A proof of the Landsberg-Schaar relation by finite methods

Moore, Ben

doi:10.48550/arxiv.1810.06172

Cited by 2 publications

(1 citation statement)

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“…In Riemann and Mumford's notations, ϑ(z; τ ) ≡ ϑ00(z; τ ) is denoted as θ3(z; q), where q ≡ e 2πiτ 2. The Landsberg-Schaar identity was also recently proved using a non-analytical method[32]. For the history of development of various proofs other than Gauss's after 1801, see[33].…”

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Double-Janus Linear Sigma Models and Generalized Reciprocity for Gauss Sums

Ganor,

Sun,

Torres-Chicon

2019

Preprint

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We study the supersymmetric partition function of a 2d linear σ-model whose target space is a torus with a complex structure that varies along one worldsheet direction and a Kähler modulus that varies along the other. This setup is inspired by the dimensional reduction of a Janus configuration of 4d N = 4 U (1) Super-Yang-Mills theory compactified on a mapping torus (T 2 fibered over S 1 ) times a circle with an SL(2, Z) duality wall inserted on S 1 , but our setup has minimal supersymmetry. The partition function depends on two independent elements of SL(2, Z), one describing the duality twist, and the other describing the geometry of the mapping torus. It is topological and can be written as a multivariate quadratic Gauss sum. By calculating the partition function in two different ways, we obtain identities relating different quadratic Gauss sums, generalizing the Landsberg-Schaar relation. These identities are a subset of a collection of identities discovered by F. Deloup. Each identity contains a phase which is an eighth root of unity, and we show how it arises as a Berry phase in the supersymmetric Janus-like configuration. Supersymmetry requires the complex structure to vary along a semicircle in the upper half-plane, as shown by Gaiotto and Witten in a related context, and that semicircle plays an important role in reproducing the correct Berry phase.

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Double-Janus Linear Sigma Models and Generalized Reciprocity for Gauss Sums

Ganor,

Sun,

Torres-Chicon

2019

Preprint

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Number of zero-modes on magnetized T4/ZN orbifolds analyzed by modular transformation

Kikuchi

Kobayashi

Nasu

et al. 2023

J. High Energ. Phys.

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We study fermion zero-mode wavefunctions on T4/ZN orbifold with background magnetic fluxes. The number of zero-modes is analyzed by use of Sp(4, ℤ) modular transformation. Conditions needed to realize three generation models are clarified. We also study parity transformation in the compact space which leads to better understanding of relationship between positive and negative chirality wavefunctions.

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A proof of the Landsberg-Schaar relation by finite methods

Cited by 2 publications

References 4 publications

Double-Janus Linear Sigma Models and Generalized Reciprocity for Gauss Sums

Double-Janus Linear Sigma Models and Generalized Reciprocity for Gauss Sums

Number of zero-modes on magnetized T4/ZN orbifolds analyzed by modular transformation

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