Recent zbMATH articles in MSC 58https://www.zbmath.org/atom/cc/582021-04-16T16:22:00+00:00WerkzeugThree applications of delooping to \(h\)-principles.https://www.zbmath.org/1456.580092021-04-16T16:22:00+00:00"Kupers, Alexander"https://www.zbmath.org/authors/?q=ai:kupers.alexanderIn his book [Partial differential relations. Berlin etc.: Springer (1986; Zbl 0651.53001)] \textit{M. Gromov} formulated a very general \(h\)-principle for invariant topological sheaves of smooth functions on manifolds, by which many geometric problems can be reduced to more tractable homotopy-theoretic problems.
In the present paper, the author applies the machinery of Gromov to give general conditions under which
an \(h\)-principle holds on closed manifolds, and check that these conditions are satisfied in three examples: (i) a homotopical version of Vassiliev's \(h\)-principle, (ii) the contractibility of the space of framed functions, (iii) a version of Mather-Thurston theory.
Reviewer: Vagn Lundsgaard Hansen (Lyngby)Introduction to Lipschitz geometry of singularities. Lecture notes of the international school on singularity theory and Lipschitz geometry, Cuernavaca, Mexico, June 11--22, 2018.https://www.zbmath.org/1456.580022021-04-16T16:22:00+00:00"Neumann, Walter (ed.)"https://www.zbmath.org/authors/?q=ai:neumann.walter-d"Pichon, Anne (ed.)"https://www.zbmath.org/authors/?q=ai:pichon.annePublisher's description: This book presents a broad overview of the important recent progress which led to the emergence of new ideas in Lipschitz geometry and singularities, and started to build bridges to several major areas of singularity theory. Providing all the necessary background in a series of introductory lectures, it also contains Pham and Teissier's previously unpublished pioneering work on the Lipschitz classification of germs of plane complex algebraic curves.
While a real or complex algebraic variety is topologically locally conical, it is in general not metrically conical; there are parts of its link with non-trivial topology which shrink faster than linearly when approaching the special point. The essence of the Lipschitz geometry of singularities is captured by the problem of building classifications of the germs up to local bi-Lipschitz homeomorphism. The Lipschitz geometry of a singular space germ is then its equivalence class in this category.
The book is aimed at graduate students and researchers from other fields of geometry who are interested in studying the multiple open questions offered by this new subject.
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Aguilar-Cabrera, Haydée; Cisneros-Molina, José Luis}, Geometric viewpoint of Milnor's fibration theorem, 1-43 [Zbl 07303679]
\textit{Snoussi, Jawad}, A quick trip into local singularities of complex curves and surfaces, 45-71 [Zbl 07303680]
\textit{Neumann, Walter D.}, 3-manifolds and links of singularities, 73-86 [Zbl 07303681]
\textit{Trotman, David}, Stratifications, equisingularity and triangulation, 87-110 [Zbl 07303682]
\textit{Ruas, Maria Aparecida Soares}, Basics on Lipschitz geometry, 111-155 [Zbl 07303683]
\textit{Birbrair, Lev; Gabrielov, Andrei}, Surface singularities in \(\mathbb{R}^4\) : first steps towards Lipschitz knot theory, 157-166 [Zbl 07303684]
\textit{Pichon, Anne}, An introduction to Lipschitz geometry of complex singularities, 167-216 [Zbl 07303685]
\textit{Giles Flores, Arturo; da Silva, Otoniel Nogueira; Teissier, Bernard}, The biLipschitz geometry of complex curves: an algebraic approach, 217-271 [Zbl 07303686]
\textit{Popescu-Pampu, Patrick}, Ultrametrics and surface singularities, 273-308 [Zbl 07303687]
\textit{Pham, Frédéric; Teissier, Bernard}, Lipschitz fractions of a complex analytic algebra and Zariski saturation, 309-337 [Zbl 07303688]On the existence of three solutions for some classes of two-point semi-linear and quasi-linear differential equations.https://www.zbmath.org/1456.340232021-04-16T16:22:00+00:00"Saiedinezhad, Somayeh"https://www.zbmath.org/authors/?q=ai:saiedinezhad.somayehThe paper under review deals with the semi-linear boundary-value problem which consists the differential equation
\[
u''(t)+\lambda f(u)=0\, t\in(0,1),
\]
and Dirichlet type boundary conditions \(u(0)=u(1)=0\).
The author prove the existence of three solutions for this two-point boundary value problem in an appropriate Sobolev space. Furthermore, some existence results for a quasi-linear Dirichlet problem is obtained.
The results in the paper can be considered as an extension and generalizations of results of the paper [\textit{G. Bonannao}, Appl. Math. Lett. 13, No. 5, 53--57 (2000; Zbl 1009.34019)].
Reviewer: Erdogan Sen (Tekirdağ)Lie 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.The method of averaging for Poisson connections on foliations and its applications.https://www.zbmath.org/1456.530642021-04-16T16:22:00+00:00"Avendaño-Camacho, Misael"https://www.zbmath.org/authors/?q=ai:avendano-camacho.misael"Hasse-Armengol, Isaac"https://www.zbmath.org/authors/?q=ai:hasse-armengol.isaac"Velasco-Barreras, Eduardo"https://www.zbmath.org/authors/?q=ai:velasco-barreras.eduardo"Vorobiev, Yury"https://www.zbmath.org/authors/?q=ai:vorobiev.yuriiSummary: On a Poisson foliation equipped with a canonical and cotangential action of a compact Lie group, we describe the averaging method for Poisson connections. In this context, we generalize some previous results on Hannay-Berry connections for Hamiltonian and locally Hamiltonian actions on Poisson fiber bundles. Our main application of the averaging method for connections is the construction of invariant Dirac structures parametrized by the 2-cocycles of the de Rham-Casimir complex of the Poisson foliation.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.Global existence of Landau-Lifshitz-Gilbert equation and self-similar blowup of harmonic map heat flow on \(\mathbb{S}^2\).https://www.zbmath.org/1456.350672021-04-16T16:22:00+00:00"Zhong, Penghong"https://www.zbmath.org/authors/?q=ai:zhong.penghong"Yang, Ganshan"https://www.zbmath.org/authors/?q=ai:yang.ganshan"Ma, Xuan"https://www.zbmath.org/authors/?q=ai:ma.xuanSummary: Under the plane wave setting, the existence of small Cauchy data global solution (or local solution) of Landau-Lifshitz-Gilbert equation is proved. Some variable separation type solutions (include some small data global solution) and self-similar type solutions are constructed for the Harmonic map heat flow on \(\mathbb{S}^2\). As far as we know, there is not any literature that presents the exact blowup solution of this equation. Some explicit solutions which include some finite time gradient-blowup solutions are provided. These blowup examples indicate a finite time blowup will happen in any spacial dimension.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].Introduction to differential and Riemannian geometry.https://www.zbmath.org/1456.530022021-04-16T16:22:00+00:00"Sommer, Stefan"https://www.zbmath.org/authors/?q=ai:sommer.stefan"Fletcher, Tom"https://www.zbmath.org/authors/?q=ai:fletcher.tom"Pennec, Xavier"https://www.zbmath.org/authors/?q=ai:pennec.xavierThis article is the first chapter of the book by \textit{X. Pennec} (ed.) et al. [Riemannian geometric statistics in medical image analysis. Amsterdam: Elsevier/Academic Press (2020; Zbl 1428.92004)] with 16 chapters.
