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.

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 Endoscopic decompositions and the Hausel–Thaddeus conjecture

2021 ◽  
Vol 9 ◽  
Author(s):  
Davesh Maulik ◽  
Junliang Shen
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Mirror Symmetry ◽  
Decomposition Theorem ◽  
Higgs Bundles ◽  
Hitchin Fibration ◽  
Vanishing Cycle ◽  
Natural Operators

Abstract We construct natural operators connecting the cohomology of the moduli spaces of stable Higgs bundles with different ranks and genera which, after numerical specialisation, recover the topological mirror symmetry conjecture of Hausel and Thaddeus concerning $\mathrm {SL}_n$ - and $\mathrm {PGL}_n$ -Higgs bundles. This provides a complete description of the cohomology of the moduli space of stable $\mathrm {SL}_n$ -Higgs bundles in terms of the tautological classes, and gives a new proof of the Hausel–Thaddeus conjecture, which was also proven recently by Gröchenig, Wyss and Ziegler via p-adic integration. Our method is to relate the decomposition theorem for the Hitchin fibration, using vanishing cycle functors, to the decomposition theorem for the twisted Hitchin fibration, whose supports are simpler.

Geometry and Physics: Volume II

2018 ◽  
Keyword(s):  
Moduli Spaces ◽  
Mirror Symmetry ◽  
Geometric Analysis ◽  
Mathematical Physics ◽  
Poisson Geometry ◽  
Special Holonomy ◽  
Low Dimensional Topology ◽  
Generalized Complex Structures ◽  
Wide Range ◽  
Low Dimensional

These volumes contain the proceedings of the conference held at Aarhus, Oxford and Madrid in September 2016 to mark the seventieth birthday of Nigel Hitchin, one of the world’s foremost geometers and Savilian Professor of Geometry at Oxford. The proceedings contain twenty-nine articles, including three by Fields medallists (Donaldson, Mori and Yau). The articles cover a wide range of topics in geometry and mathematical physics, including the following: Riemannian geometry, geometric analysis, special holonomy, integrable systems, dynamical systems, generalized complex structures, symplectic and Poisson geometry, low-dimensional topology, algebraic geometry, moduli spaces, Higgs bundles, geometric Langlands programme, mirror symmetry and string theory. These volumes will be of interest to researchers and graduate students both in geometry and mathematical physics.

scholarly journals On the Voevodsky motive of the moduli space of Higgs bundles on a curve

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Victoria Hoskins ◽  
Simon Pepin Lehalleur
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Characteristic Zero ◽  
Rational Point ◽  
Triangulated Category ◽  
Higgs Bundles ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
De Rham

AbstractWe study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky’s triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.

scholarly journals Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Song He ◽  
Zhenjie Li ◽  
Prashanth Raman ◽  
Chi Zhang
Keyword(s):  
Large Class ◽  
Configuration Space ◽  
Moduli Space ◽  
Cluster Algebras ◽  
Configuration Spaces ◽  
Minkowski Sum ◽  
Canonical Forms ◽  
Infinite Class ◽  
Special Cases ◽  
Generalized Associahedra

Abstract Stringy canonical forms are a class of integrals that provide α′-deformations of the canonical form of any polytopes. For generalized associahedra of finite-type cluster algebras, there exist completely rigid stringy integrals, whose configuration spaces are the so-called binary geometries, and for classical types are associated with (generalized) scattering of particles and strings. In this paper, we propose a large class of rigid stringy canonical forms for another class of polytopes, generalized permutohedra, which also include associahedra and cyclohedra as special cases (type An and Bn generalized associahedra). Remarkably, we find that the configuration spaces of such integrals are also binary geometries, which were suspected to exist for generalized associahedra only. For any generalized permutohedron that can be written as Minkowski sum of coordinate simplices, we show that its rigid stringy integral factorizes into products of lower integrals for massless poles at finite α′, and the configuration space is binary although the u equations take a more general form than those “perfect” ones for cluster cases. Moreover, we provide an infinite class of examples obtained by degenerations of type An and Bn integrals, which have perfect u equations as well. Our results provide yet another family of generalizations of the usual string integral and moduli space, whose physical interpretations remain to be explored.

scholarly journals EXTREMAL CASES OF RAPOPORT–ZINK SPACES

2021 ◽  
pp. 1-56
Author(s):  
Ulrich Görtz ◽  
Xuhua He ◽  
Michael Rapoport
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Complete List ◽  
Qualitative Properties ◽  
Model Case ◽  
Divisible Groups

Abstract We investigate qualitative properties of the underlying scheme of Rapoport–Zink formal moduli spaces of p-divisible groups (resp., shtukas). We single out those cases where the dimension of this underlying scheme is zero (resp., those where the dimension is the maximal possible). The model case for the first alternative is the Lubin–Tate moduli space, and the model case for the second alternative is the Drinfeld moduli space. We exhibit a complete list in both cases.

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 THE HYPERELLIPTIC THETA MAP AND OSCULATING PROJECTIONS

2020 ◽  
pp. 1-23
Author(s):  
MICHELE BOLOGNESI ◽  
NÉSTOR FERNÁNDEZ VARGAS
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Hyperelliptic Curve ◽  
Geometric Description ◽  
Birational Model ◽  
Rank 2 ◽  
Semistable Vector Bundles ◽  
Ramification Locus

Abstract Let C be a hyperelliptic curve of genus $g \geq 3$ . In this paper, we give a new geometric description of the theta map for moduli spaces of rank 2 semistable vector bundles on C with trivial determinant. In order to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb {P}^1)^{2g}//\text {PGL(2)}$ . Then, we identify the restriction of the theta map to these GIT quotients with some explicit degree 2 osculating projection. As a corollary of this construction, we obtain a birational inclusion of a fibration in Kummer $(g-1)$ -varieties over $\mathbb {P}^g$ inside the ramification locus of the theta map.

scholarly journals Topological mirror symmetry for rank two character varieties of surface groups

2021 ◽  
Author(s):  
Mirko Mauri
Keyword(s):  
Moduli Spaces ◽  
Mirror Symmetry ◽  
Surface Groups ◽  
Character Varieties ◽  
Hitchin System ◽  
Global Topology ◽  
Hodge Numbers

AbstractThe moduli spaces of flat $${\text{SL}}_2$$ SL 2 - and $${\text{PGL}}_2$$ PGL 2 -connections are known to be singular SYZ-mirror partners. We establish the equality of Hodge numbers of their intersection (stringy) cohomology. In rank two, this answers a question raised by Tamás Hausel in Remark 3.30 of “Global topology of the Hitchin system”.

scholarly journals A note on moduli spaces of conformal classes for flat tori of higher dimension and on their conformal multiplication

2021 ◽  
Author(s):  
Anna Gori ◽  
Alberto Verjovsky ◽  
Fabio Vlacci
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Abelian Varieties ◽  
Complex Multiplication ◽  
Higher Dimension ◽  
Geometric Constructions ◽  
Flat Torus ◽  
Flat Tori

AbstractMotivated by the theory of complex multiplication of abelian varieties, in this paper we study the conformality classes of flat tori in $${\mathbb {R}}^{n}$$ R n and investigate criteria to determine whether a n-dimensional flat torus has non trivial (i.e. bigger than $${\mathbb {Z}}^{*}={\mathbb {Z}}{\setminus }\{0\}$$ Z ∗ = Z \ { 0 } ) semigroup of conformal endomorphisms (the analogs of isogenies for abelian varieties). We then exhibit several geometric constructions of tori with this property and study the class of conformally equivalent lattices in order to describe the moduli space of the corresponding tori.

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