=ai:wu.ruijunLet \((M,g)\) be a closed Riemannian surface endowed with a genus bigger than one and \(K_g\) stands for the Gauss curvature of \(M\). The authors consider the functional energy defined by
\[\displaystyle J_\rho(u,\psi)=\int_M\left(|\nabla_g u|^2+2K_gu+\exp(2u)+2\langle({D}_g-\rho\exp(u))\psi,\psi\rangle\right)dv_g,\]
such that \(u\in C^\infty(M)\), \(\rho\) is a positive parameter, \(\psi\) is a spinor field on \(M\), and \({D_g}\) represents the Dirac operator on spinors. The Euler-Lagrange equation associated to \(J_\rho\) is defined by
\[ (*):\ \Delta_gu=\exp(2u)+K_g-\rho\exp(u)|\psi|^2\text{ and }{D}_g\psi=\rho\exp(u)\psi.\]
Then the authors state that \((*)\) has a non-zero solution whenever zero and \(\rho\) do not belong to the spectrum of \({D}_{g_0}\) (where \(g_0\) is a conformal metric to \(g\)) and \(K_{g_0}=-1\) (Theorem 1.1). The proof is essentially based on looking for a critical point of \(J_\rho\).
Reviewer: Mohammed El Aïdi (Bogotá)Analysis and stochastic processes on metric measure spaces.https://www.zbmath.org/1456.580192021-04-16T16:22:00+00:00"Grigor'yan, Alexander"https://www.zbmath.org/authors/?q=ai:grigoryan.alexanderThe purpose of the author is to survey some known results of the Laplacian operator on a geodesically complete and non-compact Riemannian manifold. Precisely, the overview contains, e.g., Semi-linear elliptic inequalities, Negative eigenvalues of Schrödinger, Estimates of the Green function, Heat kernels on connected sums, of Schrödinger operator, and of operators with singular drift, and so on. Likewise, the author deals with sections on Analysis on metric measure spaces and on Homology theory on graphs.
For the entire collection see [Zbl 1416.60012].
Reviewer: Mohammed El Aïdi (Bogotá)Dualities for three-dimensional \(\mathcal{N} = 2 \) \( \mathrm{SU} (N_c)\) chiral adjoint SQCD.https://www.zbmath.org/1456.814172021-04-16T16:22:00+00:00"Amariti, Antonio"https://www.zbmath.org/authors/?q=ai:amariti.antonio"Fazzi, Marco"https://www.zbmath.org/authors/?q=ai:fazzi.marcoSummary: We study dualities for 3d \(\mathcal{N} = 2 \) \( \mathrm{SU} (N_c)\) SQCD at Chern-Simons level \(k\) in presence of an adjoint with polynomial superpotential. The dualities are dubbed \textit{chiral} because there is a different amount of fundamentals \(N_f\) and antifundamentals \(N_a \). We build a complete classification of such dualities in terms of \( | N_f - N_a | \) and \(k\). The classification is obtained by studying the flow from the non-chiral case, and we corroborate our proposals by matching the three-sphere partition functions. Finally, we revisit the case of \( \mathrm{SU} (N_c)\) SQCD without the adjoint, comparing our results with previous literature.CURE: curvature regularization for missing data recovery.https://www.zbmath.org/1456.621272021-04-16T16:22:00+00:00"Dong, Bin"https://www.zbmath.org/authors/?q=ai:dong.bin|dong.bin.1"Ju, Haocheng"https://www.zbmath.org/authors/?q=ai:ju.haocheng"Lu, Yiping"https://www.zbmath.org/authors/?q=ai:lu.yiping"Shi, Zuoqiang"https://www.zbmath.org/authors/?q=ai:shi.zuoqiangOn the convergence and regularity of Aumann-Pettis integrable multivalued martingales.https://www.zbmath.org/1456.600972021-04-16T16:22:00+00:00"El Allali, Mohammed"https://www.zbmath.org/authors/?q=ai:el-allali.mohammed"El-Louh, M'hamed"https://www.zbmath.org/authors/?q=ai:el-louh.mhamed"Ezzaki, Fatima"https://www.zbmath.org/authors/?q=ai:ezzaki.fatimaSummary: We prove a representation of Aumann-Pettis integrable multivalued martingales by Pettis integrable martingale selectors. Regularity of Aumann-Pettis integrable multivalued martingales and their convergence in Mosco sense, Wijsman topology, and linear topology are established.About bounds for