Recent zbMATH articles in MSC 58Jhttps://www.zbmath.org/atom/cc/58J2022-05-16T20:40:13.078697ZWerkzeugSmooth manifold of one-dimensional lattices and shifted latticeshttps://www.zbmath.org/1483.111362022-05-16T20:40:13.078697Z"Smirnova, Elena Nikolaevna"https://www.zbmath.org/authors/?q=ai:smirnova.elena-nikolaevna"Pikhtil'kova, Ol'ga Aleksandrovna"https://www.zbmath.org/authors/?q=ai:pikhtilkova.olga-aleksandrovna"Dobrovol'skiĭ, Nikolaĭ Nikolaevich"https://www.zbmath.org/authors/?q=ai:dobrovolskii.n-n"Rebrova, Irina Yur'evna"https://www.zbmath.org/authors/?q=ai:rebrova.irina-yurevna"Rodionov, Aleksandr Valer'evich"https://www.zbmath.org/authors/?q=ai:rodionov.aleksandr-valerevich"Dobrovol'skiĭ, Nikolaĭ Mikhaĭlovich"https://www.zbmath.org/authors/?q=ai:dobrovolskii.n-mSummary: In the previous work, the authors laid the foundations of the theory of smooth varieties of number-theoretic lattices. The simplest case of one-dimensional lattices is considered.
This article considers the case of one-dimensional shifted lattices. First of all, we consider the construction of a metric space of shifted lattices by mapping one-dimensional shifted lattices to the space of two-dimensional lattices.
In this paper, we define a homeomorphic mapping of the space of one-dimensional shifted lattices to an infinite two-dimensional cylinder. Thus, it is established that the space of one-dimensional shifted lattices \(CPR_2\) is locally a Euclidean space of dimension 2.
Since the metric on these spaces is not Euclidean, but is ``logarithmic'' , unexpected results are obtained in the one-dimensional case about derivatives of basic functions, such as the determinant of the lattice, the hyperbolic lattice parameter, the norm at least, the Zeta function and lattice hyperbolic Zeta function of lattices.
The paper considers the relationship of these functions with the issues of studying the error of approximate integration over parallelepipedal grids as the determinant of the lattice, the hyperbolic lattice parameter, the norm at least, the Zeta function and lattice hyperbolic Zeta function of lattices.
Note that the geometry of metric spaces of multidimensional lattices and shifted multidimensional lattices is much more complex than the geometry of an ordinary Euclidean space. This can be seen from the paradox of nonadditivity of the length of a segment in the space of shifted one-dimensional lattices. From the presence of this paradox, it follows that there is an open problem of describing geodesic lines in the spaces of multidimensional lattices and multidimensional shifted lattices, as well as in finding a formula for the length of the arcs of lines in these spaces. Naturally, it would be interesting not only to describe these objects, but also to obtain a number-theoretic interpretation of these concepts.
A further direction of research may be the study of the analytical continuation of the hyperbolic zeta function on the spaces of lattices and multidimensional lattices. As is known, an analytical continuation of the hyperbolic zeta function of lattices is constructed for an arbitrary Cartesian lattice. Even the question of the continuity of these analytic continuations in the left half-plane on the lattice space has not been studied. All these, in our opinion, are relevant areas for further research.From torus bundles to particle-hole equivariantizationhttps://www.zbmath.org/1483.180202022-05-16T20:40:13.078697Z"Cui, Shawn X."https://www.zbmath.org/authors/?q=ai:cui.shawn-xingshan"Gustafson, Paul"https://www.zbmath.org/authors/?q=ai:gustafson.paul-p|gustafson.paul"Qiu, Yang"https://www.zbmath.org/authors/?q=ai:qiu.yang"Zhang, Qing"https://www.zbmath.org/authors/?q=ai:zhang.qing.4|zhang.qing.2|zhang.qing|zhang.qing.3|zhang.qing.1Quantum topology emerged from the discovery of the Jones polynomials [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)] and the formulation of topological quantum field theory (TQFT) [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053); \textit{E. Witten}, Adv. Ser. Math. Phys. 9, 239--329 (1989; Zbl 0726.57010)] in the 1980s, revealing deep connections between the algebraic/quantum arena of tensor categories and the topoogical/classical arena of \(3\)-dimensional manifolds. Precisely speaking, quantum invariants of \(3\)-dimensional manifolds and \((2+1)\)-dimensional TQFTs are to be constructed from modular tensor categories, two fundamental families in \((2+1)\)-dimensions being the Reshetikhin-Turaev [Zbl 0725.57007] and Turaev-Viro [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)] TQFTs, both of which are based on certain tensor categories.
The following two works lie at the backdrop of this paper.
\begin{itemize}
\item Inspired by \(M\)-theory in physics, the authors in [\textit{G. Y. Cho} et al., J. High Energy Phys. 2020, No. 11, Paper No. 115, 58 p. (2020; Zbl 1456.81341)] proposed another relation between tensor categories and \(3\)-dimensional manifolds in the opposite direction, outlining a program to construct modular tensor categories from certain classes of closed oriented \(3\)-dimensional manifolds. A central structure under study is an \(\mathrm{SL}(2,\mathbb{C})\) flat connection corresponding to a conjugacy class of morphisms from the fundamental group to \(\mathrm{SL}(2,\mathbb{C})\). The manifolds are required to have finitely many non-Abelian \(\mathrm{SL}(2,\mathbb{C})\) flat connections, each of which must be gauge equivalent to an \(\mathrm{SL}(2,\mathbb{R})\) or \(\mathrm{SU}(2)\) flat connection. Classical invariants such as the Chern-Simons invariant and twisted Reidemeister torsion also play a significant role in the construction.
\item In [\textit{S. X. Cui}, ``From three dimensional manifolds to modular tensor categories'', Preprint, \url{arXiv:2101.01674}], the authors mathematically explored the program in greater detail, systematically studying two infinite families of \(3\)-dimensional manifolds, namely Seifert fibered spaces with three singular fibers and torus bundles over the circle whose monodromy matrix \ has odd trace. It was shown that the first family being related to the Temperley-Lieb-Jones category [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] realizes modular tensor categories, while the second family is related to the quantum group category of type \(B\).
