Flat Affine Research Articles

On the running and the UV limit of Wilsonian renormalization group flows

In nonperturbative formulation of quantum field theory, the vacuum state is characterized by the Wilsonian renormalization group (RG) flow of Feynman type field correlators. Such a flow is a parametric family of ultraviolet (UV) regularized field correlators, the parameter being the strength of the UV regularization, and the instances with different strength of UV regularizations are linked by the renormalization group equation. Important RG flows are those which reach out to any UV regularization strengths. In this paper it is shown that for these flows a natural, mathematically rigorous generally covariant definition can be given, and that they form a topological vector space which is Hausdorff, locally convex, complete, nuclear, semi-Montel, Schwartz. That is, they form a generalized function space having favorable properties, similar to multivariate distributions. The other theorem proved in the paper is that for Wilsonian RG flows reaching out to all UV regularization strengths, a simple factorization formula holds in case of bosonic fields over flat (affine) spacetime: the flow always originates from a regularization-independent distributional correlator, and its running satisfies an algebraic ansatz. The conjecture is that this factorization theorem should generically hold, which is worth future investigations.

Classification of the conformally flat Tchebychev affine Kähler hypersurfaces

Tchebychev affine Kähler hypersurfaces were investigated firstly by Xu and Li, who classified the case of dimension 2. As for the case of higher dimension, Xu-Li also proved a classification for the Tchebychev affine Kähler hypersurfaces with complete Calabi metric and nonnegative Ricci curvature. In this paper, the classification of Tchebychev affine Kähler hypersurfaces is obtained under the assumption that Weyl curvature tensor vanishes. As corollaries, we completely determine 3-dimensional Tchebychev affine Kähler hypersurfaces in local sense and obtain the classification of Calabi complete Tchebychev affine Kähler hypersurfaces in R4, which partially improves the results of Xu-Li. In addition, a Bernstein-type result is got for the entire Tchebychev affine Kähler hypersurface in R4.

Conelike radiant structures

Analogues of the classical affine-projective correspondence are developed in the context of statistical manifolds compatible with a radiant vector field. These utilize a formulation of Einstein equations for special statistical structures that generalizes the usual Einstein equations for pseudo-Riemannian metrics and is of independent interest. A conelike radiant structure is a not necessarily flat affine connection equipped with a family of surfaces that behave like the intersections of the planes through the origin with a convex cone in a real vector space. A radiant structure is a torsion-free affine connection and a vector field whose covariant derivative is the identity endomorphism. A radiant structure is conelike if for every point and every two-dimensional subspace containing the radiant vector field there is a totally geodesic surface passing through the point and tangent to the subspace. Such structures exist on the total space of any principal bundle with one-dimensional fiber and on any Lie group with a quadratic structure on its Lie algebra. The affine connection of a conelike radiant structure can be normalized in a canonical way to have antisymmetric Ricci tensor. Applied to a conelike radiant structure on the total space of a principal bundle with one-dimensional fiber this yields a generalization of the classical Thomas connection of a projective structure. The compatibility of radiant and conelike structures with metrics is investigated and yields a construction of connections for which the symmetrized Ricci curvature is a constant multiple of a compatible metric that generalizes well-known constructions of Riemannian and Lorentzian Einstein–Weyl structures over Kähler–Einstein manifolds having nonzero scalar curvature. A formulation of Einstein equations for special statistical manifolds is given that generalizes the Einstein–Weyl equations and encompasses these more general examples. There are constructed left-invariant conelike radiant structures on a Lie group endowed with a left-invariant nondegenerate bilinear form, and the case of three-dimensional unimodular Lie groups is described in detail.

