The jet-space of a Frobenius manifold and higher-genus Gromov-Witten invariants

The Nielsen-Thurston Classification

10.23943/princeton/9780691147949.003.0014 ◽

2017 ◽

Author(s):

Benson Farb ◽

Dan Margalit

Keyword(s):

Hyperbolic Geometry ◽

Mapping Class Groups ◽

Classification Theorem ◽

Mapping Class ◽

Fundamental Result ◽

Class Groups ◽

Mapping Torus ◽

Higher Genus ◽

Manifold Theory

This chapter explains and proves the Nielsen–Thurston classification of elements of Mod(S), one of the central theorems in the study of mapping class groups. It first considers the classification of elements for the torus of Mod(T² before discussing higher-genus analogues for each of the three types of elements of Mod(T². It then states the Nielsen–Thurston classification theorem in various forms, as well as a connection to 3-manifold theory, along with Thurston's geometric classification of mapping torus. The rest of the chapter is devoted to Bers' proof of the Nielsen–Thurston classification. The collar lemma is highlighted as a new ingredient, as it is also a fundamental result in the hyperbolic geometry of surfaces.

Download Full-text

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

Forum of Mathematics Pi ◽

10.1017/fmp.2021.3 ◽

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.

Download Full-text

Padé approximants on Riemann surfaces and KP tau functions

Analysis and Mathematical Physics ◽

10.1007/s13324-021-00585-2 ◽

2021 ◽

Vol 11 (4) ◽

Author(s):

Marco Bertola

Keyword(s):

Riemann Surface ◽

Riemann Surfaces ◽

Hilbert Problem ◽

Line Bundles ◽

Fixed Degree ◽

Tau Function ◽

Tau Functions ◽

Deformation Parameters ◽

Higher Genus ◽

Geometric Solutions

AbstractThe paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.

Download Full-text

AdS3/CFT2 at higher genus

Journal of High Energy Physics ◽

10.1007/jhep05(2020)150 ◽

2020 ◽

Vol 2020 (5) ◽

Cited By ~ 1

Author(s):

Lorenz Eberhardt

Keyword(s):

Higher Genus

Download Full-text

Modular orbits at higher genus

Journal of High Energy Physics ◽

10.1007/jhep02(2020)113 ◽

2020 ◽

Vol 2020 (2) ◽

Author(s):

Daniel Robbins ◽

Thomas Vandermeulen

Keyword(s):

Higher Genus

Download Full-text

On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5

Open Mathematics ◽

10.1515/math-2019-0115 ◽

2019 ◽

Vol 17 (1) ◽

pp. 1631-1651

Author(s):

Ick Sun Eum ◽

Ho Yun Jung

Keyword(s):

Modular Forms ◽

Fourier Coefficients ◽

Singular Values ◽

Congruence Subgroups ◽

Maass Form ◽

Weakly Holomorphic Modular Forms ◽

Reciprocity Law ◽

Class Invariants ◽

Higher Genus ◽

Holomorphic Modular Forms

Abstract After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura’s reciprocity law.

Download Full-text