Periodic Reporting for period 1 - Quivers (AdS(2)/CFT(1) holography via quiver quantum mechanics)
Reporting period: 2023-04-11 to 2025-04-10
Black holes are key to understanding the deep connection between gravity and quantum mechanics. While general relativity predicts the macroscopic behavior of black holes, it does not explain their microscopic structure. Quantum theory suggests that black holes carry entropy, implying the existence of countless hidden microstates, but a precise description of these states is still missing. This is known as the Black Hole Microstructure Problem (BHMP), a long-standing and fundamental challenge in theoretical physics.
The QUIVERS project tackled this problem by studying the quantum mechanics of D-brane bound states—objects arising in string theory that can be used to model black holes at the microscopic level. Specifically, it focused on a class of supersymmetric systems known as Type-B superconformal quantum mechanics, which are believed to capture the physics near the horizon of extremal black holes.
The central objective of the project was to compute a refined mathematical object called the superconformal index, which counts the number of quantum ground states protected by symmetry. These indices were calculated for models with singular and non-compact geometries, using advanced localization and regularization techniques. The results were then compared with entropy predictions from supergravity theories, providing a direct test of the AdS(2)/CFT(1) holographic correspondence, a conjectured duality between gravitational systems and quantum mechanics.
QUIVERS thus provided new tools to quantify black hole microstates and offered one of the first concrete examples of AdS(2)/CFT(1) duality beyond idealized or simplified setups. This contributes not only to quantum gravity research but also to mathematical physics, as the methods developed intersect with topics in geometry and index theory.
In line with the EU’s goals to support frontier science and interdisciplinary collaboration, QUIVERS enhances our understanding of quantum black holes and strengthens Europe's role in fundamental theoretical research.
The QUIVERS project tackled this problem by studying the quantum mechanics of D-brane bound states—objects arising in string theory that can be used to model black holes at the microscopic level. Specifically, it focused on a class of supersymmetric systems known as Type-B superconformal quantum mechanics, which are believed to capture the physics near the horizon of extremal black holes.
The central objective of the project was to compute a refined mathematical object called the superconformal index, which counts the number of quantum ground states protected by symmetry. These indices were calculated for models with singular and non-compact geometries, using advanced localization and regularization techniques. The results were then compared with entropy predictions from supergravity theories, providing a direct test of the AdS(2)/CFT(1) holographic correspondence, a conjectured duality between gravitational systems and quantum mechanics.
QUIVERS thus provided new tools to quantify black hole microstates and offered one of the first concrete examples of AdS(2)/CFT(1) duality beyond idealized or simplified setups. This contributes not only to quantum gravity research but also to mathematical physics, as the methods developed intersect with topics in geometry and index theory.
In line with the EU’s goals to support frontier science and interdisciplinary collaboration, QUIVERS enhances our understanding of quantum black holes and strengthens Europe's role in fundamental theoretical research.
The QUIVERS project was structured around the goal of computing superconformal indices for Type-B supersymmetric quantum mechanical models, which describe the dynamics of D-brane bound states relevant to black hole microphysics. These models are highly non-trivial due to their singular, non-compact target space geometries and the reduced amount of supersymmetry they preserve.
The work proceeded in three main stages:
Algebraic foundation and representation theory:
The project began with a systematic classification of irreducible representations of the relevant superconformal algebras, su(1,1∣1) and D(2,1;α), identifying the short (BPS) multiplets that contribute to the ground state spectrum. This established the algebraic structure needed to define the superconformal index.
Localization and regularization techniques:
A major achievement was the development and implementation of localization methods tailored to Type-B models, allowing for exact index computations even on singular and non-Kähler spaces. These techniques adapted known geometric tools to a new class of supersymmetric quantum systems and allowed for reliable extraction of physical degeneracies.
