Powers of the Eta-Function and Hecke Operators

Carney, Alexander; Etropolski, Anastassia; Pitman, Sarah

doi:10.1142/s1793042112500339

Cited by 4 publications

(2 citation statements)

References 6 publications

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“…It is in fact possible to give concise expressions for the expansion of Z K N [T 2 ](τ ) in terms of J(τ ) in the case that N is prime, as a special case of the results in [99]. Define B(x; q) := E 2 4 (q)E 6 (q) q(j(q) − x) ,…”

Section: D/4d Correspondences and Hecke Transformsmentioning

confidence: 99%

IIB flux non-commutativity and the global structure of field theories

García-Etxebarria

Heidenreich

Regalado

2019

J. High Energ. Phys.

181

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We discuss the origin of the choice of global structure for six dimensional (2, 0) theories and their compactifications in terms of their realization from IIB string theory on ALE spaces. We find that the ambiguity in the choice of global structure on the field theory side can be traced back to a subtle effect that needs to be taken into account when specifying boundary conditions at infinity in the IIB orbifold, namely the known non-commutativity of RR fluxes in spaces with torsion. As an example, we show how the classification of N = 4 theories by Aharony, Seiberg and Tachikawa can be understood in terms of choices of boundary conditions for RR fields in IIB. Along the way we encounter a formula for the fractional instanton number of N = 4 ADE theories in terms of the torsional linking pairing for rational homology spheres. We also consider six-dimensional (1, 0) theories, clarifying the rules for determining commutators of flux operators for discrete 2-form symmetries. Finally, we analyze the issue of global structure for four dimensional theories in the presence of duality defects. arXiv:1908.08027v2 [hep-th] 9 Oct 20191 The separation into background and intrinsic data is sometimes arbitrary: if we restrict ourselves to fourdimensional Yang-Mills theories with constant coupling τ we could view τ as part of the data defining T . However, if we wish to allow for the possibility that τ varies across M then we must include it as part of the background data to be specified for each manifold. The second interpretation will be more natural from the point of view in this paper, and such configurations will play an interesting role below.2 In this paper we will take M6 to be closed, Spin and orientable, and furthermore we will assume that the cohomology groups of M6 are freely generated, so there is no torsion. 3 The free, or "abelian", (2, 0) theory can be obtained by replacing C 2 /Γ by a single-centered Taub-NUT space.5 The two groups are related by the short exact sequence 0 → W 4 → H 4 (M6 × S 3 /Zn; U (1)) → Tor(H 5 (M6 × S 3 /ZN ; Z)) → 0 , with W 4 the group of topologically trivial C4 Wilson lines on M6 × S 3 /Zn.

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Section: D/4d Correspondences and Hecke Transformsmentioning

confidence: 99%

IIB flux non-commutativity and the global structure of field theories

García-Etxebarria

Heidenreich

Regalado

2019

J. High Energ. Phys.

181

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“…More precisely, Ono derived a closed formula of half-integral Hecke operators T l 2 acting on η −1 (24z). More recently, [12] obtained closed formulae of half-integral Hecke operators T l 2 acting on eta-powers η r with the variable z replaced by some N z of suitable integer N . Such closed formulae are useful in studying congruence properties of Fourier coefficients of eta-powers.…”

Section: Introduction 1hecke Operators and Double Coset Operatorsmentioning

confidence: 99%

Double coset operators and eta quotients

Zhou¹,

Zhu²

2021

Preprint

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We study a sort of generalized double coset operators, which feature changing the characters of modular forms. For any pair of characters v1 and v2, we describe explicitly those operators mapping modular forms of character v1 to those of v2. We give three applications, concerned with eta-quotients. For the first application, we give many pairs of etaquotients of small weights and levels, such that there are operators maps one eta-quotient to another. We also find out these operators. For the second application, we apply the operators to eta-powers whose exponents are positive integers not greater than 24. This results in recursive formulae of the coefficients of these functions, generalizing Newman's theorem. For the third application, we decribe a criterion and an algorithm of whether and how an eta-power of arbitrary integral exponent can be expressed as a linear combination of certain eta-quotients.

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On the Best Linear Approximation of Holomorphic Functions

Farkov

2016

J Math Sci

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Powers of the Eta-Function and Hecke Operators

Cited by 4 publications

References 6 publications

IIB flux non-commutativity and the global structure of field theories

IIB flux non-commutativity and the global structure of field theories

Double coset operators and eta quotients

On the Best Linear Approximation of Holomorphic Functions

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