Chirality and Gravity

Chirality and classical gravity may seem to have nothing to do with each other. Indeed, chirality is a property of "handedness" of spinning particles, while classical gravity is a theory which determines the large scale structure of spacetime (viz, solar system, galaxies, cosmology, etc). However, as we shall explain, the marriage of these two things lead to remarkable simplifications of the theories of gravity and lead to surprising conclusions! Lets then begin.....

The metric formulation of Einstein gravity is the most studied one and in fact the best understood. However, the perturbative treatment of this formulation becomes complicated as we go to high loop orders or increase the number of external legs in Feynman diagrams. To give an example, when the action is expanded around flat space, the quartic order term in the Lagrangian occupies half a page. Thus, computing any physical quantity of interest becomes a daunting task in this prescription. One of the major motivations of alternative formulations of General Relativity is to simplify computations at the perturbative level. In almost all such formulations, the role played by the metric becomes secondary while the connection becomes the main object of interest. Another motivation to consider alternative formulations is about coupling matter to gravity in a consistent way. In the usual metric formulation, it is not possible to couple fermions because the group of diffeomorphisms does not admit spinor representations. One thus constructs orthonormal frames at each point in spacetime and the metric can then be expressed in terms of the basis vectors of these frames, which are called tetrads. The orthonormal frames over the spacetime manifold constitute what is called the frame bundle. In the frame bundle, one associates a connection such that covariant derivatives can be defined. In the tetradic Palatini action, the tetrad and spin connection thus become independent variables and the metric is understood as a second order construct. 

The interesting aspect of the Palatini formulation is that the action becomes first order in the independent variables, in contrast to the Einstein-Hilbert case where the action is second order in the metric. Also, the action becomes polynomial in the fields in the case of zero cosmological constant. However, in the case of non-zero cosmological constant, the action no longer stays polynomial. Thus, a better first order formulation is required which keeps the action polynomial even in the case when the cosmological constant is non-zero. The Einstein-Cartan first order formulation is the one which precisely does the same. Even though the Einstein-Cartan action gives a better formulation for perturbative computations, it is not very economical. This is because in addition to tetrad components, the Lagrangian depends on 24 connection components per spacetime points. The gauge fixing of the theory becomes complicated and in addition to the propagator of the tetrad with itself, there exists other propagators involving the connection, which makes computations difficult. The resolution to all this is provided by the chiral formulation

The basic reason behind the chiral formulations of four dimensional GR is that in this many dimensions, there happens to be some 'accidental' isomorphisms in the Lie algebra of four dimensional Lorentz groups. The underlying reason for these isomorphisms to exist lie in the fact that the Hodge dual operator in four dimensions maps any 2-form to another 2-form and thus decomposes the space of 2-forms into self-dual (SD) and anti-self-dual (ASD) parts.

The eigenvalues of the Hodge dual operator are +-1 in the case of Euclidean or split signatures, whereas +-i in the case of Lorentzian signature. The space of such 2-forms thus gets split into the eigenspaces of this operator, which is just the SD and ASD decomposition. 

This leads to an elegant decomposition of the Riemann curvature. The Riemann curvature is a symmetric matrix. We can decompose it into the SD and ASD parts and get the following block  

where X is the self-dual self-dual component, Z is the anti-self-dual anti-self dual component and Y is the self-dual anti-self-dual component. These components can also be obtained by applying appropriate SD/ASD projectors to the Riemann curvature. The main point of the chiral formulations is that in view of the above decomposition, the Einstein condition is equivalent to which implies that for a metric to be Einstein, it is enough to have access to just half of the Riemann curvature. Indeed, in the above perspective we find that the ASD-ASD part Z is not constrained by the Einstein equations. Thus, in four spacetime dimensions, the full dynamical theory of gravity can be analyzed by just accessing the half (either SD or ASD) of the curvature. Let us then apply this to the Einstein-Cartan formulation. The Riemann curvature as we said, is encoded in the curvature of the spin connection. It is then possible to impose SD or ASD projectors to this curvature and build an action which just contains the SD part of the full curvature. This is what is done in the chiral Einstein-Cartan action.

Much of what has been described here is analogous to the Yang-Mills story. In that case, one uses a self-dual auxiliary field as a projector on the full curvature and this makes the action to be chiral. A beautiful gauge fixing procedure then gives rise to very simple Feynman rules in spinor notations. Also, it becomes convenient to pass to self-dual Yang-Mills by a truncation of the full Yang-Mills. An analogous thing happens in gravity.