1,721,006 research outputs found
G2-moduli spaces: geometry, periods and degenerate limits
Get PDFThe overarching theme of the present dissertation is the geometry of compact ₂-manifolds and their moduli spaces. This thesis can be roughly divided into two parts: the first one is concerned with the spectral properties of twisted connected sums, whilst the second one studies the geometry of ₂-moduli spaces.
In the first part, we prove a deformation theorem allowing to construct torsion-free ₂-structures with ᵏ-estimates (for any ≥ 1), and use it to deduce improved estimates for the twisted connected sum construction. We then study the mapping properties of differential operators in 'neck-stretching' problems, where two asymptotically cylindrical manifolds are glued to form a family of compact manifolds containing a cylindrical neck region whose length is stretched to infinity. In this limit, we construct approximate Fredholm inverses for a large class of “adapted” elliptic operators, with good control on the growth of their norm. Our results are then refined in the case of the Laplacian operator, and we derive a precise description of the asymptotic behaviour of its lower spectrum. Specified to twisted connected sum ₂-manifolds, our results give mathematical support for the so-called swampland distance conjecture in physics.
The second part of this dissertation is concerned with the geometry of the moduli space ℳ of torsion-free ₂-structures on a compact ₂-manifold , endowed with the Riemannian metric induced by the volume-normalised ²-inner product. When the first Betti number of vanishes, this metric has the remarkable property of being Hessian and admits a global potential ℱ. Using this observation, we derive a formula for the energy (the integral of the squared velocity) of a path in ℳ. This allows us to give sufficient geometric and topological conditions for a path of torsion-free ₂-structures on to have finite energy and length in the moduli space. By considering paths that degenerate to a singular limit, we deduce that ₂-moduli spaces may be incomplete: indeed we show that ₂-manifolds constructed by the generalised Kummer construction, by resolution of isolated conical singularities or by the Joyce–Karigiannis construction all have incomplete moduli spaces.
In the final chapters, we study the local geometry of ℳ and give an alternative description of the metric through the introduction of a new notion of period map for ₂-manifolds, mimicking the classical notion of period map introduced by Griffiths for Kähler manifolds. More specifically, we show using Hodge theory that there is a natural immersion Φ : ℳ → of the moduli space into a homogeneous space diffeomorphic to (+1)/( {±1} × ₙ() ), where +1 = ³(), in such a way that coincides with the restriction of a (degenerate) homogeneous quadratic form defined on . Motivated by the question of understanding the curvature of , we also compute the derivatives of the potential function ℱ up to order 4 and relate our formulas to the second fundamental form of Φ(ℳ) ⊂ . In particular, we deduce that the map Φ is a totally geodesic immersion and is locally symmetric when ℳ = ⁷/Γ or = (³ × K3)/Γ. Finally, we use the theory of exterior differential systems to give some complements on the properties of the map Φ and relate it to the more classical notion of period map for ₂-manifolds, as a Lagrangian immersion of ℳ into ³() ⊕ ⁴()
Spin(7) Instantons on asymptotically conical Calabi-Yau Fourfolds
Get PDFThe present thesis examines the role of Hermitian Yang-Mills (HYM) connections in Spin(7) instanton moduli spaces over asymptotically conical (AC) Calabi-Yau (CY) fourfolds. The starting point is Lewis' energy estimate: Lewis: if a principal G-bundle P over a closed CY fourfold X8} admits HYM solutions, then all Spin(7) instantons on P are HYM. The question we are interested in is whether this result persists in the AC setting.
A first-order approach is to pass to the (Sasaki-Einstein) asymptotic link, which is not Ricci flat and could thus be homogeneous. The role of the Spin(7) instanton equation is assumed by the G2 instanton equation and that of the HYM equation by the contact instanton equation. It is easier to explore the relationship between these 7-dimensional systems instead. Imposing equivariance reduces the PDEs to representation theory. This allows us to exhibit an explicit non-contact G2 instanton on S7. This example agrees with the limiting connection of the standard octonionic instanton of Fubini and Nicolai. Prior to the results of this thesis, this was the only known non-HYM Spin(7) instanton on a CY fourfold. It originally appeared in the physics literature. We provide an alternative construction, in line with the modern framework for equivariant gauge theory. Because its limiting connection is not contact, its moduli space cannot contain HYM connections. Consequently, it does not help resolve the question we set out to answer.
We extend Lewis's theorem to the AC setting, conditioning on decay rates.
