Geometry and Topology Commons^™

Bi-Harmonic Maps' Existence And P-Balanced Energy And Connections To Harmonic Maps, Lina Wu

Publications and Research

We discover a bi-harmonic map’ existence, its energy growth, and its connections to a harmonic map. First, we prove the existence of a nontrivial bi-harmonic map in a unit sphere. Second, we investigate the p-balanced energy growth of biharmonic maps. As the most important energy technical breakthroughs, we propose an innovative energy algorithm called p-balanced energy technique to break the constraints of the existing L^q-energy technique in detecting L^q-energy growth towards boundlessness. The disadvantage of the finite L^q -energy technique in the L ^q spaces is not effective in dealing with infinite L^q- energy in Non-L^q spaces. Third, we study …

Trapped Surfaces, Topology Of Black Holes, And The Positive Mass Theorem, Lan-Hsuan Huang, Dan A. Lee

Publications and Research

No abstract provided.

The Octagon At Large M, Riccardo Argurio, Matteo Bertolini, Sebastían Franco, Eduardo García-Valdecasas, Shani Meynet, Antoine Pasternak, Valdo Tatitscheff

Publications and Research

Recently, the first instance of a model of D-branes at Calabi-Yau singularities where supersymmetry is broken dynamically into stable vacua has been proposed. This construction was based on a system of N regular and M = 1 fractional branes placed at the tip of the so-called (orientifolded) Octagon singularity. In this paper we show that this model admits a large M generalization, having the same low energy effective dynamics. This opens up the possibility that the effect on geometry is smooth, and amenable to describing the gauge theory all along the RG flow, including the deep IR, in terms of …

Spin(7) Orientifolds And 2d N = (0, 1) Triality, Sebastían Franco, Alessandro Mininno, Ángel M. Uranga, Xingyang Yu

Publications and Research

We present a new, geometric perspective on the recently proposed triality of 2d N = (0, 1) gauge theories, based on its engineering in terms of D1-branes probing Spin(7) orientifolds. In this context, triality translates into the fact that multiple gauge theories correspond to the same underlying orientifold. We show how Spin(7) orientifolds based on a particular involution, which we call the universal involution, give rise to precisely the original version of N = (0, 1) triality. Interestingly, our work also shows that the space of possibilities is significantly richer. Indeed, general Spin(7) orientifolds extend triality to theories that can …

Extractable Entanglement From A Euclidean Hourglass, Takanori Anegawa, Norihiro Iizuka, Daniel Kabat

Publications and Research

We previously proposed that entanglement across a planar surface can be obtained from the partition function on a Euclidean hourglass geometry. Here we extend the prescription to spherical entangling surfaces in conformal field theory. We use the prescription to evaluate log terms in the entropy of a conformal field theory in two dimensions, a conformally coupled scalar in four dimensions, and a Maxwell field in four dimensions. For Maxwell we reproduce the extractable entropy obtained by Soni and Trivedi. We take this as evidence that the hourglass prescription provides a Euclidean technique for evaluating extractable entropy in quantum field theory.

Bft2: A General Class Of 2d N = (0, 2) Theories, 3-Manifolds And Toric Geometry, Sebastían Franco, Xingyang Yu

Publications and Research

We introduce and initiate the study of a general class of 2d N = (0, 2) quiver gauge theories, defined in terms of certain 2-dimensional CW complexes on oriented 3- manifolds. We refer to this class of theories as BFT₂’s. They are natural generalizations of Brane Brick Models, which capture the gauge theories on D1-branes probing toric Calabi-Yau 4-folds. The dynamics and triality of the gauge theories translate into simple transformations of the underlying CW complexes. We introduce various combinatorial tools for analyzing these theories and investigate their connections to toric Calabi-Yau manifolds, which arise as their …

2d N = (0, 1) Gauge Theories And Spin(7) Orientifolds, Sebastían Franco, Alessandro Mininno, Ángel M. Uranga, Xingyang Yu

Publications and Research

We initiate the geometric engineering of 2d N = (0, 1) gauge theories on D1-branes probing singularities. To do so, we introduce a new class of backgrounds obtained as quotients of Calabi-Yau 4-folds by a combination of an anti-holomorphic involution leading to a Spin(7) cone and worldsheet parity. We refer to such constructions as Spin(7) orientifolds. Spin(7) orientifolds explicitly realize the perspective on 2d N = (0, 1) theories as real slices of N = (0, 2) ones. Remarkably, this projection is geometrically realized as Joyce’s construction of Spin(7) manifolds via quotients of Calabi-Yau 4-folds by antiholomorphic …

Calabi-Yau Products: Graded Quivers For General Toric Calabi-Yaus, Sebastián Franco, Azeem Hasan

