Moduli Space Metric Research Articles

Modular metric spaces : Some fixed-point theorems and application of secure dynamic routing for WSN

Data exchange within a Wireless Sensor Network (WSN) needs to adhere to high security protocols. The shortest possible route for the transfer of data between two interconnected packets is selected by dynamic routing algorithms, which increases privacy while requesting additional data to be transmitted. Several applications involving remote sensing of environmental requirements, such as WSN, are becoming more prevalent. Highly dense WSNs are evolving into a vital sensor framework enabling the Internet of Things (IoT). WSNs have the potential to synthesize a great deal of actual data, and in order to guarantee that data communication is being performed successfully, they need network architectures that have successful links between Sensor Nodes (SN). The collection of data is a term WSN utilises to define a data fusion/compression technique to minimize the volume of data sent throughout the network’s bandwidth. The process of iteration of Fixed Points (FP) of contractions will be examined in the present paper, which is scheduled to take part under a Modular Metric Space (M-MS) context. The main objective of this research is to arrive at an algorithm for secure dynamic data transmission routing in WSN by integrating the demonstrated results of the Functional Equation (FE) with randomization. An implementation is a tool for evaluating and modelling a network’s architecture in which sensors can be aggregated with a high level of security and least risk in a more secure area (Sensor Field), thereby improving the accuracy of the findings collected by the entire network.

Fixed point theorems of new generalized C-conditions for ( psi; gamma)-mappings in modular metric spaces and its applications

This paper introduces new generalizations of the C-condition for (ψ; γ)-mappings in modular metric spaces. We extend the fixed point results for such mappings yielding the generalized C-condition in metric spaces to modular ones.We proved the existence and uniqueness of solutions in modular metric spaces for these kinds of mappings. We give an example to emphasize that our results work in the difference between modularmetric spaces and usual ones. Moreover, we consider some initial and boundary value problems to support the results obtained here. We examine the existence and uniqueness of the solutions for the problems in modular metric spaces.

Engineering small flux superpotentials and mass hierarchies

We study the stabilization of complex structure moduli in Type IIB flux compactifications by using recent general results about the form of the superpotential and Kähler potential near the boundaries of the moduli space. In this process we show how vacua with an exponentially small vacuum superpotential can be realized systematically and understood conceptually within asymptotic Hodge theory. We distinguish two types of vacua realizing such superpotentials that differ by the mass scales of the stabilized moduli. Masses polynomially depending on the moduli arise if the superpotential contains exponential corrections whose existence is required to ensure the non-degeneracy of the moduli space metric. We use the fact that such essential corrections are controlled by asymptotic Hodge theory and have recently been constructed for all one- and two-moduli asymptotic regimes. These insights allow us to obtain new vacua near boundaries in complex structure moduli space that include Seiberg-Witten points. In these examples we find that the scale of the vacuum superpotential can be bounded from below through the exponential of the negative D3-brane tadpole.

Heterotic quantum cohomology

It is believed, but not demonstrated, that the large radius massless spectrum of a heterotic string theory compactified to four-dimensional Minkowski space should obey equations that split into ‘F-terms’ and ‘D-terms’ in ways analogous to that of four-dimensional supersymmetric field theories. This is not easy to do directly as string theory is first quantised. Nonetheless, in this paper we demonstrate this splitting. We construct an operator overline{mathcal{D}} whose kernel amounts to deformations solving ‘F-term’ type equations. In many previous works in this field, the spin connection is treated as an independent degree of freedom (and so is spurious or fake); here our results apply on the physical moduli space in which these fake degrees of freedom are eliminated. We utilise the moduli space metric, constructed in previous work, to define an adjoint operator overline{mathcal{D}} †. The kernel of overline{mathcal{D}} † amounts to ‘D-term’ type equations. Put together, we show there is a overline{mathcal{D}} -operator in which the massless spectrum are harmonic representatives of overline{mathcal{D}} . We conjecture that one could better study the moduli space of heterotic theories by studying the corresponding cohomology, a natural counterpart to studying the overline{partial} -cohomology groups relevant to moduli of Calabi-Yau manifolds.

The moduli space of nonnegatively curved metrics on quotients of S2 × S3 by involutions

We show that for an orientable non-spin manifold with fundamental group Z2 and universal cover S2×S3, the moduli space of metrics of nonnegative sectional curvature has infinitely many path components. The representatives of the components are quotients of the standard metric on S3×S3 or metrics on Brieskorn varieties previously constructed using cohomogeneity one actions. The components are distinguished using the relative η invariant of the spinc Dirac operator computed by means of a Lefschetz fixed point theorem.

