The tadpole problem

We examine the mechanism of moduli stabilization by fluxes in the limit of a large number of moduli. We conjecture that one cannot stabilize all complex-structure moduli in F-theory at a generic point in moduli space (away from singularities) by fluxes that satisfy the bound imposed by the tadpole cancellation condition. More precisely, while the tadpole bound in the limit of a large number of complex-structure moduli goes like 1/4 of the number of moduli, we conjecture that the amount of charge induced by fluxes stabilizing all moduli grows faster than this, and is therefore larger than the allowed amount. Our conjecture is supported by two examples: K3 × K3 compactifications, where by using evolutionary algorithms we find that moduli stabilization needs fluxes whose induced charge is 44% of the number of moduli, and Type IIB compactifications on $$ \mathbbm{CP} $$ CP 3, where the induced charge of the fluxes needed to stabilize the D7-brane moduli is also 44% of the number of these moduli. Proving our conjecture would rule out de Sitter vacua obtained via antibrane uplift in long warped throats with a hierarchically small supersymmetry breaking scale, which require a large tadpole.

Lifting 1/4-BPS states in AdS3× S3 × T4

We establish a framework for doing second order conformal perturbation theory for the symmetric orbifold SymN(T4) to all orders in N. This allows us to compute how 1/4-BPS states of the D1-D5 system on AdS3 × S3 × T4 are lifted as we move away from the orbifold point. As an application we confirm a previous observation that in the large N limit not all 1/4-BPS states that can be lifted do get lifted. This provides evidence that the supersymmetric index actually undercounts the number of 1/4-BPS states at a generic point in the moduli space.

A nilpotency index of conformal manifolds

We show that exactly marginal operators of Supersymmetric Conformal Field Theories (SCFTs) with four supercharges cannot obtain a vacuum expectation value at a generic point on the conformal manifold. Exactly marginal operators are therefore nilpotent in the chiral ring. This allows us to associate an integer to the conformal manifold, which we call the nilpotency index of the conformal manifold. We discuss several examples in diverse dimensions where we demonstrate these facts and compute the nilpotency index.

F-theory and heterotic duality, Weierstrass models from Wilson lines

We present how to construct elliptically fibered K3 surfaces via Weierstrass models which can be parametrized in terms of Wilson lines in the dual heterotic string theory. We work with a subset of reflexive polyhedras that admit two fibers whose moduli spaces contain the ones of the $$E_{8}\times E_{8}$$ E 8 × E 8 or $$\frac{Spin(32)}{{\mathbb {Z}}_{2}}$$ S p i n ( 32 ) Z 2 heterotic theory compactified on a two torus without Wilson lines. One can then interpret the additional moduli as a particular Wilson line content in the heterotic strings. A convenient way to find such polytopes is to use graphs of polytopes where links are related to inclusion relations of moduli spaces of different fibers. We are then able to map monomials in the defining equations of particular K3 surfaces to Wilson line moduli in the dual theories. Graphs were constructed developing three different programs which give the gauge group for a generic point in the moduli space, the Weierstrass model as well as basic enhancements of the gauge group obtained by sending coefficients of the hypersurface equation defining the K3 surface to zero.

On the prime spectrum of an le-module

Here, we continue to characterize a recently introduced notion, le-modules [Formula: see text] over a commutative ring [Formula: see text] with unity [A. K. Bhuniya and M. Kumbhakar, Uniqueness of primary decompositions in Laskerian le-modules, Acta Math. Hunga. 158(1) (2019) 202–215]. This paper introduces and characterizes Zariski topology on the set Spec[Formula: see text] of all prime submodule elements of [Formula: see text]. Thus, we extend many results on Zariski topology for modules over a ring to le-modules. The topological space Spec[Formula: see text] is connected if and only if [Formula: see text] contains no idempotents other than [Formula: see text] and [Formula: see text]. Open sets in the Zariski topology for the quotient ring [Formula: see text] induces a base of quasi-compact open sets for the Zariski topology on Spec[Formula: see text]. Every irreducible closed subset of Spec[Formula: see text] has a generic point. Besides, we prove a number of different equivalent characterizations for Spec[Formula: see text] to be spectral.

Examples of adic spaces

This chapter discusses various examples of adic spaces. These examples include the adic closed unit disc; the adic affine line; the closure of the adic closed unit disc in the adic affine line; the open unit disc; the punctured open unit disc; and the constant adic space associated to a profinite set. The chapter focuses on one example: the adic open unit disc over Zp. The adic spectrum Spa Zp consists of two points, a special point and a generic point. The chapter then studies the structure of analytic points. It also clarifies the relations between analytic rings and Tate rings.