From the cover of the book:
``Over the past 15 years, there has been a growing need in the medical image computing community for principled methods to process nonlinear geometric data. Riemannian geometry has emerged as one of the most powerful mathematical and computational frameworks for analyzing such data.
Riemannian Geometric Statistics in Medical Image Analysis is a complete reference on statistics on Riemannian manifolds and more general nonlinear spaces with applications in medical image analysis. It provides an introduction to the core methodology followed by a presentation of state-of-the-art methods.
As such, the presented core methodology takes its place in the field of geometric statistics, the statistical analysis of data being elements of nonlinear geometric spaces.''
From the introduction of this article: ``The following sections describe these foundational concepts and how they lead to notions commonly associated with geometry: curves, length, distances, geodesics, curvature, parallel transport, and volume form. In addition to the differential and Riemannian structure, we describe one extra layer of structure, Lie groups that are manifolds equipped with smooth group structure. Lie groups and their quotients are examples of homogeneous spaces. The group structure provides relations between distant points on the group and thereby additional ways of constructing Riemannian metrics and deriving geodesic equations.''
The article is structured in 8 sections:
1. Introduction -- 2. Manifolds (2.1 Embedded submanifolds, 2.2 Charts and local euclideaness, 2.3 Abstract manifolds and atlases, 2.4 Tangent vectors and tangent space, 2.5 Differentials and pushforward) -- 3. Riemannian manifolds (3.1 Riemannian metric, 3.2 Curve length and Riemannian distance, 3.3 Geodesics, 3.4 Levi-Cività connection, 3.5 Completeness, 3.6 Exponential and logarithm maps, 3.7 Cut locus) -- 4. Elements of analysis in Riemannian manifolds (4.1 Gradient and musical isomorphisms, 4.2 Hessian and Taylor expansion, 4.3 Riemannian measure or volume form, 4.4 Curvature) -- 5. Lie groups and homogeneous manifolds (5.1 One-parameter subgroups, 5.2 Actions, 5.3 Homogeneous spaces, 5.4 Invariant metrics and geodesics) -- 6. Elements of computing on Riemannian manifolds -- 7. Examples (7.1 The sphere, 7.2 2D Kendall shape space, 7.3 Rotations) -- 8. Additional references -- References (26 references).
For the entire collection see [Zbl 1428.92004].
Reviewer: Ludwig Paditz (Dresden)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.Brasselet number and Newton polygons.https://www.zbmath.org/1456.140622021-04-16T16:22:00+00:00"Dalbelo, Thaís M."https://www.zbmath.org/authors/?q=ai:dalbelo.thais-maria"Hartmann, Luiz"https://www.zbmath.org/authors/?q=ai:hartmann.luizAuthors' abstract: We present a formula to compute the Brasselet number of \( f:(Y,0)\rightarrow (\mathbb{C},0)\) where \(Y\subset X\) is a non-degenerate complete intersection in a toric variety \(X\). As applications we establish several results concerning invariance of the Brasselet number for families of non-degenerate complete intersections. Moreover, when \((X,0)=(\mathbb{C} ^{n},0)\) we derive sufficient conditions to obtain the invariance of the Euler obstruction for families of complete intersections with an isolated singularity which are contained in \(X\).
Reviewer: Tadeusz Krasiński (Łódź)Cyclic cohomology and Chern-Connes pairing of some crossed product algebras.https://www.zbmath.org/1456.580082021-04-16T16:22:00+00:00"Quddus, Safdar"https://www.zbmath.org/authors/?q=ai:quddus.safdarSummary: We compute the cyclic and Hochschild cohomology groups for the algebras \(\mathcal{A}_\theta^{alg} \rtimes \mathbb{Z}_3\), \(\mathcal{A}_\theta^{alg} \rtimes \mathbb{Z}_4\) and \(\mathcal{A}_\theta^{alg} \rtimes \mathbb{Z}_6\). We also compute the partial Chern-Connes index table for each of these algebras.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.Results on the homotopy type of the spaces of locally convex curves on $\mathbb{S}^3$.https://www.zbmath.org/1456.570242021-04-16T16:22:00+00:00"Alves, Emília"https://www.zbmath.org/authors/?q=ai:alves.emilia"Saldanha, Nicolau C."https://www.zbmath.org/authors/?q=ai:saldanha.nicolau-corcaoA smooth curve \(\gamma:[0,1]\to \mathbb{S}^3\) in \(4\)-dimensional Euclidean space \(\mathbb{R}^4\)
with image on the sphere \(\mathbb{S}^3\), is said to be locally convex, if the set of vectors
\({\gamma}(t),{\gamma}'(t), {\gamma}''(t),{\gamma}'''(t)\) is a positive basis in \(\mathbb{R}^4\)
for all \(t\in[0,1]\). By the Gram-Schmidt procedure, we can turn this basis into an orthonormal basis and thereby associate a Frenet frame curve \(\mathcal{F}_{\gamma}: [0,1]\to SO_4\) in the special orthogonal
group \(SO_4\) to a locally convex curve.