\end{itemize}
Although the efforts in the above two works suggest a far-reaching connection between \(3\)-dimensional manifolds and premodular tensor categories, there remain many questions to be resolved.
\begin{itemize}
\item[1.] The program currently only provides an algorithm in computing the modular \(S\)- and \(T\)-matrices, leaving other data such as \(F\)-symbols and \(R\)-symbols untouched.
\item[2.] Even for the modular data, the computation for the \(S\)-matrix essentially follows a try-and-error procedure.
\item[3.] There are a number of subtleties in choosing the correct set of characters as simple objects, determining the proper unit object, etc.
\end{itemize}
The principal objective in this paper is to apply the program to torus bundles over the circle with SOL geometry [Zbl 0561.57001]. The examples of Seifert fibered spaces in the second work covered six of the eight geometries, the ones left being the hyperbolic and SOL. Since the program concerns closed manifolds whose Chern-Simons invariants are all real, hyperbolic manifolds are surely excluded. A torus bundle over the circle is uniquely determined by the isotopy class of gluing diffeomorphism, called the monodromy matrix, which is an element of \(\mathrm{SL}(2,\mathbb{Z})\). Torus bundles whose monodromy is Anosov are of SOL geometry.
For a finite Abelian group \(G\) and a quadratic form \(q:G\rightarrow\mathbb{C}\), \(\mathcal{C}(G,q)\) denotes the pointed premodular category whose isomorphism classes of simple objects are \(G\) and whose topological twist is given by \(q\). There is a \(\mathbb{Z}_{2}\)-action, called the particle-hole symmetry, on \(\mathcal{C}(G,q)\) defined by sending each simple object to its dual or its inverse viewed as a group element. \(\mathcal{C}(G,q)^{\mathbb{Z}_{2}}\) denotes the \(\mathbb{Z}_{2}\)-equivariantization of \(\mathcal{C}(G,q)\) with respect to the particle-hole symmetry. The principal result is the following theorem.
Theorem 1. For each torus bundle over the circle \(M_{A}\) with monodromy matrix \(A\), \(N:=\left\vert Tr(A)+2\right\vert \), there is an associated finite Abelian group \(G_{A}\) isomorphic to \(\mathbb{Z}_{r}\times\mathbb{Z}_{N/r}\) for some integer \(r\geq1\) (Lemma 1) and a quadratic form \(q_{A}(x):=\exp(\frac{2\pi i\widetilde{q}(x)}{N})\), \(\widetilde{q}:G\rightarrow\mathbb{Z}_{N}\) (Lemma 3) such that the modular data realized by \(M_{A}\) coincide with those of \(\mathcal{C}(G_{A},q_{A})^{\mathbb{Z}_{2}}\).
A synopsis of the paper goes as follows.
\begin{itemize}
\item[\S 2] reviews some basic facts about premodular categories, recalling the program of constructing premodular categories from \(3\)-dimensional manifolds.
\item[\S 3] is devoted to computing the modular data of the equivariantization of a pointed premodular category under the particle-hole symmetry.
\item[\S 4] states and proves the main theorem concerning the construction of premodular categories from torus bundles.
\end{itemize}
Reviewer: Hirokazu Nishimura (Tsukuba)Decay rates for Kelvin-Voigt damped wave equations. II: The geometric control conditionhttps://www.zbmath.org/1483.350262022-05-16T20:40:13.078697Z"Burq, Nicolas"https://www.zbmath.org/authors/?q=ai:burq.nicolas"Sun, Chenmin"https://www.zbmath.org/authors/?q=ai:sun.chenminAuthors' abstract: ``We study in this article decay rates for Kelvin-Voigt damped wave equations under a geometric control condition. When the damping coefficient is sufficiently smooth (C1 vanishing nicely, see the following equation: \(|\nabla a| \leq C a^{1/2}\)) we show that exponential decay follows from geometric control conditions (see [\textit{N. Burq} and \textit{H. Christianson}, Commun. Math. Phys. 336, No. 1, 101--130 (2015; Zbl 1320.35062); \textit{L. Tebou}, C. R., Math., Acad. Sci. Paris 350, No. 11--12, 603--608 (2012; Zbl 1255.35039)] for similar results under stronger assumptions on the damping function).''
For Part I, see [\textit{N. Burq}, SIAM J. Control Optim. 58, No. 4, 1893--1905 (2020; Zbl 1452.35030)].
Reviewer: Kaïs Ammari (Monastir)Finite time blow-up for the heat flow of \(H\)-surface with constant mean curvaturehttps://www.zbmath.org/1483.350482022-05-16T20:40:13.078697Z"Li, Haixia"https://www.zbmath.org/authors/?q=ai:li.haixiaSummary: We consider an initial boundary value problem for the heat flow of the equation of surfaces with constant mean curvature which was investigated in [\textit{T. Huang} et al., Manuscr. Math. 134, No. 1--2, 259--271 (2011; Zbl 1210.53012)], where global well-posedness and finite time blow-up of regular solutions were obtained. Their results are complemented in this paper in the sense that some new conditions on the initial data are provided for the solutions to develop finite time singularity.Chiti-type reverse Hölder inequality and torsional rigidity under integral Ricci curvature conditionhttps://www.zbmath.org/1483.350512022-05-16T20:40:13.078697Z"Chen, Hang"https://www.zbmath.org/authors/?q=ai:chen.hang|chen.hang.1Summary: In this paper, we prove a reverse Hölder inequality for the eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with the integral Ricci curvature condition. We also prove an isoperimetric inequality for the torsional rigidity of such domains. These results extend some recent work of \textit{N. Gamara} et al. [Open Math. 13, 557--570 (2015; Zbl 06632233)] and \textit{D. Colladay} et al. [J. Geom. Anal. 28, No. 4, 3906--3927 (2018; Zbl 1410.58016)] from the pointwise lower Ricci curvature bound to the integral Ricci curvature condition. We also extend the results from Laplacian to \(p\)-Laplacian.Li-Yau multipLier set and optimal Li-Yau gradient estimate on hyperboLic spaceshttps://www.zbmath.org/1483.350592022-05-16T20:40:13.078697Z"Yu, Chengjie"https://www.zbmath.org/authors/?q=ai:yu.chengjie"Zhao, Feifei"https://www.zbmath.org/authors/?q=ai:zhao.feifeiThis paper is motivated by finding sharp Li-Yau-type gradient estimate for positive solution of heat equations on complete Riemannian manifolds with negative Ricci curvature lower bound. To reach this aim, the authors first introduce the notion of Li-Yau multiplier set and show that it can be computed by heat kernel of the manifold, then, an optimal Li-Yau-type gradient estimate is obtained on hyperbolic spaces by using recurrence relations of heat kernels on hyperbolic spaces. Lastly, as an application of the previous results, a sharp and interesting Harnack inequalities on hyperbolic spaces is shown.