eigenvalues of the Laplacian with density.https://www.zbmath.org/1456.351482021-04-16T16:22:00+00:00"Ndiaye, Aïssatou Mossèle"https://www.zbmath.org/authors/?q=ai:ndiaye.aissatou-mosseleSummary: Let \(M\) denote a compact, connected Riemannian manifold of dimension \(n\in{\mathbb N}\). We assume that \(M\) has a smooth and connected boundary. Denote by \(g\) and \({d}v_g\) respectively, the Riemannian metric on \(M\) and the associated volume element. Let \(\Delta\) be the Laplace operator on \(M\) equipped with the weighted volume form \({d}m:= {e}^{-h}\,{d}v_g\). We are interested in the operator \(L_h\cdot:={e}^{-h(\alpha-1)} (\Delta\cdot +\alpha g(\nabla h,\nabla\cdot))\), where \(\alpha > 1\) and \(h\in C^2(M)\) are given. The main result in this paper states about the existence of upper bounds for the eigenvalues of the weighted Laplacian \(L_h\) with the Neumann boundary condition if the boundary is non-empty.Elliptic genera of ADE type singularities.https://www.zbmath.org/1456.580012021-04-16T16:22:00+00:00"Hou, Yuhang"https://www.zbmath.org/authors/?q=ai:hou.yuhang(no abstract)Integrability characteristics of a novel (2+1)-dimensional nonlinear model: Painlevé analysis, soliton solutions, Bäcklund transformation, Lax pair and infinitely many conservation laws.https://www.zbmath.org/1456.350732021-04-16T16:22:00+00:00"Lü, Xing"https://www.zbmath.org/authors/?q=ai:lu.xing"Hua, Yan-Fei"https://www.zbmath.org/authors/?q=ai:hua.yan-fei"Chen, Si-Jia"https://www.zbmath.org/authors/?q=ai:chen.sijia"Tang, Xian-Feng"https://www.zbmath.org/authors/?q=ai:tang.xian-fengSummary: The (2+1)-dimensional Kadomtsev-Petviashvili type equations describe the nonlinear phenomena and characteristics in oceanography, fluid dynamics and shallow water. In the literature, a novel (2+1)-dimensional nonlinear model is proposed, and the localized wave interaction solutions are studied including lump-kink and lump-soliton types. Hereby, it is of further value to investigate the integrability characteristics of this model. In this paper, we firstly conduct the Painlevé analysis and find it fails to pass the Painlevé test due to a non-vanishing compatibility condition at the highest resonance level. Then we derive the soliton solutions and give the formula of the \(N\)-soliton solution, which is proved by means of the Hirota condition. The criterion for the linear superposition principle is also given to generate the resonant solutions. Bäcklund transformation, Lax pair and infinitely many conservation laws are derived through the Hirota bilinear method and Bell polynomial approach. As a result, we have a more overall understanding of the integrability characteristics of this novel (2+1)-dimensional nonlinear model.Lower order eigenvalues for the bi-drifting Laplacian on the Gaussian shrinking soliton.https://www.zbmath.org/1456.530372021-04-16T16:22:00+00:00"Zeng, Lingzhong"https://www.zbmath.org/authors/?q=ai:zeng.lingzhongSummary: : It may very well be difficult to prove an eigenvalue inequality of Payne-Pólya-Weinberger type for the bi-drifting Laplacian on the bounded domain of the general complete metric measure spaces. Even though we suppose that the differential operator is bi-harmonic on the standard Euclidean sphere, this problem still remains open. However, under certain condition, a general inequality for the eigenvalues of bi-drifting Laplacian is established in this paper, which enables us to prove an eigenvalue inequality of Ashbaugh-Cheng-Ichikawa-Mametsuka type (which is also called an eigenvalue inequality of Payne-Pólya-Weinberger type) for the eigenvalues with lower order of bi-drifting Laplacian on the Gaussian shrinking soliton.Averaging over Narain moduli