For any matrix \(Q\in SO_4\), let \(\mathcal{L}\mathbb{S}^3(Q)\) denote the space of all locally
convex curves \(\gamma:[0,1]\to \mathbb{S}^3\) where \(\mathcal{F}_{\gamma}(0)=I\) (the identity matrix)
and \(\mathcal{F}_{\gamma}(1)=Q\). It was proved by [\textit{N. C. Saldanha} and \textit{B. Shapiro}, J. Singul. 4, 1--22 (2012; Zbl 1292.58002)] that there are at most 3 different homeomorphism types among the path components of \(\mathcal{L}\mathbb{S}^3(Q)\). But otherwise very little seems to be known about these components and their generalizations to higher dimensional spheres \(\mathbb{S}^n\).
In the present paper the authors prove several interesting theorems on the homotopy and homology of these path components, in particular for the case \(Q=-I\). The results are technical and cannot be given in detail.
Reviewer: Vagn Lundsgaard Hansen (Lyngby)Podleś spheres for the braided quantum \(\mathrm{SU}(2)\).https://www.zbmath.org/1456.580062021-04-16T16:22:00+00:00"Sołtan, Piotr M."https://www.zbmath.org/authors/?q=ai:soltan.piotr-mikolajPodleś quantum sphere \(\mathsf{S}^2_q=\mathrm{SU}_q(2)/\mathbb{T}\), \(q\) is a real number with \(0<|q|\le 1\) is extended to the quotient of braded quantum \(\mathrm{SU}_q(2)\), \(q\) is a complex number with \(0<|q)\le 1\) [\textit{P. Kasprzak} et al., J. Noncommut. Geom. 10, No. 4, 1611--1625 (2016; Zbl 1358.81128)]. That is define (braided) quantum sphere \(\mathbb{S}^2_q\) by \(\mathbb{S}^2_q=\mathrm{SU}_q(2)/\mathbb{T}\), \(q\) a complex number \(0<|q|\le 1\).
Let \(\mathbb{S}^2_q\) be the braided quantum sphere and \(\mathsf{S}^2_q\) be the Pofleś' quantum sphere Then it is shown \(\mathbb{S}^2_q=\mathsf{S}^2_{|q|}\) (\S4. \S7, Cor. 7.3). An axiomatic definition of braided quantum sphere is also given (\S6. Remark 6.1, \S7. Def.7.2).
To define and study braded quantum \(\mathrm{SU}_q(2)\) use braided tensor product \(A\boxtimes B\). It is explained in\S2 together with related topics following [loc. cit.] and [\textit{R. Meyer} et al., Int. J. Math. 25, No. 2, Article ID 1450019, 37 p. (2014; Zbl 1315.46076)]. Braided quantum \(\mathrm{SU}_q(2)\) is explained in \S3, and shwo the algebra of functions \(C(\mathrm{SU}_q(2)\) is the uiversal \(C^\ast\)-algebras generated by two elements \(\alpha, \gamma\), together with their relations ((3.1),(3.2)) Braided quantum sphere \(\mathbb{S}^2_q\) is defined as the quotient \(\mathrm{SU}_q(2)/\mathbb{T}\) in \S4. It is shown \(C(\mathbb{S}^2_q)\) is the closed unital \(\ast\)-subalgebra of \(C(\mathrm{SU}_q(2))\) generated by \(\alpha\gamma^\ast\) and \(\gamma^\ast\gamma\) (Cor.4.3). From this corollary and [\textit{P. Podleś}, Lett. Math. Phys. 14, 193--202 (1987; Zbl 0634.46054)] \(\mathbb{S}^2_q=\mathsf{S}^2_{|q|}\) is derived.
To obtain axiomatic definition \(\mathbb{S}^2_q\), that says a compact quantum space with \(\mathbb{T}\)-action is a braided quantum sphere, if and only if there exists \(\Gamma:\mathbb{C}(\mathbb{X})\to C(\mathrm{SU}_q(2))\boxtimes_\zeta C(\mathbb{X})\);
\[(\Gamma\boxtimes_\zeta\circ\Gamma)=(\mathrm{id})\circ\Gamma=(\mathrm{id}\boxtimes_\zeta \Delta_{\mathrm{SU}_q(2)})\circ\Gamma, \]
with 4 conditions, stated in the beginning of \S6, the three dimensional irreducible representation of \(\mathrm{SU}_q(2)\) found as an irreducible subrepresentation of the tensor square of the fundamental representation is defined and studied in \S5. Then the four conditions found in \S6, from discussions in \S5 and \S4. \S7, the last Section, show that \(C^\ast\)-algebra defined by relations in \S6 indeed carry and action of the braided \(\mathrm{SU}_q(2)\) and that they coincide with Podleś sphere (Def.7,2. Cor. 7.3).
Some hard calculus in the study of the tensor product of the fundamental representation are given in Appendix.
Reviewer: Akira Asada (Takarazuka)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.A singular radial connection over \(\mathbb B^5\) minimizing the Yang-Mills energy.https://www.zbmath.org/1456.580152021-04-16T16:22:00+00:00"Petrache, Mircea"https://www.zbmath.org/authors/?q=ai:petrache.mirceaSummary: We prove that the pullback of the \(\mathrm{SU}(n)\)-soliton of Chern number \(c_2=1\) over \(\mathbb S^4\) via the radial projection \(\pi :\mathbb B^5{\setminus }\{0\}\to \mathbb S^4\) minimizes the Yang-Mills energy under a topologically fixed boundary trace constraint. In particular this shows that stationary Yang-Mills connections in high dimension can have singular sets of codimension 5.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.zuoqiangTopological classification of Hamiltonian systems on two-dimensional noncompact manifolds.https://www.zbmath.org/1456.370622021-04-16T16:22:00+00:00"Nikolaenko, Stanislav S."https://www.zbmath.org/authors/?q=ai:nikolaenko.stanislav-sThis paper presents a topological classification of Hamiltonian systems on two-dimensional noncompact symplectic manifolds. The author continues along the line of research established in [\textit{D. A. Fedoseev} and \textit{A. T. Fomenko}, Fundam. Prikl. Mat., 21, No. 6, 217---243 (2016)].
