Equivariant Localization in Supergravity.

Area of Science:

• Theoretical physics
• Mathematical physics
• String theory

Background:

• Supersymmetric supergravity theories are crucial for understanding quantum gravity.
• R-symmetry Killing vectors play a significant role in classifying these solutions.
• Evaluating physical observables often requires solving complex differential equations.

Purpose of the Study:

• To develop a method for calculating physical observables in supersymmetric supergravity solutions.
• To demonstrate that these calculations can be performed without solving the full supergravity equations.
• To connect topological data with physical observables via equivariant forms.

Main Methods:

• Identifying equivariantly closed forms associated with R-symmetry Killing vectors.
• Expressing physical observables as integrals of these closed forms.
• Applying the Berline-Vergne-Atiyah-Bott fixed point theorem for evaluation.

Main Results:

• Supersymmetric supergravity solutions with R-symmetry Killing vectors possess a set of equivariantly closed forms.
• Physical observables such as on-shell actions, black hole entropies, central charges, and operator scaling dimensions can be expressed using these forms.
• The evaluation of these observables depends solely on topological data and the R-symmetry vector.

Conclusions:

• A novel, efficient method is established for computing physical observables in specific supergravity solutions.
• This approach bypasses the need to solve complex supergravity equations, relying instead on topological invariants.
• The findings offer a powerful tool for analyzing holographic models and black hole physics.