Publications and Research

The open string sector of the topological B-model on CY (m + 2)-folds is described by m-graded quivers with superpotentials. This correspondence generalizes the connection between CY (m + 2)-folds and gauge theories on the worldvolume of D(5 − 2m)-branes for m = 0, . . . , 3 to arbitrary m. In this paper we introduce the Calabi-Yau product, a new algorithm that starting from the known quiver theories for a pair of toric CY_m+2 and CY_n+2 produces the quiver theory for a related CY_{m+n+3 …}

Dimers, Orientifolds And Stability Of Supersymmetry Breaking Vacua, Riccardo Argurio, Matteo Bertolini, Sebastían Franco, Eduardo García-Valdecasas, Shani Meynet, Antoine Pasternak, Valdo Tatitscheff

Publications and Research

We study (orientifolded) toric Calabi-Yau singularities in search for D-brane configurations which lead to dynamical supersymmetry breaking at low energy. By exploiting dimer techniques we are able to determine that while most realizations lead to a Coulomb branch instability, a rather specific construction admits a fully stable supersymmetry breaking vacuum. We describe the geometric structure that a singularity should have in order to host such a construction, and present its simplest example, the Octagon.

Light-Ray Moments As Endpoint Contributions To Modular Hamiltonians, Daniel Kabat, Gilad Lifschytz, Phuc Nguyen, Debajyoti Sarkar

Publications and Research

We consider excited states in a CFT, obtained by applying a weak unitary perturbation to the vacuum. The perturbation is generated by the integral of a local operator J⁽ⁿ⁾ of modular weight n over a spacelike surface passing through x = 0. For |n| ≥ 2 the modular Hamiltonian associated with a division of space at x = 0 picks up an endpoint contribution, sensitive to the details of the perturbation (including the shape of the spacelike surface) at x = 0. The endpoint contribution is a sum of light-ray moments of the perturbing operator …

Dimers, Orientifolds And Anomalies, Riccardo Argurio, Matteo Bertolini, Sebastían Franco, Eduardo Garcia-Valdecasas, Shani Meynet, Antoine Pasternak, Valdo Tatitscheff

Publications and Research

We study 4d N = 1 gauge theories engineered via D-branes at orientifolds of toric singularities, where gauge anomalies are cancelled without the introduction of non-compact flavor branes. Using dimer model techniques, we derive geometric criteria for establishing whether a given singularity can admit anomaly-free D-brane configurations purely based on its toric data and the type of orientifold projection. Our results therefore extend the dictionary between geometric properties of singularities and physical properties of the corresponding gauge theories.

Winding Numbers And Full Extendability In Holomorphic Motions, Yunping Jiang

Publications and Research

We construct an example of a holomorphic motion of a five-point subset of the Riemann sphere over an annulus such that it satisfies the zero winding number condition but is not fully extendable.

"Lagrangian Disks" In M-Theory, Sebastián Franco, Sergei Gukov, Sangmin Lee, Rak-Kyeong Seong, James Sparks

Publications and Research

While the study of bordered (pseudo-)holomorphic curves with boundary on Lagrangian submanifolds has a long history, a similar problem that involves (special) Lagrangian submanifolds with boundary on complex surfaces appears to be largely overlooked in both physics and math literature. We relate this problem to geometry of coassociative submanifolds in G₂ holonomy spaces and to Spin(7) metrics on 8-manifolds with T² fibrations. As an application to physics, we propose a large class of brane models in type IIA string theory that generalize brane brick models on the one hand and 2d theories T[M₄] on …

On The Classification Of Duality Webs For Graded Quivers, Sebastían Franco, Azeem Hasan, Xingyang Yu

Publications and Research

We study the m-graded quiver theories associated to CY (m+2)-folds and their order (m + 1) dualities. We investigate how monodromies give rise to mutation invariants, which in turn can be formulated as Diophantine equations characterizing the space of dual theories associated to a given geometry. We discuss these ideas in general and illustrate them in the case of orbifold theories. Interestingly, we observe that even in this simple context the corresponding Diophantine equations may admit an infinite number of seeds for m ≥ 2, which translates into an infinite number of disconnected duality webs. Finally, …

Graded Quivers, Generalized Dimer Models And Toric Geometry, Sebastián Franco, Azeem Hasan

Publications and Research

The open string sector of the topological B-model on CY (m+2)-folds is described by m-graded quivers with superpotentials. This correspondence extends to general m the well known connection between CY (m+2)-folds and gauge theories on the world-volume of D(5-2m)-branes for m = 0, ..., 3. We introduce m-dimers, which fully encode the m-graded quivers and their superpotentials, in the case in which the CY (m+2)-folds are toric. Generalizing the well known m = 1,2 cases, m-dimers significantly simplify the connection between geometry and m-graded quivers. A key …