space.https://www.zbmath.org/1456.830672021-04-16T16:22:00+00:00"Maloney, Alexander"https://www.zbmath.org/authors/?q=ai:maloney.alexander"Witten, Edward"https://www.zbmath.org/authors/?q=ai:witten.edwardSummary: Recent developments involving JT gravity in two dimensions indicate that under some conditions, a gravitational path integral is dual to an average over an ensemble of boundary theories, rather than to a specific boundary theory. For an example in one dimension more, one would like to compare a random ensemble of two-dimensional CFT's to Einstein gravity in three dimensions. But this is difficult. For a simpler problem, here we average over Narain's family of two-dimensional CFT's obtained by toroidal compactification. These theories are believed to be the most general ones with their central charges and abelian current algebra symmetries, so averaging over them means picking a random CFT with those properties. The average can be computed using the Siegel-Weil formula of number theory and has some properties suggestive of a bulk dual theory that would be an exotic theory of gravity in three dimensions. The bulk dual theory would be more like \(\mathrm{U}(1)^{2D}\) Chern-Simons theory than like Einstein gravity.Theta functions and Brownian motion.https://www.zbmath.org/1456.580252021-04-16T16:22:00+00:00"Duncan, Tyrone E."https://www.zbmath.org/authors/?q=ai:duncan.tyrone-eSummary: A theta function for an arbitrary connected and simply connected compact simple Lie group is defined as an infinite determinant that is naturally related to the transformation of a family of independent Gaussian random variables associated with a pinned Brownian motion in the Lie group. From this definition of a theta function, the equality of the product and the sum expressions for a theta function is obtained. This equality for an arbitrary connected and simply connected compact simple Lie group is known as a Macdonald identity which generalizes the Jacobi triple product for the elliptic theta function associated with \textit{su}(2).Subleading corrections to the free energy in a theory with \(N^{5/3}\) scaling.https://www.zbmath.org/1456.814392021-04-16T16:22:00+00:00"Liu, James T."https://www.zbmath.org/authors/?q=ai:liu.james-t"Lu, Yifan"https://www.zbmath.org/authors/?q=ai:lu.yifanSummary: We numerically investigate the sphere partition function of a Chern-Simons-matter theory with \(\mathrm{SU} (N)\) gauge group at level \(k\) coupled to three adjoint chiral multiplets that is dual to massive IIA theory. Beyond the leading order \(N^{5/3}\) behavior of the free energy, we find numerical evidence for a term of the form \((2/9) \log N\). We conjecture that this term may be universal in theories with \(N^{5/3}\) scaling in the large-\(N\) limit with the Chern-Simons level \(k\) held fixed.Linear instability for periodic orbits of non-autonomous Lagrangian systems.https://www.zbmath.org/1456.580132021-04-16T16:22:00+00:00"Portaluri, Alessandro"https://www.zbmath.org/authors/?q=ai:portaluri.alessandro"Wu, Li"https://www.zbmath.org/authors/?q=ai:wu.li"Yang, Ran"https://www.zbmath.org/authors/?q=ai:yang.ranThe spectral properties of transversally elliptic operators and some applications.https://www.zbmath.org/1456.580222021-04-16T16:22:00+00:00"Morimoto, Masahiro"https://www.zbmath.org/authors/?q=ai:morimoto.masahiroSummary: This is a survey of my paper [``The heat operator of a transversally elliptic operator'', \url{arXiv:1705.09039}] (which is based mainly on [\textit{M. A. Shubin}, Tr. Semin. Im. I. G. Petrovskogo 8, 239--258 (1982; Zbl 0521.47003)]). Following [\url{arXiv:1705.09039}], we introduce the spectral properties of transversally elliptic operators and their applications, with additional information such as historical remarks and the motivation.
