Periodic Reporting for period 2 - QGBoot (Scattering Strings and Other Things: A Modern Approach to Quantum Gravity and the Conformal Bootstrap)
Reporting period: 2022-06-01 to 2023-11-30
A holy grail of modern theoretical physics is to find a theory of quantum gravity. This would merge the two pillars of 20th century physics – namely, the theory of gravity as originally canonized in Einstein’s general relativity, and quantum mechanics – into a single consistent framework. Whereas the language of quantum field theory (a descendant and generalization of quantum mechanics) is well-suited to describe the standard model of particle physics, it fails to account for gravitational physics, and especially so for interesting cosmological or astrophysical phenomena, such as black hole formation or collision.
A tantalizing relation between gravity and quantum mechanics of a different sort was discovered just over 25 years ago. This is known as the AdS/CFT Correspondence. It draws a certain equivalence between two apparently distinct theories - one with gravity (AdS), the other without (CFT) – furnishing a precise map between them. This “holographic” relation has been the dominant thread of 21st century theoretical physics. By leveraging our knowledge of gravity, holography has given insight into strongly coupled dynamics of quantum field theories; conversely, we have understood many fundamental things about gravity by applying the rapidly-developing toolkit of quantum field theory.
This toolkit has grown most robustly for conformal field theories, or CFTs, in particular. These special quantum field theories are endowed with a self-similarity at all scales; they exist both formally and literally, including in the physics of everyday life, describing the critical points of various natural phenomena (e.g. boiling water). The idea that one can constrain the space and properties of these theories by suitably imposing symmetries and abstract consistency condition is known as the “conformal bootstrap”, a growing subfield that has been immensely successful. This becomes doubly impactful when combined with the AdS/CFT Correspondence: as one learns about conformal field theories, so one likewise constrains the space of theories of quantum gravity. This latter pursuit is an overarching goal of this project.
Within this picture, years of intensive study of the AdS/CFT Correspondence have supported the existence of string theory as a fundamental ingredient of our universe. String theory is a candidate theory of quantum gravity, a well-developed and formally consistent theory in myriad ways. Its repeated appearance on the “gravity side” of the AdS/CFT Correspondence provides a “dual” description of many conformal field theories. In this way, one can co-opt the results of conformal bootstrap to indirectly derive the structure of string theory and quantum gravity, eschewing direct approaches to the latter which are either very challenging or lack an explicit framework.
A tantalizing relation between gravity and quantum mechanics of a different sort was discovered just over 25 years ago. This is known as the AdS/CFT Correspondence. It draws a certain equivalence between two apparently distinct theories - one with gravity (AdS), the other without (CFT) – furnishing a precise map between them. This “holographic” relation has been the dominant thread of 21st century theoretical physics. By leveraging our knowledge of gravity, holography has given insight into strongly coupled dynamics of quantum field theories; conversely, we have understood many fundamental things about gravity by applying the rapidly-developing toolkit of quantum field theory.
This toolkit has grown most robustly for conformal field theories, or CFTs, in particular. These special quantum field theories are endowed with a self-similarity at all scales; they exist both formally and literally, including in the physics of everyday life, describing the critical points of various natural phenomena (e.g. boiling water). The idea that one can constrain the space and properties of these theories by suitably imposing symmetries and abstract consistency condition is known as the “conformal bootstrap”, a growing subfield that has been immensely successful. This becomes doubly impactful when combined with the AdS/CFT Correspondence: as one learns about conformal field theories, so one likewise constrains the space of theories of quantum gravity. This latter pursuit is an overarching goal of this project.
Within this picture, years of intensive study of the AdS/CFT Correspondence have supported the existence of string theory as a fundamental ingredient of our universe. String theory is a candidate theory of quantum gravity, a well-developed and formally consistent theory in myriad ways. Its repeated appearance on the “gravity side” of the AdS/CFT Correspondence provides a “dual” description of many conformal field theories. In this way, one can co-opt the results of conformal bootstrap to indirectly derive the structure of string theory and quantum gravity, eschewing direct approaches to the latter which are either very challenging or lack an explicit framework.
Much of this project’s work so far has been driven by the discovery of novel ways in which the fundamental symmetries of a conformal field theory, or a quantum theory of gravity, constrain its observables, such as correlation functions and scattering amplitudes. By making certain symmetries manifest, profound hidden structures are revealed. This work makes satisfying use of established tools in the mathematics, namely, those of spectral analysis for the modular group.
There have been two main prongs of our work in this direction, distinguished by their spacetime dimension.
In two-dimensional conformal field theories (that is theories with one space and one time dimension) and their holographically dual three-dimensional gravity theories, there is a symmetry under relabeling which dimension is space and which dimension is time. This symmetry known as “modular invariance.” Importing techniques from the mathematical world of spectral analysis, one can make this symmetry manifest, in a way that implies a strong underlying redundancy in the spectra of these theories, thus greatly reducing the set of building blocks needed to fully characterize them.
