Reduction, Induction and Ricci Flat Symplectic Connections

SYMPLECTIC CONNECTIONS

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s021988780600117x ◽

2006 ◽

Vol 03 (03) ◽

pp. 375-420 ◽

Cited By ~ 20

Author(s):

PIERRE BIELIAVSKY ◽

MICHEL CAHEN ◽

SIMONE GUTT ◽

JOHN RAWNSLEY ◽

LORENZ SCHWACHHÖFER

Keyword(s):

Variational Principle ◽

Critical Points ◽

Ricci Tensor ◽

Symmetric Spaces ◽

Complex Geometry ◽

Ricci Flat ◽

Symplectic Connections

This article is an overview of the results obtained in recent years on symplectic connections. We present what is known about preferred connections (critical points of a variational principle). The class of Ricci-type connections (for which the curvature is entirely determined by the Ricci tensor) is described in detail, as well as its far-reaching generalization to special connections. A twistorial construction shows a relation between Ricci-type connections and complex geometry. We give a construction of Ricci-flat symplectic connections. We end up by presenting, through an explicit example, an approach to non-commutative symplectic symmetric spaces.

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Existence of left invariant Ricci flat metrics on nilpotent Lie groups

Archiv der Mathematik ◽

10.1007/s00013-021-01645-6 ◽

2021 ◽

Author(s):

Yujian Xiang ◽

Zaili Yan

Keyword(s):

Lie Groups ◽

Nilpotent Lie Groups ◽

Ricci Flat

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Long way to Ricci flatness

Journal of High Energy Physics ◽

10.1007/jhep10(2020)059 ◽

2020 ◽

Vol 2020 (10) ◽

Author(s):

Jin Chen ◽

Chao-Hsiang Sheu ◽

Mikhail Shifman ◽

Gianni Tallarita ◽

Alexei Yung

Keyword(s):

Sigma Model ◽

Ultra Violet ◽

The Other ◽

Target Space ◽

Linear Sigma Model ◽

Close Relative ◽

Two Dimensional ◽

World Sheet ◽

Ricci Flat ◽

The World

Abstract We study two-dimensional weighted $$ \mathcal{N} $$ N = (2) supersymmetric ℂℙ models with the goal of exploring their infrared (IR) limit. 𝕎ℂℙ(N,$$ \tilde{N} $$ N ˜ ) are simplified versions of world-sheet theories on non-Abelian strings in four-dimensional $$ \mathcal{N} $$ N = 2 QCD. In the gauged linear sigma model (GLSM) formulation, 𝕎ℂℙ(N,$$ \tilde{N} $$ N ˜ ) has N charges +1 and $$ \tilde{N} $$ N ˜ charges −1 fields. As well-known, at $$ \tilde{N} $$ N ˜ = N this GLSM is conformal. Its target space is believed to be a non-compact Calabi-Yau manifold. We mostly focus on the N = 2 case, then the Calabi-Yau space is a conifold. On the other hand, in the non-linear sigma model (NLSM) formulation the model has ultra-violet logarithms and does not look conformal. Moreover, its metric is not Ricci-flat. We address this puzzle by studying the renormalization group (RG) flow of the model. We show that the metric of NLSM becomes Ricci-flat in the IR. Moreover, it tends to the known metric of the resolved conifold. We also study a close relative of the 𝕎ℂℙ model — the so called zn model — which in actuality represents the world sheet theory on a non-Abelian semilocal string and show that this zn model has similar RG properties.

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The Global Sections of Chiral de Rham Complexes on Compact Ricci-Flat Kähler Manifolds

Communications in Mathematical Physics ◽

10.1007/s00220-021-03975-6 ◽

2021 ◽

Vol 382 (1) ◽

pp. 351-379

Author(s):

Bailin Song

Keyword(s):

Kähler Manifolds ◽

Kahler Manifolds ◽

Ricci Flat ◽

De Rham

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On supersymmetric AdS4 orientifold vacua

Journal of High Energy Physics ◽

10.1007/jhep08(2020)087 ◽

2020 ◽

Vol 2020 (8) ◽

Cited By ~ 1

Author(s):

Fernando Marchesano ◽

Eran Palti ◽

Joan Quirant ◽

Alessandro Tomasiello

Keyword(s):

String Theory ◽

Cosmological Constant ◽

Bianchi Identities ◽

Expansion Parameter ◽

First Order ◽

Ricci Flat ◽

Key Features ◽

One To One ◽

Calabi Yau Manifolds

Abstract In this work we study ten-dimensional solutions to type IIA string theory of the form AdS4 × X6 which contain orientifold planes and preserve $$ \mathcal{N} $$ N = 1 supersymmetry. In particular, we consider solutions which exhibit some key features of the four-dimensional DGKT proposal for compactifications on Calabi-Yau manifolds with fluxes, and in this sense may be considered their ten-dimensional uplifts. We focus on the supersymmetry equations and Bianchi identities, and find solutions to these that are valid at the two-derivative level and at first order in an expansion parameter which is related to the AdS cosmological constant. This family of solutions is such that the background metric is deformed from the Ricci-flat one to one exhibiting SU(3) × SU(3)-structure, and dilaton gradients and warp factors are induced.

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