Reviewer: Vincenzo Vespri (Firenze)Asymptotic behavior of solutions of the Dirichlet problem for the Poisson equation on model Riemannian manifoldshttps://www.zbmath.org/1483.350782022-05-16T20:40:13.078697Z"Losev, Alexander Georgievoch"https://www.zbmath.org/authors/?q=ai:losev.alexander-georgievoch"Mazepa, Elena Alexeevna"https://www.zbmath.org/authors/?q=ai:mazepa.elena-alexeevnaSummary: The paper is devoted to estimating the speed of approximation of solutions of the Dirichlet problem for the Poisson equation on non-compact model Riemannian manifolds to their boundary data at ``infinity''. Quantitative characteristics that estimate the speed of the approximation are found in terms of the metric of the manifold and the smoothness of the inhomogeneity in the Poisson equation.Maximal regularity of parabolic transmission problemshttps://www.zbmath.org/1483.351062022-05-16T20:40:13.078697Z"Amann, Herbert"https://www.zbmath.org/authors/?q=ai:amann.herbertSummary: Linear reaction-diffusion equations with inhomogeneous boundary and transmission conditions are shown to possess the property of maximal \(L_{\mathrm{p}}\) regularity. The new feature is the fact that the transmission interface is allowed to intersect the boundary of the domain transversally.Qualitative properties of stationary solutions of the NLS on the hyperbolic space without and with external potentialshttps://www.zbmath.org/1483.352262022-05-16T20:40:13.078697Z"Selvitella, Alessandro"https://www.zbmath.org/authors/?q=ai:selvitella.alessandro|selvitella.alessandro-mariaSummary: In this paper, we prove some qualitative properties of stationary solutions of the NLS on the Hyperbolic space. First, we prove a variational characterization of the ground state and give a complete characterization of the spectrum of the linearized operator around the ground state. Then we prove some rigidity theorems and necessary conditions for the existence of solutions in weighted spaces. Finally, we add a slowly varying potential to the homogeneous equation and prove the existence of non-trivial solutions concentrating on the critical points of a reduced functional. The results are the natural counterparts of the corresponding theorems on the Euclidean space. We produce also the natural virial identity on the Hyperbolic space for the complete evolution, which however requires the introduction of a weighted energy, which is not conserved and so does not lead directly to finite time blow-up as in the Euclidean case.Approximation of the fractional Schrödinger propagator on compact manifoldshttps://www.zbmath.org/1483.353172022-05-16T20:40:13.078697Z"Chen, Jie Cheng"https://www.zbmath.org/authors/?q=ai:chen.jiecheng"Fan, Da Shan"https://www.zbmath.org/authors/?q=ai:fan.dashan"Zhao, Fa You"https://www.zbmath.org/authors/?q=ai:zhao.fayouSummary: Let \(\mathcal{L}\) be a second order positive, elliptic differential operator that is self-adjoint with respect to some \(C^\infty\) density \(dx\) on a compact connected manifold \(\mathbb{M}\). We proved that if \(0<\alpha<1\), \(\alpha/2<s<\alpha\) and \(f\in H^s(\mathbb{M})\) then the fractional Schrödinger propagator \(\mathrm{e}^{\mathrm{i}t\mathcal{L}^{\alpha/2}}\) on \(\mathbb{M}\) satisfies \(\mathrm{e}^{\mathrm{i}t\mathcal{L}^{\alpha/2}}f(x)-f(x)=o(t^{s/\alpha-\varepsilon})\) almost everywhere as \(t\rightarrow 0^+\), for any \(\varepsilon>0\).An ODE reduction method for the semi-Riemannian Yamabe problem on space formshttps://www.zbmath.org/1483.530592022-05-16T20:40:13.078697Z"Fernández, Juan Carlos"https://www.zbmath.org/authors/?q=ai:fernandez.juan-carlos"Palmas, Oscar"https://www.zbmath.org/authors/?q=ai:palmas.oscarThe authors prove the existence of blowing-up and globally defined solutions of Yamabe-type partial differential equations on semi-Euclidean space and on the pseudosphere of dimension at least 3. In the proof they use isoparametric functions which allow the reduction to a generalized Emden-Fowler ordinary differential equation.
Reviewer: Hans-Bert Rademacher (Leipzig)Lower bounds for Cauchy data on curves in a negatively curved surfacehttps://www.zbmath.org/1483.530602022-05-16T20:40:13.078697Z"Galkowski, Jeffrey"https://www.zbmath.org/authors/?q=ai:galkowski.jeffrey"Zelditch, Steve"https://www.zbmath.org/authors/?q=ai:zelditch.steveSummary: We prove a uniform lower bound on Cauchy data on an arbitrary curve on a negatively curved surface using the Dyatlov-Jin(-Nonnenmacher) observability estimate on the global surface. In the process, we prove some further results about defect measures of restrictions of eigenfunctions to a hypersurface.Canonical identification at infinity for Ricci-flat manifoldshttps://www.zbmath.org/1483.530692022-05-16T20:40:13.078697Z"Park, Jiewon"https://www.zbmath.org/authors/?q=ai:park.jiewonSummary: We give a natural way to identify between two scales, potentially arbitrarily far apart, in a non-compact Ricci-flat manifold with Euclidean volume growth when a tangent cone at infinity has smooth cross section. The identification map is given as the gradient flow of a solution to an elliptic equation.A formula for the heat kernel coefficients of the Dirac Laplacians on spin manifoldshttps://www.zbmath.org/1483.530722022-05-16T20:40:13.078697Z"Nagase, Masayoshi"https://www.zbmath.org/authors/?q=ai:nagase.masayoshi"Shirakawa, Takumi"https://www.zbmath.org/authors/?q=ai:shirakawa.takumiSummary: Based on Getzler's rescaling transformation, we obtain a formula for the heat kernel coefficients of the Dirac Laplacian on a spin manifold. One can compute them explicitly up to an arbitrarily high order by using only a basic knowledge of calculus added to the formula.On center of mass and foliations by constant spacetime mean curvature surfaces for isolated systems in general relativityhttps://www.zbmath.org/1483.530862022-05-16T20:40:13.078697Z"Cederbaum, Carla"https://www.zbmath.org/authors/?q=ai:cederbaum.carla"Sakovich, Anna"https://www.zbmath.org/authors/?q=ai:sakovich.annaThe authors define a new total center of mass for an ``isolated system'': the ``Universe'' is a 4-dimensional Lorentzian manifold \((\mathfrak M^{1,3}, \mathfrak{g})\), endowed with an energy-momentum tensor field \(\mathfrak T\), with an ``initial data set'' given by a spacelike hypersuface \((M^3,g)\), with the second fundamental form \(K\), the scalar local energy density \(\mu\) and the (1-form) local momentum density \(J\). When this configuration is ``asymptotically Euclidean'' and with non-vanishing energy, it gives rise to a (unique) foliation by 2-spheres of constant spacetime mean curvature. This foliation is the main tool for constructing the total center of mass. It is shown that this center of mass behaves as a point particle in Special Relativity (i.e. it transforms equivariantly under the asymptotic Poincaré group of \({\mathfrak M^{1,3}}\)). In particular, it evolves in time under the Einstein evolution equations like a point particle in Special Relativity.