For the entire collection see [Zbl 1390.53003].Fermi gas approach to general rank theories and quantum curves.https://www.zbmath.org/1456.813432021-04-16T16:22:00+00:00"Kubo, Naotaka"https://www.zbmath.org/authors/?q=ai:kubo.naotakaSummary: It is known that matrix models computing the partition functions of three-dimensional \(\mathcal{N} = 4\) superconformal Chern-Simons theories described by circular quiver diagrams can be written as the partition functions of ideal Fermi gases when all the nodes have equal ranks. We extend this approach to rank deformed theories. The resulting matrix models factorize into factors depending only on the relative ranks in addition to the Fermi gas factors. We find that this factorization plays a critical role in showing the equality of the partition functions of dual theories related by the Hanany-Witten transition. Furthermore, we show that the inverses of the density matrices of the ideal Fermi gases can be simplified and regarded as quantum curves as in the case without rank deformations. We also comment on four nodes theories using our results.Spectral embedding norm: looking deep into the spectrum of the graph Laplacian.https://www.zbmath.org/1456.621182021-04-16T16:22:00+00:00"Cheng, Xiuyuan"https://www.zbmath.org/authors/?q=ai:cheng.xiuyuan"Mishne, Gal"https://www.zbmath.org/authors/?q=ai:mishne.galHyperbolic harmonic functions and hyperbolic Brownian motion.https://www.zbmath.org/1456.601942021-04-16T16:22:00+00:00"Eriksson, Sirkka-Liisa"https://www.zbmath.org/authors/?q=ai:eriksson.sirkka-liisa"Kaarakka, Terhi"https://www.zbmath.org/authors/?q=ai:kaarakka.terhiSummary: We study harmonic functions with respect to the Riemannian metric
\[ds^2=\frac{dx_1^2+\cdots +dx_n^2}{x_n^{\frac{2\alpha}{n-2}}}\] in the upper half space \(\mathbb{R}_+^n=\{(x_1,\dots,x_n) \in \mathbb{R}^n :x_n>0\}\). They are called \(\alpha\)-hyperbolic harmonic. An important result is that a function \(f\) is \(\alpha\)-hyperbolic harmonic íf and only if the function \(g(x) =x_n^{-\frac{2-n+\alpha}{2}}f(x)\) is the eigenfunction of the hyperbolic Laplace operator \(\triangle_h=x_n^2\triangle -(n-2) x_n\frac{\partial}{\partial x_n}\) corresponding to the eigenvalue \(\frac{1}{4} ((\alpha+1)^2-(n-1)^2)=0\). This means that in case \(\alpha =n-2\), the \(n-2\)-hyperbolic harmonic functions are harmonic with respect to the hyperbolic metric of the Poincaré upper half-space. We are presenting some connections of \(\alpha\)-hyperbolic functions to the generalized hyperbolic Brownian motion. These results are similar as in case of harmonic functions with respect to usual Laplace and Brownian motion.Analytic Pontryagin duality.https://www.zbmath.org/1456.190022021-04-16T16:22:00+00:00"Lim, Johnny"https://www.zbmath.org/authors/?q=ai:lim.johnnyThis paper describes a geometric realisation of \(K^0(X,\mathbb{R}/\mathbb{Z})\)
and an index pairing
\(
K^0(X,\mathbb{R}/\mathbb{Z}) \times K_0(X)
\rightarrow
\mathbb{R}/\mathbb{Z}
\).
The pairing is an even analogue of
the odd \(\mathbb{R}/\mathbb{Z}\) index pairing
explained by Lott
[\textit{J. Lott},
Commun. Anal. Geom. 2,
No. 2,
279--311
(1994; Zbl 0840.58044)],
which
realises \(K^1(X,\mathbb{R}/\mathbb{Z})\) following Karoubi
[\textit{M. Karoubi},
{Astérisque} 149,
(1987; Zbl 0648.18008)]
and
involves the Atiyah-Patodi-Singer eta-invariant
[\textit{M. F. Atiyah} et al.,
Math. Proc. Camb. Philos. Soc. 79, 71--99
(1976; Zbl 0325.58015)].
After the introduction in Section 1,
Section 2 examines a pairing
\(H^2(X,\mathbb{R}/\mathbb{Z}) \times H_2(X) \rightarrow \mathbb{R}/\mathbb{Z}\)
that can be considered as a special case of the main result.
This pairing is described
using
a modified eta-invariant
whose definition involves
the index of a projective Dirac operator
[\textit{V. Mathai} et al., Geom. Topol. 9, 341--373 (2005; Zbl 1083.58021)].
