Global anomalies in chiral lattice gauge theory

Abstract

As first realized by Witten an SU (2) gauge theory coupled to a single Weyl fermion suffers from a global anomaly. This problem is addressed here in the context of the recent developments on chiral gauge theories on the lattice. We find Witten's anomaly manifests in the impossibility of defining globally a fermion measure that reproduces the proper continuum limit. Moreover, following Witten's original argument, we check numerically the crossing of the lowest eigenvalues of Neuberger's operator along a path connecting two gauge fields that differ by a topologically non-trivial gauge transformation.

On Witten's global anomaly for higher SU(2) representations

“…Suitable means that the Dirac operator should satisfy the Ginsparg-Wilson relation [3] in order to preserve exact chiral symmetry even for nonzero lattice spacing. This method has already been employed for the fundamental representation [4,5], and n 1/2 was found to be 1. Here we report about a similar calculation for the 3/2 and 5/2 representation and present the results for n 3/2 and n 5/2 .…”

Families index theory for Overlap lattice Dirac operator. I

“…However, there is another obstruction, namely Witten's global gauge anomaly [27]. 7 The presence of this obstruction in the SU(2) theory in the lattice setting has been demonstrated both numerically [6,8] and analytically (in the classical continuum limit) in [9]. This obstruction also has a natural description in the context of families index theory for the Dirac operator: There is a canonical trivialisation of the U(1) determinant line bundle over 1-dimensional balls (i.e.…”

Real representation in chiral gauge theories on the lattice

“…This implies that the symplectic measure is 9 We define ξ ≡ J −1 ξ * . If ξ is an eigenvector of Q, Qξ = e iθ ξ, then ξ has the eigenvalue e −iθ .…”

“…The first is the local gauge anomaly that corresponds to the gauge anomaly in the continuum theory, but requires a control with finite lattice spacings [1]- [7] (see also [8]). The second is the global topological obstruction [1,3,9], which is a lattice counterpart of the Witten anomaly [10]. 2 The local anomaly is absent from real representations, so we expect that global issues in the formulation are highlighted.…”

“…3 For pseudo-real representations of SU (2), from which the local anomaly is also absent, it has been shown [9] that a globally consistent definition of the fermion integration measure is impossible.…”