Reviewer: Gabriel Teodor Pripoae (Bucureşti)Evolution of the first eigenvalue of the Laplace operator and the \(p\)-Laplace operator under a forced mean curvature flowhttps://www.zbmath.org/1483.531102022-05-16T20:40:13.078697Z"Qi, Xuesen"https://www.zbmath.org/authors/?q=ai:qi.xuesen"Liu, Ximin"https://www.zbmath.org/authors/?q=ai:liu.ximinSummary: In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the \(p\)-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the \(p\)-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao's work. Moreover, we give an example to specify applications of conclusions obtained above.On traces of operators associated with actions of compact Lie groupshttps://www.zbmath.org/1483.580032022-05-16T20:40:13.078697Z"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-yuLet \((X,M)\) be a smooth pair consisting of a manifold \(M\) and its sub-manifold \(X\), \(G\) be a compact Lie group acting on manifold \(M\), \(T_g\) be a shift operator
\[
T_gu=g^{-1^*}u
\]
which is induced by the diffeomorhism \(g\). The \(G\)-operator \(D\) under consideration is the following
\[
D=\int\limits_GD_gT_gdg, \tag{1}
\]
where \(dg\) is Haar measure on the group \(G\), \(T_g\) is a smooth family of pseudo-differential operators.
The authors study the trace operator
\[
i^*Di_*: H^s(X)\rightarrow H^{s-d-\nu}(X),
\]
where \(i^*: H^s(M)\rightarrow H^{s-\nu/2}(X)\) and \( i_*: H^s(X)\rightarrow H^{s-\nu/2}(X)\) are boundary and co-boundary operators induced by the imbedding \(i: X\rightarrow M\), \(d\) is an order of the operator \(D\), \(\nu\) is a codimension of \(X\). They introduce the set
\[
X_G=\{x\in X: Gx\subset X\}
\]
and prove their main result ``on a localization''. It asserts that the trace operator (1) is supported on the set \(X_G\subset X\), and particularly if \(X_G\) is an empty set then the operator (1) is compact.
There are a lot of examples in the paper illustrating these conclusions.
Reviewer: Vladimir Vasilyev (Belgorod)A weighted Trudinger-Moser inequality on a closed Riemann surface with a finite isometric group actionhttps://www.zbmath.org/1483.580042022-05-16T20:40:13.078697Z"Yang, Jie"https://www.zbmath.org/authors/?q=ai:yang.jie.4|yang.jie.3|yang.jie.1|yang.jie.2Summary: Let \((\Sigma, g)\) be a closed Riemann surface, \(G\) be a finite isometric group acting on \((\Sigma, g)\) and \(H^{1, 2}(\Sigma)\) be the standard Sobolev space. Taking a positive smooth function \(f\) which is \(G\)-invariant, we define a function space \(\mathcal{H}_f^G\) by
\[
\mathcal{H}_f^G=\left\{ u\in H^{1,2}(\Sigma)\left| u(\sigma(x))=u(x), \int_\Sigma uf dv_g=0,\, \forall x\in \Sigma ,\, \forall \sigma \in G \right.\right\}.
