Exact Equivalence of the D=4 Gauged Wess-Zumino-Witten Term and the D=5 Yang-Mills Chern-Simons Term

TL;DR

This paper demonstrates that the D=4 gauged Wess-Zumino-Witten term can be exactly derived from a D=5 Yang-Mills Chern-Simons term in a compactified setting with boundary quarks, providing a systematic approach for theories with extra dimensions.

Contribution

It establishes a direct, systematic derivation of the Wess-Zumino-Witten term from a five-dimensional Yang-Mills Chern-Simons action, extending the understanding of anomaly inflow and boundary effects.

Findings

Derived the Wess-Zumino-Witten term from D=5 Yang-Mills Chern-Simons theory.

Provided a systematic method for generalizing to extra-dimensional theories.

Presented a novel form of the Wess-Zumino-Witten term for massless fermions.

Abstract

We derive the full Wess-Zumino-Witten term of a gauged chiral lagrangian in D=4 by starting from a pure Yang-Mills theory of gauged quark flavor in a flat, compactified D=5. The theory is compactified such that there exists a B_5 zero mode, and supplemented with quarks that are ``chirally delocalized'' with q_L (q_R) on the left (right) boundary (brane). The theory then necessarily contains a Chern-Simons term (anomaly flux) to cancel the fermionic anomalies on the boundaries. The constituent quark mass represents chiral symmetry breaking and is a bilocal operator in D=5 of the form: \bar{q}_LWq_R+h.c, where W is the Wilson line spanning the bulk, 0\leq x^5 \leq R and is interpreted as a chiral meson field, W=\exp(2i\tilde{\pi}/f_\pi), where f_\pi \sim 1/R. The quarks are integrated out, yielding a Dirac determinant which takes the form of a ``boundary term'' (anomaly flux return), and is equivalent to Bardeen's counterterm that connects consistent and covariant anomalies. The Wess-Zumino-Witten term then emerges straightforwardly, from the Yang-Mills Chern-Simons term, plus boundary term. The method is systematic and allows generalization of the Wess-Zumino-Witten term to theories of extra dimensions, and to express it in alternative and more compact forms. We give a novel form appropriate to the case of (unintegrated) massless fermions.