The author investigates topological invariants of foliations of finite type defined by smooth functions on two-dimensional noncompact orientable manifolds. The goal is to describe a complete topological classification of noncompact bifurcations for foliations like this.
Fomenko's approach uses the compactness of leaves in the Liouville foliation (the partition of the phase manifold into connected components of common level surfaces of the first integrals, known as Liouville leaves). Liouville foliations with noncompact leaves appear in many systems in mechanics, so there is motivation to extend the theory to systems with noncompact leaves.
This paper completely solves the problem of trajectory classification for Hamiltonian systems with noncompact foliations in the case of systems with one degree of freedom of finite type. Such Hamiltonian systems are said to be of finite type if the foliation defined by the Hamiltonian \(H\) is of finite type, and this means that the number of bifurcation values of \(H\) is finite and that the atoms corresponding to the bifurcation values are atoms of finite type. The author does not assume that the system is nondegenerate or that the Hamiltonian flow is complete.
In systems with one degree of freedom almost all Liouville leaves are one-dimensional; they are the level curves of the Hamiltonian, and each leaf consists of one or more trajectories. Liouville and trajectory classification are essentially identical in this case except for the orientation of trajectories. The symplectic structure is not relevant, except for enforcing orientability.
Special difficulties arise with noncompact manifolds. Much of the paper is devoted to handling those issues. In the end the author provides a natural one-to-one correspondence between the set of noncompact bifurcations of foliations and a set of oriented colored graphs that have a special form.
Reviewer: William J. Satzer Jr. (St. Paul)On 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.Spinor modules for Hamiltonian loop group spaces.https://www.zbmath.org/1456.580052021-04-16T16:22:00+00:00"Loizides, Yiannis"https://www.zbmath.org/authors/?q=ai:loizides.yiannis"Meinrenken, Eckhard"https://www.zbmath.org/authors/?q=ai:meinrenken.eckhard"Song, Yanli"https://www.zbmath.org/authors/?q=ai:song.yanli.1|song.yanliThis paper studies the spinor modules theory of loop groups.
Let \( G \) be a compact, connected Lie group and let the loop group \( LG \) be the Banach Lie group of \(G\)-valued loops of a fixed Sobolev class \( S > 1/2 \). The authors prove that the tangent bundle of any proper Hamiltonian loop group space \(M\) possesses a canonically defined \(LG-\)equivariant completion \(\overline{T}M\), such
that any weakly symplectic 2-form \(\omega\) of any proper Hamiltonian loop group space extends to a strongly symplectic 2-form on \(\overline{T}M\).
Furthermore, it is proved that the bundle \(\overline{T}M\) possesses a distinguished \(LG-\)invariant polarization and a global \(LG-\)invariant \(\omega-\)compatible complex structure \(J\)
within this polarization class, unique up to homotopy. This leads to the definition
of \( LG-\)equivariant spinor bundle \( \mathrm{S}_{\overline{T}M} \),
which is used to construct the twisted \( \mathrm{Spin}_c \)-structure for the associated quasi-Hamiltonian \(G\)-space \(M\). This is is a way to get a finite-dimensional version of the spinor module \( \mathrm{S}_{\overline{T}M} \).
The authors also discuss \textquoteleft abelianization procedure\textquoteright\, which is another way to get a finite-dimensional version of \( \mathrm{S}_{\overline{T}M} \). The idea is to shift
to a finite-dimensional maximal torus \(T \subseteq LG-\)invariant submanifold of \(M,\) and construct an equivalent
\(\mathrm{Spin}_c \)-structure on that
submanifold. More precisely, if the moment map \(\Phi\) of a proper Hamiltonian
\(LG\)-space is transverse to the Lie algebra \( \mathfrak{t}^* \) (as a space of constant connections valued
in the Lie algebra of the maximal torus \( T \)), then the pre-image \(\Phi^{-1} (\mathfrak{t}^*)\)
is a finite-dimensional pre-symplectic manifold that
inherits a \(T\)-equivalent \(\mathrm{Spin}_c \)-structure.
Reviewer: Kaveh Eftekharinasab (Kyiv)A new projection-type method for solving multi-valued mixed variational inequalities without monotonicity.https://www.zbmath.org/1456.490192021-04-16T16:22:00+00:00"Wang, Zhong-bao"https://www.zbmath.org/authors/?q=ai:wang.zhongbao"Chen, Zhang-you"https://www.zbmath.org/authors/?q=ai:chen.zhangyou"Xiao, Yi-bin"https://www.zbmath.org/authors/?q=ai:xiao.yibin"Zhang, Cong"https://www.zbmath.org/authors/?q=ai:zhang.congSummary: In this paper, a new projection-type algorithm for solving multi-valued mixed variational inequalities without monotonicity is presented. Under some suitable assumptions, it is showed that the sequence generated by the proposed algorithm converges globally to a solution of the multi-valued mixed variational inequality considered. The algorithm exploited in this paper is based on the generalized \(f\)-projection operator due to \textit{K. Wu} and \textit{N. Huang} [Bull. Aust. Math. Soc. 73, No. 2, 307--317 (2006; Zbl 1104.47053)] rather than the well-known resolvent operator. Preliminary computational experience is also reported. The results presented in this paper generalize and improve some known results given in the literature.Shifted derived Poisson manifolds associated with Lie pairs.https://www.zbmath.org/1456.530652021-04-16T16:22:00+00:00"Bandiera, Ruggero"https://www.zbmath.org/authors/?q=ai:bandiera.ruggero"Chen, Zhuo"https://www.zbmath.org/authors/?q=ai:chen.zhuo"Stiénon, Mathieu"https://www.zbmath.org/authors/?q=ai:stienon.mathieu"Xu, Ping"https://www.zbmath.org/authors/?q=ai:xu.pingSummary: We study the shifted analogue of the ``Lie-Poisson'' construction for \(L_\infty\) algebroids and we prove that any \(L_\infty\) algebroid naturally gives rise to shifted derived Poisson manifolds. We also investigate derived Poisson structures from a purely algebraic perspective and, in particular, we establish a homotopy transfer theorem for derived Poisson algebras. As an application, we prove that, given a Lie pair \((L, A)\), the space \(\mathrm{tot}\Omega^{\bullet}_A(\Lambda^\bullet (L/A))\) admits a degree \((+1)\) derived Poisson algebra structure with the wedge product as associative multiplication and the Chevalley-Eilenberg differential \(d_A^{\mathrm{Bott}}:\Omega^{\bullet}_A(\Lambda^\bullet(L/A))\rightarrow\Omega^{\bullet+1}_A(\Lambda^\bullet (L/A))\) as unary \(L_\infty\) bracket. This degree \((+1)\) derived Poisson algebra structure on \(\mathrm{tot}\Omega^{\bullet}_A(\Lambda^\bullet(L/A))\) is unique up to an isomorphism having the identity map as first Taylor coefficient. Consequently, the Chevalley-Eilenberg hypercohomology \(\mathbb{H}(\mathrm{tot}\Omega^{\bullet}_A(\Lambda^\bullet (L/A)),d_A^{\mathrm{Bott}})\) admits a canonical Gerstenhaber algebra structure.Noncommutative fiber products and lattice models.https://www.zbmath.org/1456.580072021-04-16T16:22:00+00:00"Hartwig, Jonas T."https://www.zbmath.org/authors/?q=ai:hartwig.jonas-tThis paper studies the representation theory of certain
noncommutative singular varieties using two-dimensional
lattice models. In more detail, the first main result
of the paper describes categories of weight modules over a
noncommutative biparametric deformation \(\mathcal{A}\) of
the fiber product of two Kleinian singularities of type \(A\)
in terms of periodic higher spin vertex configurations.