\]
Using blow-up analysis, we prove that for any \(\alpha <\lambda_1^f\), the supremum
\[
\sup_{u\in\mathcal{H}_f^G, \int_\Sigma |\nabla_g u|^2fdv_g-\alpha \int_\Sigma u^2fdv_g\le 1}\int_\Sigma e^{4\pi \ell u^2f}dv_g
\]
is attained, where \(\lambda_1^f\) is the first eigenvalue of the \(f\)-Laplacian \(\Delta_f=-\operatorname{div}_g(f\nabla_g)\) on the space \(\mathcal{H}_f^G\), \(\ell =\min_{x\in \Sigma}\sharp G(x)\) and \(\sharp G(x)\) denotes the number of all distinct points of \(G(x)\). Moreover, we consider the case of higher order eigenvalues. Our results generalized those of \textit{Y. Yang} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 65, No. 3, 647--659 (2006; Zbl 1095.58005); J. Differ. Equations 258, No. 9, 3161--3193 (2015; Zbl 1339.46041)] and \textit{Y. Fang} and \textit{Y. Yang} [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 20, No. 4, 1295--1324 (2020; Zbl 1471.30005)].Fredholm conditions and index for restrictions of invariant pseudodifferential operators to isotypical componentshttps://www.zbmath.org/1483.580052022-05-16T20:40:13.078697Z"Baldare, Alexandre"https://www.zbmath.org/authors/?q=ai:baldare.alexandre"Côme, Rémi"https://www.zbmath.org/authors/?q=ai:come.remi"Lesch, Matthias"https://www.zbmath.org/authors/?q=ai:lesch.matthias"Nistor, Victor"https://www.zbmath.org/authors/?q=ai:nistor.victorSummary: Let \(\Gamma\) be a compact group acting on a smooth, compact manifold \(M\), let \(P\in\psi^m(M;E_0,E_1)\) be a \(\Gamma\)-invariant, classical pseudodifferential operator acting between sections of two equivariant vector bundles \(E_i\to M\), \(i=0,1\), and let \(\alpha\) be an irreducible representation of the group \(\Gamma\). Then \(P\) induces a map \(\pi_\alpha(P):H^s(M;E_0)_\alpha\to H^{s-m}(M;E_1)_\alpha\) between the \(\alpha\)-isotypical components of the corresponding Sobolev spaces of sections. When \(\Gamma\) is finite, we explicitly characterize the operators \(P\) for which the map \(\pi_\alpha(P)\) is Fredholm in terms of the principal symbol of \(P\) and the action of \(\Gamma\) on the vector bundles \(E_i\). When \(\Gamma=\{1\}\), that is, when there is no group, our result extends the classical characterization of Fredholm (pseudo)differential operators on compact manifolds. The proof is based on a careful study of the symbol \(C^*\)-algebra and of the topology of its primitive ideal spectrum. We also obtain several results on the structure of the norm closure of the algebra of invariant pseudodifferential operators and their relation to induced representations. As an illustration of the generality of our results, we provide some applications to Hodge theory and to index theory of singular quotient spaces.Eigenvalue asymptotics for weighted Laplace equations on rough Riemannian manifolds with boundaryhttps://www.zbmath.org/1483.580062022-05-16T20:40:13.078697Z"Bandara, Lashi"https://www.zbmath.org/authors/?q=ai:bandara.lashi"Nursultanov, Medet"https://www.zbmath.org/authors/?q=ai:nursultanov.medet"Rowlett, Julie"https://www.zbmath.org/authors/?q=ai:rowlett.julieSummary: Our topological setting is a smooth compact manifold of dimension two or higher with smooth boundary. Although this underlying topological structure is smooth, the Riemannian metric tensor is only assumed to be bounded and measurable. This is known as a \textit{rough Riemannian manifold}. For a large class of boundary conditions we demonstrate a Weyl law for the asymptotics of the
eigenvalues of the Laplacian associated to a rough metric. Moreover, we obtain eigenvalue asymptotics for weighted Laplace equations associated to a rough metric. Of particular novelty is that the weight function is not assumed to be of fixed sign, and thus the eigenvalues may be both positive and negative. Key ingredients in the proofs were demonstrated by Birman and Solomjak nearly fifty
years ago in their seminal work on eigenvalue asymptotics. In addition to determining the eigenvalue asymptotics in the rough Riemannian manifold setting for weighted Laplace equations, we also wish to promote their achievements which may have further applications to modern problems.Compact manifolds with fixed boundary and large Steklov eigenvalueshttps://www.zbmath.org/1483.580072022-05-16T20:40:13.078697Z"Colbois, Bruno"https://www.zbmath.org/authors/?q=ai:colbois.bruno"El Soufi, Ahmad"https://www.zbmath.org/authors/?q=ai:el-soufi.ahmad"Girouard, Alexandre"https://www.zbmath.org/authors/?q=ai:girouard.alexandreSummary: Let \((M,g)\) be a compact Riemannian manifold with boundary. Let \(b>0\) be the number of connected components of its boundary. For manifolds of dimension \(\geq 3\), we prove that for \(j=b+1\) it is possible to obtain an arbitrarily large Steklov eigenvalue \(\sigma_j(M,e^\delta g)\) using a conformal perturbation \(\delta \in C^\infty (M)\) which is supported in a thin neighbourhood of the boundary, with \(\delta =0\) on the boundary. For \(j\leq b\), it is also possible to obtain arbitrarily large eigenvalues, but the conformal factor must spread throughout the interior of \(M\). In fact, when working in a fixed conformal class and for \(\delta =0\) on the boundary, it is known that the volume of \((M,e^\delta g)\) has to tend to infinity in order for some \(\sigma _j\) to become arbitrarily large. This is in stark contrast with the situation for the eigenvalues of the Laplace operator on a closed manifold, where a conformal factor that is large enough for the volume to become unbounded results in the spectrum collapsing to 0. We also prove that it is possible to obtain large Steklov eigenvalues while keeping different boundary components arbitrarily close to each other, by constructing a convenient Riemannian submersion.Geometric hypoelliptic Laplacian and orbital integrals [after Bismut, Lebeau and Shen]https://www.zbmath.org/1483.580082022-05-16T20:40:13.078697Z"Ma, Xiaonan"https://www.zbmath.org/authors/?q=ai:ma.xiaonanSummary: Hodge theory for a hypoelliptic Laplacian acting on the total space of the cotangent bundle of a Riemannian manifold. This operator interpolates between the classical elliptic Laplacian on the base and the generator of the geodesic flow. We will describe recent developments of the theory of hypoelliptic Laplacians, in particular the explicit formula obtained by Bismut for orbital integrals and the recent solution by Shen of Fried's conjecture (dating back to 1986) for locally symmetric spaces. The conjecture predicts the equality of the analytic torsion and the value at 0 of the dynamic zeta function.
For the entire collection see [Zbl 1416.00029].Renormalization of stochastic continuity equations on Riemannian manifoldshttps://www.zbmath.org/1483.580092022-05-16T20:40:13.078697Z"Galimberti, Luca"https://www.zbmath.org/authors/?q=ai:galimberti.luca"Karlsen, Kenneth H."https://www.zbmath.org/authors/?q=ai:karlsen.kenneth-hvistendahlThe authors study initial value problems for stochastic continuity equations on smooth closed Riemannian manifolds \(M\) with metric \(h\), of the form \[ \partial_{t}\rho + \operatorname{div}_{h}\left[ \rho \left( u(t,x) + \sum_{i=1}^{N} a_{i}(x) \circ \frac{d W^{i}}{dt} \right) \right]=0, \tag{1} \] for Sobolev velocity fields \(u\), perturbed by Gaussian noise terms driven by that independent Wiener processes \(W^{i}\), where \(a_{i}\) are smooth spatially dependent vector fields on M (with the stochastic integrals interpreted in the Stratonovich sense), supplemented with initial data \(\rho(0)=\rho_{0} \in L^2(M)\).
