Ricci-Flat Metrics

What is Ricci-Flatness?

A metric on a manifold assigns a length to any curve connecting a pair of points. Two nearby points with separation $dx^i$ ($i=1,2,\ldots,d$) are assigned a distance $ds$ defined by

\begin{align} ds^2 = g_{ij} \ dx^idx^j \ , \end{align}

where the Einstein summation convention is assumed. We usually refer to $g_{ij}$ as the metric. The matrix inverse of $g_{ij}$ is written as $g^{ij}$.

The Ricci tensor is a second-rank symmetric tensor obtained from the Riemann curvature tensor by contracting a pair of indices $R_{ij} = g^{kl} R_{ikjl}$. In 3 dimensions, the Ricci tensor is sufficient to reconstruct the Riemann curvature tensor. Thus the vanishing of the Ricci tensor in 3 dimensions implies the vanishing of the Riemann curvature tensor. This is not true in 4 and higher dimensions. A manifold is called Ricci-flat if the Ricci tensor vanishes at all points on the manifold.

Uses of Ricci-Flat manifolds

In string theory, manifolds that admit metrics for which the Ricci tensor vanishes play a special role. Such metrics are called Ricci-Flat. Here are some applications of such manifolds.

  • Superstring theory requires spacetime to be ten-dimensional. However, spacetime as we perceive it at low energies is four-dimensional. A simple and effective way to get around this is to assume that six of the dimensions are compact and small enough to be invisible at low energies. String compactification assumes that spacetime is assumed to be of the form $\mathbb{R}^{1,3}\times M^{(6)}$, where $M$ is a compact six-dimensional manifold. Consistency of string propagation (conformal invariance) requires $M$ to be Ricci-Flat to leading order.
  • Non-compact Ricci-Flat manifolds make an appearance in the context of the AdS-CFT correspondence (more generally, the gravity-gauge correspondence). In its simplest form, this correspondence relates type IIB string theory on a spacetime $AdS_5\times X^{(5)}$ to a four-dimensional superconformal field theory (CFT). Let $M^{(6)}$ be a non-compact six-dimensional manifold obtained as a cone over $X^{(5)}$ i.e., consider a six-dimensional metric obtained from the five-dimensional metric on $X$ (which we write as $ds^2_X$):
\begin{align} ds^2_M = dr^2 + r^2 ds^2_X\ \ ,\ r\in [0,\infty)\ . \end{align}

Again, consistency of string propagation on $AdS_5\times X^{(5)}$ translates into the condition that $M$ be Ricci-Flat.

  • Ricci-Flat metrics appear as the fixed-points of the dynamical system, called Ricci-Flow
\begin{align} \frac{dg_{ij}}{dt} = R_{ij}\ , \end{align}

where $g_{ij}$ are the components of the metric, i.e., $ds^2 = g_{ij}dx^idx^j$. This is an area being actively pursued in mathematics. The proof of the Poincare conjecture by Perelman makes use of this dynamical system.

This page is devoted to explicit examples of Ricci-Flat metrics, typically but not necessarily in six-dimensions. The methods used fall into two categories:

  • Analytical: Here one produces an analytical expression for the metric and one can verify that the Ricci tensor vanishes. This is expected to work only for non-compact examples.
  • Numerical: Here one numerically solves the non-linear partial differential equation implied by the vanishing of the Ricci-tensor — this is the only method that works for compact manifolds. It appears to be necessary for some non-compact examples as well.

Extensions to Ricci-Flatness

There are two different generalisations or extensions of Ricci-flatness that we will describe below:

  • In string theory, Ricci-Flatness of the manifold $M$ is only the condition for conformal invariance at leading order. It obtains corrections, the so-called $\alpha'$ corrections as well as string-loop corrections. For instance, it has been shown that given a Ricci-Flat metric on a Kahler manifold, it is possible to find a metric, order by order in $\alpha'$, that satisfies the condition for conformal invariance. The argument is a formal one and not too many explicit examples exist for which this has been carried out. However, see Higher-Order Corrections to Non-Compact Calabi-Yau Manifolds in String Theory by Lu, Pope and Stelle for some explicit examples.
  • In the context of string theory, the metric is one of several fields in theory. For instance, there is a second-rank antisymmetric tensor, called the $B$-field, $B=\tfrac12 B_{ij}dx^i\wedge dx^j$, a scalar called the dilaton, $\Phi$, that is common to all string theories. There are also other $p$-form gauge fields,$C^{(p)}=\tfrac1{p!}C_{i_1\ldots i_p}\ dx^{i_1}\wedge \cdots \wedge dx^{i_p}$, that appear. So the condition for conformal invariance gives rise to a much more complicated system of coupled partial differential equations involving all these fields. In the limit that these fields vanish or take constant values, on recovers the condition that the metric must be Ricci-flat. The term manifolds is typically used for spaces that are Riemannian and have a metric. A generalised manifold can be defined to be a space with the various massless fields of string theory and a generalised geometry on them is given by imposing the conditions for conformal invariance of string theory. Clearly, a string theorist like me would be interested in understanding these generalised geometries.
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