Applying P-Balanced Energy Technique To Solve Liouville-Type Problems In Calculus, Lina Wu

Publications and Research

We are interested in solving Liouville-type problems to explore constancy properties for maps or differential forms on Riemannian manifolds. Geometric structures on manifolds, the existence of constancy properties for maps or differential forms, and energy growth for maps or differential forms are intertwined. In this article, we concentrate on discovery of solutions to Liouville-type problems where manifolds are Euclidean spaces (i.e. flat Riemannian manifolds) and maps become real-valued functions. Liouville-type results of vanishing properties for functions are obtained. The original work in our research findings is to extend the  q-energy for a function from finite in L^q space to infinite …

Higher Cluster Categories And Qft Dualities, Sebastián Franco, Gregg Musiker

Publications and Research

We introduce a unified mathematical framework that elegantly describes minimally supersymmetry gauge theories in even dimensions, ranging from six dimensions to zero dimensions, and their dualities. This approach combines and extends recent developments on graded quivers with potentials, higher Ginzburg algebras, and higher cluster categories (also known as m-cluster categories). Quiver mutations studied in the context of mathematics precisely correspond to the order-(m + 1) dualities of the gauge theories. Our work indicates that these equivalences of quiver gauge theories sit inside an infinite family of such generalized dualities.

Bipartite Field Theories And D-Brane Instantons, Sebastían Franco, Eduardo García-Valdecasas, Angel M. Uranga

Publications and Research

We study D-brane instantons in systems of D3-branes at toric CY 3-fold singularities. The instanton effect can be described as a backreaction modifying the geometry of the mirror configuration, in which the breaking of U(1) symmetries by the instanton translates into the recombination of gauge D-branes, which also directly generates the instanton-induced charged field theory operator. In this paper we describe the D-brane instanton backreaction in terms of a combinatorial operation in the bipartite dimer diagram of the original theory. Interestingly, the resulting theory is a general Bipartite Field Theory (BFT), defined by a bipartite graph tiling a general (possibly …

3d Printing Of 2d N = (0, 2) Gauge Theories, Sebastían Franco, Azeem Hasan

Publications and Research

We introduce 3d printing, a new algorithm for generating 2d N = (0, 2) gauge theories on D1-branes probing singular toric Calabi-Yau 4-folds using 4d N = 1 gauge theories on D3-branes probing toric Calabi-Yau 3-folds as starting points. Equivalently, this method produces brane brick models starting from brane tilings. 3d printing represents a significant improvement with respect to previously available tools, allowing a straightforward determination of gauge theories for geometries that until now could only be tackled using partial resolution. We investigate the interplay between triality, an IR equivalence between different 2d N = …

Solving Liouville-Type Problems On Manifolds With Poincar´E-Sobolev Inequality By Broadening Q-Energy From Finite To Infinite, Lina Wu

Publications and Research

The aim of this article is to investigate Liouville-type problems on complete non-compact Riemannian manifolds with Poincar´e-Sobolev Inequality. Two significant technical breakthroughs are demonstrated in research findings. The first breakthrough is an extension from non-flat manifolds with non-negative Ricci curvatures to curved manifolds with Ricci curvatures varying among negative values, zero, and positive values. Poincar´e-Sobolev Inequality has been applied to overcome difficulties of an extension on manifolds. Poincar´e-Sobolev Inequality has offered a special structure on curved manifolds with a mix of Ricci curvature signs. The second breakthrough is a generalization of q-energy from finite to infinite. At this point, a …

Equivalence Between A Harmonic Form And A Closed Co-Closed Form In Both Lq And Non-Lq Spaces, Lina Wu

Publications and Research

For a differential k-form ω on a complete non-compact manifold, we establish an equivalent relation between a harmonic form and a closed co-closed form. We extend this equivalence from ω in L^2 spaces to ω with 2-balanced growth including L^2 spaces and non-L^2 spaces. Especially for a simple differential k-form ¯ω on a complete non-compact manifold, we generalize this equivalence from ¯ω in L^q spaces to ¯ω with 2-balanced growth including L^q spaces and non-L^q spaces for 2 ≤ q < 3. Our research findings recapture the work of Andreotti and Vesentini. Our ideas and calculation methods in this paper could provide a new way of broadening L^q spaces to non-L^q spaces in a variety of energy for differential forms.

