Holomorphic Coordinates Research Articles

Peaked Stokes waves as solutions of Babenko’s equation

Babenko’s equation describes traveling water waves in holomorphic coordinates. It has been used in the past to obtain properties of Stokes waves with smooth profiles analytically and numerically. We show in the deep-water limit that properties of Stokes waves with peaked profiles can also be recovered from the same Babenko’s equation. In order to develop the local analysis of singularities, we rewrite Babenko’s equation as a fixed-point problem near the maximal elevation level. As a by-product, our results rule out a corner point singularity in the holomorphic coordinates, which has been obtained in a local version of Babenko’s equation.

Reconstructing Atiyah-Hitchin manifold in the generalized Legendre transform

We revisit the Atiyah-Hitchin manifold using the generalized Legendre transform approach. Originally it is examined by Ivanov and Roček, and it has been further explored by Ionaș, with a particular focus on calculating the explicit forms of the Kähler potential and the Kähler metric. Notably, there exists a distinction between the former study and the latter. In the framework of the generalized Legendre transform approach, a Kähler potential is formulated through the contour integration of a specific function with holomorphic coordinates. It’s essential to note that the choice of the contour in the latter differs from that in the former. This discrepancy in contour selection may result in variations in both the Kähler potential and, consequently, the Kähler metric. Our findings demonstrate that the former exclusively yields the real Kähler potential, aligning with its defined properties. In contrast, the latter produces a complex Kähler potential. We present the derivation of the Kähler potential and metric for the Atiyah-Hitchin manifold in terms of holomorphic coordinates, considering the contour specified by Ivanov and Roček.

On string one-loop correction to the Einstein-Hilbert term and its implications on the Kähler potential

To compute the string one-loop correction to the Kähler potential of moduli fields of string compactifications in Einstein-frame, one must compute: the string one-loop correction to the Einstein-Hilbert action, the string one-loop correction to the moduli kinetic terms, the string one-loop correction to the definition of the holomorphic coordinates. In this note, in the small warping limit, we compute the string one-loop correction to the Einstein-Hilbert action of type II string theory compactified on orientifolds of Calabi-Yau threefolds. We find that the one-loop correction is determined by the new supersymmetric index studied by Cecotti, Fendley, Intriligator, and Vafa and the Witten index. As a simple application, we apply our results to estimate the size of the one-loop corrections around a conifold point in the Kähler moduli space.

Revisiting atiyah-hitchin manifold in the generalized legendre transform

Abstract We revisit construction of the Atiyah-Hitchin manifold in the generalized Legendre transform approach. This is originally studied by Ivanov and Rocek and is subsequently investigated more by Ionas, in the latter of which the explicit forms of the Kähler potential and the Kähler metric are calculated. There is a difference between the former and the latter. In the generalized Legendre transform approach, a Kähler potential is constructed from the contour integration of one function with holomorphic coordinates. The choice of the contour in the latter is different from the former’s one, whose difference may yield a discrepancy in the Kähler potential and eventually in the Kähler metric. We show that the former only gives the real Kähler potential, which is consistent with its definition, while the latter yields the complex one. We derive the Kähler potential and the metric for the Atiyah-Hitchin manifold in terms of holomorphic coordinates for the contour considered by Ivanov and Roček for the first time.

A hyperkähler geometry associated to the BPS structure of the resolved conifold

We associate to the resolved conifold an affine special Kähler (ASK) manifold of complex dimension 1, and an instanton corrected hyperkähler (HK) manifold of complex dimension 2. We describe these geometries explicitly, and show that the instanton corrected HK geometry realizes an Ooguri-Vafa-like smoothing of the semi-flat HK metric associated to the ASK geometry. On the other hand, the instanton corrected HK geometry associated to the resolved conifold can be described in terms of a twistor family of two holomorphic Darboux coordinates. We study a certain conformal limit of the twistor coordinates, and conjecture a relation to a solution of a Riemann-Hilbert problem previously considered by T. Bridgeland.

