Gromov-Witten Theory and Automorphic Forms

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.

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

Journal of High Energy Physics ◽

10.1007/jhep09(2020)133 ◽

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.

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On the Bloch–Kato conjecture for Hilbert modular forms

Mathematische Zeitschrift ◽

10.1007/s00209-020-02689-0 ◽

2021 ◽

Author(s):

Matteo Tamiozzo

Keyword(s):

Modular Forms ◽

Automorphic Forms ◽

Euler System ◽

Central Point ◽

Shimura Curves ◽

Hilbert Modular Forms ◽

Reciprocity Laws ◽

Hilbert Modular ◽

Special Points

AbstractThe aim of this paper is to prove inequalities towards instances of the Bloch–Kato conjecture for Hilbert modular forms of parallel weight two, when the order of vanishing of the L-function at the central point is zero or one. We achieve this implementing an inductive Euler system argument which relies on explicit reciprocity laws for cohomology classes constructed using congruences of automorphic forms and special points on several Shimura curves.

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Sums of k-th powers and the Whittaker–Fourier coefficients of automorphic forms

The Ramanujan Journal ◽

10.1007/s11139-020-00332-4 ◽

2021 ◽

Author(s):

Wenzhi Luo

Keyword(s):

Fourier Coefficients ◽

Automorphic Forms

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Regular and residual Eisenstein series and the automorphic cohomology of Sp(2,2)

Compositio Mathematica ◽

10.1112/s0010437x09004266 ◽

2009 ◽

Vol 146 (1) ◽

pp. 21-57 ◽

Cited By ~ 1

Author(s):

Harald Grobner

Keyword(s):

Algebraic Group ◽

Eisenstein Series ◽

Symplectic Group ◽

Automorphic Forms ◽

General Theorem ◽

Simple Algebraic Group ◽

Ramanujan Conjecture ◽

Eisenstein Cohomology ◽

Derivatives Of

AbstractLetGbe the simple algebraic group Sp(2,2), to be defined over ℚ. It is a non-quasi-split, ℚ-rank-two inner form of the split symplectic group Sp8of rank four. The cohomology of the space of automorphic forms onGhas a natural subspace, which is spanned by classes represented by residues and derivatives of cuspidal Eisenstein series. It is called Eisenstein cohomology. In this paper we give a detailed description of the Eisenstein cohomologyHqEis(G,E) ofGin the case of regular coefficientsE. It is spanned only by holomorphic Eisenstein series. For non-regular coefficientsEwe really have to detect the poles of our Eisenstein series. SinceGis not quasi-split, we are out of the scope of the so-called ‘Langlands–Shahidi method’ (cf. F. Shahidi,On certainL-functions, Amer. J. Math.103(1981), 297–355; F. Shahidi,On the Ramanujan conjecture and finiteness of poles for certainL-functions, Ann. of Math. (2)127(1988), 547–584). We apply recent results of Grbac in order to find the double poles of Eisenstein series attached to the minimal parabolicP0ofG. Having collected this information, we determine the square-integrable Eisenstein cohomology supported byP0with respect to arbitrary coefficients and prove a vanishing result. This will exemplify a general theorem we prove in this paper on the distribution of maximally residual Eisenstein cohomology classes.

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