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 DONALDSON–THOMAS INVARIANTS OF LOCAL ELLIPTIC SURFACES VIA THE TOPOLOGICAL VERTEX

2019 ◽  
Vol 7 ◽  
Author(s):  
JIM BRYAN ◽  
MARTIJN KOOL
Keyword(s):  
Partition Function ◽  
Elliptic Surface ◽  
Computational Technique ◽  
Partition Functions ◽  
K3 Surface ◽  
Jacobi Forms ◽  
Topological Vertex ◽  
Elliptic Surfaces ◽  
Special Case ◽  
Product Formulas

We compute the Donaldson–Thomas invariants of a local elliptic surface with section. We introduce a new computational technique which is a mixture of motivic and toric methods. This allows us to write the partition function for the invariants in terms of the topological vertex. Utilizing identities for the topological vertex proved in Bryan et al. [‘Trace identities for the topological vertex’, Selecta Math. (N.S.)24 (2) (2018), 1527–1548, arXiv:math/1603.05271], we derive product formulas for the partition functions. The connected version of the partition function is written in terms of Jacobi forms. In the special case where the elliptic surface is a K3 surface, we get a derivation of the Katz–Klemm–Vafa formula for primitive curve classes which is independent of the computation of Kawai–Yoshioka.

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 On alienation of two functional equations of quadratic type

2021 ◽  
Author(s):  
Roman Ger
Keyword(s):  
Functional Equation ◽  
Functional Equations ◽  
Quadratic Functional Equation ◽  
Quadratic Functional ◽  
Real Numbers ◽  
Alpha Beta ◽  
Beta 2 ◽  
Special Case

Abstract  We deal with an alienation problem for an Euler–Lagrange type functional equation $$\begin{aligned} f(\alpha x + \beta y) + f(\alpha x - \beta y) = 2\alpha ^2f(x) + 2\beta ^2f(y) \end{aligned}$$ f ( α x + β y ) + f ( α x - β y ) = 2 α 2 f ( x ) + 2 β 2 f ( y ) assumed for fixed nonzero real numbers $$\alpha ,\beta ,\, 1 \ne \alpha ^2 \ne \beta ^2$$ α , β , 1 ≠ α 2 ≠ β 2 , and the classic quadratic functional equation $$\begin{aligned} g(x+y) + g(x-y) = 2g(x) + 2g(y). \end{aligned}$$ g ( x + y ) + g ( x - y ) = 2 g ( x ) + 2 g ( y ) . We were inspired by papers of Kim et al. (Abstract and applied analysis, vol. 2013, Hindawi Publishing Corporation, 2013) and Gordji and Khodaei (Abstract and applied analysis, vol. 2009, Hindawi Publishing Corporation, 2009), where the special case $$g = \gamma f$$ g = γ f was examined.

scholarly journals Hodge integrals and Gromov-Witten theory

2000 ◽  
Vol 139 (1) ◽  
pp. 173-199 ◽  
Author(s):  
C. Faber ◽  
R. Pandharipande
Keyword(s):  
Hodge Integrals ◽  
Witten Theory

scholarly journals Are even maps on surfaces likely to be bipartite?

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Guillaume Chapuy
Keyword(s):  
Asymptotic Behaviour ◽  
Planar Map ◽  
Generating Series ◽  
Finite Set ◽  
Positive Genus ◽  
International Audience ◽  
Special Case

International audience It is well known that a planar map is bipartite if and only if all its faces have even degree (what we call an even map). In this paper, we show that rooted even maps of positive genus $g$ chosen uniformly at random are bipartite with probability tending to $4^{−g}$ when their size goes to infinity. Loosely speaking, we show that each of the $2g$ fundamental cycles of the surface of genus $g$ contributes a factor $\frac{1}{2}$ to this probability.We actually do more than that: we obtain the explicit asymptotic behaviour of the number of even maps and bipartite maps of given genus with any finite set of allowed face degrees. This uses a generalisation of the Bouttier-Di Francesco-Guitter bijection to the case of positive genus, a decomposition inspired by previous works of Marcus, Schaeffer and the author, and some involved manipulations of generating series counting paths. A special case of our results implies former conjectures of Gao.