The second main result provides a combinatorial classification
of simple weight \(\mathcal{A}\)-modules. Finally, the third main
result describes the center of \(\mathcal{A}\), it turns our
that in some cases the center is trivial, while in some other
cases it is isomorphic to the algebra of Laurent polynomials
in one variable.
Reviewer: Volodymyr Mazorchuk (Uppsala)Monotone Sobolev functions in planar domains: level sets and smooth approximation.https://www.zbmath.org/1456.260152021-04-16T16:22:00+00:00"Ntalampekos, Dimitrios"https://www.zbmath.org/authors/?q=ai:ntalampekos.dimitriosThe aim of the present work is to describe the structure of the level sets of Sobolev functions defined in a planar domain. The author establishes that almost every level set of a Sobolev function in a planar domain consists of points, Jordan curves, or homeomorphic copies of an interval. Furthermore, a stronger result is proved in the case that the Sobolev function in the plane is monotone in the Lebesgue sense (i.e., the maximum and minimum of the function in an open set are attained at the boundary), namely almost every level set is an embedded 1-dimensional (in the topological sense) submanifold of the plane. The obtained result is an analog of Sard's theorem for \(C^2\)-smooth functions in a planar domain, which asserts that almost every value is a regular value. Using the theory of \(p\)-harmonic functions, as an application of the obtained results, he proves that monotone Sobolev functions in planar domains can be approximated uniformly and in the Sobolev norm by smooth monotone functions.
Reviewer: Andrey Zahariev (Plovdiv)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.Nonlinear flag manifolds as coadjoint orbits.https://www.zbmath.org/1456.370602021-04-16T16:22:00+00:00"Haller, Stefan"https://www.zbmath.org/authors/?q=ai:haller.stefan"Vizman, Cornelia"https://www.zbmath.org/authors/?q=ai:vizman.corneliaIn [Math. Ann. 329, No. 4, 771--785 (2004; Zbl 1071.58005)], the present authors introduced the notion of a nonlinear Grassmannian and studied the Fréchet manifold \(\mathrm{Gras}_n(M)\) of all \(n\)-dimensional oriented compact submanifolds of a smooth closed connected \(m\)-dimensional manifold \(M\).
They showed that every closed \((n+2)\)-form \(\alpha\) on \(M\) defines a closed 2-form \(\widetilde{\alpha}\) on \(\mathrm{Gras}_n(M)\), and if \(\alpha\) is integrable, then \(\widetilde{\alpha}\) is the curvature form of a principal connection on a principal \(S^1\)-bundle over \(\mathrm{Gras}_n(M)\). In the case \(\alpha\) is a closed, integrable volume form, then every connected component \(\mathcal{M}\) of \(\mathrm{Gras}_{m-2}(M)\), equipped with the symplectic form \(\widetilde{\alpha}\), is a prequantizable coadjoint orbit of some central extension of the Hamiltonian group \(\text{Ham}(M,\alpha)\) by \(S^1\).
In this paper, the authors generalize the notion of a nonlinear Grassmannian to the notion of a nonlinear flag manifold.
If \(M\) is a smooth manifold, \(S_1,\dots,S_r\) are closed smooth manifolds, then a sequence of nested embedded submanifolds \(N_1\subseteq\dots\subseteq N_r\subseteq M\) such that \(N_i\) is diffeomorphic to \(S_i\) for all \(i=1,\dots,r\) is called a nonlinear flag of type \(\mathscr{S}=(S_1,\dots,S_r)\) in \(M\).
The space of all nonlinear flags of type \(\mathscr{S}\) in \(M\) can be equipped with the structure of a Fréchet manifold in a natural way and is denoted by \(\mathrm{Flag}_{\mathscr{S}}(M)\).
The main goal of this paper is to study the geometry of this space.
A nonlinear Grassmannian is a special case of a nonlinear flag and corresponds to the case \(r=1\).
The authors present some applications of nonlinear flag manifolds by using them to describe certain coadjoint orbits of the Hamiltonian group.
If \(M\) is a closed symplectic manifold, \(\mathrm{Flag}_{\mathscr{S}}^{\mathrm{symp}}(M)\) is the open subset in \(\mathrm{Flag}_{\mathscr{S}}(M)\) consisting of all symplectic flags of type \(\mathscr{S}\), then the symplectic form on \(M\) induces by transgression a symplectic form on the manifold of symplectic nonlinear flags. The Hamiltonian group \(\mathrm{Ham}(M)\) acts on \(\mathrm{Flag}_{\mathscr{S}}^{\mathrm{symp}}(M)\) in a Hamiltonian fashion with equivariant moment map \(J:\mathrm{Flag}_{\mathscr{S}}^{\mathrm{symp}}(M)\to\mathfrak{ham}(M)^*\).
