On the M2-Brane Index on Noncommutative Crepant Resolutions

Area of Science:

• String theory and M-theory
• Algebraic geometry
• Mathematical physics

Background:

• M-theory backgrounds involving circle fibration over Calabi-Yau manifolds.
• The established connection between M2 brane quantum theory and K-theoretic Donaldson-Thomas theory.

Purpose of the Study:

• To extend the relationship between M-theory and K-theoretic Donaldson-Thomas theory to noncommutative crepant resolutions.
• To investigate the geometric and algebraic structures arising from singularities in these resolutions.
• To compute K-theoretic quantities on quiver representation moduli spaces.

Main Methods:

• Utilizing K-theoretic Donaldson-Thomas theory.
• Applying noncommutative geometry to singular Calabi-Yau threefolds.
• Employing quiver representation theory and moduli spaces.
• Leveraging toric localization techniques for computations.

Main Results:

• Demonstrated that K-theoretic quantities on quiver representation moduli spaces can be calculated using toric localization.
• These computations yield specific rational functions of toric parameters.
• The study analyzes the conifold and certain orbifold singularities within this framework.

Conclusions:

• The findings provide a bridge between M-theory, noncommutative geometry, and representation theory.
• The methods offer a computational approach to K-theoretic invariants in singular spaces.
• This work deepens the understanding of dualities in theoretical physics and mathematics.