Moduli space of non-negative sectional or positive Ricci curvature metrics on sphere bundles over spheres and their quotients

We show that the moduli space of positive Ricci curvature metrics on all the total spaces of S^7-bundles over S^8 which are rational homology spheres has infinitely many path components. Furthermore, we carry out the diffeomorphism classification of quotients of Milnor spheres by a certain involution and show that the moduli space of metrics of non-negative sectional curvature on them has infinitely many path components. Finally, a diffeomorphism finiteness result is obtained on quotients of Shimada spheres by the same type of involution and we show that for the types that can be expressed by an infinite family of manifolds, the moduli space of positive Ricci curvature metrics has infinitely many path components.

S-Duality and the Universal Isometries of q-Map Spaces

The tree-level q-map assigns to a projective special real (PSR) manifold of dimension n-1ge 0, a quaternionic Kähler (QK) manifold of dimension 4n+4. It is known that the resulting QK manifold admits a (3n+5)-dimensional universal group of isometries (i.e. independently of the choice of PSR manifold). On the other hand, in the context of Calabi–Yau compactifications of type IIB string theory, the classical hypermultiplet moduli space metric is an instance of a tree-level q-map space, and it is known from the physics literature that such a metric has an mathrm {SL}(2,{mathbb {R}}) group of isometries related to the mathrm {SL}(2,{mathbb {Z}}) S-duality symmetry of the full 10d theory. We present a purely mathematical proof that any tree-level q-map space admits such an mathrm {SL}(2,{mathbb {R}}) action by isometries, enlarging the previous universal group of isometries to a (3n+6)-dimensional group G. As part of this analysis, we describe how the (3n+5)-dimensional subgroup interacts with the mathrm {SL}(2,{mathbb {R}})-action, and find a codimension one normal subgroup of G that is unimodular. By taking a quotient with respect to a lattice in the unimodular group, we obtain a quaternionic Kähler manifold fibering over a projective special real manifold with fibers of finite volume, and compute the volume as a function of the base. We furthermore provide a mathematical treatment of results from the physics literature concerning the twistor space of the tree-level q-map space and the holomorphic lift of the (3n+6)-dimensional group of universal isometries to the twistor space.

Moduli space of nonnegatively curved metrics on manifolds of dimension 4k + 1

In each dimension $4k+1\geq 9$, we exhibit infinite families of closed manifolds with fundamental group $\mathbb Z_2$ for which the moduli space of metrics of nonnegative sectional curvature has infinitely many path components. Examples of closed manifolds with finite fundamental group with this property were known before only in dimension $5$ and dimensions $4k+3\geq 7$.

Quaternionic Kähler Metrics Associated to Special Kähler Manifolds with Mutually Local Variations of BPS Structures

We construct a quaternionic Kähler manifold from a conical special Kähler manifold with a certain type of mutually local variation of BPS structures. We give global and local explicit formulas for the quaternionic Kähler metric and specify under which conditions it is positive-definite. Locally, the metric is a deformation of the 1-loop corrected Ferrara–Sabharwal metric obtained via the supergravity c-map. The type of quaternionic Kähler metrics we obtain is related to work in the physics literature by S. Alexandrov and S. Banerjee, where they discuss the hypermultiplet moduli space metric of type IIA string theory, with mutually local D-instanton corrections.

Small Gauge Transformations and Universal Geometry in Heterotic Theories

The first part of this paper describes in detail the action of small gauge transformations in heterotic supergravity. We show a convenient gauge fixing is 'holomorphic gauge' together with a condition on the holomorphic top form. This gauge fixing, combined with supersymmetry and the Bianchi identity, allows us to determine a set of non-linear PDEs for the terms in the Hodge decomposition. Although solving these in general is highly non-trivial, we give a prescription for their solution perturbatively in α and apply this to the moduli space metric. The second part of this paper relates small gauge transformations to a choice of connection on the moduli space. We show holomorphic gauge is related to a choice of holomorphic structure and Lee form on a 'universal bundle'. Connections on the moduli space have field strengths that appear in the second order deformation theory and we point out it is generically the case that higher order deformations do not commute.

Moduli space of metrics of nonnegative sectional or positive Ricci curvature on homotopy real projective spaces

We show that the moduli space of metrics of nonnegative sectional curvature on every homotopy R P 5 \mathbb {R} P^5 has infinitely many path components. We also show that in each dimension 4 k + 1 4k+1 there are at least 2 2 k 2^{2k} homotopy R P 4 k + 1 \mathbb {R} P^{4k+1} ’s of pairwise distinct oriented diffeomorphism type for which the moduli space of metrics of positive Ricci curvature has infinitely many path components. Examples of closed manifolds with finite fundamental group with these properties were known before only in dimensions 4 k + 3 ≥ 7 4k+3\geq 7 .