General teleparallel metrical geometries

In the conventional formulation of general relativity, gravity is represented by the metric curvature of Riemannian geometry. There are also alternative formulations in flat affine geometries, wherein the gravitational dynamics is instead described by torsion and nonmetricity. These so-called general teleparallel geometries may also have applications in material physics, such as the study of crystal defects. In this work, we explore the general teleparallel geometry in the language of differential forms. We discuss the special cases of metric and symmetric teleparallelisms, clarify the relations between formulations with different gauge fixings and without gauge fixing, and develop a method of recasting Riemannian into teleparallel geometries. As illustrations of the method, exact solutions are presented for the generic quadratic theory in 2, 3 and 4 dimensions.

Locally conformally flat affine hyperspheres with parallel Ricci tensor

Trying to extend the result of Vrancken et al. (1991) [20] which established the complete classification of locally strongly convex affine hyperspheres with constant sectional curvature, we consider the classification problem of n-dimensional locally strongly convex affine hyperspheres with locally conformally flat affine metrics. As the main result of this paper, we solved the problem under the additional condition that the affine metric has parallel Ricci tensor with respect to its Levi-Civita connection.

Gauge and Poincaré properties of the UV cutoff and UV completion in quantum field theory

The ultraviolet (UV) cutoff on a quantum field theory (QFT) can explicitly break or conserve the Poincare (translation) symmetry. And the very same cutoff can explicitly break or conserve the gauge symmetry. In the present work, we perform a systematic study of the UV cutoff in regard to its gauge and Poincare properties, and construct UV completions restoring the broken gauge symmetry. In the case of Poincare-conserving UV cutoff, we find that the gauge symmetry gets restored via the Higgs mechanism. In the case of Poincare-breaking UV cutoff, however, we find that the flat spacetime affine curvature takes the place of the Higgs field and, when taken to curved spacetime, gauge symmetry gets restored at the extremum of the metric-affine action. We also find that gravity emerges at the extremum if the QFT under concern consists of new particles beyond the known ones. The resulting emergent gravity plus renormalized QFT setup has the potential to reveal itself in various astrophysical, cosmological and collider phenomena.

On curves in K-theory and TR

We prove that TR is corepresentable by the reduced topological Hochschild homology of the flat affine line \mathbf{S}[t] as a functor defined on the \infty -category of cyclotomic spectra with values in the \infty -category of spectra with Frobenius lifts, refining a result of Blumberg–Mandell. We define the notion of an integral topological Cartier module using Barwick’s formalism of spectral Mackey functors on orbital \infty -categories, extending the work of Antieau–Nikolaus in the p -typical setting. As an application, we show that TR evaluated on a connective \mathbf{E}_1 -ring admits a description in terms of the spectrum of curves on algebraic K-theory, generalizing the work of Hesselholt and Betley–Schlichtkrull.

On conformally flat affine hypersurfaces with pseudo-parallel cubic form

In this paper, we study locally strongly convex affine hypersurfaces with vanishing Weyl curvature tensor and semi-parallel cubic form relative to the Levi-Civita connection of affine metric. As the main result, we classify such hypersurfaces being not of flat affine metric. In particular, $2, 3$-dimensional locally strongly convex affine hypersurfaces with semi-parallel cubic form are completely determined.

Motivic cohomology and infinitesimal group schemes

For $k$ a perfect field of characteristic $p>0$ and $G/k$ a split reductive group with $p$ a non-torsion prime for $G,$ we compute the mod $p$ motivic cohomology of the geometric classifying space $BG_{(r)}$, where $G_{(r)}$ is the $r$th Frobenius kernel of $G.$ Our main tool is a motivic version of the Eilenberg-Moore spectral sequence, due to Krishna. For a flat affine group scheme $G/k$ of finite type, we define a cycle class map from the mod $p$ motivic cohomology of the classifying space $BG$ to the mod $p$ \'etale motivic cohomology of the classifying stack $\mathcal{B}G.$ This also gives a cycle class map into the Hodge cohomology of $\mathcal{B}G.$ We study the cycle class map for some examples, including Frobenius kernels.

λ-Deformation: A Canonical Framework for Statistical Manifolds of Constant Curvature.