Discovering Liouville-Type Problems For P-Energy Minimizing Maps In Closed Half-Ellipsoids By Calculus Variation Method, Lina Wu

Publications and Research

The goal of this project is to investigate constant properties (called the Liouville-type Problem) for a p-stable map as a local or global minimum of a p-energy functional where the domain is a Euclidean space and the target space is a closed half-ellipsoid. The First and Second Variation Formulas for a p-energy functional has been applied in the Calculus Variation Method as computation techniques. Stokes’ Theorem, Cauchy-Schwarz Inequality, Hardy-Sobolev type Inequalities, and the Bochner Formula as estimation techniques have been used to estimate the lower bound and the upper bound of the derived p-Harmonic Stability Inequality. One challenging point in …

Hyperplanes That Intersect Each Ray Of A Cone Once And A Banach Space Counterexample, Chris Mccarthy

Publications and Research

Suppose � is a cone contained in real vector space �. When does � contain a hyperplane � that intersects each of the 0-rays in �\{0} exactly once? We build on results found in Aliprantis, Tourky, and Klee Jr.’s work to give a partial answer to this question.We also present an example of a salient, closed Banach space cone � for which there does not exist a hyperplane that intersects each 0-ray in � \ {0} exactly once.

Experimental Demonstration Of Topological Effects In Bianisotropic Metamaterials, Alexey P. Slobozhanyuk, Alexander B. Khanikaev, Dmitry S. Filonov, Daria A. Smirnova, Andrey E. Miroshnichenko, Yuri S. Kivshar

Publications and Research

Existence of robust edge states at interfaces of topologically dissimilar systems is one of the most fascinating manifestations of a novel nontrivial state of matter, a topological insulator. Such nontrivial states were originally predicted and discovered in condensed matter physics, but they find their counterparts in other fields of physics, including the physics of classical waves and electromagnetism. Here, we present the first experimental realization of a topological insulator for electromagnetic waves based on engineered bianisotropic metamaterials. By employing the near-field scanning technique, we demonstrate experimentally the topologically robust propagation of electromagnetic waves around sharp corners without backscattering effects.

Locally Anisotropic Toposes, Jonathon Funk, Pieter Hofstra

Publications and Research

This paper continues the investigation of isotropy theory for toposes. We develop the theory of isotropy quotients of toposes, culminating in a structure theorem for a class of toposes we call locally anisotropic. The theory has a natural interpretation for inverse semigroups, which clarifies some aspects of how inverse semigroups and toposes are related.

Discovering Geometric And Topological Properties Of Ellipsoids By Curvatures, Lina Wu, Shihshu Walter Wei, Jia Liu, Ye Li

Publications and Research

Aims/ Objectives: We are interested in discovering the geometric, topological and physical properties of ellipsoids by analyzing curvature properties on ellipsoids. We begin with studying ellipsoids as a starting point. Our aim is to find a way to study geometric, topological and physical properties from the analytic curvature properties for convex hyper-surfaces in the general setting.

Study Design: Multiple-discipline study between Differential Geometry, Topology and Mathematical Physics.

Place and Duration of Study: Department of Mathematics (Borough of Manhattan Community College-The City University of New York), Department of Mathematics (University of Oklahoma), Department of Mathematics and Statistics (University of West Florida), …

Non-Planar On-Shell Diagrams, Sebastían Franco, Daniele Galloni, Brenda Penante, Congkao Wen

Publications and Research

We initiate a systematic study of non-planar on-shell diagrams in N = 4 SYM and develop powerful technology for doing so. We introduce canonical variables generalizing face variables, which make the d log form of the on-shell form explicit. We make significant progress towards a general classification of arbitrary on-shell diagrams by means of two classes of combinatorial objects: generalized matching and matroid polytopes. We propose a boundary measurement that connects general on-shell diagrams to the Grassmannian. Our proposal exhibits two important and non-trivial properties: positivity in the planar case and it matches the combinatorial description of the diagrams in …

On Groups Of Homological Dimension One, Jonathan Cornick

Publications and Research

It has been conjectured that the groups of homological dimension one are precisely the nontrivial locally free groups. Some algebraic, geometric and analytic properties of any potential counter example to the conjecture are discussed.

On C^1 Robust Singular Transitive Sets For Three-Dimensional Flows, Carlos Arnoldo Morales, Maria José Pacífico, Enrique Ramiro Pujals

Publications and Research

Abstract:

The main goal of this paper is to study robust invariant transitive sets containing singularities for C¹ flows on three-dimensional compact boundaryless manifolds:they are partially hyperbolic with volume expanding central direction. Moreover, they are either attractors or repellers. Robust here means that this property cannot be destroyed by small C¹-perturbations of the flow.

Résumé:

Le but de ce travail est d'étudier des ensembles invariants robustes ayant des singularités pour des flots C¹ sur des variétés tridimensionelles : ce sont des ensembles hyperboliques singuliers. << Robuste >> veut dire ici que cette propriété ne peut être détruite par des …

Finite-Difference Approach To The Hodge Theory Of Harmonic Forms, Jozef Dodziuk

Publications and Research

No abstract provided.