Section 3 discusses a pairing
\(H^1(X,\mathbb{R}/\mathbb{Z}) \times H_1(X) \rightarrow \mathbb{R}/\mathbb{Z}\)
that is a special case of Lott's pairing.
Section 4 realises \(K^0(X,\mathbb{R}/\mathbb{Z})\)
using triples \((g,(d,g^{-1} d g), \mu)\),
where \(g\) is an element of \(K^1(X)\),
\((d,g^{-1} d g)\) is a pair of flat connections and
\(\mu\) is a differential form
related to \(g\) and \(d\) by an exactness condition
involving the odd Chern Character
[\S 1.8, \textit{W. Zhang}, Lectures on Chern-Weil theory and Witten deformations. Singapore: World Scientific (2001; Zbl 0993.58014)].
Section 5 states the main theorem.
The pairing
\(
K^0(X,\mathbb{R}/\mathbb{Z}) \times K_0(X)
\rightarrow
\mathbb{R}/\mathbb{Z}
\)
in this theorem
consists of
an eta-type-invariant
that
appears in the
Dai-Zhang Toeplitz index theorem
for odd-dimensional manifolds with boundary
[\textit{X. Dai} and \textit{W. Zhang},
J. Funct. Anal. 238, No. 1, 1--26
(2006; Zbl 1114.58011)]
and a topological term involving \(\mu\).
Section 6 proves that the pairing in the main theorem
is well-defined and non-degenerate.
Reviewer: Simon Kitson (Lismore)Holographic spin liquids and Lovelock Chern-Simons gravity.https://www.zbmath.org/1456.830762021-04-16T16:22:00+00:00"Gallegos, A. D."https://www.zbmath.org/authors/?q=ai:gallegos.a-d"Gürsoy, U."https://www.zbmath.org/authors/?q=ai:gursoy.umutSummary: We explore the role of torsion as source of spin current in strongly interacting conformal fluids using holography. We establish the constitutive relations of the basic hydrodynamic variables, the energy-momentum tensor and the spin current based on the classification of the spin sources in irreducible Lorentz representations. The fluids we consider are assumed to be described by the five dimensional Lovelock-Chern-Simons gravity with independent vielbein and spin connection. We construct a hydrodynamic expansion that involves the stress tensor and the spin current and compute the corresponding one-point functions holographically. As a byproduct we find a class of interesting analytic solutions to the Lovelock-Chern-Simons gravity, including blackholes, by mapping the equations of motion into non-linear algebraic constraints for the sources. We also derive a Lee-Wald entropy formula for these black holes in Chern-Simons theories with torsion. The black hole solutions determine the thermodynamic potentials and the hydrodynamic constitutive relations in the corresponding fluid on the boundary. We observe novel spin induced transport in these holographic models: a dynamical version of the Barnett effect where vorticity generates a spin current and anomalous vortical transport transverse to a vector-like spin source.Heat semigroups on Weyl algebra.https://www.zbmath.org/1456.580202021-04-16T16:22:00+00:00"Avramidi, Ivan G."https://www.zbmath.org/authors/?q=ai:avramidi.ivan-gSummary: We study the algebra of semigroups of Laplacians on the Weyl algebra. We consider first-order partial differential operators \(\nabla_i^\pm\) forming the Lie algebra \([\nabla_j^\pm,\nabla_k^\pm]=i\mathscr{R}_{jk}^\pm\) and \([\nabla_j^+,\nabla_k^-]=i\frac{1}{2}(\mathscr{R}_{j k}^++\mathscr{R}_{j k}^-)\) with some anti-symmetric matrices \(\mathscr{R}_{ij}^\pm\) and define the corresponding Laplacians \(\Delta_\pm=g_\pm^{ij}\nabla_i^\pm\nabla_j^\pm\) with some positive matrices \(g_\pm^{i j} \). We show that the heat semigroups \(\exp(t\varDelta_\pm)\) can be represented as a Gaussian average of the operators \(\exp\left< \xi , \nabla^\pm\right>\) and use these representations to compute the product of the semigroups, \(\exp(t\varDelta_+) \exp(s\varDelta_-)\) and the corresponding heat kernel.A \(U(1)_{B- L}\)-extension of the standard model from noncommutative geometry.https://www.zbmath.org/1456.814712021-04-16T16:22:00+00:00"Besnard, Fabien"https://www.zbmath.org/authors/?q=ai:besnard.fabienSummary: We derive a \(U(1)_{B- L}\)-extension of the standard model from a generalized Connes-Lott model with algebra \(\mathbb{C} \oplus \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})\). This generalization includes the Lorentzian signature, the presence of a real structure, and the weakening of the order 1 condition. In addition to the SM fields, it contains a \(Z_{B- L}\)' boson and a complex scalar field \(\sigma\) that spontaneously breaks the new symmetry. This model is the smallest one that contains the SM fields and is compatible with both the Connes-Lott theory and the algebraic background framework.
