Axial Anomaly in SU(N) Yang-Mills Matrix Models.

Area of Science:

• High Energy Physics
• Quantum Field Theory
• Mathematical Physics

Background:

• The SU(N) Yang-Mills matrix model features self-dual and anti-self-dual instantons.
• Coupling to N_{f} massless quarks introduces chirality zero modes in instanton backgrounds.

Purpose of the Study:

• To establish a matrix model analog of the Atiyah-Singer index theorem.
• To investigate the chiral and axial symmetries in the presence of instantons.

Main Methods:

• Analysis of the Euclidean Dirac equation in instanton backgrounds.
• Examination of vacuum structure and tunneling between representations.
• Investigation of path integral measure invariance under chiral rotation.

Main Results:

• An "instanton charge" T_{new} = ±[r_{0}^{(2)}-r_{0}^{(1)}] is defined, equaling the index (n_{+}-n_{-}).
• The path integral measure's non-invariance is linked to the Dirac operator index.
• Axial symmetry is anomalously broken, resulting in finite residual symmetries Z_{2N_{f}} or Z_{4N_{f}}.

Conclusions:

• The study presents a matrix model version of the Atiyah-Singer index theorem.
• Chiral and axial symmetries exhibit anomalous breaking, with specific residual symmetries identified for different quark types.