A simple presentation of the Siegel modular groups

Lu, Ning

doi:10.1016/0024-3795(92)90276-g

Cited by 5 publications

(6 citation statements)

References 2 publications

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“…Ref. [104] yields an economic set of generators for Sp(2m+2, Z) given by (at most) 3 explicit matrices. We get a set of 3m + 3 functional equations for the F I .…”

Section: Jhep09(2020)147mentioning

confidence: 99%

Special geometry and the swampland

Cecotti

2020

J. High Energ. Phys.

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In the context of 4d effective gravity theories with 8 supersymmetries, we propose to unify, strenghten, and refine the several swampland conjectures into a single statement: the structural criterion, modelled on the structure theorem in Hodge theory. In its most abstract form the new swampland criterion applies to all 4d $$ \mathcal{N} $$ N = 2 effective theories (having a quantum-consistent UV completion) whether supersymmetry is local or rigid: indeed it may be regarded as the more general version of Seiberg-Witten geometry which holds both in the rigid and local cases. As a first application of the new swampland criterion we show that a quantum-consistent $$ \mathcal{N} $$ N = 2 supergravity with a cubic pre-potential is necessarily a truncation of a higher-$$ \mathcal{N} $$ N sugra. More precisely: its moduli space is a Shimura variety of ‘magic’ type. In all other cases a quantum-consistent special Kähler geometry is either an arithmetic quotient of the complex hyperbolic space SU(1, m)/U(m) or has no local Killing vector. Applied to Calabi-Yau 3-folds this result implies (assuming mirror symmetry) the validity of the Oguiso-Sakurai conjecture in Algebraic Geometry: all Calabi-Yau 3-folds X without rational curves have Picard number ρ = 2, 3; in facts they are finite quotients of Abelian varieties. More generally: the Kähler moduli of X do not receive quantum corrections if and only if X has infinite fundamental group. In all other cases the Kähler moduli have instanton corrections in (essentially) all possible degrees.

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“…Ref. [104] yields an economic set of generators for Sp(2m+2, Z) given by (at most) 3 explicit matrices. We get a set of 3m + 3 functional equations for the F I .…”

Section: Jhep09(2020)147mentioning

confidence: 99%

Special geometry and the swampland

Cecotti

2020

J. High Energ. Phys.

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“…At genus g = 2, the minimal number of generators of Sp(4, Z) is two [43]; these are reviewed in appendix B. For Sp(2g, Z) with g ≥ 3 the minimal number of generators is three [44]. Some elements of Γ g act trivially on the canonical homology basis, leaving it invariant.…”

Section: Review Of the Relevant Conceptsmentioning

confidence: 99%

“…with X ≡ KL 5 KL 7 K = 1 2 ⊗ σ x , L 6 = 1 2 ⊗ σ z , where σ i denote the standard three Pauli matrices. Its generalization to arbitrary genus Sp(2g, Z) with at most 3 generators and 3g + 5 relations can be found in [44]. In the following basis of Riemann theta functions…”

Section: B Generators For Sp(4 Z) and The Algorithmmentioning

confidence: 99%

Establishing strongly-coupled 3D AdS quantum gravity with Ising dual using all-genus partition functions

Jian

Ludwig

Luo

et al. 2020

J. High Energ. Phys.

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We study 3D pure Einstein quantum gravity with negative cosmological constant, in the regime where the AdS radius l is of the order of the Planck scale. Specifically, when the Brown-Henneaux central charge c = 3l/2GN (GN is the 3D Newton constant) equals c = 1/2, we establish duality between 3D gravity and 2D Ising conformal field theory by matching gravity and conformal field theory partition functions for AdS spacetimes with general asymptotic boundaries. This duality was suggested by a genus-one calculation of Castro et al. [Phys. Rev. D85 (2012) 024032]. Extension beyond genus-one requires new mathematical results based on 3D Topological Quantum Field Theory; these turn out to uniquely select the c = 1/2 theory among all those with c < 1, extending the previous results of Castro et al. Previous work suggests the reduction of the calculation of the gravity partition function to a problem of summation over the orbits of the mapping class group action on a “vacuum seed”. But whether or not the summation is well-defined for the general case was unknown before this work. Amongst all theories with Brown-Henneaux central charge c < 1, the sum is finite and unique only when c = 1/2, corresponding to a dual Ising conformal field theory on the asymptotic boundary.

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“…(ii) The symplectic group Sp(2g, Z), and thus all of its quotients, is a quotient of the mapping class group Γ g , which is perfect for g ≥ 3 (see for example [14,Theorems 5.2,6.4]). For g = 2, one can abelianize the presentation of Sp(4, Z) given in [3] or [25,Theorem 2].…”

Section: 2mentioning

confidence: 99%

Signature cocycles on the mapping class group and symplectic groups

Benson

Campagnolo

Ranicki

et al. 2020

Journal of Pure and Applied Algebra

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Werner Meyer constructed a cocycle in H 2 (Sp(2g, Z); Z) which computes the signature of a closed oriented surface bundle over a surface, with fibre a surface of genus g. By studying properties of this cocycle, he also showed that the signature of such a surface bundle is a multiple of 4. In this paper, we study signature cocycles both from the geometric and algebraic points of view. We present geometric constructions which are relevant to the signature cocycle and provide an alternative to Meyer's decomposition of a surface bundle. Furthermore, we discuss the precise relation between the Meyer and Wall-Maslov index. The main theorem of the paper, Theorem 7.2, provides the necessary group cohomology results to analyze the signature of a surface bundle modulo any integer N . Using these results, we are able to give a complete answer for N = 2, 4, and 8, and based on a theorem of Deligne, we show that this is the best we can hope for using this method.2010 Mathematics Subject Classification. 20J06 (primary) ; 55R10, 20C33 (secondary).

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A simple presentation of the Siegel modular groups

Cited by 5 publications

References 2 publications

Special geometry and the swampland

Special geometry and the swampland

Establishing strongly-coupled 3D AdS quantum gravity with Ising dual using all-genus partition functions

Signature cocycles on the mapping class group and symplectic groups

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