{\copyright 2021 American Institute of Physics}Positive harmonic functions on groups and covering spaces.https://www.zbmath.org/1456.530532021-04-16T16:22:00+00:00"Polymerakis, Panagiotis"https://www.zbmath.org/authors/?q=ai:polymerakis.panagiotisSummary: We show that if \(p : M \to N\) is a normal Riemannian covering, with \(N\) closed, and \(M\) has exponential volume growth, then there are non-constant, positive harmonic functions on \(M\). This was conjectured by \textit{T. Lyons} and \textit{D. Sullivan} [J. Differ. Geom. 19, 299--323 (1984; Zbl 0554.58022)].On a gravity dual to flavored topological quantum mechanics.https://www.zbmath.org/1456.830632021-04-16T16:22:00+00:00"Feldman, Andrey"https://www.zbmath.org/authors/?q=ai:feldman.andreySummary: In this paper, we propose a generalization of the \(\mathrm{AdS}_2/\mathrm{CFT}_1\) correspondence constructed by \textit{M. Mezei} in [``A 2d/1d holographic duality'', Preprint, \url{arXiv:1703.08749}], which is the duality between 2d Yang-Mills theory with higher derivatives in the \(\mathrm{AdS}_2\) background, and 1d topological quantum mechanics of two adjoint and two fundamental \(\mathrm{U}(N)\) fields, governing certain protected sector of operators in 3d ABJM theory at the Chern-Simons level \(k = 1\). We construct a holographic dual to a flavored generalization of the 1d quantum mechanics considered in [loc. cit.], which arises as the effective field theory living on the intersection of stacks of \(N\) D2-branes and \(k\) D6-branes in the \(\Omega\)-background in Type IIA string theory, and describes the dynamics of the protected sector of operators in \(\mathcal{N} = 4\) theory with \(k\) fundamental hypermultiplets, having a holographic description as M-theory in the \(\mathrm{AdS}_4 \times S^7/ \mathbb{Z}_k\) background. We compute the structure constants of the bulk theory gauge group, and construct a map between the observables of the boundary theory and the fields of the bulk theory.More on Wilson toroidal networks and torus blocks.https://www.zbmath.org/1456.830542021-04-16T16:22:00+00:00"Alkalaev, Konstantin"https://www.zbmath.org/authors/?q=ai:alkalaev.konstantin"Belavin, Vladimir"https://www.zbmath.org/authors/?q=ai:belavin.vladimir-aSummary: We consider the Wilson line networks of the Chern-Simons \(3d\) gravity theory with toroidal boundary conditions which calculate global conformal blocks of degenerate quasi-primary operators in torus \(2d\) CFT. After general discussion that summarizes and further extends results known in the literature we explicitly obtain the one-point torus block and two-point torus blocks through particular matrix elements of toroidal Wilson network operators in irreducible finite-dimensional representations of \(sl (2, \mathbb{R})\) algebra. The resulting expressions are given in two alternative forms using different ways to treat multiple tensor products of \(sl (2, \mathbb{R})\) representations: (1) 3\textit{mj} Wigner symbols and intertwiners of higher valence, (2) totally symmetric tensor products of the fundamental \(sl (2, \mathbb{R})\) representation.Tasks