Chiral solution to the Ginsparg-Wilson equation

We present a chiral solution of the Ginsparg-Wilson equation. This work is motivated by our recent proposal for nonperturbatively regulating chiral gauge theories, where five-dimensional domain wall fermions couple to a four-dimensional gauge field that is extended into the extra dimension as the solution to a gradient flow equation. Mirror fermions at the far surface decouple from the gauge field as if they have form factors that become infinitely soft as the distance between the two surfaces is increased. In the limit of an infinite extra dimension we derive an effective four-dimensional chiral overlap operator which is shown to obey the Ginsparg-Wilson equation, and which correctly reproduces a number of properties expected of chiral gauge theories in the continuum.

Figure 1

Left: The conventional domain wall fermion construction, with negative and positive chiral modes bound to the $s=0$ and $s=L$ surfaces of the extra dimension respectively; the gauge field $A$ is constant over the extra dimension. Center: The chiral construction where the gauge field in the extra dimension is a solution of the gradient flow equation, flowing from initial value $A$ on the left to the corresponding fixed point value $A_{\star }$ on the right. Right: Flow which makes an abrupt transition from $A$ to $A_{\star }$, a case that is more easily treated analytically. When assuming this scenario we place hats on our lattice operators, such as ${\widehat{\mathcal{D}}}_{\chi }$.

Figure 2

The four regions in the domain wall construction of Ref. [19].