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Transverse Kähler–Ricci Solitons of Five-Dimensional Sasaki–Einstein Spaces Y^{p,q} and T^{1,1}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Sasakian Manifolds

**Definition**

**1.**

- g is Einstein with $Ri{c}_{g}=2ng$;
- the metric cone $C(M,\overline{g})$ is a Ricci-flat Kähler manifold (i.e., Calabi–Yau manifold); and
- the transverse Kähler metric ${g}^{T}$ satisfies $Ri{c}_{{g}^{T}}=(2n+2){g}^{T}$.

**Definition**

**2.**

#### 2.2. Deformations of Sasaki Structures

**Lemma**

**1.**

**Definition**

**3.**

- 1.
- $d{\pi}_{\alpha}\left(X\right)$is a holomorphic vector field on${V}_{\alpha}$and
- 2.
- the complex vector field$${u}_{X}=i\eta \left(X\right)$$$${\overline{\partial}}_{B}{u}_{X}=-\frac{i}{2}\iota \left(X\right)d\eta \phantom{\rule{0.166667em}{0ex}}.$$

**Remark**

**1.**

#### 2.3. Sasaki–Ricci Flow

**Remark**

**2.**

## 3. Sasaki–Ricci Flow on Spaces ${\mathsf{T}}^{1,1}$ and ${\mathsf{Y}}^{p,q}$

#### 3.1. Sasaki–Ricci flow on Sasaki–Einstein Space ${T}^{1,1}$

**Proposition**

**1.**

**Remark**

**3.**

**Proposition**

**2.**

#### 3.2. Sasaki–Ricci flow on Sasaki–Einstein Space ${Y}^{p,q}$

**Remark**

**4.**

**Proposition**

**3.**

**Remark**

**5.**

**Proposition**

**4.**

## 4. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Kehagias, A. New type IIB vacua and their F-theory interpretation. Phys. Lett. B
**1998**, 435, 337. [Google Scholar] [CrossRef][Green Version] - Klebanov, I.R.; Witten, E. Superconformal field theory on threebranes at a Calabi-Yau singularity. Nucl. Phys. B
**1998**, 536, 199. [Google Scholar] [CrossRef][Green Version] - Gauntlett, J.P.; Martelli, D.; Sparks, J.; Waldram, D. Supersymmetric AdS(5) solutions of M-theory. Class. Quantum. Grav.
**2004**, 21, 4335. [Google Scholar] [CrossRef][Green Version] - Gauntlett, J.P.; Martelli, D.; Sparks, J.; Waldram, D. Sasaki-Einstein metrics on S(2)×S(3). Adv. Theor. Math. Phys.
**2004**, 8, 711. [Google Scholar] [CrossRef][Green Version] - Gauntlett, J.P.; Martelli, D.; Sparks, J.; Waldram, D. A new infinite class of Sasaki-Einstein manifolds. Adv. Theor. Math. Phys.
**2004**, 8, 987. [Google Scholar] [CrossRef][Green Version] - Hamilton, R. Three-manifolds with positive Ricci curvature. J. Differ. Geom.
**1982**, 17, 255. [Google Scholar] [CrossRef] - Smoczyk, K.; Wang, G.; Zhang, Y. The Sasaki-Ricci flow. Int. J. Math.
**2010**, 21, 951. [Google Scholar] [CrossRef] - Futaki, A.; Ono, H.; Wang, G. Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds. J. Differ. Geom.
**2009**, 83, 585. [Google Scholar] [CrossRef] - Giataganas, D. Semiclassical strings in marginally deformed toric AdS/CFT. JHEP
**2011**, 12, 051. [Google Scholar] [CrossRef][Green Version] - Visinescu, M. Sasaki-Ricci flow on Sasaki-Einstein space T
^{1,1}and deformations. Int. J. Mod. Phys. A**2018**, 33, 1845014. [Google Scholar] [CrossRef] - Slesar, S.; Visinescu, M.; Vîlcu, G.-E. Transverse Kähler-Ricci flow and deformations of the metric on the Sasaki space T
^{1,1}. arXiv**2019**, arXiv:1905.05024. [Google Scholar] - Visinescu, M. Sasaki-Ricci flow equation on five-dimensional Sasaki-Einstein space Y
^{p,q}. arXiv**2019**, arXiv:1910.12495. [Google Scholar] - Boyer, C.P.; Galicki, K. Sasakian Geometry; Oxford Mathematical Monograph; Oxford University Press: Oxford, UK, 2008. [Google Scholar]
- Godliński, M.; Kopczyński, W.; Nurowski, P. Locally Sasakian manifolds. Class. Quantum Grav.
**2000**, 17, L105. [Google Scholar] [CrossRef] - Candelas, P.; de la Ossa, X.C. Comments on conifolds. Nucl. Phys. B
**1990**, 342, 246. [Google Scholar] [CrossRef] - Martelli, D.; Sparks, J. Toric geometry, Sasaki-Einstein manifolds and a new infinite class of AdS/CFT duals. Commun. Math. Phys.
**2006**, 262, 51. [Google Scholar] [CrossRef][Green Version] - Burrington, B.A.; Liu, J.T.; Mahato, M.; Pando Zayas, L.A. Towards supergravity duals of chiral symmetry breaking in Sasaki-Einstein cascading quiver theories. JHEP
**2005**, 7, 19. [Google Scholar] [CrossRef] - Babalic, E.M.; Visinescu, M. Complete integrability of geodesic motion in Sasaki-Einstein toric Y
^{p,q}spaces. Mod. Phys. Lett. A**2015**, 33, 1550180. [Google Scholar] [CrossRef][Green Version] - Bravetti, A. Contact Hamiltonian dynamics: The concept and its use. Entropy
**2017**, 19, 535. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Visinescu, M.
Transverse Kähler–Ricci Solitons of Five-Dimensional Sasaki–Einstein Spaces Y^{p,q} and T^{1,1}. *Symmetry* **2020**, *12*, 330.
https://doi.org/10.3390/sym12030330

**AMA Style**

Visinescu M.
Transverse Kähler–Ricci Solitons of Five-Dimensional Sasaki–Einstein Spaces Y^{p,q} and T^{1,1}. *Symmetry*. 2020; 12(3):330.
https://doi.org/10.3390/sym12030330

**Chicago/Turabian Style**

Visinescu, Mihai.
2020. "Transverse Kähler–Ricci Solitons of Five-Dimensional Sasaki–Einstein Spaces Y^{p,q} and T^{1,1}" *Symmetry* 12, no. 3: 330.
https://doi.org/10.3390/sym12030330