Geometry and Physics: Volume I

Generalized Complex Structures ◽

Wide Range ◽

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.

Operator Algebras, Mathematical Physics, and Low Dimensional Topology

10.1201/9781439863510 ◽

1993 ◽

Keyword(s):

Operator Algebras ◽

Mathematical Physics ◽

QFT, STRINGS AND LOW-DIMENSIONAL TOPOLOGY

Modern Physics Letters A ◽

10.1142/s0217732302008964 ◽

2002 ◽

Vol 17 (35) ◽

pp. 2289-2295

Author(s):

HU SEN

Keyword(s):

Mirror Symmetry ◽

Theoretical Physics ◽

Yang Mills ◽

Witten Theory ◽

Close Relationship ◽

Jones Polynomials ◽

Recent Developments ◽

Low Dimensional ◽

Chern Simons

In between the 80's and 90's we witnessed deep interactions between mathematics and theoretical physics, especially in the understanding of low-dimensional topology in terms of quantum field theory. For example, Jones polynomials (Chern–Simons–Witten theory), Donaldson and Seiberg–Witten invariants (SUSY Yang–Mills theory) and mirror symmetry (T duality in strings) are all naturally understood in terms of QFT and strings. Recent developments indicate a close relationship between gauge theory and gravity theory both in physics and in low-dimensional topology. We shall survey these developments and report some of our work. We shall also find that the keys to connect geometric and physical objects are through symmetry and quantization.

Introduction

10.1093/oso/9780198794899.003.0001 ◽

2017 ◽

Author(s):

Dusa McDuff ◽

Dietmar Salamon

Keyword(s):

Mirror Symmetry ◽

Complex Geometry ◽

Equations Of Motion ◽

Classical Mechanics ◽

Hamiltonian Formalism ◽

Theoretical Physics ◽

Lagrange Equations ◽

The Subject ◽

Symplectic topology has a long history. It has its roots in classical mechanics and geometric optics and in its modern guise has many connections to other fields of mathematics and theoretical physics ranging from dynamical systems, low-dimensional topology, algebraic and complex geometry, representation theory, and homological algebra, to classical and quantum mechanics, string theory, and mirror symmetry. One of the origins of the subject is the study of the equations of motion arising from the Euler–Lagrange equations of a one-dimensional variational problem. The Hamiltonian formalism arising from a Legendre transformation leads to the notion of a ...

Counting curves on surfaces

International Journal of Mathematics ◽

10.1142/s0129167x17500124 ◽

2017 ◽

Vol 28 (02) ◽

pp. 1750012

Author(s):

Norman Do ◽

Musashi A. Koyama ◽

Daniel V. Mathews

Keyword(s):

Moduli Spaces ◽

Symplectic Geometry ◽

Differential Forms ◽

Combinatorial Problem ◽

Partition Functions ◽

Moduli Spaces Of Curves ◽

Finite Set ◽

Curves On Surfaces ◽

We consider an elementary, and largely unexplored, combinatorial problem in low-dimensional topology: for a compact surface [Formula: see text], with a finite set of points [Formula: see text] fixed on its boundary, how many configurations of disjoint arcs are there on [Formula: see text] whose boundary is [Formula: see text]? We find that this enumerative problem, counting curves on surfaces, has a rich structure. We show that such curve counts obey an effective recursion, in the general spirit of topological recursion, and exhibit quasi-polynomial behavior. This “elementary curve-counting” is in fact related to a more advanced notion of “curve-counting” from algebraic geometry or symplectic geometry. The asymptotics of this enumerative problem are closely related to the asymptotics of volumes of moduli spaces of curves, and the quasi-polynomials governing the enumerative problem encode intersection numbers on moduli spaces. Among several other results, we show that generating functions and differential forms for these curve counts exhibit structure that is reminiscent of the mathematical physics of free energies, partition functions and quantum curves.

A chiral switchable photovoltaic ferroelectric 1D perovskite

Science Advances ◽

10.1126/sciadv.aay4213 ◽

2020 ◽

Vol 6 (9) ◽

pp. eaay4213 ◽

Cited By ~ 6

Author(s):

Yang Hu ◽

Fred Florio ◽

Zhizhong Chen ◽

W. Adam Phelan ◽

Maxime A. Siegler ◽

...

Keyword(s):

Electric Fields ◽

Mirror Symmetry ◽

Degrees Of Freedom ◽

Electrical Measurements ◽

Temperature Dependent ◽

Paraelectric Phase Transition ◽

Strongly Coupled ◽

Yield Phenomena ◽

Hybrid Perovskite ◽

Spin and valley degrees of freedom in materials without inversion symmetry promise previously unknown device functionalities, such as spin-valleytronics. Control of material symmetry with electric fields (ferroelectricity), while breaking additional symmetries, including mirror symmetry, could yield phenomena where chirality, spin, valley, and crystal potential are strongly coupled. Here we report the synthesis of a halide perovskite semiconductor that is simultaneously photoferroelectricity switchable and chiral. Spectroscopic and structural analysis, and first-principles calculations, determine the material to be a previously unknown low-dimensional hybrid perovskite (R)-(−)-1-cyclohexylethylammonium/(S)-(+)-1 cyclohexylethylammonium) PbI3. Optical and electrical measurements characterize its semiconducting, ferroelectric, switchable pyroelectricity and switchable photoferroelectric properties. Temperature dependent structural, dielectric and transport measurements reveal a ferroelectric-paraelectric phase transition. Circular dichroism spectroscopy confirms its chirality. The development of a material with such a combination of these properties will facilitate the exploration of phenomena such as electric field and chiral enantiomer–dependent Rashba-Dresselhaus splitting and circular photogalvanic effects.

Topological mirror symmetry for rank two character varieties of surface groups

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg ◽

10.1007/s12188-021-00246-y ◽

2021 ◽

Cited By ~ 1

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”.