Common Fixed Point Results on Fuzzy Metric Spaces and Modular Metric Spaces via Simulation Function

In this paper, we prove common fixed point theorems for two map- pings by using simulation function on fuzzy metric spaces. We also deduce some consequences in modular metric spaces.

On the domain of moduli fields

The concept of the moduli space allows for a simple, universally applicable description of the low-energy dynamics of topological solitons. This description is remarkably insensitive to the properties of the underlying theory, whose details only manifest themselves via the moduli space metric. This article presents a generalization of this concept, which allows to transfer its most intriguing features to configurations of any energy captured by the theory giving rise to the soliton, given that these are localized sufficiently close to the soliton’s center. The resulting theory is capable of describing all dynamics within its range of applicability by just one family of fields, with all the information about the underlying theory entering via a finite number of background functions, which can be linked to physical properties of the present soliton.

Moduli Spaces of Metrics of Positive Scalar Curvature on Topological Spherical Space Forms

AbstractLet $M$ be a topological spherical space form, i.e., a smooth manifold whose universal cover is a homotopy sphere. We determine the number of path components of the space and moduli space of Riemannian metrics with positive scalar curvature on $M$ if the dimension of $M$ is at least 5 and $M$ is not simply-connected.

On the moduli spaces of metrics with nonnegative sectional curvature

The Kreck–Stolz \(s\) invariant is used to distinguish connected components of the moduli space of positive scalar curvature metrics. We use a formula of Kreck and Stolz to calculate the \(s\) invariant for metrics on \(S^n\) bundles with nonnegative sectional curvature. We then apply it to show that the moduli spaces of metrics with nonnegative sectional curvature on certain 7-manifolds have infinitely many path components. These include certain positively curved Eschenburg and Aloff–Wallach spaces.

Instantons and infinite distances

We consider geodesics of infinite length and with constant 4d dilaton in the (classical) hypermultiplet moduli space of type II Calabi-Yau compactifications. When approaching such infinite distance points, a large amount of D-instantons develop an exponentially suppressed action, substantially modifying the moduli space metric. We consider a particular large volume/strong coupling trajectory for which, in the corrected metric, the path length becomes finite. The instanton effects also modify the cllassical 4d dilaton such that, in order to keep the 4d Planck mass finite, the string scale has to be lowered. Our results can be related, via the c-map, to the physics around points of infinite distance in the vector multiplet moduli space where the Swampland Distance Conjecture and the Emergence Proposal have been discussed, and provide further evidence for them.

Special geometry on the 101 dimesional moduli space of the quintic threefold

A new method for explicit computation of the CY moduli space metric was proposed by the authors recently. The method makes use of the connection of the moduli space with a certain Frobenius algebra. Here we clarify this approach and demonstrate its efficiency by computing the Special geometry of the 101-dimensional moduli space of the quintic threefold around the orbifold point.

Incompleteness of the pressure metric on the Teichmüller space of a bordered surface

We prove that the pressure metric on the Teichmüller space of a bordered surface is incomplete and that a completion can be given by the moduli space of metrics on a graph (dual to a special ideal triangulation of the same bordered surface) equipped with pressure metric. In contrast to the closed surface case, we obtain as a corollary that the pressure metric is not bi-Lipschitz to the Weil–Petersson metric.

The effective action of warped M-theory reductions with higher derivative terms — part I

We study the three-dimensional effective action obtained by reducing eleven-dimensional supergravity with higher-derivative terms on a background solution including a warp-factor, an eight-dimensional compact manifold, and fluxes. The dynamical fields are Kahler deformations and vectors from the M-theory three-form. We show that the potential is only induced by fluxes and the naive contributions obtained from higher-curvature terms on a Calabi-Yau background vanish once the back-reaction to the full solution is taken into account. For the resulting three-dimensional action we analyse the Kahler potential and complex coordinates and show compatibility with $$ \mathcal{N}=2 $$ supersymmetry. We argue that the higher-order result is also compatible with a no-scale condition. We find that the complex coordinates should be formulated as divisor integrals for which a non-trivial interplay between the warp-factor terms and the higher-curvature terms allow a derivation of the moduli space metric. This leads us to discuss higher-derivative corrections to the M5-brane action.

Path-connectedness of the moduli spaces of metrics with positive isotropic curvature on four-manifolds

In the paper we prove the path-connectedness of the moduli spaces of metrics with positive isotropic curvature on certain compact four-dimensional manifolds.