scholarly journals Quantized Moduli Spaces of the Bundles on the Elliptic Curve and Their Applications

2001 ◽  
pp. 123-137 ◽  
Author(s):  
B. L. Feigin ◽  
A. V. Odesskii
Keyword(s):  
Elliptic Curve ◽  
Moduli Spaces

scholarly journals Holomorphic anomaly equation for and the Nekrasov-Shatashvili limit of local

2021 ◽  
Vol 9 ◽  
Author(s):  
Pierrick Bousseau ◽  
Honglu Fan ◽  
Shuai Guo ◽  
Longting Wu
Keyword(s):  
Elliptic Curve ◽  
Moduli Spaces ◽  
Topological String ◽  
Holomorphic Anomaly ◽  
Holomorphic Anomaly Equation ◽  
Anomaly Equation ◽  
Witten Theory ◽  
Generating Series ◽  
Maximal Contact ◽  
Higher Genus

Abstract We prove a higher genus version of the genus $0$ local-relative correspondence of van Garrel-Graber-Ruddat: for $(X,D)$ a pair with X a smooth projective variety and D a nef smooth divisor, maximal contact Gromov-Witten theory of $(X,D)$ with $\lambda _g$ -insertion is related to Gromov-Witten theory of the total space of ${\mathcal O}_X(-D)$ and local Gromov-Witten theory of D. Specializing to $(X,D)=(S,E)$ for S a del Pezzo surface or a rational elliptic surface and E a smooth anticanonical divisor, we show that maximal contact Gromov-Witten theory of $(S,E)$ is determined by the Gromov-Witten theory of the Calabi-Yau 3-fold ${\mathcal O}_S(-E)$ and the stationary Gromov-Witten theory of the elliptic curve E. Specializing further to $S={\mathbb P}^2$ , we prove that higher genus generating series of maximal contact Gromov-Witten invariants of $({\mathbb P}^2,E)$ are quasimodular and satisfy a holomorphic anomaly equation. The proof combines the quasimodularity results and the holomorphic anomaly equations previously known for local ${\mathbb P}^2$ and the elliptic curve. Furthermore, using the connection between maximal contact Gromov-Witten invariants of $({\mathbb P}^2,E)$ and Betti numbers of moduli spaces of semistable one-dimensional sheaves on ${\mathbb P}^2$ , we obtain a proof of the quasimodularity and holomorphic anomaly equation predicted in the physics literature for the refined topological string free energy of local ${\mathbb P}^2$ in the Nekrasov-Shatashvili limit.

scholarly journals Moduli Space in Homological Mirror Symmetry

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Matsuo Sato
Keyword(s):  
Elliptic Curve ◽  
Configuration Space ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Mirror Symmetry ◽  
Feynman Diagrams ◽  
Holomorphic Curves ◽  
Homological Mirror Symmetry

We prove that the moduli space of the pseudo holomorphic curves in the A-model on a symplectic torus is homeomorphic to a moduli space of Feynman diagrams in the configuration space of the morphisms in the B-model on the corresponding elliptic curve. These moduli spaces determine the A∞ structure of the both models.

Integrable Systems Associated to a Hopf Surface

2003 ◽  
Vol 55 (3) ◽  
pp. 609-635 ◽  
Author(s):  
Ruxandra Moraru
Keyword(s):  
Elliptic Curve ◽  
Moduli Spaces ◽  
Symplectic Geometry ◽  
Vector Bundles ◽  
Integrable Hamiltonian System ◽  
Complex Surface ◽  
Point Of View ◽  
Infinite Cyclic Group ◽  
Completely Integrable Hamiltonian System ◽  
Hopf Surface

AbstractA Hopf surface is the quotient of the complex surface by an infinite cyclic group of dilations of . In this paper, we study the moduli spaces of stable -bundles on a Hopf surface , from the point of view of symplectic geometry. An important point is that the surface is an elliptic fibration, which implies that a vector bundle on can be considered as a family of vector bundles over an elliptic curve. We define a map that associates to every bundle on a divisor, called the graph of the bundle, which encodes the isomorphism class of the bundle over each elliptic curve. We then prove that the map G is an algebraically completely integrable Hamiltonian system, with respect to a given Poisson structure on . We also give an explicit description of the fibres of the integrable system. This example is interesting for several reasons; in particular, since the Hopf surface is not Kähler, it is an elliptic fibration that does not admit a section.

