Anomaly Theories Advance with 5D Geometry: Efficient Extraction from Superconformal Field Data

Fundamental inconsistencies in quantum field theory, known as anomalies, represent crucial information about the underlying physics, and scientists now demonstrate a new way to calculate these anomalies by examining the geometry of extra dimensions. Mirjam Cvetič from the University of Pennsylvania and University of Maribor, Ron Donagi from the University of Pennsylvania, and Jonathan J. Heckman from the University of Pennsylvania, along with Max Hübner from Uppsala University and Harvard University, reveal that these anomalies can be efficiently extracted from mathematical quantities called eta-invariants. This approach bypasses complex computational methods previously required to analyse these anomalies in five-dimensional superconformal field theories, offering a powerful new tool for understanding the relationship between geometry and quantum physics. The team’s results apply to a broad range of theoretical scenarios, including those with complex, non-isolated singularities, and extend beyond simplified theoretical models to encompass more realistic backgrounds.

String Theory, Calabi-Yau Manifolds, and Anomalies

A comprehensive body of work spans string theory, Calabi-Yau manifolds, cobordism, anomalies, generalized symmetries, and various aspects of theoretical physics and mathematics. Core string theory and Calabi-Yau manifolds form the foundation for much of this research, with established references providing introductions to string theory and detailed explorations of mirror symmetry and techniques for constructing string vacua. These studies also cover toric varieties, crucial for understanding Calabi-Yau compactifications. A prominent theme focuses on cobordism, a topological invariant used to constrain possible consistent theories and identify those residing in the swampland.

Recent work introduces the cobordism conjecture and explores its connection to anomalies and the string lamppost principle. Researchers investigate the swampland cobordism conjecture, dynamical tadpoles, and stringy cobordism, seeking to understand spontaneous compactification. Further studies explore the anomaly that was not meant to be, structure within the cobordism conjecture, and the interplay between cobordism, singularities, and the Ricci flow conjecture. Recent advances chronicle IIBordia, exploring dualities, bordisms, and the swampland, while investigations into Cobordism Utopia examine reflection branes, bordisms, and U-dualities.

Beyond standard symmetries, researchers explore generalized symmetries in F-theory and the topology of elliptic fibrations, as well as generalized global symmetries of T[M] theories. These investigations build upon foundational work in Chern-Simons theories. Studies also focus on the construction and classification of superconformal field theories in lower dimensions, often arising from compactifications of higher-dimensional theories on Calabi-Yau manifolds. Mathematical foundations and tools, including cohomology of twisted projective spaces, Gorenstein quotient singularities, and finite group theory, underpin these investigations. These studies provide the necessary mathematical background for understanding the physics and exploring connections between different theoretical frameworks.

Anomalies from Extra Dimensional Geometry

Scientists have achieved a breakthrough in understanding the anomalies of superconformal field theories (SCFTs) by directly extracting them from the geometry of extra dimensions. This work bypasses computationally intensive methods previously required for analyzing these complex systems, offering a more efficient approach to determine key characteristics of SCFTs. The research demonstrates that anomalies, fundamental non-perturbative data defining these theories, can be determined by analyzing η-invariants derived from the boundaries of the extra-dimensional geometry. The team illustrated this method using 5D SCFTs engineered within M-theory using non-compact geometries, specifically those defined by subgroups within SU(3).

Results show that the anomalies are precisely determined by the η-invariants calculated on the asymptotic boundary of the geometry, denoted as S5/Γ. This approach applies equally well to both Abelian and non-Abelian subgroups, and is effective for both isolated and non-isolated singularities within the geometry. Further analysis focused on non-isolated singularities, revealing how anomaly structures interact across different strata of the singular locus. Measurements confirm that this method extends beyond simple geometries, successfully applying to backgrounds that are not global orbifolds and do not necessarily preserve supersymmetry. The team demonstrated computations for specific examples, including C3/Z2n+1(1, 1, 2n −1) and C3/Z2n+2(1, 1, 2n), confirming the accuracy and versatility of the approach. This breakthrough delivers a powerful new tool for exploring the intricate properties of SCFTs and their underlying symmetries.

Geometry Defines Anomalies in Superconformal Field Theories

This research presents a new method for extracting anomalies from five-dimensional superconformal field theories (SCFTs) directly from the geometry of the extra dimensions in which they are embedded. The team demonstrates that these anomalies, fundamental characteristics of the theories, can be efficiently calculated using geometric invariants, bypassing computationally intensive techniques previously required. This approach applies to a broad range of SCFTs, including those with both Abelian and non-Abelian symmetries, and accommodates both isolated and more complex, non-isolated singularities within the extra-dimensional geometry. The key achievement lies in establishing a direct link between the geometry of the extra dimensions and the symmetry data of the SCFTs, effectively reducing the problem of calculating anomalies to a geometric one. By focusing on invariants of the asymptotic boundary of the extra-dimensional space, the researchers provide a streamlined method for determining these crucial characteristics of the SCFTs. The method extends beyond simple geometric backgrounds, remaining valid even when supersymmetry is not preserved, broadening its applicability to a wider range of theoretical models.