In four-dimensional conformal field theories and their holographically dual five-dimensional theories of gravity, the very same modular symmetry appears albeit with a completely different physical meaning. We focused on a particularly well-controlled model which is endowed with extra (unrelated) symmetry known as supersymmetry. This theory has a constant parameter, a “coupling constant” on which its observables depend. Somewhat magically, the theory looks the same whether the coupling is small or large. This so-called self-duality is precisely the modular symmetry described above. Again applying the same spectral techniques, we learn that the underlying structures of the theory are very strongly constrained. This led to tantalizing connections with certain notions of averaging over the space of theories.
The set of observables in which we are interested consists of scattering amplitudes and correlation functions. To this end, we studied a particularly computable set of correlation functions in this theory, finding beautiful simplicity in an a priori complicated object. We leveraged this to study string theory, via holographic duality, in a quantum regime in which there are relatively few existing computations. This is the kind of computation that hints at exact properties of string theory.
There have been two main prongs of our work in this direction, distinguished by their spacetime dimension.
In two-dimensional conformal field theories (that is theories with one space and one time dimension) and their holographically dual three-dimensional gravity theories, there is a symmetry under relabeling which dimension is space and which dimension is time. This symmetry known as “modular invariance.” Importing techniques from the mathematical world of spectral analysis, one can make this symmetry manifest, in a way that implies a strong underlying redundancy in the spectra of these theories, thus greatly reducing the set of building blocks needed to fully characterize them.
In four-dimensional conformal field theories and their holographically dual five-dimensional theories of gravity, the very same modular symmetry appears albeit with a completely different physical meaning. We focused on a particularly well-controlled model which is endowed with extra (unrelated) symmetry known as supersymmetry. This theory has a constant parameter, a “coupling constant” on which its observables depend. Somewhat magically, the theory looks the same whether the coupling is small or large. This so-called self-duality is precisely the modular symmetry described above. Again applying the same spectral techniques, we learn that the underlying structures of the theory are very strongly constrained. This led to tantalizing connections with certain notions of averaging over the space of theories.
The set of observables in which we are interested consists of scattering amplitudes and correlation functions. To this end, we studied a particularly computable set of correlation functions in this theory, finding beautiful simplicity in an a priori complicated object. We leveraged this to study string theory, via holographic duality, in a quantum regime in which there are relatively few existing computations. This is the kind of computation that hints at exact properties of string theory.
The approaches described above represent a shift in our understanding of modular symmetry in conformal field theory. This state-of-the-art technique will have many further applications in theoretical physics, given the ubiquity of this symmetry.
One connection currently being developed in this project is to a yet-separate field of physics, namely that of Random Matrix Theory. This is the study of matrices whose entries are random numbers, drawn from some specified distribution. Despite this simple characterization it turns out that random matrices are highly capable of modeling the intricate properties of a diverse set of mathematical and physical systems. This famously includes the distribution of zeros of the Riemann zeta function, but also the dynamics of chaotic quantum systems: suitably understood, both of these can be very well described by pretending they are random matrices.
What is the connection of random matrix theory to the spectral theory of the modular group? For two-dimensional conformal field theories in particular, we are discovering that these spectral techniques lead directly to a formalism for understanding their chaotic dynamics – a notoriously hard problem, given the many constraints that two-dimensional conformal field theories must satisfy over and above those of a one-dimensional chaotic quantum system. Moreover, in the context of the AdS/CFT Correspondence, these results hold a key to illuminating the precise sense in which simple theories of three-dimensional gravity do – and do not – capture the dynamics of conformal field theories and ensembles thereof. Work in these directions will appear.
In the realm of four-dimensional conformal field theories and their dual scattering amplitudes, we hope to gain a deeper understanding of how the S-duality symmetry plays with the other axiomatic constraints implied by the conformal bootstrap. This would have the potential to constrain not just the observables in individual theories, but the space of theories itself.
Overall, the structure of conformal field theory observables, and the landscape of gravity theories allowed by symmetries and the conformal bootstrap, should hold many further surprises which we hope to discover.
One connection currently being developed in this project is to a yet-separate field of physics, namely that of Random Matrix Theory. This is the study of matrices whose entries are random numbers, drawn from some specified distribution. Despite this simple characterization it turns out that random matrices are highly capable of modeling the intricate properties of a diverse set of mathematical and physical systems. This famously includes the distribution of zeros of the Riemann zeta function, but also the dynamics of chaotic quantum systems: suitably understood, both of these can be very well described by pretending they are random matrices.
What is the connection of random matrix theory to the spectral theory of the modular group? For two-dimensional conformal field theories in particular, we are discovering that these spectral techniques lead directly to a formalism for understanding their chaotic dynamics – a notoriously hard problem, given the many constraints that two-dimensional conformal field theories must satisfy over and above those of a one-dimensional chaotic quantum system. Moreover, in the context of the AdS/CFT Correspondence, these results hold a key to illuminating the precise sense in which simple theories of three-dimensional gravity do – and do not – capture the dynamics of conformal field theories and ensembles thereof. Work in these directions will appear.
In the realm of four-dimensional conformal field theories and their dual scattering amplitudes, we hope to gain a deeper understanding of how the S-duality symmetry plays with the other axiomatic constraints implied by the conformal bootstrap. This would have the potential to constrain not just the observables in individual theories, but the space of theories itself.
Overall, the structure of conformal field theory observables, and the landscape of gravity theories allowed by symmetries and the conformal bootstrap, should hold many further surprises which we hope to discover.