Gravitational helicity in connection variables

We study the gravitational helicity using the covariant phase space method. Starting from the covariant phase space of general relativity expressed in terms of connection variables, we construct the symplectic form and identify a duality transformation based on the internal Hodge dual. To make the symmetry manifest, we employ the complex self-dual Ashtekar variables, where the duality becomes a simple $U(1)$ phase rotation. The Noether charge associated with this duality yields a conserved quantity which is interpreted as the gravitational helicity. Remarkably, this helicity can be related to the Nieh-Yan topological term, when expressed in terms of the real variables, establishing a direct link between the gravitational helicity and a topological invariant.