  Given an essentially embedded torus in a homology $S^1 \times S^3$,
Echeverria and Ruberman defined two equivariant signature invariants from gauge
theory and geometric topology respectively as generalizations of the
Levine-Tristram signature for knots. We prove the equivalence of these two
invariants by introducing the periodic rho invariant associated to the ASD
DeRham complex over a cohomology $T^2 \times S^2$ as a bridge. We also prove a
surgery formula for the torus signature.