This moment map is injective and identifies each connected component of \(\mathrm{Flag}_{\mathscr{S}}^{\mathrm{symp}}(M)\) with a coadjoint orbit of \(\mathrm{Ham}(M)\).
The main result of the paper states that the restriction of the moment map \(J:\mathrm{Flag}_{\mathscr{S}}^{\mathrm{symp}}(M)\to\mathfrak{ham}(M)^*\) to any connected component is one-to-one onto a coadjoint orbit cf the Hamiltonian group \(\mathrm{Ham}(M)\). The Kostant-Kirillov-Souriau symplectic form \(\omega_{\mathrm{KKS}}\) on the coadjoint orbit satisfies \(J^*\omega_{\mathrm{KKS}}=\Omega\), where \(\Omega\) is a natural symplectic form.
Reviewer: Andrew Bucki (Edmond)Renormalization 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.\(G_2\)-manifolds and the ADM formalism.https://www.zbmath.org/1456.580102021-04-16T16:22:00+00:00"Chihara, Ryohei"https://www.zbmath.org/authors/?q=ai:chihara.ryoheiA \(G_2\)-manifold is a \(7\)-dimensional Riemannian manifold with holonomy group contained in the exceptional Lie group \(G_2\). The author regards the present paper as a continuation of previous work [``\(G_2\)-metrics arising from non-integrable special Lagrangian fibrations'', Preprint, \url{arXiv:1801.05540}], in which the main result gives a characterization of a certain dynamical system as a constraint Hamiltonian dynamical system related to \(G_2\).
The paper is adequately described in the abstract: ``In this paper we study a Hamiltonian function on the cotangent bundle of the space of Riemannian metrics on a \(3\)-manifold \(M\) and prove the orbits of the constrained Hamiltonian dynamical system correspond to \(G_2\)-manifolds foliated by hypersurfaces diffeomorphic to \(M\times SO(3)\).''
Reviewer: Vagn Lundsgaard Hansen (Lyngby)Differential geometry and Lie groups. A second course.https://www.zbmath.org/1456.530012021-04-16T16:22:00+00:00"Gallier, Jean"https://www.zbmath.org/authors/?q=ai:gallier.jean-h"Quaintance, Jocelyn"https://www.zbmath.org/authors/?q=ai:quaintance.jocelynThis book is written as a second course on differential geometry. So the reader is supposed to be familiar with some themes from the first course on differential geometry -- the theory of manifolds and some elements of Riemannian geometry.
In the first two chapters here some topics from linear algebra are provided -- a detailed exposition of tensor algebra and symmetric algebra, exterior tensor products and exterior algebra. These chapters may be useful when studying the material of this book for those students, who did not study these topics in their algebraic course.
Some themes, which are covered in this book, are rather standard for books on differential geometry - they are differential forms, de Rham cohomology, integration on manifolds, connections and curvature in vector bundles, fibre bundles, principal bundles and metrics on bundles. But a number of topics discussed in this book are not always included in courses on differential geometry and are rarely contained in textbooks on differential geometry. The presence of these topics makes this book especially interesting for modern students. Here is a list of some such topics: an introduction to Pontrjagin
classes, Chern classes, and the Euler class, distributions and the Frobenius theorem. Three chapters need to be highlighted separately. Chapter 7 -- spherical harmonics and an introduction to the representations of compact Lie groups. Chapter 8 -- operators on Riemannian manifolds: Hodge Laplacian, Laplace-Beltrami Laplacian, Bochner
Laplacian. Chapter 11 -- Clifford algebras and groups, groups Pin\((n)\), Spin\((n)\).
Not all statements in this book are given with proofs, for some only links to other textbooks are given. But the most important results are given here with complete proofs and accompanied by examples. Each chapter of this book ends with a list of interesting and sometimes very important problems. At the end of the book there is a very detailed list of the notation used (symbol index) and a detailed list (index) of the terms used.
Reviewer: V. V. Gorbatsevich (Moskva)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.Homoclinic solutions for a class of nonlinear fourth order \(p\)-Laplacian differential equations.https://www.zbmath.org/1456.340442021-04-16T16:22:00+00:00"Dimitrov, Nikolay D."https://www.zbmath.org/authors/?q=ai:dimitrov.nikolay-d"Tersian, Stepan A."https://www.zbmath.org/authors/?q=ai:tersian.stepan-agopIn this paper, the authors deal with a class of fourth-order differential equation involving \(p\)-Laplacian
\[
|u''(x)|^{p-2}u''(x))''+\omega (|u'(x)|^{p-2}u'(x))'+\lambda V(x) |u(x)|^{p-2} u(x)=f(x,u(x)
\]
where \(\omega\) is a constant, \(\lambda\) is a parameter and \(f\in C (\mathbb{R},\mathbb{R})\). with the aid of critical point theory and variational methods, they prove that under suitable growth conditions, the above equation possesses at least one nontrivial homoclinic solution, i.e., a nontrivial solution satisfying \(u(x)\longrightarrow 0\) as \(x\longrightarrow \mp\infty\).
Reviewer: Mohsen Timoumi (Monastir)Fermions and loops on graphs. I: Loop calculus for determinants.https://www.zbmath.org/1456.820382021-04-16T16:22:00+00:00"Chernyak, Vladimir Y."https://www.zbmath.org/authors/?q=ai:chernyak.vladimir-y"Chertkov, Michael"https://www.zbmath.org/authors/?q=ai:chertkov.michaelTheta 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).Fermions and loops on graphs. II: A monomer-dimer model as a series of determinants.https://www.zbmath.org/1456.820392021-04-16T16:22:00+00:00"Chernyak, Vladimir Y."https://www.zbmath.org/authors/?q=ai:chernyak.vladimir-y"Chertkov, Michael"https://www.zbmath.org/authors/?q=ai:chertkov.michaelOn LA-Courant algebroids and Poisson Lie 2-algebroids.https://www.zbmath.org/1456.580042021-04-16T16:22:00+00:00"Jotz Lean, M."https://www.zbmath.org/authors/?q=ai:jotz.madeleineT.J. Courant discovered a skew-symmetric Lie bracket on \(TM \oplus T^* M\). The more general structure of a Courant algebroid, links the study of constrained Hamiltonian systems with generalised complex geometry. They were studied extensively throughout the 1990s by Zhang-Ju Liu, Alan Weinstein and Ping Xu, as well as Severa and Roytenberg. The associated integrability problem is an open question to this day.
