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Register a new userAxial anomaly in the reduced model: Higher representations
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Authors
Teruaki Inagaki
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The axial anomaly arising from the fermion sector of $U(N)$ or $SU(N)$ reduced model is studied under a certain restriction of gauge field configurations (the ``$U(1)$ embedding with $N=L^d$). We use the overlap-Dirac operator and consider how the anomaly changes as a function of a gauge-group representation of the fermion. A simple argument shows that the anomaly vanishes for an irreducible representation expressed by a Young tableau whose number of boxes is a multiple of $L^2$ (such as the adjoint representation) and for a tensor-product of them. We also evaluate the anomaly for general gauge-group representations in the large $N$ limit. The large $N$ limit exhibits expected algebraic properties as the axial anomaly. Nevertheless, when the gauge group is $SU(N)$, it does not have a structure such as the trace of a product of traceless gauge-group generators which is expected from the corresponding gauge field theory.
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The topological charge in the $U(N)$ vector-like reduced model can be defined by using the overlap Dirac operator. We obtain its large $N$ limit for a fermion in a general gauge-group representation under a certain restriction of gauge field configurations which is termed $U(1)$ embedding.
The spectral flow of the overlap operator is computed numerically along a particular path in gauge field space. The path connects two gauge equivalent configurations which differ by a gauge transformation in the non-trivial class of pi_4(SU(2)). The computation is done with the SU(2) gauge field in the fundamental, the 3/2, and the 5/2 representation. The number of eigenvalue pairs that change places along this path is established for these three representations and an even-odd pattern predicted by Witten is verified.
The spectral flow of the overlap operator is computed numerically along a path connecting two gauge fields which differ by a topologically non-trivial gauge transformation. The calculation is performed for SU(2) in the 3/2 and 5/2 representation. An even-odd pattern for the spectral flow as predicted by Witten is verified. The results are, however, more complicated than naively expected.
The Abelian dominance for the string tension was shown for the fundamental sources in MA gauge in the lattice simulations. For higher representations, however, it is also known that the naive Abelian Wilson loop, which is defined by using the diagonal part of the gauge field, does not reproduce the correct behavior. To solve this problem, for an arbitrary representation of an arbitrary compact gauge group, we propose to redefine the Abelian Wilson loop. By using this redefined operator, we demonstrate the Abelian dominance for sources in the adjoint representation and the sextet representation of $SU(3)$ gauge group in lattice simulations.
In previous works, we have proposed a new formulation of Yang-Mills theory on the lattice so that the so-called restricted field obtained from the gauge-covariant decomposition plays the dominant role in quark confinement. This framework improves the Abelian projection in the gauge-independent manner. For quarks in the fundamental representation, we have demonstrated some numerical evidences for the restricted field dominance in the string tension, which means that the string tension extracted from the restricted part of the Wilson loop reproduces the string tension extracted from the original Wilson loop. However, it is known that the restricted field dominance is not observed for the Wilson loop in higher representations if the restricted part of the Wilson loop is extracted by adopting the Abelian projection or the field decomposition naively in the same way as in the fundamental representation. In this paper, therefore, we focus on confinement of quarks in higher representations. By virtue of the non-Abelian Stokes theorem for the Wilson loop operator, we propose suitable gauge-invariant operators constructed from the restricted field to reproduce the correct behavior of the original Wilson loop averages for higher representations. Moreover, we perform lattice simulations to measure the static potential for quarks in higher representations using the proposed operators. We find that the proposed operators well reproduce the behavior of the original Wilson loop average, namely, the linear part of the static potential with the correct value of the string tension, which overcomes the problem that occurs in naively applying Abelian-projection to the Wilson loop operator for higher representations.
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