scholarly journals Application of Divisors on a Hyperelliptic Curve in Python

2021 ◽  
Vol 3 ◽  
pp. 11-24
Author(s):  
Denys Boiko
Keyword(s):  
Programming Language ◽  
Hyperelliptic Curve ◽  
Polynomial Function ◽  
Hyperelliptic Curves ◽  
Unique Form ◽  
Python Programming Language ◽  
Definition Of ◽  
Mumford Representation ◽  
Python Programming ◽  
Special Case

The paper studies hyperelliptic curves of the genus g > 1, divisors on them and their applications in Python programming language. The basic necessary definitions and known properties of hyperelliptic curves are demonstrated, as well as the notion of polynomial function, its representation in unique form, also the notion of rational function, norm, degree and conjugate to a polynomial are presented. These facts are needed to calculate the order of points of desirable functions, and thus to quickly and efficiently calculate divisors. The definition of a divisor on a hyperelliptic curve is shown, and the main known properties of a divisor are given. There are also an example of calculating a divisor of a polynomial function, reduced and semi-reduced divisors are described, theorem of the existence of such a not unique semi-reduced divisor, and theorem of the existence of a unique reduced divisor, which is equivalent to the initial one, are proved. In particular, a semi-reduced divisor can be represented as an GCD of divisors of two polynomial functions. It is also demonstrated that each reduced divisor can be represented in unique form by pair of polynomials [a(x), b(x)], which is called Mumford representation, and several examples of its representation calculation are given. There are shown Cantor’s algorithms for calculating the sum of two divisors: its compositional part, by means of which a not unique semi-reduced divisor is formed, and the reduction part, which gives us a unique reduced divisor. In particular, special case of the compositional part of Cantor’s algorithm, doubling of the divisor, is described: it significantly reduces algorithm time complexity. Also the correctness of the algorithms are proved, examples of applications are given. The main result of the work is the implementation of the divisor calculation of a polynomial function, its Mumford representation, and Cantor’s algorithm in Python programming language. Thus, the aim of the work is to demonstrate the possibility of e↵ective use of described algorithms for further work with divisors on the hyperelliptic curve, including the development of cryptosystem, digital signature based on hyperelliptic curves, attacks on such cryptosystems.

HIGHER INTEGRALS OF MOTION IN A PERTURBED k = 1 SU(2) WESS-ZUMINO-WITTEN THEORY

1990 ◽  
Vol 05 (11) ◽  
pp. 823-830 ◽  
Author(s):  
KEN-ICHIRO KOBAYASHI ◽  
TSUNEO UEMATSU
Keyword(s):  
Perturbation Theory ◽  
Integrals Of Motion ◽  
Perturbed System ◽  
Witten Theory ◽  
Special Case ◽  
Conserved Currents ◽  
Relevant Operator

We investigate higher integrals of motion in the k = 1 SU(2) Wess-Zumino-Witten (WZW) model perturbed by a certain relevant operator. While the perturbed system is a special case of a Sine-Gordon theory, it is shown to the lowest order in perturbation theory that there exist extra conserved currents due to the SU(2) symmetry in the original WZW model.

scholarly journals Motivic decompositions for the Hilbert scheme of points of a K3 surface

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Andrei Neguţ ◽  
Georg Oberdieck ◽  
Qizheng Yin
Keyword(s):  
Lie Algebra ◽  
Hilbert Scheme ◽  
K3 Surface ◽  
Hilbert Scheme Of Points

Abstract We construct an explicit, multiplicative Chow–Künneth decomposition for the Hilbert scheme of points of a K3 surface. We further refine this decomposition with respect to the action of the Looijenga–Lunts–Verbitsky Lie algebra.

scholarly journals Polyfolds and Fredholm Theory

2017 ◽  
Author(s):  
Helmut H. W. Hofer
Keyword(s):  
Fredholm Theory ◽  
Stable Maps ◽  
Witten Theory ◽  
Recent Developments ◽  
Special Case

This paper is based on a lecture given at the Clay Mathematics Institute in 2088, but has been rewritten to take account of recent developments. It focuses on a special case of the theory of Fredholm theory in polyfolds, which allows for boundaries with corners, it focuses on a special and illustrates it with a discussion of stable maps, a topic closely related to Gromov-Witten theory.

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