To this end, it is important to understand better these structures. Courant algebroids are ``higher'' geometric structures. This can be made precise in the following ways: Roytenberg and Severa (independently) understood them in a graded sense, namely they described them as symplectic Lie 2-algebroids. On the other hand, Courant's example fits into \textit{K. C. H. Mackenzie}'s study of multiple structures, in particular it is an example of a double Lie algebroid [J. Reine Angew. Math. 658, 193--245 (2011; Zbl 1246.53112)]. \textit{D. Li-Bland} in his PhD thesis [LA-Courant algebroids and their applications. \url{arXiv:1204.2796}] introduced a more general class of Courant algebroids (LA-Courant algebroids) which are Courant algebroid structures in the category of vector bundles. They too fit in the multiple structures studied by Mackenzie.
The paper under review studies the correspondence between LA-Courant algebroids and Poisson Lie 2-algebroids (the latter generalize symplectic Lie 2-algebroids), using the author's earlier results on split Lie 2-algebroids and self-dual 2-representations.
Reviewer: Iakovos Androulidakis (Athína)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].Morse theory for minimal surfaces in manifolds.https://www.zbmath.org/1456.580122021-04-16T16:22:00+00:00"Kim, Hwajeong"https://www.zbmath.org/authors/?q=ai:kim.hwajeongSummary: A Morse theory of a given function gives information of the numbers of critical points of some topological type. A minimal surface, bounded by a given curve in a manifold, is characterized as a harmonic extension of a critical point of the functional \(\mathcal{E}\) which corresponds to the Dirichlet integral. We want to obtain Morse theories for minimal surfaces in Riemannian manifolds. We first investigate the higher differentiabilities of \(\mathcal{E}\). We then develop a Morse inequality for minimal surfaces of annulus type in a Riemannian manifold. Furthermore, we also construct body handle theories for minimal surfaces of annulus type as well as of disc type. Here we give a setting where the functional \(\mathcal{E}\) is non-degenerated.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-hamidThe geometry of synchronization problems and learning group actions.https://www.zbmath.org/1456.051052021-04-16T16:22:00+00:00"Gao, Tingran"https://www.zbmath.org/authors/?q=ai:gao.tingran"Brodzki, Jacek"https://www.zbmath.org/authors/?q=ai:brodzki.jacek"Mukherjee, Sayan"https://www.zbmath.org/authors/?q=ai:mukherjee.sayanSummary: We develop a geometric framework, based on the classical theory of fibre bundles, to characterize the cohomological nature of a large class of synchronization-type problems in the context of graph inference and combinatorial optimization. We identify each synchronization problem in topological group \(G\) on connected graph \(\Gamma\) with a flat principal \(G\)-bundle over \(\Gamma\), thus establishing a classification result for synchronization problems using the representation variety of the fundamental group of \(\Gamma\) into \(G\). We then develop a twisted Hodge theory on flat vector bundles associated with these flat principal \(G\)-bundles, and provide a geometric realization of the graph connection Laplacian as the lowest-degree Hodge Laplacian in the twisted de Rham-Hodge cochain complex. Motivated by these geometric intuitions, we propose to study the problem of learning group actions -- partitioning a collection of objects based on the local synchronizability of pairwise correspondence relations -- and provide a heuristic synchronization-based algorithm for solving this type of problems. We demonstrate the efficacy of this algorithm on simulated and real datasets.Hyperbolic 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.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.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)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.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.Topology change of level sets in Morse theory.https://www.zbmath.org/1456.370612021-04-16T16:22:00+00:00"Knauf, Andreas"https://www.zbmath.org/authors/?q=ai:knauf.andreas"Martynchuk, Nikolay"https://www.zbmath.org/authors/?q=ai:martynchuk.nikolayThis paper considers Morse functions \(f \in C^2(M, \mathbb{R})\) on a smooth \(m\)-dimensional manifold without boundary. For such functions and for every critical point \(x \in M\) of \(f\), the Hessian of \(f\) at \(x\) is nondegenerate.
The authors are interested in the topology of the level sets \(f^{-1}(a) = \partial{M^a}\) where \(M^a = \{ x \in M : f(x) \le a\}\) is a so-called sublevel set. The authors address the question of under what conditions the topology of \(\partial{M^a}\) can change when the function \(f\) passes a critical level with one or more critical points. ``Change in topology'' here means roughly that if \(a\) and \(b\) are regular values with \(a < b\), then \(H_k(\partial{M^a};G) \ne H_k(\partial{M^b};G)\) where \(k\) is an integer and the \(H_k\) are homology groups over some abelian group \(G\) when the function \(f\) passes through a level with one or more critical points. A motivation for this question derives from the \(n\)-body problem: does the topology of the integral manifolds always change when passing through a bifurcation level?
The first part of the paper considers level sets of abstract Morse functions that satisfy the Palais-Smale condition and whose level sets have finitely generated homology groups. They show that for such functions the topology of \(f^{-1}(a)\) changes when passing a single critical point if the index of the critical point is different from \(m/2\) where \(m\) is the dimension of the manifold \(M\). Then they move on to consider the case where \(M\) is a vector bundle of rank \(n\) over a manifold \(N\) of dimension \(n\), and where (up to a translation) \(f\) is a Morse function that is the sum of a positive definite quadratic form on the fibers and a potential function that is constant on the fibers. Here the authors have in mind Hamiltonian functions \(H\) on the cotangent bundle of the base manifold \(N\). In this case they show that the topology of \(H^{-1}(h)\) always changes when passing a single critical point if the Euler characteristic of \(N\) is not \(\pm 1\).