with fast oscillating data. Two examples of asymptotics construction.https://www.zbmath.org/1456.350192021-04-16T16:22:00+00:00"Ivleva, N. S."https://www.zbmath.org/authors/?q=ai:ivleva.n-sSummary: For two specific problems with rapidly oscillating data in time -- the semilinear parabolic system with two spatial variables and the Navier-Stokes system that simulates the fluid flow in the flat case -- the question of constructing asymptotic expansions of their time-periodic solutions is solved. Both problems are considered in the cylinder, infinite in time, the axis of which is the temporary numerical axis, and the basis is the two-dimensional unit circle. The Dirichlet conditions are taken as boundary conditions. The construction of these asymptotic expansions is based on two algorithms developed, justified and obtained earlier by the author and V. B. Levenstam.Classification of a modified de Sitter metric by variational symmetries and conservation laws.https://www.zbmath.org/1456.580112021-04-16T16:22:00+00:00"Beesham, A."https://www.zbmath.org/authors/?q=ai:beesham.aroonkumar"Gadjagboui, B. B. I."https://www.zbmath.org/authors/?q=ai:gadjagboui.b-b-i"Kara, A. H."https://www.zbmath.org/authors/?q=ai:kara.abdul-hamidRenormalization group flow of Chern-Simons boundary conditions and generalized Ricci tensor.https://www.zbmath.org/1456.813172021-04-16T16:22:00+00:00"Pulmann, Ján"https://www.zbmath.org/authors/?q=ai:pulmann.jan"Ševera, Pavol"https://www.zbmath.org/authors/?q=ai:severa.pavol"Youmans, Donald R."https://www.zbmath.org/authors/?q=ai:youmans.donald-rSummary: We find a Chern-Simons propagator on the ball with the chiral boundary condition. We use it to study perturbatively Chern-Simons boundary conditions related to 2-dim \(\sigma\)-models and to Poisson-Lie T-duality. In particular, we find their renormalization group flow, given by the generalized Ricci tensor. Finally we briefly discuss what happens when the Chern-Simons theory is replaced by a Courant \(\sigma\)-model or possibly by a more general AKSZ model.Higher order Cheeger inequalities for Steklov eigenvalues.https://www.zbmath.org/1456.580212021-04-16T16:22:00+00:00"Hassannezhad, Asma"https://www.zbmath.org/authors/?q=ai:hassannezhad.asma"Miclo, Laurent"https://www.zbmath.org/authors/?q=ai:miclo.laurentThe Steklov eigenvalue problem is the following boundary value problem
\[
\Delta u=0\text{ in }\Omega,\, \frac{\partial u}{\partial \nu}=\sigma u\text{ on }\partial\Omega,\tag{1}
\]
such that \(\Omega=(\Omega,g)\) is an \(n\)-dimensional compact Riemannian manifold endowed with a smooth boundary \(\partial \Omega\), \(\frac{\partial u}{\partial \nu}\) represents the directional derivative with respect to \(\nu\), the unit outward normal vector along \(\partial \Omega\), and \(\sigma\) is a real eigenvalue. The authors provide a lower bound of the \(k\)-th eigenvalue of \((1)\) in terms of the \(k\)-th Cheeger-Steklov constant. The authors also study the case when \((\Omega,g)\) is swapped by a probability measure space and by a finite state space, respectively.
