Linking forms, reciprocity for Gauss sums and invariants of 3-manifolds

Deloup, Florian

doi:10.1090/s0002-9947-99-02304-1

Cited by 21 publications

(20 citation statements)

References 21 publications

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“…In 1996, independently of Jeffrey, Deloup [87,88] geometrically generalized Krazer's formula (8.10) (as well as Jeffrey's) by essentially replacing both integers a and b in (8.9) with bilinear forms, and applied his result to calculate topological invariants of 3-manifolds, such as Witten-Reshetikhin-Turaev (WRT) invariants (i.e., Chern-Simons partition functions). For integral quadratic forms determined by two invertible, even, symmetric matrices A and B, Deloup's reciprocity theorem relates a Gauss sum with bilinear form A ⊗ B −1 to another Gauss sum with bilinear form −A −1 ⊗ B.…”

Section: Relation To Krazer's Jeffrey's Deloup's and Turaev's Reciprocity Formulaementioning

confidence: 99%

“…For integral quadratic forms determined by two invertible, even, symmetric matrices A and B, Deloup's reciprocity theorem relates a Gauss sum with bilinear form A ⊗ B −1 to another Gauss sum with bilinear form −A −1 ⊗ B. 23 This identity appears as Theorem 3 in [87], and we do not present it here, since it would require quite a few new notations and definitions. In the context of our section 8.2, A can be identified with the coupling constant matrix K (derived from M ) of abelian Chern-Simons theory, while B can be identified with a similar but independent matrix derived from…”

Section: Relation To Krazer's Jeffrey's Deloup's and Turaev's Reciprocity Formulaementioning

confidence: 99%

“…Deloup's reciprocity relation is actually more general, allowing non-even A and B by introducing arithmetic "Wu classes". For a quadratic form x t Ax (with x ∈ Z n ), a Wu class is realized by a constant vector w ∈ Z n such that x t Ax+w t x ∈ 2Z for all x ∈ Z n [87]. By adding such linear terms (also similarly in Jeffrey's), one overcomes the ambiguity in the definition of Gauss sums in (8.10) for a non-even A.…”

Section: Relation To Krazer's Jeffrey's Deloup's and Turaev's Reciprocity Formulaementioning

confidence: 99%

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Double-Janus linear sigma models and generalized reciprocity for Gauss sums

Ganor

Sun

Torres-Chicon

2021

J. High Energ. Phys.

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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 $$ \mathcal{N} $$ N = 4 U(1) Super-Yang-Mills theory compactified on a mapping torus (T2 fibered over S1) times a circle with an SL(2, ℤ) duality wall inserted on S1, 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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Section: Relation To Krazer's Jeffrey's Deloup's and Turaev's Reciprocity Formulaementioning

confidence: 99%

Section: Relation To Krazer's Jeffrey's Deloup's and Turaev's Reciprocity Formulaementioning

confidence: 99%

Section: Relation To Krazer's Jeffrey's Deloup's and Turaev's Reciprocity Formulaementioning

confidence: 99%

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Double-Janus linear sigma models and generalized reciprocity for Gauss sums

Ganor

Sun

Torres-Chicon

2021

J. High Energ. Phys.

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“…We state here a reciprocity formula due to Deloup [6] and deduce it from (3). We need the following construction.…”

Section: •4•3 the Deloup Formulamentioning

confidence: 99%

“…Recently, Florian Deloup [6] found a new and most beautiful reciprocity formula for multivariate Gauss sums. His formula is a far reaching generalization of Krazer's result.…”

mentioning

confidence: 99%

Reciprocity for Gauss sums on finite abelian groups

Turaev

1998

Math. Proc. Camb. Phil. Soc.

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A classical reciprocity formula for Gauss sums due to Cauchy, Dirichlet, and Kronecker states thatformula herewhere a, b are nonzero integers and w∈ℚ such that ab+2aw∈2ℤ. For a detailed proof and historical background, see [4, chapter IX]. Various versions of formula (1) have been extensively studied in connection with transformation properties of theta functions. A version of (1) for multivariate Gauss sums was first obtained by A. Krazer [10] in 1912, see also [2, 5, 11, 14]. Krazer's formula generalizes the case w=0 of (1) via replacing one of the numbers a, b by an integer quadratic form of several variables.Recently, Florian Deloup [6] found a new and most beautiful reciprocity formula for multivariate Gauss sums. His formula is a far reaching generalization of Krazer's result. Roughly speaking, Deloup replaces both numbers a, b with quadratic forms. Deloup involves Wu classes of quadratic forms which allows him to remove the evenness condition appearing in Krazer's formulation. However, Deloup's formula covers only the cases w=0 and w=b/2 of (1).In this paper we establish a more general reciprocity for Gauss sums including the Krazer and Deloup formulas and formula (1) in its full generality. Our reciprocity law involves two quadratic forms and a so-called rational Wu class, as defined below.The original proofs of Cauchy, Dirichlet, Kronecker and Krazer are analytical and involve a study of a limit of a transformation formula for theta-functions. Deloup's proof goes by a reduction to the case w=b/2 of (1) based on a careful study of Witt groups of quadratic forms. Our proof is more direct and uses only (a generalization of) the van der Blij computation of Gauss sums via signatures of integer quadratic forms. In particular, our argument provides a new proof of (1).

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On Spin and Complex Spin Borromean Surgeries

Deloup

NATO Science Series

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Linking forms, reciprocity for Gauss sums and invariants of 3-manifolds

Cited by 21 publications

References 21 publications

Double-Janus linear sigma models and generalized reciprocity for Gauss sums

Double-Janus linear sigma models and generalized reciprocity for Gauss sums

Reciprocity for Gauss sums on finite abelian groups

On Spin and Complex Spin Borromean Surgeries

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