Finally the authors apply their results to examples from Hamiltonian and celestial mechanics, with emphasis on the planar three-body problem. There they show that the topology always changes for the planar three-body problem provided that the reduced Hamiltonian is a Morse function with at most two critical points on each level set.
Reviewer: William J. Satzer Jr. (St. Paul)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}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.galPositive 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)].Exterior multiplication with singularities: a Saito theorem in vector bundles.https://www.zbmath.org/1456.580032021-04-16T16:22:00+00:00"Jakubczyk, B."https://www.zbmath.org/authors/?q=ai:jakubczyk.bronislawSummary: Let \(E\) be a vector bundle over a differentiable manifold \(M\) and let \(\bigwedge^pE\) denote the \(p\)th exterior product of \(E\). Given sections \(\omega_1,\dots ,\omega_k\) of \(E\) and a section \(\eta\) of \(\bigwedge^pE\), we consider the problem of whether \(\eta\) can be written in the form \[\eta =\sum \omega_i\wedge \gamma_i,\] where \(\gamma_i\) are sections of \(\bigwedge^{p-1}E\). An obvious necessary condition \(\Omega \wedge \eta =0\), where \(\Omega =\omega_1\wedge \cdots \wedge \omega_k\), has to be supplemented with a condition that the form \(\Omega\) has sufficiently regular singularities at points where \(\Omega (x)=0\). Such a local condition is suggested by an algebraic theorem of K. Saito and is given in terms of the depth of the ideal defined by the coefficients of \(\Omega \). Working in the smooth, real analytic or holomorphic (with \(M\) a Stein manifold) category, we show that the condition is sufficient for the above property to hold. Moreover, in the smooth category it is sufficient for the existence of a continuous right inverse to the operator defined by \((\gamma_1,\dots ,\gamma_k)\mapsto \sum \omega_i\wedge \gamma_i\). All these results are also proven in the case where \(E\) is a bundle over a suitable closed subset of \(M\).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.Quasi-optimal adaptive mixed finite element methods for controlling natural norm errors.https://www.zbmath.org/1456.651642021-04-16T16:22:00+00:00"Li, Yuwen"https://www.zbmath.org/authors/?q=ai:li.yuwenSummary: For a generalized Hodge Laplace equation, we prove the quasi-optimal convergence rate of an adaptive mixed finite element method. This adaptive method can control the error in the natural mixed variational norm when the space of harmonic forms is trivial. In particular, we obtain new quasi-optimal adaptive mixed methods for the Hodge Laplace, Poisson, and Stokes equations. Comparing to existing adaptive mixed methods, the new methods control errors in both variables.Energy gap for Yang-Mills connections. I: Four-dimensional closed Riemannian manifolds.https://www.zbmath.org/1456.580142021-04-16T16:22:00+00:00"Feehan, Paul M. N."https://www.zbmath.org/authors/?q=ai:feehan.paul-m-nSummary: We extend an \(L^2\) energy gap result due to \textit{M. Min-Oo} [Compos. Math. 47, 153--163 (1982; Zbl 0519.53042), Theorem 2] and \textit{T. H. Parker} [Commun. Math. Phys. 85, 563--602 (1982; Zbl 0502.53022), Proposition 2.2] for Yang-Mills connections on principal \(G\)-bundles, \(P\), over closed, connected, four-dimensional, oriented, smooth manifolds, \(X\), from the case of positive Riemannian metrics to the more general case of good Riemannian metrics, including metrics that are generic and where the topologies of \(P\) and \(X\) obey certain mild conditions and the compact Lie group, \(G\), is \(\operatorname{SU}(2)\) or \(\operatorname{SO}(3)\).
[For Part II see Zbl 1375.58013.]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.Spectral gaps for reversible Markov processes with chaotic invariant measures: the Kac process with hard sphere collisions in three dimensions.https://www.zbmath.org/1456.602522021-04-16T16:22:00+00:00"Carlen, Eric"https://www.zbmath.org/authors/?q=ai:carlen.eric-anders"Carvalho, Maria"https://www.zbmath.org/authors/?q=ai:carvalho.maria-conceicao"Loss, Michael"https://www.zbmath.org/authors/?q=ai:loss.michael-p|loss.michaelSummary: We develop a method for producing estimates on the spectral gaps of reversible Markov jump processes with chaotic invariant measures, that is effective in the case of degenerate jump rates, and we apply it to prove the Kac conjecture for hard sphere collision in three dimensions.Conley theory for Gutierrez-Sotomayor fields.https://www.zbmath.org/1456.370242021-04-16T16:22:00+00:00"Montúfar, H."https://www.zbmath.org/authors/?q=ai:montufar.h"de Rezende, K. A."https://www.zbmath.org/authors/?q=ai:de-rezende.ketty-abaroaThe authors study, from a topological perspective, continuous flows associated to \(C^1\) structurally stable vector fields tangent to a two-dimensional compact subset \(M\) of \(\mathbb{R}^k\). They call these flows Gutierrez-Sotomayor (shortly denoted by GS) flows on manifolds \(M\) with simple singularities and they use Conley index theory to study them. The Conley indices of all simple singularities are computed and an Euler characteristic formula is obtained. By considering a stratification of \(M\) which decomposes it into a union of its regular and singular strata, certain Euler-type formulas which relate the topology of \(M\) and the dynamics on the strata are obtained. The existence of a Lyapunov function for GS flows without periodic orbits and singular cycles is established. Using long exact sequence analysis of index pairs the authors determine necessary and sufficient conditions for a GS flow to be defined on an isolating block. They organize this information combinatorially with the aid of Lyapunov graphs and using a Poincaré-Hopf equality to construct isolating blocks for all simple singularities.
The main results generalize results of the second author and \textit{R. D. Franzosa} [Trans. Am. Math. Soc. 340, No. 2, 767--784 (1993; Zbl 0806.58042)], where Morse-Smale flows and more generally continuous flows on smooth surfaces are classified.
Reviewer: Dorin Andrica (Riyadh)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.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.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)\( T\overline{T} \)-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.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.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 \((2l-1)n+(2m-1)\).