Gromov–Hausdorff limits of Kähler manifolds with Ricci curvature bounded below

We show that non-collapsed Gromov–Hausdorff limits of polarized Kähler manifolds, with Ricci curvature bounded below, are normal projective varieties, and the metric singularities of the limit space are precisely given by a countable union of analytic subvarieties. This extends a fundamental result of Donaldson–Sun, in which 2-sided Ricci curvature bounds were assumed. As a basic ingredient we show that, under lower Ricci curvature bounds, almost Euclidean balls in Kähler manifolds admit good holomorphic coordinates. Further applications are integral bounds for the scalar curvature on balls, and a rigidity theorem for Kähler manifolds with almost Euclidean volume growth.

The NLS approximation for two dimensional deep gravity waves

This article is concerned with infinite depth gravity water waves in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. Our goal is to study this problem with small wave packet data, and to show that this is well approximated by the cubic nonlinear Schrodinger equation (NLS) on the natural cubic time scale.

Sasaki-Einstein space T1,1, transverse Kähler-Ricci flow and Sasaki-Ricci soliton

We examine the compact Sasaki manifolds in view of transverse Kähler geometry and transverse Kähler-Ricci flow. In particular we study the transverse Kähler geometry of the five-dimensional Sasaki-Einstein space T1,1. For this purpose a set of local holomorphic coordinates is introduced and a Sasakian analogue of the Kähler potential is produced. There are considered deformations of the contact structure fixing the Reeb foliation while varying the Kähler metric on the transverse holomorphic structure.

Two-dimensional gravity water waves with constant vorticity, I: Cubic lifespan

This article is concerned with the incompressible, infinite depth water wave equation in two space dimensions, with gravity and constant vorticity but with no surface tension. We consider this problem expressed in position-velocity potential holomorphic coordinates, and prove local well-posedness for large data, as well as cubic lifespan bounds for small data solutions.

Sasaki–Ricci flow on Sasaki–Einstein space T1,1 and deformations

We study the transverse Kähler structure of the Sasaki–Einstein space [Formula: see text]. A set of local holomorphic coordinates is introduced and a Sasakian analogue of the Kähler potential is given. We investigate deformations of the Sasaki–Einstein structure preserving the Reeb vector field, but modifying the contact form. For this kind of deformations, we consider the Sasaki–Ricci flow which converges in a suitable sense to a Sasaki–Ricci soliton. Finally, it is described the constructions of Hamiltonian holomorphic vector fields and Hamiltonian function on the [Formula: see text] manifold.

The Lifespan of Small Data Solutions in Two Dimensional Capillary Water Waves

This article is concerned with the incompressible, irrotational infinite depth water wave equation in two space dimensions, without gravity but with surface tension. We consider this problem expressed in position–velocity potential holomorphic coordinates, and prove that small data solutions have at least cubic lifespan while small localized data leads to global solutions.

Finite Depth Gravity Water Waves in Holomorphic Coordinates

In this article we consider irrotational gravity water waves with finite bottom. Our goal is two-fold. First, we represent the equations in holomorphic coordinates and discuss the local well-posedness of the problem in this context. Second, we consider the small data problem and establish cubic lifespan bounds for the solutions. Our results are uniform in the infinite depth limit, and match our earlier infinite depth result in Hunter et al. (Commun Math Phys, 2014).

Two Dimensional Water Waves in Holomorphic Coordinates

This article is concerned with the infinite depth water wave equation in two space dimensions. We consider this problem expressed in position-velocity potential holomorphic coordinates. Viewing this problem as a quasilinear dispersive equation, we establish two results: (i) local well-posedness in Sobolev spaces, and (ii) almost global solutions for small localized data. Neither of these results are new; they have been recently obtained by Alazard–Burq–Zuily (Invent Math 198:71–163, 2014), respectively by Wu (Invent Math 177:45–135, 2009) using different coordinates and methods. Instead our goal is to improve the understanding of this problem by providing a single setting for both problems, by proving sharper versions of the above results, as well as presenting new, simpler proofs. This article is self contained.

Two dimensional water waves in holomorphic coordinates II: global solutions

Author(s): Ifrim, Mihaela; Tataru, Daniel | Abstract: This article is concerned with the infinite depth water wave equation in two space dimensions. We consider this problem expressed in position-velocity potential holomorphic coordinates,and prove that small localized data leads to global solutions. This article is a continuation of authors' earlier paper arXiv:1401.1252.