We construct the moduli space of SO(5)-invariant Spin(7) instantons with structure group SO(3) on the Stenzel space. These new instantons sit exactly at the slow-rate cut-off point of our extension of Lewis's theorem. They provide a negative answer to the question we set out to answer: the moduli space is one-dimensional and contains precisely two HYM connections. One of these is the epicentre of a removable singularity/ bubbling phenomenon and the development of a corresponding Fueter section. We compute this explicitly and verify (after suitable modifications) an infinite-energy version of Tian's energy conservation identity. This phenomenon hints at a possible relationship between the AC Spin(7) instanton and HYM systems
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Hermitian and G2-structures with large symmetry groups
Get PDFIn the context of Hermitian geometry, the Hull--Strominger system is a system of non-linear PDEs on heterotic string theory, over a six-dimensional manifold endowed with an SU(3)-structure. Its seven-dimensional analogue, the heterotic G2 system, is a system for both geometric fields and gauge fields over a manifold with a G2-structure. In this thesis, we study manifolds with geometric structures compatible with the Hull--Strominger system and the heterotic G2 system in the cohomogeneity one setting. In the former case, we develop a case-by-case analysis to provide a non-existence result for balanced non-Kähler SU(3)-structures which are invariant under a cohomogeneity one action on a simply connected six-manifold. In the latter case, we study two different SU(2)2-invariant cohomogeneity one manifolds, one non-compact M = R4 x S3, and one compact M = S4 x S3. For R4 x S3, we prove the existence of a family of coclosed (but not necessarily torsion-free) G2-structures which is given by three smooth functions satisfying certain boundary conditions around the singular orbit and a non-zero parameter. Moreover, any coclosed G2-structure constructed from a half-flat SU(3)-structure is in this family. For S4 x S3, we prove that there are no SU(2)2-invariant coclosed G2-structures constructed from half-flat SU(3)-structures. Then, we study the existence of SU(2)2-invariant G2-instantons on R4 x S3 manifold with the coclosed G2-structures found. We find two 1-parameter families of smooth SU(2)3-invariant G2-instantons with gauge group SU(2) on R4 x S3 and study its ``bubbling'' behaviour. We also provide existence results for locally defined SU(2)2-invariant G2-instantons
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Get PDFWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
Associative submanifolds of G2-manifolds
Get PDFDie hier dargelegte Dissertation ist motiviert durch die Vorschläge von Joyce, Doan und Walpuski zur Definitionen enumerativer Invarianten für G2-Mannigfaltigkeit, durch das Zählen gewisser kalibrierter Untermannigfaltigkeiten, sogenannter assoziativen Untermannigfaltigkeiten.
In Kapitel 1, werde ich Definitionen und grundlegende Fakten über G2-Mannigfaltigkeit und deren assoziative Untermannigfaltigkeit wiederholen. Darüber hinaus erläutere ich die Konstruktion von G2-Mannigfaltigkeit als verdrehte verbundener Summe.
Kapitel 2 schafft die nötige Grundlage für das darauf folgende dritte Kapitel. Hier definiere ich den Modul-Raum der asymptotisch zylindrischen assoziativen Untermannigfaltigkeiten zusammen mit seiner natürlichen Topologie und zeige, dass der Modul-Raum lokal homeomorph zur Urbild-Menge der Null einer glatten Abbildung zwischen zwei endlich-dimensionalen Räu- men ist. In besonderen Fällen ist dieser Modul-Raum eine Lagrangesche Untermannigfaltigkeit des Modul Raums der holomorphen Kurven einer asymptotisch zylindrischen Calabi-Yau Man- nigfaltigkeit.
In Kapitel 3 beweise ich ein Klebe-Theorem für ein Paar von asymptotisch zylindrischen as- soziativen Untermannigfaltigkeiten in einem zusammenpassenden Paar von asymptotisch zylin- drischen G2-Mannigfaltigkeiten. Hiermit konstruiere ich neue geschlossene und starre (rigid) assoziative Untermannigfaltigkeiten in verdrehten verbundenen Summe G2-Mannigfaltigkeiten.