This type of equation is very interesting both from the mathematical point of view as well as from the point of view of applications (e.g. in fluid mechanics). The deterministic case (\(a_{i}=0\)) has been studied and existence of weak solution was shown using the DiPerna-Lions theory of renormalized solutions [\textit{R. J. DiPerna} and \textit{P. L. Lions}, Invent. Math. 98, No. 3, 511--547 (1989; Zbl 0696.34049)], both in the Euclidean and the smooth closed manifold case and there are important extensions by \textit{L. Ambrosio} [ibid. 158, No. 2, 227--260 (2004; Zbl 1075.35087)] in the case of BV velocity fields). We recall that a renormalized solution \(\rho\) is a weak solution such that \(S(\rho)\) is also a weak solution for any ``reasonable'' \(S : {\mathbb R} \to {\mathbb R}\). The stochastic case for Lipschitz type coefficients has been studied by \textit{H. Kunita} [Stochastic flows and stochastic differential equations. Cambridge etc.: Cambridge University Press (1990; Zbl 0743.60052)] in the Euclidean case, whereas results by \textit{S. Attanasio} and \textit{F. Flandoli} [Commun. Partial Differ. Equations 36, No. 7--9, 1455--1474 (2011; Zbl 1237.60048)] establish the renormalization property for BV velocity field and constant \(a_{i}\), revealing an interesting regularization by noise property of the equation (in the sense that the renormalization property implies uniqueness without the usual \(L^{\infty}\) assumption on the divergence of \(u\)) which has become a recurrent theme in the study of stochastic transport or continuity equations. Extensions in the case of stochastic continuity equations in Itō form in the Euclidean domain with spatially dependent noise coefficients were obtained in [\textit{S. Punshon-Smith} and \textit{S. Smith}, Arch. Ration. Mech. Anal. 229, No. 2, 627--708 (2018; Zbl 1394.35313)] and [\textit{J. A. Rossmanith} et al., J. Comput. Phys. 199, No. 2, 631--662 (2004; Zbl 1126.76350)].
In this paper the authors study problem (1) in the generalized setting already mentioned above, and their main result is the renormalization property for weak \(L^2\) solutions of (1). As a corollary they deduce the uniqueness of weak solutions and an a priori estimate under the usual condition that \(\operatorname{div}_{h} u \in L_{t}^{1}L^{\infty}\). The proof consists of a systematic procedure for regularizing tensor fields on a manifold, a convenient choice of atlas to simplify technical computations linked to the Christoffel symbols, and several DiPerna-Lions type commutators between (first/second order) geometric differential operators and the regularization device.
Reviewer: Athanasios Yannacopoulos (Athína)Preserving invariance properties of reaction-diffusion systems on stationary surfaceshttps://www.zbmath.org/1483.651552022-05-16T20:40:13.078697Z"Frittelli, Massimo"https://www.zbmath.org/authors/?q=ai:frittelli.massimo"Madzvamuse, Anotida"https://www.zbmath.org/authors/?q=ai:madzvamuse.anotida"Sgura, Ivonne"https://www.zbmath.org/authors/?q=ai:sgura.ivonne"Venkataraman, Chandrasekhar"https://www.zbmath.org/authors/?q=ai:venkataraman.chandrasekharSummary: We propose and analyse a lumped surface finite element method for the numerical approximation of reaction-diffusion systems on stationary compact surfaces in \(\mathbb{R}^3\). The proposed method preserves the invariant regions of the continuous problem under discretization and, in the special case of scalar equations, it preserves the maximum principle. On the application of a fully discrete scheme using the implicit-explicit Euler method in time, we prove that invariant regions of the continuous problem are preserved (i) at the spatially discrete level with no restriction on the meshsize and (ii) at the fully discrete level under a timestep restriction. We further prove optimal error bounds for the semidiscrete and fully discrete methods, that is, the convergence rates are quadratic in the meshsize and linear in the timestep. Numerical experiments are provided to support the theoretical findings. We provide examples in which, in the absence of lumping, the numerical solution violates the invariant region leading to blow-up.Schrödinger dynamics and optimal transport of measureshttps://www.zbmath.org/1483.810712022-05-16T20:40:13.078697Z"Zanelli, Lorenzo"https://www.zbmath.org/authors/?q=ai:zanelli.lorenzoOne parameter family of rationally extended isospectral potentialshttps://www.zbmath.org/1483.810812022-05-16T20:40:13.078697Z"Yadav, Rajesh Kumar"https://www.zbmath.org/authors/?q=ai:yadav.rajesh-kumar"Banerjee, Suman"https://www.zbmath.org/authors/?q=ai:banerjee.suman"Kumari, Nisha"https://www.zbmath.org/authors/?q=ai:kumari.nisha"Khare, Avinash"https://www.zbmath.org/authors/?q=ai:khare.avinash"Mandal, Bhabani Prasad"https://www.zbmath.org/authors/?q=ai:mandal.bhabani-prasadSummary: We start from a given one dimensional rationally extended shape invariant potential associated with \(X_m\) exceptional orthogonal polynomials and using the idea of supersymmetry in quantum mechanics, we obtain one continuous parameter \((\lambda)\) family of rationally extended strictly isospectral potentials. We illustrate this construction by considering three well known rationally extended potentials, two with pure discrete spectrum (the extended radial oscillator and the extended Scarf-I) and one with both the discrete and the continuous spectrum (the extended generalized Pöschl-Teller) and explicitly construct the corresponding one continuous parameter family of rationally extended strictly isospectral potentials. Further, in the special case of \(\lambda=0\) and \(-1\), we obtain two new exactly solvable rationally extended potentials, namely the rationally extended Pursey and the rationally extended Abraham-Moses potentials respectively. We illustrate the whole procedure by discussing in detail the particular case of the \(X_1\) rationally extended one parameter family of potentials including the corresponding Pursey and the Abraham Moses potentials.Batalin-Vilkovisky quantization of fuzzy field theorieshttps://www.zbmath.org/1483.811062022-05-16T20:40:13.078697Z"Nguyen, Hans"https://www.zbmath.org/authors/?q=ai:nguyen.hans"Schenkel, Alexander"https://www.zbmath.org/authors/?q=ai:schenkel.alexander"Szabo, Richard J."https://www.zbmath.org/authors/?q=ai:szabo.richard-jSummary: We apply the modern