Calabi–Yau Manifolds with Torsion and Geometric Flows

Picard, Sébastien

doi:10.1007/978-3-030-25883-2_2

Cited by 9 publications

(6 citation statements)

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“…These include: characterization of the blow-up time, dynamical stability, and real analyticity. The Laplacian flow has also been studied with symmetry (various dimensionally reduced situations) in [20,39,46].…”

Section: A Brief History Of Flows Of G 2 -Structuresmentioning

confidence: 99%

A Gradient Flow of Isometric $$\mathrm {G}_2$$-Structures

2019

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We study a flow of G2-structures that all induce the same Riemannian metric. This isometric flow is the negative gradient flow of an energy functional. We prove Shi-type estimates for the torsion tensor T along the flow. We show that at a finite-time singularity the torsion must blow up, so the flow will exist as long as the torsion remains bounded. We prove a Cheeger-Gromov type compactness theorem for the flow. We describe an Uhlenbeck-type trick which together with a modification of the underlying connection yields a nice reaction-diffusion equation for the torsion along the flow. We define a scale-invariant quantity Θ for any solution of the flow and prove that it is almost monotonic along the flow. Inspired by the work of Colding-Minicozzi [CM12] on the mean curvature flow, we define an entropy functional and after proving an ε-regularity theorem, we show that low entropy initial data lead to solutions of the flow that exist for all time and converge smoothly to a G2-structure with divergence-free torsion. We also study finite-time singularities and show that at the singular time the flow converges to a smooth G2-structure outside a closed set of finite 5-dimensional Hausdorff measure. Finally, we prove that if the singularity is of Type-I then a sequence of blow-ups of a solution admits a subsequence that converges to a shrinking soliton for the flow.

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Section: A Brief History Of Flows Of G 2 -Structuresmentioning

confidence: 99%

A Gradient Flow of Isometric $$\mathrm {G}_2$$-Structures

2019

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“…4 However, there are backgrounds where the G-structures reduce further to SU (3) or SU (2) and the flow does preserve supersymmetry. This is similar in spirit to [40] where the flow of a G-structure simplifies to a flow for a reduced G-structure given an appropriate choice of initial data. In particular, upon reducing to SU (2), the flow simplifies to a scalar evolution equation for the dilaton.…”

Section: Introductionmentioning

confidence: 87%

Geometric flows and supersymmetry

Ashmore¹,

Minasian²,

Proto³

2023

Preprint

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We study the relation between supersymmetry and geometric flows driven by the Bianchi identity for the three-form flux H in heterotic supergravity. We describe how the flow equations can be derived from a functional that appears in a rewriting of the bosonic action in terms of squares of supersymmetry operators.On a complex threefold, the resulting equations match what is known in the mathematics literature as "anomaly flow". We generalise this to seven-and eightmanifolds with G 2 or Spin(7) structures and discuss examples where the manifold is a torus fibration over a K3 surface. In the latter cases, the flow simplifies to a single scalar equation, with the existence of the supergravity solution implied by the long-time existence and convergence of the flow. We also comment on the α ′ expansion and highlight the importance of using the proper connection in the Bianchi identity to ensure that the flow's fixed points satisfy the supergravity equations of motion.

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“…Note that the construction of X differs from the construction of the Iwasawa manifold only in the introduction of the complex conjugate of a in the action on z 3 . Nevertheless, X is a smooth, complex, Calabi-Yau manifold and non-Kähler [76]. As in the case of the Iwasawa manifold there is a holomorphic fibration structure…”

Section: Examples: Manifolds Of Iwasawa-typementioning

confidence: 99%