Gauged Quiver Mechanics Realize Superconformal Symmetry, Revealing Black Hole Microscopics in Four Dimensions

The fundamental nature of black holes and the underlying geometry of spacetime remain central questions in theoretical physics, and recent research sheds new light on these complex systems. Canberk Şanlı, from CEICO at the Institute of Physics of the Czech Academy of Sciences, and colleagues investigate the geometry of ‘conformal quivers’, mathematical models that describe the behaviour of black holes formed from multiple interacting sources. This work provides a crucial link between theoretical string theory and the observed properties of black holes, offering a detailed description of their microscopic structure. By developing a coordinate system that maps the geometry of these models, the researchers successfully decompose the complex interactions within the system and demonstrate a new method for resolving inconsistencies that arise in traditional calculations, ultimately advancing our understanding of gravity and quantum mechanics.

Near-horizon microscopies of multi-centred black holes within four-dimensional N = 2 supergravity form the basis of this work. The research employs a gauged sigma model formulation utilising off-shell (4, 4, 0) multiplets, enabling the introduction of a coordinate system specifically adapted to the complex structure of the gauged sigma model target space. This coordinate system assigns a radial-angular frame to each quiver node within the physical configuration space, providing an explicit decomposition of the sigma model quiver metric into sectors governed by couplings between each node.

Gauging Superconformal Symmetry for Mechanics

This research investigates the intersection of conformal and superconformal mechanics, gauged superconformal mechanics, and black hole physics. Conformal and superconformal mechanics study quantum mechanical systems possessing powerful symmetry constraints, while gauged superconformal mechanics extends this by introducing interactions. A central motivation is understanding the microstates of black holes, particularly those that are nearly extremal, and counting these states to interpret black hole entropy quantum mechanically. The team utilizes techniques like localization to compute superconformal indices, crucial for counting these microstates, and connects the physics to geometric structures using quiver diagrams.

Superconformal symmetry combines conformal symmetry, involving scaling, rotations, and translations, with supersymmetry, relating bosons and fermions. Gauging promotes a symmetry to a local symmetry by introducing gauge fields, thereby introducing interactions. The superconformal index is a protected quantity, remaining stable under small changes to the theory, and counts the number of BPS states, which preserve some supersymmetry. Localization is a powerful technique for evaluating path integrals by focusing on fixed points of certain symmetries. Quiver diagrams graphically represent gauge theories, with nodes representing gauge groups and edges representing interactions. Microstates of black holes are the fundamental quantum states that, when counted, give the entropy of a black hole, and BPS states are easier to count due to their supersymmetry preservation.

Gauging Conical Singularities in Black Hole Models

Scientists have achieved a breakthrough in theoretical physics by developing a novel approach to understanding superconformal symmetry within the context of multi-centered black holes. Their work focuses on “quiver mechanics,” models describing the near-horizon properties of these black holes and rooted in string theory. The team successfully implemented the D(2, 1; 0) index to gauged superconformal mechanics, demonstrating that this gauging process naturally resolves conical singularities, a long-standing problem. This resolution simplifies calculations and provides a well-defined quantum theory.

The researchers discovered that introducing a “gauge” into the system effectively shifts the localization locus, pushing fixed points away from problematic singularities and enabling a more robust mathematical framework. This is achieved through a novel geometric description of the system, utilizing a coordinate system adapted to the gauged sigma model target space. This coordinate system assigns radial and angular directions to each node within the “quiver,” allowing for a clear decomposition of the complex target space metric into manageable sectors governed by couplings between nodes. The team evaluated the quiver target space metric for configurations with two and three nodes, extending naturally to symmetric N-node crystal quivers, demonstrating a closed-form decomposition of the metric detailing radial-angular couplings.

Quiver Geometry and Resolution of Singularities

This research investigates the geometry of gauged quiver mechanics, providing a framework for understanding multi-centered black holes and their microscopic structure within string theory. The team developed a coordinate system simplifying the complex geometry of these systems, allowing for an explicit decomposition of the target space metric into manageable components, clarifying relationships between nodes within the quiver configuration and extending naturally to more complex arrangements. The study implements the D(2,1;0) index to resolve conical singularities, complications often arising in these calculations, and accurately determine relevant configurations. By combining this index with the new coordinate system, researchers can more precisely map the geometry of the gauged sigma model, offering insights into the underlying physics of black holes and potentially paving the way for applications in holography. The authors acknowledge that their current work focuses on abelian gauge groups, and future research could explore non-abelian scenarios.