Abstract
We derive the vector-like four-dimensional overlap Dirac operator starting from a five-dimensional Dirac action in the presence of a delta-function space–time defect. The effective operator is obtained by first integrating out all the fermionic modes in the fixed gauge background, and then identifying the contribution from the localized modes as the determinant of an operator in one dimension less. We define physically relevant degrees of freedom on the defect by introducing an auxiliary defect-bound fermion field and integrating out the original five-dimensional bulk fields.
Highlights
A real valued functional of the four dimensional gauge field which is invariant under gauge transformations
In this note we present a non-rigorous derivation of the overlap formula, from a starting point where the two chiral components of the Dirac field live on the same four dimensional manifold M
We say that the derivation is non-rigorous because we do not employ an explicit UV regularization, M can be taken to be compact. This approach makes it possible to understand the formula for the overlap Dirac operator in just a few lines
Summary
Thinking in terms of the setup reviewed at the beginning of this note, we wish to replace the segment L with the two Weyl fields living at its opposite endpoints, by first gluing the two ends to each other, cutting L in the middle, and letting the two (new) endpoints created by the cutting go to ±∞, respectively. In general, be glued continuously to each other. The defect can be realized by adding a suitable singular mass term to the d + 1 dimensional Dirac operator Dd+1, so that the eigenfunctions of Dd+1 have a discontinuity at s = 0. Where D = ∂+ A is the d-dimensional Dirac operator in the presence of a gauge field whose components Aμ depend only on x and our Lie algebra generators are such that Aμ(x) = −Aμ(x)†. We can determine ξ by ensuring that Dd+1 has exact zero modes when the d-dimensional gauge field on M is from a nontrivial bundle and as a consequence, for the chosen background, D has chiral zero modes ψd(x).
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