This paper systematically presents the -deformation as the canonical framework of deformation to the dually flat (Hessian) geometry, which has been well established in information geometry. We show that, based on deforming the Legendre duality, all objects in the Hessian case have their correspondence in the -deformed case: -convexity, -conjugation, -biorthogonality, -logarithmic divergence, -exponential and -mixture families, etc. In particular, -deformation unifies Tsallis and Rényi deformations by relating them to two manifestations of an identical -exponential family, under subtractive or divisive probability normalization, respectively. Unlike the different Hessian geometries of the exponential and mixture families, the -exponential family, in turn, coincides with the -mixture family after a change of random variables. The resulting statistical manifolds, while still carrying a dualistic structure, replace the Hessian metric and a pair of dually flat conjugate affine connections with a conformal Hessian metric and a pair of projectively flat connections carrying constant (nonzero) curvature. Thus, -deformation is a canonical framework in generalizing the well-known dually flat Hessian structure of information geometry.

Black holes in f(Q) gravity

We systematically study the field equations of $f(\mathbb Q)$ gravity for spherically symmetric and stationary metric-affine spacetimes. Such spacetimes are described by a metric as well as a flat and torsionless affine connection. In the Symmetric Teleparallel Equivalent of GR (STEGR), the connection is pure gauge and hence unphysical. However, in the non-linear extension $f(\Q)$, it is promoted to a dynamical field which changes the physics. Starting from a general metric-affine geometry, we construct the most general static and spherically symmetric forms of the metric and the affine connection. We then use these symmetry reduced geometric objects to prove that the field equations of $f(\Q)$ gravity admit GR solutions as well as beyond-GR solutions, contrary to what has been claimed in the literature. We formulate precise criteria, under which conditions it is possible to obtain GR solutions and under which conditions it is possible to obtain beyond-GR solutions. We subsequently construct several perturbative corrections to the Schwarzschild solution for different choices of $f(\Q)$, which in particular include a hair stemming from the now dynamical affine connection. We also present an exact beyond-GR vacuum solution. Lastly, we apply this method of constructing spherically symmetric and stationary solutions to $f(\T)$ gravity, which reproduces similar solutions but without a dynamical connection.

The geometry of filtrations

We display a symmetric monoidal equivalence between the stable ∞-category of filtered spectra, and quasi-coherent sheaves on A 1 / G m , the quotient in the setting of spectral algebraic geometry of the flat affine line by the canonical action of the flat multiplicative group scheme. Via a Tannaka duality argument, we identify the underlying spectrum and associated graded functors with pull-backs of quasi-coherent sheaves along certain morphisms of stacks.

Symmetric products, linear representations and trace identities

We give the equations of the n-th symmetric product X^n/S_n of a flat affine scheme X=mathrm {Spec},A over a commutative ring F. As a consequence, we find a closed immersion into the coarse moduli space parameterizing n-dimensional linear representations of A. This is done by exhibiting an isomorphism between the ring of symmetric tensors over A and the ring generated by the coefficients of the characteristic polynomial of polynomials in commuting generic matrices giving representations of A. Using this we derive an isomorphism of the associated reduced schemes over an infinite field. When the characteristic is zero we show that this isomorphism is an isomorphism of schemes and we express it in term of traces.

Transformation groups of certain flat affine manifolds

In this paper we characterize the group of affine transformations of a flat affine simply connected manifold whose developing map is a diffeomorphism. This is proved by making use of some simple facts about homeomorphisms of $\mathbb{R}^n$ preserving open connected sets. We show some examples where the characterization is useful.

Algebraic approach to Rump’s results on relations between braces and pre-lie algebras

In 2014, Wolfgang Rump showed that there exists a correspondence between left nilpotent right [Formula: see text]-braces and pre-Lie algebras. This correspondence, established using a geometric approach related to flat affine manifolds and affine torsors, works locally. In this paper, we explain Rump’s correspondence using only algebraic formulae. An algebraic interpretation of the correspondence works for fields of sufficiently large prime characteristic as well as for fields of characteristic zero.