Reviewer: Mohammed El Aïdi (Bogotá)Global analysis of quasilinear wave equations on asymptotically de Sitter spaces.https://www.zbmath.org/1456.351412021-04-16T16:22:00+00:00"Hintz, Peter"https://www.zbmath.org/authors/?q=ai:hintz.peterSummary: We establish the small data solvability of suitable quasilinear wave and Klein-Gordon equations in high regularity spaces on a geometric class of spacetimes including asymptotically de Sitter spaces. We obtain our results by proving the global invertibility of linear operators with coefficients in high regularity \(L^2\)-based function spaces and using iterative arguments for the nonlinear problems. The linear analysis is accomplished in two parts: Firstly, a regularity theory is developed by means of a calculus for pseudodifferential operators with non-smooth coefficients, similar to the one developed by Beals and Reed, on manifolds with boundary. Secondly, the asymptotic behavior of solutions to linear equations is studied using resonance expansions, introduced in this context by Vasy using the framework of Melrose's \(b\)-analysis.\( T\overline{T} \)-deformation of \(q\)-Yang-Mills theory.https://www.zbmath.org/1456.830692021-04-16T16:22:00+00:00"Santilli, Leonardo"https://www.zbmath.org/authors/?q=ai:santilli.leonardo"Szabo, Richard J."https://www.zbmath.org/authors/?q=ai:szabo.richard-j"Tierz, Miguel"https://www.zbmath.org/authors/?q=ai:tierz.miguelSummary: We derive the \(T\overline{T} \)-perturbed version of two-dimensional \(q\)-deformed Yang-Mills theory on an arbitrary Riemann surface by coupling the unperturbed theory in the first order formalism to Jackiw-Teitelboim gravity. We show that the \(T\overline{T} \)-deformation results in a breakdown of the connection with a Chern-Simons theory on a Seifert manifold, and of the large \(N\) factorization into chiral and anti-chiral sectors. For the \( \mathrm{U} (N)\) gauge theory on the sphere, we show that the large \(N\) phase transition persists, and that it is of third order and induced by instantons. The effect of the \(T\overline{T} \)-deformation is to decrease the critical value of the 't Hooft coupling, and also to extend the class of line bundles for which the phase transition occurs. The same results are shown to hold for \( (q,t) \)-deformed Yang-Mills theory. We also explicitly evaluate the entanglement entropy in the large \(N\) limit of Yang-Mills theory, showing that the \(T\overline{T} \)-deformation decreases the contribution of the Boltzmann entropy.On three-point functions in ABJM and the latitude Wilson loop.https://www.zbmath.org/1456.813532021-04-16T16:22:00+00:00"Bianchi, Marco S."https://www.zbmath.org/authors/?q=ai:bianchi.marco-sSummary: I consider three-point functions of twist-one operators in ABJM at weak coupling. I compute the structure constant of correlators involving one twist-one un-protected operator and two protected ones for a few finite values of the spin, up to two-loop order. As an application I enforce a limit on the gauge group ranks, in which I relate the structure constant for three chiral primary operators to the expectation value of a supersymmetric Wilson loop. Such a relation is then used to perform a successful five-loop test on the matrix model conjectured to describe the supersymmetric Wilson loop.Spectrum of the Laplacian and the Jacobi operator on rotational CMC hypersurfaces of spheres.https://www.zbmath.org/1456.530512021-04-16T16:22:00+00:00"Perdomo, Oscar M."https://www.zbmath.org/authors/?q=ai:perdomo.oscar-marioSummary: Let \(M\subset \mathbb{S}^{n+1}\subset\mathbb{R}^{n+2}\) be a compact CMC rotational hypersurface of the \((n+1)\)-dimensional Euclidean unit sphere. Denote by \(|A|^2\) the square of the norm of the second fundamental form and \(J(f)=-\Delta f-nf-|A|^2f\) the stability or Jacobi operator. In this paper we compute the spectra of their Laplace and Jacobi operators in terms of eigenvalues of second order Hill's equations. For the minimal rotational examples, we prove that the stability index -- the numbers of negative eigenvalues of the Jacobi operator counted with multiplicity -- is greater than \(3 n+4\) and we also prove that there are at least 2 positive eigenvalues of the Laplacian of \(M\) smaller than \(n\). When \(H\) is not zero, we have that every nonflat CMC rotational immersion is generated by rotating a planar profile curve along a geodesic called the axis of rotation. We assume that the coordinates of this plane has been set up so that the axis of rotation goes through the origin. The planar profile curve is made up of \(m\) copies, each one of them is a is rigid motion of a single curve that we will call the fundamental piece. For this reason every nonflat rotational CMC hypersurface has \(Z_m\) in its group of isometries. If \(\theta\) denotes the change of the angle of the fundamental piece when written in polar coordinates, then \(l=\frac{m\theta}{2 \pi}\) is a nonnegative integer. For unduloids (a subfamily of the rotational CMC hypersurfaces that include all the known embedded examples), we show that the number of negative eigenvalues of the operator \(J\) counted with multiplicity is at least \((2l-1)n+(2m-1)\).