In Kapitel 4 untersuche ich den Modul-Raum der konisch singulären assoziativen Un- termannigfaltigkeiten in G2-Mannigfaltigkeiten. Durch das Umformulieren des Indexes des Operators, der die Deformationstheorie kontrolliert, in bestimmte Stabilität-Indizes des zu- grundeliegenden assoziativen Kegels begründe ich, dass in einem generischen Pfad in dem Raum der ko-geschlossenen G2-Strukturen keine asymptotisch konischen assoziative Unter- mannigfaltigkeiten existieren, die mindestens eine Singularität besitzen, die auf einem Kegel mit Stabiltätsindex größer als eins modeliert werden. Dieses Resultat lässt sich auf alle speziellen Lagrangesche-Kegel außer den Harvey-Lawson-T2-Kegel und die Vereinigung zweier speziellen Lagrangesche-Flächen anwenden. Zusätzlich lässt sich das Ergebnis auch auf alle konischen assoziativen Untermannigfaltigkeiten anwenden, deren zugrundeliegende Verschlingung (link) holomorphe Kurven mit Null-Torsion in S6 sind. Des Weiteren dienen Teile des vierten Kapitels als Grundlage für das darauf folgende Kapitel 5.
Aufgrund einiger Übergangsphänomene entlang eines generischen Pfades von G2-Strukturen, führt das naive Zählen von assoziativen Untermannigfaltigkeiten zu keiner Invariante. Tat- sächlich wurde vermutet, dass a) eine assoziative Untermannigfaltigkeit aus einer assoziativen Untermannigfaltigkeit mit Selbstsschnitt (self-intersection) geboren werden kann, und, dass b) drei assoziative Untermannigfaltigkeiten aus einer konisch singulären assoziativen Un- termannigfaltigkeit, deren Singularität durch den Harvey-Lawson-T2-Kegel modelliert wird, entspringen. In Kapitel 5, beweise ich ein Desingularitätstheorem für konisch singulären assoziative Untermannigfaltigkeit entlang eines Pfades von ko-geschlossenen G2-Strukturen. Somit verifiziere ich Vermutung b) bewiesen und teilweise auch Vermutung a).The dissertation presented here is motivated from the proposals made by Joyce, Doan and Walpuski to define enumerative invariants of G2-manifolds by counting certain calibrated submanifolds, called associative submanifolds.
In Chapter 1, I review the definitions and basic facts of G2-manifolds and associative submanifolds. Moreover, I explain the construction of G2-manifolds as twisted connected sums. Chapter 2 serves as a necessary groundwork for Chapter 3. Here, I define the moduli space
of asymptotically cylindrical associative submanifolds with its natural topology and prove that the moduli space is locally homeomorphic to the zero set of a smooth map between two finite-dimensional spaces. In the best scenario, this moduli space is a Lagrangian submanifold of the moduli space of holomorphic curves in the asymptotic Calabi-Yau 3-fold.
In Chapter 3, I prove a gluing theorem for a pair of asymptotically cylindrical associative submanifolds in a matching pair of asymptotically cylindrical G2-manifolds. Using this I construct new closed and rigid associative submanifolds of twisted connected sum G2-manifolds.
In Chapter 4, I study the moduli space of conically singular associative submanifolds in G2-manifolds. By reformulating the index of the operator that controls the deformation theory in terms of certain stability-index of the associative cones, I establish that in a generic path of co-closed G2-structures there are no conically singular associative submanifolds that have at least one singularity modeled on a cone of stability-index greater than one. This result applies to all special Lagrangian cones, except the Harvey-Lawson T2-cone and a union of two special Lagrangian planes. Additionally, it applies to all associative cones whose links are null-torsion holomorphic curves in S6. Furthermore, parts of Chapter 4 also serve as a necessary groundwork for Chapter 5.
The naive counting of associative submanifolds does not lead to an invariant due to several transitions that may occur along a generic path of G2-structures. In fact it was conjectured that a) an associative submanifold born out of an associative submanifold with self intersection, and b) three associative submanifolds arise from a conically singular associative submanifold whose singularity is modeled on Harvey-Lawson T2-cone. In Chapter 5, I prove a desingularization theorem for conically singular associative submanifolds along a path of co-closed G2-structures. Consequently, I verify conjecture b) and partially confirm conjecture a)
Dispelling the Myths Behind First-author Citation Counts
Get PDFWe conducted a full-scale evaluative citation analysis study of scholars in the XML research field to explore just how different from each other author rankings resulting from different citation counting methods actually are, and to demonstrate the capability of emerging data and tools on the Web in supporting more realistic citation counting methods. Our results contest some common arguments for the continued
use of first-author citation counts in the evaluation of scholars, such as high correlations between author rankings by first-author citation counts and other citation
counting methods, and high costs of using more realistic citation counting methods that are not well-supported by the ISI databases. It is argued that increasingly available digital full text research papers make it possible for citation analysis studies to go beyond what the ISI databases have directly supported and to employ more
sophisticated methods
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