Batalin-Vilkovisky quantization techniques of Costello and Gwilliam to noncommutative field theories in the finite-dimensional case of fuzzy spaces. We further develop a generalization of this framework to theories that are equivariant under a triangular Hopf algebra symmetry, which in particular leads to quantizations of finite-dimensional analogues of the field theories proposed recently through the notion of `braided \(L_{\infty}\)-algebras'. The techniques are illustrated by computing perturbative correlation functions for scalar and Chern-Simons theories on the fuzzy 2-sphere, as well as for braided scalar field theories on the fuzzy 2-torus.Chern-Simons perturbative series revisitedhttps://www.zbmath.org/1483.811082022-05-16T20:40:13.078697Z"Lanina, E."https://www.zbmath.org/authors/?q=ai:lanina.elena|lanina.e-g"Sleptsov, A."https://www.zbmath.org/authors/?q=ai:sleptsov.alexey"Tselousov, N."https://www.zbmath.org/authors/?q=ai:tselousov.nSummary: A group-theoretical structure in a perturbative expansion of the Wilson loops in the 3d Chern-Simons theory with \(SU(N)\) gauge group is studied in symmetric approach. A special basis in the center of the universal enveloping algebra \(ZU(\mathfrak{sl}_N)\) is introduced. This basis allows one to present group factors in an arbitrary irreducible finite-dimensional representation. Developed methods have wide applications, the most straightforward and evident ones are mentioned. Namely, Vassiliev invariants of higher orders are computed, a conjecture about existence of new symmetries of the colored HOMFLY polynomials is stated, and the recently discovered tug-the-hook symmetry of the colored HOMFLY polynomial is proved.Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structurehttps://www.zbmath.org/1483.811092022-05-16T20:40:13.078697Z"Lanina, E."https://www.zbmath.org/authors/?q=ai:lanina.e-g|lanina.elena"Sleptsov, A."https://www.zbmath.org/authors/?q=ai:sleptsov.alexey"Tselousov, N."https://www.zbmath.org/authors/?q=ai:tselousov.nSummary: We have recently proposed [Phys. Lett., B 823, Article ID 136727, 8 p. (2021; Zbl 1483.81108)] a powerful method for computing group factors of the perturbative series expansion of the Wilson loop in the Chern-Simons theory with \(SU(N)\) gauge group. In this paper, we apply the developed method to obtain and study various properties, including nonperturbative ones, of such vacuum expectation values.
First, we discuss the computation of Vassiliev invariants. Second, we discuss the Vogel theorem of not distinguishing chord diagrams by weight systems coming from semisimple Lie (super)algebras. Third, we provide a method for constructing linear recursive relations for the colored Jones polynomials considering a special case of torus knots \(T[2, 2 k + 1]\). Fourth, we give a generalization of the one-hook scaling property for the colored Alexander polynomials. And finally, for the group factors we provide a combinatorial description, which has a clear dependence on the rank \(N\) and the representation \(R\).Three-dimensional Maxwellian Carroll gravity theory and the cosmological constanthttps://www.zbmath.org/1483.830112022-05-16T20:40:13.078697Z"Concha, Patrick"https://www.zbmath.org/authors/?q=ai:concha.patrick"Peñafiel, Diego"https://www.zbmath.org/authors/?q=ai:penafiel.diego-m"Ravera, Lucrezia"https://www.zbmath.org/authors/?q=ai:ravera.lucrezia"Rodríguez, Evelyn"https://www.zbmath.org/authors/?q=ai:rodriguez.evelynSummary: In this work, we present the three-dimensional Maxwell Carroll gravity by considering the ultra-relativistic limit of the Maxwell Chern-Simons gravity theory defined in three spacetime dimensions. We show that an extension of the Maxwellian Carroll symmetry is necessary in order for the invariant tensor of the ultra-relativistic Maxwellian algebra to be non-degenerate. Consequently, we discuss the origin of the aforementioned algebra and theory as a flat limit. We show that the theoretical setup with cosmological constant yielding the extended Maxwellian Carroll Chern-Simons gravity in the vanishing cosmological constant limit is based on a new enlarged extended version of the Carroll symmetry. Indeed, the latter exhibits a non-degenerate invariant tensor allowing the proper construction of a Chern-Simons gravity theory which reproduces the extended Maxwellian Carroll gravity in the flat limit.Entropy of Reissner-Nordström-like black holeshttps://www.zbmath.org/1483.830362022-05-16T20:40:13.078697Z"Blagojević, M."https://www.zbmath.org/authors/?q=ai:blagojevic.milutin"Cvetković, B."https://www.zbmath.org/authors/?q=ai:cvetkovic.branislavSummary: In Poincaré gauge theory, black hole entropy is defined canonically by the variation of a boundary term \(\Gamma_H\), located at horizon. For a class of static and spherically symmetric black holes in vacuum, the explicit formula reads \(\delta \Gamma_H = T \delta S\), where \(T\) is black hole temperature and \(S\) entropy. Here, we analyze a new member of the same class, the Reissner-Nordström-like black hole with torsion [\textit{ J. A.R. Cembranos} and [\textit{J. G. Valcarcel}, ``New torsion black hole solutions in Poincaré gauge theory,'' J. Cosmol. Astropart. Phys., 01, Article 014 (2017; \url{doi:10.1088/1475-7516/2017/01/014})], where the electric charge of matter is replaced by a gravitational parameter, induced by the existence of torsion. This parameter affects \(\delta \Gamma_H\) in a way that ensures the validity of the first law.An alternative to the Teukolsky equationhttps://www.zbmath.org/1483.830472022-05-16T20:40:13.078697Z"Hatsuda, Yasuyuki"https://www.zbmath.org/authors/?q=ai:hatsuda.yasuyukiSummary: We conjecture a new ordinary differential equation exactly isospectral to the radial component of the homogeneous Teukolsky equation. We find this novel relation by a hidden symmetry implied from a four-dimensional \(\mathcal{N}=2\) supersymmetric quantum chromodynamics. Our proposal is powerful both in analytical and in numerical studies. As an application, we derive high-order perturbative series of quasinormal mode frequencies in the slowly rotating limit. We also test our result numerically by comparing it with a known technique.Holographic complexity in charged accelerating black holeshttps://www.zbmath.org/1483.830492022-05-16T20:40:13.078697Z"Jiang, Shun"https://www.zbmath.org/authors/?q=ai:jiang.shun"Jiang, Jie"https://www.zbmath.org/authors/?q=ai:jiang.jie.2Summary: