d-elliptic Loci in Genus 2 and 3

2020 ◽  
Author(s):  
Carl Lian
Keyword(s):  
Witten Theory ◽  
Higher Genus ◽  
Genus 2 ◽  
Fixed Genus

Abstract We consider the loci of curves of genus 2 and 3 admitting a $d$-to-1 map to a genus 1 curve. After compactifying these loci via admissible covers, we obtain formulas for their Chow classes, recovering results of Faber–Pagani and van Zelm when $d=2$. The answers exhibit quasimodularity properties similar to those in the Gromov–Witten theory of a fixed genus 1 curve; we conjecture that the quasimodularity persists in higher genus and indicate a number of possible variants.

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 topological symmetric orbifold

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Songyuan Li ◽  
Jan Troost
Keyword(s):  
Complex Surface ◽  
Conformal Field Theories ◽  
Hurwitz Numbers ◽  
Conformal Field ◽  
Witten Theory ◽  
Higher Genus ◽  
Chiral Ring ◽  
Physics Literature ◽  
Structure Constants ◽  
Genus One

Abstract We analyze topological orbifold conformal field theories on the symmetric product of a complex surface M. By exploiting the mathematics literature we show that a canonical quotient of the operator ring has structure constants given by Hurwitz numbers. This proves a conjecture in the physics literature on extremal correlators. Moreover, it allows to leverage results on the combinatorics of the symmetric group to compute more structure constants explicitly. We recall that the full orbifold chiral ring is given by a symmetric orbifold Frobenius algebra. This construction enables the computation of topological genus zero and genus one correlators, and to prove the vanishing of higher genus contributions. The efficient description of all topological correlators sets the stage for a proof of a topological AdS/CFT correspondence. Indeed, we propose a concrete mathematical incarnation of the proof, relating Gromow-Witten theory in the bulk to the cohomology of the Hilbert scheme on the boundary.

scholarly journals SPHERICAL TRIANGLES AND THE TWO COMPONENTS OF THE SO(3)-CHARACTER SPACE OF THE FUNDAMENTAL GROUP OF A CLOSED SURFACE OF GENUS 2

2011 ◽  
Vol 22 (09) ◽  
pp. 1261-1364 ◽  
Author(s):  
SUHYOUNG CHOI
Keyword(s):  
Fundamental Group ◽  
Closed Surface ◽  
Branch Locus ◽  
Character Space ◽  
Branch Covering ◽  
Higher Genus ◽  
Genus 2 ◽  
Spherical Triangles ◽  
Geometric Techniques ◽  
Future Work

We use geometric techniques to explicitly find the topological structure of the space of SO (3)-representations of the fundamental group of a closed surface of genus two quotient by the conjugation action by SO (3). There are two components of the space. We will describe the topology of both components and describe the corresponding SU (2)-character spaces by parametrizing them by spherical triangles. There is the sixteen to one branch-covering for each component, and the branch locus is a union of two-spheres or two-tori. Along the way, we also describe the topology of both spaces. We will later relate this result to future work into higher-genus cases and the SL (3, ℝ)-representations.

scholarly journals Counting Partitions of a Fixed Genus

10.37236/7632 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Robert Cori ◽  
Gábor Hetyei
Keyword(s):  
Generating Function ◽  
Computer Assisted ◽  
Genus 2 ◽  
Fixed Genus ◽  
Element Set

We show that, for any fixed genus $g$, the ordinary generating function for the genus $g$ partitions of an $n$-element set into $k$ blocks is algebraic. The proof involves showing that each such partition may be reduced in a unique way to a primitive partition and that the number of primitive partitions of a given genus is finite. We illustrate our method by finding the generating function for genus $2$ partitions, after identifying all genus $2$ primitive partitions, using a computer-assisted search.

scholarly journals HIGHER GENUS GROMOV–WITTEN THEORY OF AND ASSOCIATED TO LOCAL CURVES

2019 ◽  
Vol 7 ◽  
Author(s):  
RAHUL PANDHARIPANDE ◽  
HSIAN-HUA TSENG
Keyword(s):  
Hilbert Scheme ◽  
American Mathematical Society ◽  
Quantum Cohomology ◽  
Stable Curves ◽  
Hilbert Scheme Of Points ◽  
Pure Mathematics ◽  
Witten Theory ◽  
Cohomological Field Theories ◽  
Higher Genus ◽  
Crepant Resolution Conjecture

We study the higher genus equivariant Gromov–Witten theory of the Hilbert scheme of $n$ points of $\mathbb{C}^{2}$ . Since the equivariant quantum cohomology, computed by Okounkov and Pandharipande [Invent. Math. 179 (2010), 523–557], is semisimple, the higher genus theory is determined by an $\mathsf{R}$ -matrix via the Givental–Teleman classification of Cohomological Field Theories (CohFTs). We uniquely specify the required $\mathsf{R}$ -matrix by explicit data in degree $0$ . As a consequence, we lift the basic triangle of equivalences relating the equivariant quantum cohomology of the Hilbert scheme $\mathsf{Hilb}^{n}(\mathbb{C}^{2})$ and the Gromov–Witten/Donaldson–Thomas correspondence for 3-fold theories of local curves to a triangle of equivalences in all higher genera. The proof uses the analytic continuation of the fundamental solution of the QDE of the Hilbert scheme of points determined by Okounkov and Pandharipande [Transform. Groups 15 (2010), 965–982]. The GW/DT edge of the triangle in higher genus concerns new CohFTs defined by varying the 3-fold local curve in the moduli space of stable curves. The equivariant orbifold Gromov–Witten theory of the symmetric product $\mathsf{Sym}^{n}(\mathbb{C}^{2})$ is also shown to be equivalent to the theories of the triangle in all genera. The result establishes a complete case of the crepant resolution conjecture [Bryan and Graber, Algebraic Geometry–Seattle 2005, Part 1, Proceedings of Symposia in Pure Mathematics, 80 (American Mathematical Society, Providence, RI, 2009), 23–42; Coates et al., Geom. Topol. 13 (2009), 2675–2744; Coates & Ruan, Ann. Inst. Fourier (Grenoble) 63 (2013), 431–478].

scholarly journals Refined floor diagrams from higher genera and lambda classes

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Pierrick Bousseau
Keyword(s):  
Explicit Result ◽  
Change Of Variables ◽  
Hirzebruch Surfaces ◽  
Witten Theory ◽  
Generating Series ◽  
Higher Genus ◽  
Degeneration Formula ◽  
Correspondence Theorem ◽  
Floor Diagrams

AbstractWe show that, after the change of variables $$q=e^{iu}$$ q = e iu , refined floor diagrams for $${\mathbb {P}}^2$$ P 2 and Hirzebruch surfaces compute generating series of higher genus relative Gromov–Witten invariants with insertion of a lambda class. The proof uses an inductive application of the degeneration formula in relative Gromov–Witten theory and an explicit result in relative Gromov–Witten theory of $${\mathbb {P}}^1$$ P 1 . Combining this result with the similar looking refined tropical correspondence theorem for log Gromov–Witten invariants, we obtain a non-trivial relation between relative and log Gromov–Witten invariants for $${\mathbb {P}}^2$$ P 2 and Hirzebruch surfaces. We also prove that the Block–Göttsche invariants of $${\mathbb {F}}_0$$ F 0 and $${\mathbb {F}}_2$$ F 2 are related by the Abramovich–Bertram formula.

Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2

1996 ◽  
Author(s):  
J. W. S. Cassels ◽  
E. V. Flynn
Keyword(s):  
Genus 2

The Nielsen-Thurston Classification

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.

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