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Krzysztof Galicki

Publications and source records attributed to Krzysztof Galicki.

25 records · Page 2Linked to original sources
arXiv2001

On the Geometry of Sasakian-Einstein 5-Manifolds

On simply connected five manifolds Sasakian-Einstein metrics coincide with Riemannian metrics admitting real Killing spinors which are of great interest as models of near horizon geometry for three-brane solutions in superstring theory [KW]. We expand on the recent work of Demailly and Kollár [DK] and Johnson and Kollár [JK1] who give methods for constructing Kähler-Einstein metrics on log del Pezzo surfaces. By [BG1] circle V-bundles over log del Pezzo surfaces with Kähler-Einstein metrics have Sasakian-Einstein metrics on the total space of the bundle. Here these simply connected 5-manifolds arise as links of isolated hypersurface singularities which by the well known work of Smale [Sm] together with [BG3] must be diffeomorphic to $\scriptstyle{S^5#l(S^2\times S^3)}.$ More precisely, using methods from Mori theory in algebraic geometry we prove the existence of 14 inequivalent Sasakian-Einstein structures on $\scriptstyle{S^2\times S^3}$ and infinite families of such structures on $\scriptstyle{#l(S^2\times S^3)}$ with $\scriptstyle{2\leq l\leq7}$. We also discuss the moduli problem for these Sasakian-Einstein structures.

math.DG↗
arXiv2001

Sasakian-Einstein Structures on $9#(S^2\times S^3)$

We show that $\scriptstyle{#9(S^2\times S^3)}$ admits an 8-dimensional complex family of inequivalent non-regular Sasakian-Einstein structures. These are the first known Einstein metrics on this 5-manifold. In particular, the bound $\scriptstyle{b_2(M)\leq8}$ which holds for any regular Sasakian-Einstein $\scriptstyle{M}$ does not apply to the non-regular case. We also discuss the failure of the Hitchin-Thorpe inequality in the case of 4-orbifolds and describe the orbifold version.

math.DG↗
arXiv2000

On Sasakian-Einstein Geometry

We discuss Sasakian-Einstein geometry under a quasi-regularity assumption. It is shown that the space of all quasi-regular Sasakian-Einstein orbifolds has a natural multiplication on it. Furthermore, necessary and sufficient conditions are given for the `product' of two Sasakian-Einstein manifolds to be a smooth Sasakian-Einstein manifold. Using spectral sequence arguments we work out the cohomology ring in many cases of interest. This type of geometry has recently become of interest in the physics of supersymmetric conformal field theories.

math.DG↗
arXiv2000

3-Sasakian Geometry, Nilpotent Orbits, and Exceptional Quotients

Using 3-Sasakian reduction techniques we obtain infinite families of new 3-Sasakian manifolds $\scriptstyle{{\cal M}(p_1,p_2,p_3)}$ and $\scriptstyle{{\cal M}(p_1,p_2,p_3,p_4)}$ in dimension 11 and 15 respectively. The metric cone on $\scriptstyle{{\cal M}(p_1,p_2,p_3)}$ is a generalization of the Kronheimer hyperkähler metric on the regular maximal nilpotent orbit of $\scriptstyle{{\Got s}{\Got l}(3,\bbc)}$ whereas the cone on $\scriptstyle{{\cal M}(p_1,p_2,p_3,p_4)}$ generalizes the hyperkähler metric on the 16-dimensional orbit of $\scriptstyle{{\Got s}{\Got o}(6,\bbc)}$. These are first examples of 3-Sasakian metrics which are neither homogeneous nor toric. In addition we consider some further $\scriptstyle{U(1)}$-reductions of $\scriptstyle{{\cal M}(p_1,p_2,p_3)}$. These yield examples of non-toric 3-Sasakian orbifold metrics in dimensions 7. As a result we obtain explicit families $\scriptstyle{{\cal O}(Θ)}$ of compact self-dual positive scalar curvature Einstein metrics with orbifold singularities and with only one Killing vector field.

math.DG↗
arXiv1999

A Note on Toric Contact Geometry

After observing that the well-known convexity theorems of symplectic geometry also hold for compact contact manifolds with an effective action of a torus whose Reeb vector field corresponds to an element of the Lie algebra of the torus, we use this fact together with a recent symplectic orbifold version of Delzant's theorem due to Lerman and Tolman [LT] to show that every such compact toric contact manifold can be obtained by a contact reduction from an odd dimensional sphere.

math.DG↗
arXiv1998

3-Sasakian Manifolds

This is an expository paper describing the geometry of certain Sasakian-Einstein manifolds. Such manifolds have recently become of interest due to Maldacena's AdS/CFT conjecture. They describe near-horizon geometries of branes at conical singularities.

hep-th↗