arXiv subjects
Publications and source records attributed to Elena Mirela Babalic.
We analyze the relation between connection matrices and Berry phases on the Siegel-Jacobi disk $\mathcal{D}^J_1$ and Siegel-Jacobi upper half-plane $\mathcal{X}^J_1$ and we explicitly compute the connection matrices and the Berry connections on these homogeneous spaces in different coordinates.
Using the energy function on the extended Siegel-Jacobi upper half space of order $n$, $\tilde{\mathcal{X}}^J_n$, with $n\in \mathbb{N}$, the equations of motion in the variables $(x,y,q,p,κ)$ attached to linear Hamiltonians in the generators of the real Jacobi group $G^J_n(\mathbb{R})$ are presented, where $x,y$ are symmetric matrices in $\mathcal{M}(n,\mathbb{R})$ and $p,q$ are real $n$-vectors. The case $n=1$ is presented separately.
The inverse of the metric matrices on the Siegel-Jacobi upper half space ${\mathcal{X}}^J_n$, invariant to the restricted real Jacobi group $G^J_n(\mathbb{R})_0$ and extended Siegel-Jacobi $\tilde{\mathcal{X}}^J_n$ upper half space, invariant to the action of the real Jacobi $G^J_n(\mathbb{R})$, are presented. The results are relevant for Berezin quantization of the manifolds ${\mathcal{X}}^J_ n$ and $\tilde{\mathcal{X}}^J_n$. Explicit calculations in the case $n=2$ are given.
We discuss the behavior of cosmological curves and their first order infrared approximants near critical ends of the scalar manifold $Σ$ and near interior critical points of the scalar potential for tame hyperbolizable two-field cosmological models by determining the universal forms of the asymptotic gradient flow of the classical effective potential with respect to the uniformizing metric near all these points and ends. We compare the asymptotic behavior of gradient flow curves with numerical results for cosmological curves.
We study the first order infared behavior of tame hyperbolizable two-field cosmological models, defined as those classical two-field models whose scalar manifold is a connected, oriented and topologically finite hyperbolizable Riemann surface $(Σ,\mathcal{G})$ and whose scalar potential $Φ$ admits a positive and Morse extension to the end compactification of $Σ$. We achieve this by determining the universal forms of the asymptotic gradient flow of the classical effective potential $V$ with respect to the uniformizing metric $G$ near all interior critical points and ends of $Σ$, finding that some of the latter act like fictitious but exotic stationary points of the gradient flow. We also compare these results with numerical studies of cosmological orbits. For critical cusp ends, we find that cosmological curves have transient quasiperiodic behavior but are eventually attracted or repelled by the cusp along principal geodesic orbits determined by the extended effective potential. This behavior is approximated in the infrared by that of gradient flow curves near the cusp.
The real Jacobi group $G^J_1(\mathbb{R})={\rm SL}(2,\mathbb{R})\ltimes {\rm H}_1$, where ${\rm H}_1$ denotes the 3-dimensional Heisenberg group, is parametrized by the $S$-coordinates $(x,y,θ,p,q,κ)$. We show that the parameter $η$ that appears passing from Perelomov's un-normalized coherent state vector based on the Siegel--Jacobi disk $\mathcal{D}^J_1$ to the normalized one is $η=q+\rm{i} p$. The two-parameter invariant metric on the Siegel--Jacobi upper half-plane $\mathcal{X}^J_1=\frac{G^J_1(\R)}{\rm{SO}(2)\times\mathbb{R}}$ is expressed in the variables $(x,y,\rm{Re}~η,\rm{Im}~η)$. It is proved that the five dimensional manifold $\tilde{\mathcal{X}}^J_1=\frac{G^J_1(\R)}{\rm{SO}(2)}\approx\mathcal{X}^J_1\times\mathbb{R}$, called extended Siegel--Jacobi upper half-plane, is a reductive, non-symmetric, non-naturally reductive manifold with respect to the three-parameter metric invariant to the action of $G^J_1(\mathbb{R})$, and its geodesic vectors are determined.
We summarize our work on "hidden" Noether symmetries of multifield cosmological models and the classification of those two-field cosmological models which admit such symmetries.
We determine the most general time-independent Noether symmetries of two-field cosmological models with rotationally-invariant scalar manifold metrics. In particular, we show that such models can have hidden symmetries, which arise if and only if the scalar manifold metric has Gaussian curvature $-3/8$, i.e. when the model is of elementary $α$-attractor type with a fixed value of the parameter $α$. In this case, we find explicitly all scalar potentials compatible with hidden Noether symmetries, thus classifying all models of this type. We also discuss some implications of the corresponding conserved quantity.
We study Noether symmetries in two-field cosmological $α$-attractors, investigating the case when the scalar manifold is an elementary hyperbolic surface. This encompasses and generalizes the case of the Poincare disk. We solve the conditions for the existence of a `separated' Noether symmetry and find the form of the scalar potential compatible with such, for any elementary hyperbolic surface. For this class of symmetries, we find that the $α$-parameter must have a fixed value. Using those Noether symmetries, we also obtain many exact solutions of the equations of motion of these models, which were studied previously with numerical methods.
We describe a mathematically rigorous differential model for B-type open-closed topological Landau-Ginzburg theories defined by a pair $(X,W)$, where $X$ is a non-compact Kählerian manifold with holomorphically trivial canonical line bundle and $W$ is a complex-valued holomorphic function defined on $X$ and whose critical locus is compact but need not consist of isolated points. We also show how this construction specializes to the case when $X$ is Stein and $W$ has finite critical set, in which case one recovers a simpler mathematical model.