Using the ``complexity equals action'' (CA) conjecture, for an ordinary charged system, it has been shown that the late-time complexity growth rate is given by a difference between the value of \(\Phi_H Q + \Omega_H J\) on the inner and outer horizons. In this paper, we investigate the complexity of the boundary quantum system with conical deficits. From the perspective of holography, we consider charged accelerating black holes which contain conical deficits on the north and south poles in the bulk gravitational theory and evaluate the complexity growth rate using the CA conjecture. As a result, the late-time growth rate of complexity is different from the ordinary charged black holes. It implies that complexity can carry the information of conical deficits on the boundary quantum system.Geometrothermodynamics of black holes with a nonlinear sourcehttps://www.zbmath.org/1483.830632022-05-16T20:40:13.078697Z"Sánchez, Alberto"https://www.zbmath.org/authors/?q=ai:rivadulla-sanchez.albertoSummary: We study thermodynamics and geometrothermodynamics of a particular black hole configuration with a nonlinear source. We use the mass as fundamental equation, from which it follows that the curvature radius must be considered as a thermodynamic variable, leading to an extended equilibrium space. Using the formalism of geometrothermodynamics, we show that the geometric properties of the thermodynamic equilibrium space can be used to obtain information about thermodynamic interaction, critical points and phase transitions. We show that these results are compatible with the results obtained from classical black hole thermodynamics.Holographic superconductors in 4D Einstein-Gauss-Bonnet gravity with backreactionshttps://www.zbmath.org/1483.830722022-05-16T20:40:13.078697Z"Pan, Jie"https://www.zbmath.org/authors/?q=ai:pan.jie"Qiao, Xiongying"https://www.zbmath.org/authors/?q=ai:qiao.xiongying"Wang, Dong"https://www.zbmath.org/authors/?q=ai:wang.dong.7|wang.dong|wang.dong.3|wang.dong.8|wang.dong.2|wang.dong.4|wang.dong.1|wang.dong.6|wang.dong.5"Pan, Qiyuan"https://www.zbmath.org/authors/?q=ai:pan.qiyuan"Nie, Zhang-Yu"https://www.zbmath.org/authors/?q=ai:nie.zhang-yu"Jing, Jiliang"https://www.zbmath.org/authors/?q=ai:jing.jiliangSummary: We construct the holographic superconductors away from the probe limit in the consistent \(D \to 4\) Einstein-Gauss-Bonnet gravity. We observe that, both for the ground state and excited states, the critical temperature first decreases then increases as the curvature correction tends towards the Chern-Simons limit in a backreaction dependent fashion. However, the decrease of the backreaction, the increase of the scalar mass, or the increase of the number of nodes will weaken this subtle effect of the curvature correction. Moreover, for the curvature correction approaching the Chern-Simons limit, we find that the gap frequency \(\omega_g/T_c\) of the conductivity decreases first and then increases when the backreaction increases in a scalar mass dependent fashion, which is different from the finding in the \((3 + 1)\)-dimensional superconductors that increasing backreaction increases \(\omega_g/T_c\) in the full parameter space. The combination of the Gauss-Bonnet gravity and backreaction provides richer physics in the scalar condensates and conductivity in the \((2 + 1)\)-dimensional superconductors.Non-singular non-flat universeshttps://www.zbmath.org/1483.830942022-05-16T20:40:13.078697Z"Salamanca, Andrés Felipe Estupiñán"https://www.zbmath.org/authors/?q=ai:salamanca.andres-felipe-estupinan"Medina, Sergio Bravo"https://www.zbmath.org/authors/?q=ai:bravo-medina.sergio"Nowakowski, Marek"https://www.zbmath.org/authors/?q=ai:nowakowski.marek"Batic, Davide"https://www.zbmath.org/authors/?q=ai:batic.davideSummary: The quest to understand better the nature of the initial cosmological singularity is with us since the discovery of the expanding universe. Here, we propose several non-flat models, among them the standard cosmological scenario with a critical cosmological constant, the Einstein-Cartan cosmology, the Milne-McCrea universe with quantum corrections and a non-flat universe with bulk viscosity. Within these models, we probe into the initial singularity by using different techniques. Several nonsingular universes emerge, one of the possibilities being a static non-expanding and stable Einstein universe.Inflation with Gauss-Bonnet and Chern-Simons higher-curvature-corrections in the view of GW170817https://www.zbmath.org/1483.830972022-05-16T20:40:13.078697Z"Venikoudis, S. A."https://www.zbmath.org/authors/?q=ai:venikoudis.s-a"Fronimos, F. P."https://www.zbmath.org/authors/?q=ai:fronimos.f-pSummary: Inflationary era of our Universe can be characterized as semi-classical because it can be described in the context of four-dimensional Einstein's gravity involving quantum corrections. These string motivated corrections originate from quantum theories of gravity such as superstring theories and include higher gravitational terms as, Gauss-Bonnet and Chern-Simons terms. In this paper we investigated inflationary phenomenology coming from a scalar field, with quadratic curvature terms in the view of GW170817. Firstly, we derived the equations of motion, directly from the gravitational action. As a result, formed a system of
differential equations with respect to Hubble's parameter and the inflaton field which was very complicated and cannot be solved analytically, even in the minimal coupling case. Based on the observations from GW170817 event, which have shown that the speed of the primordial gravitational wave is equal to the speed of light, \(c_{\mathcal{T}}^2=1\) in natural units, our equations of motion where simplified after applying the constraint \(c_{\mathcal{T}}^2=1\), the slow-roll approximations and neglecting the string corrections. We described the dynamics of inflationary phenomenology and proved that theories with Gauss-Bonnet term can be compatible with recent observations. Also, the Chern-Simons term leads to asymmetric generation and evolution of the two circular polarization states of gravitational wave. Finally, viable inflationary models are presented, consistent with the observational constraints. The possibility of a blue tilted tensor spectral index is briefly investigated.