Operators Integrate Heavy Sectors, Revealing Light States in Holographic Theories

Scientists have long sought to reconcile the seemingly disparate fields of two-dimensional conformal field theory and the AdS/RMT program, uncovering connections to areas such as permutation orbifold theories and ensembles of code CFTs. Nico Cooper from the Department of Physics and Astronomy, University of Kentucky, and colleagues demonstrate an equidistribution theorem for Hecke operators that reveals a significant simplification within these complex systems. Their work, achieved in collaboration with researchers across multiple institutions, shows that large sectors of the partition function effectively disappear in certain limits, focusing contributions on lighter states. This finding provides a direct holographic interpretation, linking the mathematics to a sum over semiclassical handlebody geometries and offering potential insights into ergodicity, thereby advancing our understanding of quantum gravity and the fundamental structure of spacetime.

The research focuses on the behaviour of ‘Hecke operators’, mathematical tools appearing in diverse areas of theoretical physics, from conformal field theory to holographic duality, and reveals how their application leads to an unexpected streamlining of calculations. Specifically, the study shows that as the index of the Hecke operator, denoted as ‘N’, approaches infinity, a substantial ‘heavy sector’ of the partition function is integrated out. This finding has profound implications for understanding the relationship between gravity in Anti-de Sitter space (AdS) and conformal field theories (CFTs) on its boundary, a cornerstone of modern theoretical physics. Researchers achieved this simplification by leveraging a theorem concerning the equidistribution of Hecke operators, demonstrating its applicability to a range of physical scenarios; this equidistribution theorem essentially states that, in the limit of large N, the Hecke operator converges to a modular integral, effectively smoothing out complex fluctuations. By applying this splitting in conjunction with the equidistribution theorem, scientists derived a new form for partition functions in several contexts, including code CFT averages, cyclic and symmetric product orbifolds, and a specific string theory model known as the “Zstring”. The new form highlights the emergence of a Poincaré series, naturally interpreted as a sum over semiclassical handlebody geometries in the holographic dual. This work suggests a potential connection between number theory (through Hecke operators) and quantum gravity, offering a novel perspective on the search for a consistent theory of quantum gravity, and opens avenues for exploring potential ergodicity statements within these systems. The key number driving this simplification is ‘N’, as the research shows that in the limit as N approaches infinity, the heavy sector of the partition function is integrated out, leading to the simplification described above. A detailed examination of code conformal field theories (CFTs) forms the basis of this work, leveraging their connection to error-correcting codes and modular averaging techniques. The study begins with Narain CFTs, theories dependent on charges corresponding to momentum and winding numbers on a torus, and their relationship to even self-dual lattices. These lattices are systematically linked to doubly even self-dual error correcting codes, establishing a correspondence where the code’s weight enumerator polynomial mirrors the CFT’s partition function. This analogy allows for a discrete averaging procedure over Narain CFTs at a fixed central charge, c, equal to the length of the codewords in the code. The averaging process, initially reviewed for c = 1, considers two inequivalent codes over Zp × Zp, generated by specific codewords defining compact bosons with distinct radii, R+ and R−, subject to T-duality, a symmetry transforming R to 2/R. Consequently, consideration of the moduli space requires a quotient by this duality to a half-line. The averaged partition function is then calculated as a sum over the vacuum characters, Ψab, defined in terms of the torus partition function and the code’s generators. Vacuum characters, Ψab, are constructed using a summation over integer values, n and m, incorporating terms dependent on the radius, r, and the code’s letters, a and b, directly relating to the weight enumerator and providing a concrete link between the code’s structure and the CFT’s spectral properties. Researchers demonstrate that as the index of the Hecke operator, ‘N’, approaches infinity, the heavy sector of the partition function effectively vanishes through integration. This simplification arises from an equidistribution theorem applied to Hecke operators, revealing a direct connection to sums over semiclassical handlebody geometries within a holographic framework. The core of this finding lies in the behaviour of the partition function as N becomes very large, allowing for a streamlined representation based on Poincaré series of light states. Specifically, the study establishes that the integrated contribution from the heavy sector diminishes significantly as N increases, leading to a partition function dominated by lighter states, a consequence of demanding modular invariance within the system. Calculations reveal that the spectral decomposition of Narain CFT partition functions plays a crucial role in achieving explicit forms of these calculations. For cyclic product orbifolds, the large N limit of the torus partition function maintains modular invariance, a feature absent in previous approximations that relied on limN→∞TNf(τ) ≈f(Nτ). Furthermore, the work extends to symmetric product orbifolds and the di Ubaldo-Perlmutter “Zstring”, yielding new large N forms of their partition functions expressed as Poincaré series, validating the approach and its potential for further exploration. Scientists have uncovered a surprising simplification within complex mathematical systems describing the behaviour of black holes and quantum fields. This work, rooted in modular functions and Hecke operators, reveals that seemingly intractable calculations can be streamlined by effectively discarding a vast ‘heavy’ sector of possibilities. For years, physicists have struggled to reconcile gravity, quantum mechanics, and information theory; the holographic principle offers a potential bridge, but calculations often become bogged down in an overwhelming number of variables. This new approach offers a powerful tool for cutting through that complexity, providing a clearer path towards understanding the fundamental nature of spacetime and the information it contains. The implications extend beyond theoretical physics, as the techniques employed here draw parallels with statistical mechanics and random matrix theory, suggesting potential applications in areas like data analysis and machine learning. However, the reliance on large limits and specific mathematical structures means the direct applicability to real-world scenarios remains uncertain. Crucially, this isn’t a final answer, but a refinement of the toolkit; the observed simplification hints at a deeper underlying principle, perhaps a form of ergodicity, suggesting that the system explores all possible states with equal probability. Future research will likely focus on extending this simplification to more realistic scenarios and exploring the connections to other areas of theoretical physics, potentially unlocking new insights into the nature of quantum gravity and the fate of information in black holes.