Darboux coordinates, Yang-Yang functional, and gauge theory

The moduli space of flat SL 2 connections on a punctured Riemann surface Σ with fixed conjugacy classes of the monodromies around the punctures is endowed with a system of holomorphic Darboux coordinates in which the generating function of the variety of SL 2 -opers is identified with the universal part of the effective twisted superpotential of the corresponding four-dimensional N=2 supersymmetric theory subject to the two-dimensional Ω-deformation. This allows defining the Yang-Yang functionals for the quantum Hitchin system in terms of the classical geometry of the moduli space of local systems for the dual gauge group and relating it to the instanton counting of the four-dimensional gauge theories in the rank-one case.

Quasi-coherent states for a photon in time varying dielectric media

In this study, we derive the quasi-coherent states for a photon in a medium with time varying dielectric permittivity ϵ0eηt. We introduce a new kind of holomorphic coordinates and solve the Schrödinger equation for a harmonic oscillator with the time dependent parameters by reducing it to the Riccati equation for which the nonstationary solution gives the squeezed states of a photon with non-minimum uncertainties. We also derive the quasi-coherent states and discrete, nonstationary quantum states in the configuration space of a photon.

Quasi-coherent states for harmonic oscillator with time-dependent parameters

In this study, we discuss the harmonic oscillator with the time-dependent frequency, ω(t), and the mass, M(t), by generalizing the holomorphic coordinates for the harmonic oscillator. In general cases, we solve the Schrödinger equation by reducing it into the Riccati equation and discuss the uncertainties for the quasi-coherent states of the time-dependent harmonic oscillator. In special cases, we find the following results: First, for a time-dependent harmonic oscillator, if [ω(t)M(t)] is constant, then the coherent states will evolve as the coherent states. Second, for the driven harmonic oscillator, the coherent states will evolve as the coherent states with new eigenvalues. Third, we derive quasi-coherent states for the Caldirola–Kanai Hamiltonian and show that the product of uncertainties, ΔxΔp, is larger than minimum value; however, it is constant. We also discuss the classical equations of motion for the system.

Bottom of Spectrum of Kähler Manifolds with a Strongly Pseudoconvex Boundary

Theorem 1. Suppose (M, g) is a noncompact complete Riemannian manifold with Ric ≥ − (n− 1), we have λ0 ≤ (n− 1) /4. The estimate is sharp since the spectrum is the ray [(n− 1) /4,+∞) for the hyperbolic space H. For Kahler manifolds, this estimate can be improved. On a Kahler manifold (M, g) of complex dimension n, where g is the Riemannian metric, let ω = g (J ·, ·) be the Kahler form. In local holomorphic coordinates z1, · · · , zn, we have

Darboux coordinates, Yang-Yang functional, and gauge theory

The moduli space of SL(2) flat connections on a punctured Riemann surface with the fixed conjugacy classes of the monodromies around the punctures is endowed with a system of holomorphic Darboux coordinates, in which the generating function of the variety of SL(2)-opers is identified with the universal part of the effective twisted superpotential of the corresponding four dimensional N=2 supersymmetric theory subject to the two-dimensional Omega-deformation. This allows to give a definition of the Yang-Yang functionals for the quantum Hitchin system in terms of the classical geometry of the moduli space of local systems for the dual gauge group, and connect it to the instanton counting of the four dimensional gauge theories, in the rank one case.

Flavor hierarchy from F-theory

It has recently been shown that F-theory based constructions provide a potentially promising avenue for engineering GUT models which descend to the MSSM. In this Note we show that in the presence of background fluxes, these models automatically achieve hierarchical Yukawa matrices in the quark and lepton sectors. At leading order, the existence of a U ( 1 ) symmetry which is related to phase rotations of the internal holomorphic coordinates at the brane intersection point leads to rank one Yukawa matrices. Subleading corrections to the internal wave functions from variations in the background fluxes generate small violations of this U ( 1 ) , leading to hierarchical Yukawa structures reminiscent of the Froggatt–Nielsen mechanism. The expansion parameter for this perturbation is in terms of α GUT . Moreover, we naturally obtain a hierarchical CKM matrix with V 12 ∼ V 21 ∼ ε , V 23 ∼ V 32 ∼ ε 2 , V 13 ∼ V 31 ∼ ε 3 , where ε ∼ α GUT , in excellent agreement with observation.