Chiral Symmetry and Lattice Fermions

TL;DR

The notes survey how chiral symmetry, anomalies, domain wall fermions, and overlap fermions are treated in lattice gauge theory, linking fundamental continuum structures to lattice realizations. They explain how chiral symmetry is preserved or broken at the lattice level through the Callan–Harvey mechanism and the Ginsparg–Wilson relation, and how overlap and domain-wall constructions provide practical, anomaly-consistent frameworks. A central theme is the interplay between topology (index theorem, zeromodes) and lattice Dirac operators, with concrete discussion of numerical implementations and the remaining obstacles for chiral gauge theories. The work highlights both the theoretical elegance of GW/overlap formulations and the computational challenges that must be overcome for robust nonperturbative studies of chiral dynamics in QCD and beyond.

Abstract

The subject of these summer school lectures are (i) Chiral symmetry; (ii) Anomalies; (iii) Domain wall fermions; (iv) Overlap fermions and the Ginsparg-Wilson equation

Figures (10)

• Figure 1: One-loop renormalization of the electron mass in QED due to photon exchange. A mass operator flips chirality, while gauge interactions do not. A contribution to the electron mass requires an odd number of chirality flips, and so there has to be at least one insertion of the electron mass in the diagram: the electron mass is multiplicatively renormalized. A scalar interaction flips chirality when the scalar is emitted, and flips it back when the scalar is absorbed, so replacing the photon with a scalar in the above graph again requires a fermion mass insertion to contribute to mass renormalization.
• Figure 2: One-loop additive renormalization of the scalar mass due to a quadratically divergent fermion loop.
• Figure 3: The eigenvalue flow of the Dirac operator as a function of gauge fields, and two unsatisfactory ways to define the Weyl fermion determinant $\det D_L$ as a square root of $\det \hbox{$/$ $D$}$. The expression $\sqrt{\left|\det\hbox{$/$ $D$}\right|}$ corresponds to the picture on the left, where $\det D_L$ is defined as the product of positive eigenvalues of $\hbox{$/$ $D$}$; this definition is nonanalytic at $A_*$. The picture on the right corresponds to the product of half the eigenvalues, following those which were positive at some reference gauge field $A_0$. This definition is analytic, but not necessarily local. Both definitions are gauge invariant, which is incorrect for an anomalous fermion representation.
• Figure 4: On the left: the ground state for a theory of a single massless Dirac fermion in $(1+1)$ dimensions; on the right: the theory after application of an adiabatic electric field with all states shifted to the right by $\Delta p$, given in eqn. \ref{['eq:dp']}. Filled states are indicated by the heavier blue lines.
• Figure 5: The anomaly diagram in 1+1 dimensions, with one Pauli-Villars loop and an insertion of $2iM \overline{\Phi} \Gamma \Phi$ at the X.