scholarly journals Counting curves on surfaces

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 ◽  
Low Dimensional Topology ◽  
Moduli Spaces Of Curves ◽  
Finite Set ◽  
Curves On Surfaces ◽  
Low Dimensional

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.

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.

Geometry and Physics: Volume I

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 Single-valued integration and double copy

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Francis Brown ◽  
Clément Dupont
Keyword(s):  
Moduli Spaces ◽  
Modular Forms ◽  
Differential Forms ◽  
Smooth Projective Variety ◽  
Genus Zero ◽  
Moduli Spaces Of Curves ◽  
Multiple Zeta ◽  
De Rham ◽  
Zeta Values ◽  
Double Copy

AbstractIn this paper, we study a single-valued integration pairing between differential forms and dual differential forms which subsumes some classical constructions in mathematics and physics. It can be interpreted as a p-adic period pairing at the infinite prime. The single-valued integration pairing is defined by transporting the action of complex conjugation from singular to de Rham cohomology via the comparison isomorphism. We show how quite general families of period integrals admit canonical single-valued versions and prove some general formulae for them. This implies an elementary “double copy” formula expressing certain singular volume integrals over the complex points of a smooth projective variety as a quadratic expression in ordinary period integrals of half the dimension. We provide several examples, including non-holomorphic modular forms, archimedean Néron–Tate heights on curves, single-valued multiple zeta values and polylogarithms. The results of the present paper are used in [F. Brown and C. Dupont, Single-valued integration and superstring amplitudes in genus zero, preprint 2019, https://arxiv.org/abs/1910.01107] to prove a recent conjecture of Stieberger which relates the coefficients in a Laurent expansion of two different kinds of periods of twisted cohomology on the moduli spaces of curves {\mathcal{M}_{0,n}} of genus zero with n marked points. We also study a morphism between certain rings of “motivic” periods, called the de Rham projection, which provides a bridge between complex periods and single-valued periods in many situations of interest.

Low-Dimensional Topology and Number Theory

2010 ◽  
pp. 2101-2163 ◽  
Author(s):  
Paul Gunnells ◽  
Walter Neumann ◽  
Adam Sikora ◽  
Don Zagier
Keyword(s):  
Number Theory ◽  
Low Dimensional Topology ◽  
Low Dimensional

scholarly journals Moduli spaces of curves with homology chains and c = 1 matrix models

1994 ◽  
Vol 327 (3-4) ◽  
pp. 221-225 ◽  
Author(s):  
A.S. Cattaneo ◽  
A. Gamba ◽  
M. Martellini
Keyword(s):  
Matrix Models ◽  
Moduli Spaces ◽  
Moduli Spaces Of Curves

scholarly journals Cross-sections of unknotted ribbon disks and algebraic curves

2019 ◽  
Vol 155 (2) ◽  
pp. 413-423
Author(s):  
Kyle Hayden
Keyword(s):  
Cross Sections ◽  
Algebraic Curves ◽  
American Mathematical Society ◽  
Geometric Topology ◽  
Mathematical Society ◽  
Advanced Mathematics ◽  
Low Dimensional Topology ◽  
Low Dimensional ◽  
Holomorphic Disks

We resolve parts (A) and (B) of Problem 1.100 from Kirby’s list [Problems in low-dimensional topology, in Geometric topology, AMS/IP Studies in Advanced Mathematics, vol. 2 (American Mathematical Society, Providence, RI, 1997), 35–473] by showing that many nontrivial links arise as cross-sections of unknotted holomorphic disks in the four-ball. The techniques can be used to produce unknotted ribbon surfaces with prescribed cross-sections, including unknotted Lagrangian disks with nontrivial cross-sections.

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