We study generalized two-field $α$-attractor models whose rescaled scalar manifold is the triply-punctured sphere endowed with its complete hyperbolic metric, whose underlying complex manifold is the modular curve $Y(2)$. Using an explicit embedding into the end compactification, we compute solutions of the cosmological evolution equations for a few globally well-behaved scalar potentials, displaying particular trajectories with inflationary behavior as well as more general cosmological trajectories of surprising complexity. In such models, the orientation-preserving isometry group of the scalar manifold is isomorphic with the permutation group on three elements, acting on $Y(2)$ as the group of anharmonic transformations. When the scalar potential is preserved by this action, $α$-attractor models of this type provide a geometric description of two-field `modular invariant $j$-models' in terms of gravity coupled to a non-linear sigma model with topologically non-trivial target and with a finite (as opposed to discrete but infinite) group of symmetries. The precise relation between the two perspectives is provided by the elliptic modular function $λ$, which can be viewed as a field redefinition that eliminates almost all of the countably infinite unphysical ambiguity present in the Poincaré half-plane description of such models.
We outline the geometric formulation of cosmological flows for FLRW models with scalar matter as well as certain aspects which arise in their study with methods originating from the geometric theory of dynamical systems. We briefly summarize certain results of numerical analysis which we carried out when the scalar manifold of the model is a hyperbolic surface of infinite area.
We propose a family of differential models for B-type open-closed topological Landau-Ginzburg theories defined by a pair $(X,W)$, where $X$ is any non-compact Calabi-Yau manifold and $W$ is any holomorphic complex-valued function defined on $X$ whose critical set is compact. The models are constructed at cochain level using smooth data, including the twisted Dolbeault algebra of polyvector valued forms and a twisted Dolbeault category of holomorphic factorizations of $W$. We give explicit proposals for cochain level versions of the bulk and boundary traces and for the bulk-boundary and boundary-bulk maps of the Landau-Ginzburg theory. We prove that most of the axioms of an open-closed topological field theory are satisfied on cohomology and conjecture that the remaining axioms are also satisfied.
We consider the bulk algebra and topological D-brane category arising from the differential model of the open-closed B-type topological Landau-Ginzburg theory defined by a pair $(X,W)$, where $X$ is a non-compact Calabi-Yau manifold and $W$ has compact critical set. When $X$ is a Stein manifold (but not restricted to be a domain of holomorphy), we extract equivalent descriptions of the bulk algebra and of the category of topological D-branes which are constructed using only the analytic space associated to $X$. In particular, we show that the D-brane category is described by projective matrix factorizations defined over the ring of holomorphic functions of $X$. We also discuss simplifications of the analytic models which arise when $X$ is holomorphically parallelizable and illustrate these analytic models in a few classes of examples.
We consider generalized $α$-attractor models whose scalar potentials are globally well-behaved and whose scalar manifolds are elementary hyperbolic surfaces. Beyond the Poincaré disk $\mathbb{D}$, such surfaces include the hyperbolic punctured disk $\mathbb{D}^\ast$ and the hyperbolic annuli $\mathbb{A}(R)$ of modulus $μ=2\log R>0$. For each elementary surface, we discuss its decomposition into canonical end regions and give an explicit construction of the embedding into the Kerekjarto-Stoilow compactification (which in all cases is the unit sphere), showing how this embedding allows for a universal treatment of globally well-behaved scalar potentials upon expanding their extension in real spherical harmonics. For certain simple but natural choices of extended potentials, we compute scalar field trajectories by projecting numerical solutions of the lifted equations of motion from the Poincaré half-plane through the uniformization map, thus illustrating the rich cosmological dynamics of such models.
We propose a class of two-field cosmological models derived from gravity coupled to non-linear sigma models whose target space is a non-compact and geometrically-finite hyperbolic surface, which provide a wide generalization of so-called $α$-attractor models and can be studied using uniformization theory. We illustrate cosmological dynamics in such models for the case of the hyperbolic triply-punctured sphere.
We study stratified G-structures in ${\cal N}=2$ compactifications of M-theory on eight-manifolds $M$ using the uplift to the auxiliary nine-manifold ${\hat M}=M\times S^1$. We show that the cosmooth generalized distribution ${\hat {\cal D}}$ on ${\hat M}$ which arises in this formalism may have pointwise transverse or non-transverse intersection with the pull-back of the tangent bundle of $M$, a fact which is responsible for the subtle relation between the spinor stabilizers arising on $M$ and ${\hat M}$ and for the complicated stratified G-structure on $M$ which we uncovered in previous work. We give a direct explanation of the latter in terms of the former and relate explicitly the defining forms of the $\mathrm{SU}(2)$ structure which exists on the generic locus ${\cal U}$ of $M$ to the defining forms of the $\mathrm{SU}(3)$ structure which exists on an open subset ${\hat {\cal U}}$ of ${\hat M}$, thus providing a dictionary between the eight- and nine-dimensional formalisms.
We construct explicitly the constants of motion for geodesics in the $5$-dimensional Sasaki-Einstein spaces $Y^{p,q}$. To carry out this task we use the knowledge of the complete set of Killing vectors and Killing-Yano tensors on these spaces. In spite of the fact that we generate a multitude of constants of motion, only five of them are functionally independent implying the complete integrability of geodesic flow on $Y^{p,q}$ spaces. In the particular case of the homogeneous Sasaki-Einstein manifold $T^{1,1}$ the integrals of motion have simpler forms and the relations between them are described in detail.