scholarly journals COHERENT SYSTEMS ON ELLIPTIC CURVES

2005 ◽  
Vol 16 (07) ◽  
pp. 787-805 ◽  
Author(s):  
H. LANGE ◽  
P. E. NEWSTEAD
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Moduli Spaces ◽  
Coherent Systems

In this paper we consider coherent systems (E,V) on an elliptic curve which are α-stable with respect to some value of a parameter α. We show that the corresponding moduli spaces, if non-empty, are smooth and irreducible of the expected dimension. Moreover we give precise conditions for non-emptiness of the moduli spaces. Finally we study the variation of the moduli spaces with α.

scholarly journals A Map Between Moduli Spaces of Connections

2020 ◽  
Author(s):  
Frank Loray ◽  
◽  
Valente Ramírez ◽  
Keyword(s):  
Normal Form ◽  
Elliptic Curve ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Riemann Sphere ◽  
Target Space ◽  
Transformation Rule ◽  
Rank 2 ◽  
Logarithmic Connection ◽  
Do So

We are interested in studying moduli spaces of rank 2 logarithmic connections on elliptic curves having two poles. To do so, we investigate certain logarithmic rank 2 connections defined on the Riemann sphere and a transformation rule to lift such connections to an elliptic curve. The transformation is as follows: given an elliptic curve C with elliptic quotient, and the logarithmic connection, we may pullback the connection to the elliptic curve to obtain a new connection on C. After suitable birational modifications we bring the connection to a particular normal form. The whole transformation is equivariant with respect to bundle automorphisms and therefore defines a map between the corresponding moduli spaces of connections. The aim of this paper is to describe the moduli spaces involved and compute explicit expressions for the above map in the case where the target space is the moduli space of rank 2 logarithmic connections on an elliptic curve C with two simple poles and trivial determinant.

scholarly journals Invariants for trees of non-archimedean polynomials and skeleta of superelliptic curves

2022 ◽  
Author(s):  
Paul Alexander Helminck
Keyword(s):  
Elliptic Curve ◽  
Moduli Spaces ◽  
Newton Polygon ◽  
Semistable Reduction ◽  
Plücker Coordinates ◽  
Different Types ◽  
Residue Characteristic ◽  
Superelliptic Curves

AbstractIn this paper we generalize the j-invariant criterion for the semistable reduction type of an elliptic curve to superelliptic curves X given by $$y^{n}=f(x)$$ y n = f ( x ) . We first define a set of tropical invariants for f(x) using symmetrized Plücker coordinates and we show that these invariants determine the tree associated to f(x). This tree then completely determines the reduction type of X for n that are not divisible by the residue characteristic. The conditions on the tropical invariants that distinguish between the different types are given by half-spaces as in the elliptic curve case. These half-spaces arise naturally as the moduli spaces of certain Newton polygon configurations. We give a procedure to write down their equations and we illustrate this by giving the half-spaces for polynomials of degree $$d\le {5}$$ d ≤ 5 .

scholarly journals HIGGS BUNDLES OVER ELLIPTIC CURVES FOR COMPLEX REDUCTIVE GROUPS

2018 ◽  
Vol 61 (2) ◽  
pp. 297-320
Author(s):  
EMILIO FRANCO ◽  
OSCAR GARCIA-PRADA ◽  
P. E. NEWSTEAD
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Moduli Spaces ◽  
Structure Group ◽  
Reductive Groups ◽  
Higgs Bundles ◽  
Arbitrary Degree ◽  
Hitchin Fibration

AbstractWe study Higgs bundles over an elliptic curve with complex reductive structure group, describing the (normalisation of) its moduli spaces and the associated Hitchin fibration. The case of trivial degree is covered by the work of Thaddeus in 2001. Our arguments are different from those of Thaddeus and cover arbitrary degree.

Study of Safe Elliptic Curve Cryptography over Gaussian Integer

2020 ◽  
Vol E103.A (12) ◽  
pp. 1624-1628
Author(s):  
Kazuki NAGANUMA ◽  
Takashi SUZUKI ◽  
Hiroyuki TSUJI ◽  
Tomoaki KIMURA
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curve Cryptography ◽  
Gaussian Integer
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