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.

scholarly journals Counting curves on Hirzebruch surfaces: tropical geometry and the Fock space

2021 ◽  
pp. 1-41
Author(s):  
RENZO CAVALIERI ◽  
PAUL JOHNSON ◽  
HANNAH MARKWIG ◽  
DHRUV RANGANATHAN
Keyword(s):  
Fock Space ◽  
Tropical Geometry ◽  
Geometric Interpretation ◽  
Toric Surfaces ◽  
Tropical Curves ◽  
Hirzebruch Surfaces ◽  
Witten Theory ◽  
Toric Degenerations ◽  
Correspondence Theorem ◽  
Floor Diagrams

We study the stationary descendant Gromov–Witten theory of toric surfaces by combining and extending a range of techniques – tropical curves, floor diagrams and Fock spaces. A correspondence theorem is established between tropical curves and descendant invariants on toric surfaces using maximal toric degenerations. An intermediate degeneration is then shown to give rise to floor diagrams, giving a geometric interpretation of this well-known bookkeeping tool in tropical geometry. In the process, we extend floor diagram techniques to include descendants in arbitrary genus. These floor diagrams are then used to connect tropical curve counting to the algebra of operators on the bosonic Fock space, and are showno coincide with the Feynman diagrams of appropriate operators. This extends work of a number of researchers, including Block–Göttsche, Cooper–Pandharipande and Block–Gathmann–Markwig.

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 Cohomological localization of $$ \mathcal{N} $$ = 2 gauge theories with matter

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Guido Festuccia ◽  
Anastasios Gorantis ◽  
Antonio Pittelli ◽  
Konstantina Polydorou ◽  
Lorenzo Ruggeri
Keyword(s):  
Vector Field ◽  
Extended Supersymmetry ◽  
General Framework ◽  
Gauge Theories ◽  
Killing Vector ◽  
Killing Vector Field ◽  
Explicit Result ◽  
Yang Mills ◽  
Self Duality ◽  
Witten Theory

Abstract We construct a large class of gauge theories with extended supersymmetry on four-dimensional manifolds with a Killing vector field and isolated fixed points. We extend previous results limited to super Yang-Mills theory to general $$ \mathcal{N} $$ N = 2 gauge theories including hypermultiplets. We present a general framework encompassing equivariant Donaldson-Witten theory and Pestun’s theory on S4 as two particular cases. This is achieved by expressing fields in cohomological variables, whose features are dictated by supersymmetry and require a generalized notion of self-duality for two-forms and of chirality for spinors. Finally, we implement localization techniques to compute the exact partition function of the cohomological theories we built up and write the explicit result for manifolds with diverse topologies.

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.

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 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 Counting tropical rational space curves with cross-ratio constraints

2021 ◽  
Author(s):  
Christoph Goldner
Keyword(s):  
Tropical Geometry ◽  
Rational Curves ◽  
Cross Ratio ◽  
Current Paper ◽  
Higher Dimension ◽  
Rational Space ◽  
Space Curves ◽  
Correspondence Theorem ◽  
Floor Diagrams

AbstractThis is a follow-up paper of Goldner (Math Z 297(1–2):133–174, 2021), where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in Tyomkin (Adv Math 305:1356–1383, 2017) allowed us to use tropical geometry, and, in particular, a degeneration technique called floor diagrams. This correspondence theorem also holds in higher dimension. In the current paper, we introduce so-called cross-ratio floor diagrams and show that they allow us to determine the number of rational space curves that satisfy general positioned point and cross-ratio conditions. The multiplicities of such cross-ratio floor diagrams can be calculated by enumerating certain rational tropical curves in the plane.

Gromov–Witten Theory of $\text{K3} \times {\mathbb{P}}^1$ and Quasi-Jacobi Forms

2017 ◽  
Vol 2019 (16) ◽  
pp. 4966-5011 ◽  
Author(s):  
Georg Oberdieck
Keyword(s):  
Hyperelliptic Curves ◽  
K3 Surface ◽  
Jacobi Forms ◽  
Hilbert Scheme Of Points ◽  
Hodge Integrals ◽  
Witten Theory ◽  
Generating Series ◽  
Beta 2 ◽  
Formal Properties ◽  
Special Case

Abstract Let $S$ be a K3 surface with primitive curve class $\beta$. We solve the relative Gromov–Witten theory of $S \times {\mathbb{P}}^1$ in classes $(\beta,1)$ and $(\beta,2)$. The generating series are quasi-Jacobi forms and equal to a corresponding series of genus $0$ Gromov–Witten invariants on the Hilbert scheme of points of $S$. This proves a special case of a conjecture of Pandharipande and the author. The new geometric input of the paper is a genus bound for hyperelliptic curves on K3 surfaces proven by Ciliberto and Knutsen. By exploiting various formal properties we find that a key generating series is determined by the very first few coefficients. Let $E$ be an elliptic curve. As collorary of our computations, we prove that Gromov–Witten invariants of $S \times E$ in classes $(\beta,1)$ and $(\beta,2)$ are coefficients of the reciprocal of the Igusa cusp form. We also calculate several linear Hodge integrals on the moduli space of stable maps to a K3 surface and the Gromov–Witten invariants of an abelian threefold in classes of type $(1,1,d)$.

scholarly journals Elliptic modular graph forms. Part I. Identities and generating series

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Eric D’Hoker ◽  
Axel Kleinschmidt ◽  
Oliver Schlotterer
Keyword(s):  
Eisenstein Series ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Differential Identities ◽  
Generating Series ◽  
Higher Genus ◽  
Multiple Integrals ◽  
The Green Function ◽  
Arbitrary Graphs ◽  
Genus One

Abstract Elliptic modular graph functions and forms (eMGFs) are defined for arbitrary graphs as natural generalizations of modular graph functions and forms obtained by including the character of an Abelian group in their Kronecker-Eisenstein series. The simplest examples of eMGFs are given by the Green function for a massless scalar field on the torus and the Zagier single-valued elliptic polylogarithms. More complicated eMGFs are produced by the non-separating degeneration of a higher genus surface to a genus one surface with punctures. eMGFs may equivalently be represented by multiple integrals over the torus of combinations of coefficients of the Kronecker-Eisenstein series, and may be assembled into generating series. These relations are exploited to derive holomorphic subgraph reduction formulas, as well as algebraic and differential identities between eMGFs and their generating series.

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