Dynamics of Cohomological Expanding Mappings II : Third and Fourth Main Results

Let f : V → V be a Cohomological Expanding Mapping1 of a smooth complex compact homogeneous manifold with $ dim_{\mathbb{C}}(\Vc)=k \ge 1$ and Kodaira Dimension $\leq 0$. We study the dynamics of such mapping from a probabilistic point of view, that is, we describe the asymptotic behavior of the orbit $ O_{h} (x) = \{h^{n} (x), n \in \mathbb{N} \quad \mbox{or}\quad \mathbb{Z}\}$ of a generic point. Using pluripotential methods, we have constructed in our previous paper \cite{Armand4} a natural invariant canonical probability measure of maximal Cohomological Entropy $ \nu_{h} $ such that ${\chi_{2l}^{-m}} (h^m)^\ast \Omega \to \nu_h \qquad \mbox{as} \quad m\to\infty$ for each smooth probability measure $\Omega $ in $\Vc$ . We have also studied the main stochastic properties of $ \nu_{h}$ and have shown that $ \nu_{h}$ is a smooth equilibrium measure , ergodic, mixing, K-mixing, exponential-mixing. In this paper we are interested on equidistribution problems and we show in particular that $ \nu_{h}$ reflects a property of equidistribution of periodic points by setting out the Third and Fourth Main Results in our study. Finally we conjecture that $$\nu_h:=T_l^+ \wedge T_{k-l}^-,$$ $$\dim_\HH(\nu_h)= \Psi h_{\chi}(h) , $$ $$\dim_\HH( \mbox{Supp} T_l^+) \geq 2(k-l) + \frac{\log \chi_{2l}}{\psi_l},$$ $$|\langle \nu_m^x-\nu_h,\zeta\rangle|\leq M \Big[1+\log^+{1\over D(x,\Tc)}\Big]^{\beta/2}\|\zeta\|_{\Cc^\beta} \gamma^{-\beta m/2}$$ and $$|\langle \nu_m^x-\nu_h,\zeta\rangle|\leq M \Big[1 +\log^+{1\over D(x,E_\gamma)}\Big]^{\beta/2}\|\zeta\|_{\Cc^\beta} \gamma^{-\beta m/2}.$$

Dynamics of Cohomological Expanding Mappings I: First and Second Main Results

Let $f:V\rightarrow V $ be a Cohomological Expanding Mapping1 of a smooth complex compact homogeneous manifold with dimC(V) = k ≥ 1 and Kodaira Dimension $\leq 0$. We study the dynamics of such mapping from a probabilistic point of view, that is, we describe the asymptotic behavior of the orbit $ O_{f} (x) = \{f^{n} (x), n \in \mathbb{N} \quad \mbox{or}\quad \mathbb{Z}\}$ of a generic point. Using pluripotential methods, we construct a natural invariant canonical probability measure of maximum Cohomological Entropy $ \mu_{f} $ such that ${\chi_{2l}^{-m}} (f^m)^\ast \Omega \to \mu_f \qquad \mbox{as} \quad m\to\infty$ for each smooth probability measure $\Omega $ on V . Then we study the main stochastic properties of $ \mu_{f}$ and show that $ \mu_{f}$ is a measure of equilibrium, smooth, ergodic, mixing, K-mixing, exponential-mixing and the unique measure with maximum Cohomological Entropy. We also conjectured that $\mu_f:=T_l^+ \wedge T_{k-l}^-$, $\dim_\H(\mu_f)= \Psi h_{\chi}(f) $ and $\dim_\H( \mbox{Supp} T_l^+) \geq 2(k-l) + \frac{\log \chi_{2l}}{\psi_l}.$

ESCOLA ECONHECIMENTO: EM DEFESA DOS CLÁSSICOS E DA CRÍTICA

This text is an instrument through which we question the relationship between knowledge and school education from the contributions of the lukacsian ontology and the historical-critical pedagogy. We start from the fact that the transmission of knowledge is an indispensable practice that marks the process of reproduction of societies. From these understandings, we advance to the specificity of school education and the role of teachers from a human-generic point of view, that is, that puts in the foreground the authentic human demands and not the self-centered market and business needs. We demonstrate that, from the perspective of the essential interests of students and teachers, it is essential the critical socialization of the most erudite and elaborate knowledge historically produced, from the recognition of its historical and ontological parameters.

Regularity of CR mappings of abstract CR structures

We study the [Formula: see text] regularity problem for CR maps from an abstract CR manifold [Formula: see text] into some complex Euclidean space [Formula: see text]. We show that if [Formula: see text] satisfies a certain condition called the microlocal extension property, then any [Formula: see text]-smooth CR map [Formula: see text], for some integer [Formula: see text], which is nowhere [Formula: see text]-smooth on some open subset [Formula: see text] of [Formula: see text], has the following property: for a generic point [Formula: see text] of [Formula: see text], there must exist a formal complex subvariety through [Formula: see text], tangent to [Formula: see text] to infinite order, and depending in a [Formula: see text] and CR manner on [Formula: see text]. As a consequence, we obtain several [Formula: see text] regularity results generalizing earlier ones by Berhanu–Xiao and the authors (in the embedded case).