Virtually spinning hyperbolic manifolds

Long, D. D.; Reid, Alan W.

doi:10.1017/s0013091519000324

Cited by 5 publications

(3 citation statements)

References 17 publications

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“…It is guaranteed that any finite volume hyperbolic space is virtually spinnable, i.e. a finite cover admits a spin structure [128] 19 . The antiperiodic boundary conditions for fermions that we require are natural: if a small circle is contractible, the boundary conditions must be antiperiodic, in order to smoothly match to the fact that spinor → −spinor upon 2π rotation about a point (and if it is not contractible, it is an option to assign this boundary condition consistently).…”

Section: Context and Further Directionsmentioning

confidence: 99%

Hyperbolic compactification of M-theory and de Sitter quantum gravity

2022

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We present a mechanism for accelerated expansion of the universe in the generic case of negative-curvature compactifications of M-theory, with minimal ingredients. M-theory on a hyperbolic manifold with small closed geodesics supporting Casimir energy -- along with a single classical source (7-form flux) -- contains an immediate 3-term structure for volume stabilization at positive potential energy. Hyperbolic manifolds are well-studied mathematically, with an important rigidity property at fixed volume. They and their Dehn fillings to more general Einstein spaces exhibit explicit discrete parameters that yield small closed geodesics supporting Casimir energy. The off-shell effective potential derived by M. Douglas incorporates the warped product structure via the constraints of general relativity, screening negative energy. Analyzing the fields sourced by the localized Casimir energy and the available discrete choices of manifolds and fluxes, we find a regime where the net curvature, Casimir energy, and flux compete at large radius and stabilize the volume. Further metric and form field deformations are highly constrained by hyperbolic rigidity and warping effects, leading to calculations giving strong indications of a positive Hessian, and residual tadpoles are small. We test this via explicit back reacted solutions and perturbations in patches including the Dehn filling regions, initiate a neural network study of further aspects of the internal fields, and derive a Maldacena-Nunez style no-go theorem for Anti-de Sitter extrema for a range of parameters. A simple generalization incorporating 4-form flux produces axion monodromy inflation. As a relatively simple de Sitter uplift of the large-N M2-brane theory, the construction applies to de Sitter holography as well as to cosmological modeling, and introduces new connections between mathematics and the physics of string/M theory compactifications.

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Section: Context and Further Directionsmentioning

confidence: 99%

Hyperbolic compactification of M-theory and de Sitter quantum gravity

2022

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“…It is guaranteed that any finite volume hyperbolic space is virtually spinnable, i.e. a finite cover admits a spin structure [123] 17 . The antiperiodic boundary conditions for fermions that we require are natural: if a small circle is contractible, the boundary conditions must be antiperiodic, in order to smoothly match to the fact that spinor → −spinor upon 2π rotation about a point (and if it is not contractible, it is an option to assign this boundary condition consistently).…”

Section: A Virtual Hyperbolic Yoga: Explicit Examples Of Compactifica...mentioning

confidence: 99%

Hyperbolic compactification of M-theory and de Sitter quantum gravity

De Luca,

Silverstein,

Torroba

2021

Preprint

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We present a mechanism for accelerated expansion of the universe in the generic case of negative-curvature compactifications of M-theory, with minimal ingredients. M-theory on a hyperbolic manifold with small closed geodesics supporting Casimir energy -along with a single classical source (7-form flux) -contains an immediate 3-term structure for volume stabilization at positive potential energy. Hyperbolic manifolds are well-studied mathematically, with an important rigidity property at fixed volume. They and their Dehn fillings to more general Einstein spaces exhibit explicit discrete parameters that yield small closed geodesics supporting Casimir energy. The off-shell effective potential derived by M. Douglas incorporates the warped product structure via the constraints of general relativity, screening negative energy. Analyzing the fields sourced by the localized Casimir energy and the available discrete choices of manifolds and fluxes, we find a regime where the net curvature, Casimir energy, and flux compete at large radius and stabilize the volume. Further metric and form field deformations are highly constrained by hyperbolic rigidity and warping effects, leading to calculations giving strong indications of a positive Hessian, and residual tadpoles are small. We test this via explicit back reacted solutions and perturbations in patches including the Dehn filling regions, initiate a neural network study of further aspects of the internal fields, and derive a Maldacena-Nunez style no-go theorem for Anti-de Sitter extrema for a range of parameters. A simple generalization incorporating 4-form flux produces axion monodromy inflation. As a relatively simple de Sitter uplift of the large-N M2-brane theory, the construction applies to de Sitter holography as well as to cosmological modeling, and introduces new connections between mathematics and the physics of string/M theory compactifications.

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“…Using non-spin flat 4-manifolds as cusp sections, Long and Reid have recently constructed some non-spin finite-volume cusped orientable hyperbolic n-manifolds for all n ≥ 5 in [17]. The paper also contains a nice short proof of the virtual spinness for finite-volume hyperbolic manifolds, together with an effective bound on the covering degree for many arithmetic manifolds of simplest type.…”

Section: Introductionmentioning

confidence: 99%

Compact hyperbolic manifolds without spin structures

Martelli,

Riolo,

Slavich

2019

Preprint

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We exhibit the first examples of compact orientable hyperbolic manifolds that do not have any spin structure. We show that such manifolds exist in all dimensions n ≥ 4.The core of the argument is the construction of a compact oriented hyperbolic 4-manifold M that contains a surface S of genus 3 with self-intersection 1. The 4-manifold M has an odd intersection form and is hence not spin. It is built by carefully assembling some right-angled 120-cells along a pattern inspired by the minimum trisection of CP 2 .The manifold M is also the first example of a compact orientable hyperbolic 4-manifold satisfying any of these conditions:• H 2 (M, Z) is not generated by geodesically immersed surfaces.• There is a covering M that is a non-trivial bundle over a compact surface.

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Virtually spinning hyperbolic manifolds

Cited by 5 publications

References 17 publications

Hyperbolic compactification of M-theory and de Sitter quantum gravity

Hyperbolic compactification of M-theory and de Sitter quantum gravity

Hyperbolic compactification of M-theory and de Sitter quantum gravity

Compact hyperbolic manifolds without spin structures

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