Flat affine manifolds and their transformations

We give a characterization of flat affine connections on manifolds by means of a natural affine representation of the universal covering of the Lie group of diffeomorphisms preserving the connection. From the infinitesimal point of view, this representation is determined by the 1-connection form and the fundamental form of the bundle of linear frames of the manifold. We show that the group of affine transformations of a real flat affine $n$-dimensional manifold, acts on $\mathbb{R}^n$ leaving an open orbit when its dimension is greater than $n$. Moreover, when the dimension of the group of affine transformations is $n$, this orbit has discrete isotropy. For any given Lie subgroup $H$ of affine transformations of the manifold, we show the existence of an associative envelope of the Lie algebra of $H$, relative to the connection. The case when $M$ is a Lie group and $H$ acts on $G$ by left translations is particularly interesting. We also exhibit some results about flat affine manifolds whose group of affine transformations admits a flat affine bi-invariant structure. The paper is illustrated with several examples.

On widetilde{J}-tangent Affine Hyperspheres

In this paper we study widetilde{J}-tangent affine hyperspheres, where widetilde{J} is the canonical para-complex structure on mathbb {R}^{2n+2}. The main purpose of this paper is to give a classification of widetilde{J}-tangent affine hyperspheres of an arbitrary dimension with an involutive distribution mathcal {D}. In particular, we classify all such hyperspheres in the 3-dimensional case. We also show that there is a direct relation between widetilde{J}-tangent affine hyperspheres and Calabi products. As an application we obtain certain classification results. In particular, we show that, with one exception, all odd dimensional proper flat affine hyperspheres are, after a suitable affine transformation, widetilde{J}-tangent. Some examples of widetilde{J}-tangent affine hyperspheres are also given.

On singularities of flat affine systems with n states and n−1 controls

SummaryWe study the set of intrinsic singularities of flat affine systems with n−1 controls and n states using the notion of Lie‐Bäcklund atlas, previously introduced by the authors. For this purpose, we prove two easily computable sufficient conditions to construct flat outputs as a set of independent first integrals of distributions of vector fields: the first one in a generic case, namely, in a neighborhood of a point where the n−1 control vector fields are independent and the second one at a degenerate point where p−1 control vector fields are dependent of the n−p others, with p>1. After introducing the Γ‐accessibility rank condition, we show that the set of intrinsic singularities includes the set of points where the system does not satisfy this rank condition and is included in the set where a distribution of vector fields introduced in the generic case is singular. We conclude this analysis by three examples of apparent singularities of flat systems in generic and nongeneric degenerate cases.

A study of Rényi entropy based on the information geometry formalism

In the framework of information geometry, we explore the structure of statistical manifold related to the Rényi entropy. Starting from a new family of generalized exponential distributions given by the equilibrium distribution of a canonical system described by the Rényi entropy, we introduce on a dually-flat geometry endowed by a Hessian structure derived from a generalization of the Fisher metric and a flat affine connection. This geometrical structure admits a Legendre transform that links the thermodynamic potentials of the underlying statistical mechanics to the dual potentials of the corresponding information geometry. A canonical divergence function à la Bregman is naturally derived in this framework. Further geometric structures, derived from other contrast functions strictly related to the Rényi entropy are explored and their link with the -geometry of Amari is highlighted.

Constrained Trajectory Planning for Second-Order Chained Form Systems Using Time Polynomials

This paper deals with time polynomial based trajectory planning for differentially flat affine dynamical systems that can be written as a chain of second-order controlled subsystems. An analytical approach is proposed to account for state and input constraints by adjusting the standard third-order time polynomial based considerations. For a point-to-point motion planning problem the constraints are met by properly selecting the time of motion value or/and initial or final values of some of the state variables. As an illustrative example trajectory planning for a 3-DoF Delta pick and place robot is considered.