Explicit examples and holographic interpretation:
Two representative models—one with α=0 and one with α≠0—were canonically quantized. Their spectra were analyzed independently of localization, and the resulting indices matched those from the localization framework, validating the methodology. These results were then interpreted in the context of black hole entropy, supporting the AdS(2)/CFT(1) correspondence. The project also identified a twist structure in the metrics of quiver models, which resemble the analog property of the corresponding supergravity solutions. Overall, the project produced novel analytic tools, exact index formulas, and a consistent holographic picture for extremal black holes. These achievements were documented in four peer-reviewed articles and two forthcoming submissions, marking a significant contribution to the field of quantum gravity and supersymmetric quantum mechanics.
The work proceeded in three main stages:
Algebraic foundation and representation theory:
The project began with a systematic classification of irreducible representations of the relevant superconformal algebras, su(1,1∣1) and D(2,1;α), identifying the short (BPS) multiplets that contribute to the ground state spectrum. This established the algebraic structure needed to define the superconformal index.
Localization and regularization techniques:
A major achievement was the development and implementation of localization methods tailored to Type-B models, allowing for exact index computations even on singular and non-Kähler spaces. These techniques adapted known geometric tools to a new class of supersymmetric quantum systems and allowed for reliable extraction of physical degeneracies.
Explicit examples and holographic interpretation:
Two representative models—one with α=0 and one with α≠0—were canonically quantized. Their spectra were analyzed independently of localization, and the resulting indices matched those from the localization framework, validating the methodology. These results were then interpreted in the context of black hole entropy, supporting the AdS(2)/CFT(1) correspondence. The project also identified a twist structure in the metrics of quiver models, which resemble the analog property of the corresponding supergravity solutions. Overall, the project produced novel analytic tools, exact index formulas, and a consistent holographic picture for extremal black holes. These achievements were documented in four peer-reviewed articles and two forthcoming submissions, marking a significant contribution to the field of quantum gravity and supersymmetric quantum mechanics.
The QUIVERS project has delivered several novel scientific results that go beyond the current state of the art in quantum gravity, supersymmetric quantum mechanics, and mathematical physics.
Most significantly, it provided the first exact computation of the superconformal index for Type-B supersymmetric quantum mechanical models with singular and non-compact target spaces. Unlike previously studied Type-A models, Type-B models lack a full geometric classification and present unique challenges due to torsion and reduced symmetry. By adapting and extending localization techniques to these models, QUIVERS introduced a new analytical framework applicable to a wider class of physical systems.
The project also quantized two explicit D(2,1;α) models and confirmed the consistency of localization results through direct spectral analysis—this dual validation approach had not been previously demonstrated in the literature for such systems. These findings confirm the robustness of the localization framework developed and provide reliable tools for future studies in moduli space quantization and supersymmetric sigma models.
In the context of black hole physics, QUIVERS made a major conceptual advance by linking the computed indices to black hole entropy in AdS(2)/CFT(1) holography.
Potential future directions include:
Applying these methods to non-supersymmetric extremal black holes, using attractor mechanisms to extend the results beyond BPS settings.
Generalizing the index construction to more complex geometries, including monopole moduli spaces and higher-dimensional quiver systems.
Exploring links with topological invariants and index theorems on HKT manifolds in mathematical physics.
Most significantly, it provided the first exact computation of the superconformal index for Type-B supersymmetric quantum mechanical models with singular and non-compact target spaces. Unlike previously studied Type-A models, Type-B models lack a full geometric classification and present unique challenges due to torsion and reduced symmetry. By adapting and extending localization techniques to these models, QUIVERS introduced a new analytical framework applicable to a wider class of physical systems.
The project also quantized two explicit D(2,1;α) models and confirmed the consistency of localization results through direct spectral analysis—this dual validation approach had not been previously demonstrated in the literature for such systems. These findings confirm the robustness of the localization framework developed and provide reliable tools for future studies in moduli space quantization and supersymmetric sigma models.
In the context of black hole physics, QUIVERS made a major conceptual advance by linking the computed indices to black hole entropy in AdS(2)/CFT(1) holography.
Potential future directions include:
Applying these methods to non-supersymmetric extremal black holes, using attractor mechanisms to extend the results beyond BPS settings.
Generalizing the index construction to more complex geometries, including monopole moduli spaces and higher-dimensional quiver systems.
Exploring links with topological invariants and index theorems on HKT manifolds in mathematical physics.