## Periodic Reporting for period 1 - DualNRSQNM (Fundamentals of Holographic Dualities via Non-relativistic Systems and Quasinormal Modes)

Reporting period: 2015-09-01 to 2017-08-31

"When physicists study the quantum behavior of a system, we generally use an approach called mean (as in average) quantum field theory. This approach assumes that the system is mostly in the average state, with only small variations. It works well when the small variations don’t interact much; this is known as a weakly-coupled field theory. However, for many cases we care about, even small variations interact strongly with each other. One example is the strong nuclear force; another is high-temperature superconductors. Conventional techniques don’t work well for studying these “strongly-coupled” field theories. As we discuss below, holographic dualities provide a genuinely new approach to exploring these strongly-coupled systems.

A holographic duality is a map between a field theory in d dimensions, and a gravity theory in d+1 dimensions. In these maps, any physical quantity in the field theory has a dual in the gravitational theory, and vice versa. ""Dual"" means that a calculation in the field theory will always give the same answer as a calculation in the gravitational theory (and vice versa). We can thus choose to do the calculation wherever it is easiest. We use these maps in two ways: to understand strongly-coupled field theories, or to study gravity when quantum mechanics is important and spacetime is highly curved. Since quantum gravity is one of the great open questions of our time, developing a tool that helps us study it is of profound importance.

The first example, AdS-CFT duality, was found twenty years ago via string theory. AdS or Anti-de Sitter is the name of the gravitational theory, while the field theory behaves the same at any scale which means it is called a CFT or conformal field theory. This original duality has a few features that aren’t like our real world: it assumes supersymmetry (which we haven’t seen), and it assumes a large number of different charges (in the theory of the strong nuclear force, this number is only 3).

Physicists have proposed new holographic dualities in order to use them for realistic systems. In the last decade, there has been particular interest in building duals to non-relativistic systems, where space and time enter on fundamentally different footings, just as in the Schroedinger equation which has one time derivative but two space derivatives.

The first major goal of the research in this project was to “stress-test” the original AdS-CFT duality, by subjecting it to various alterations in order to understand if those changes broke the duality somehow. When a change succeeds, then we have a whole new forum where the tools of holographic duality can be applied; even when a change fails, we still learn something— instead we learn about how dualities themselves work.

The second major goal of this project was to develop a novel computational method, known as the quasinormal mode method (QNM), for studying quantum effects in gravitational systems such as the Anti-de Sitter space part of AdS-CFT. The QNM method relies on studying the vibrational modes of the spacetime as it relaxes after being perturbed. These relaxation modes are known as quasinormal modes, and their behavior can tell us a great deal about the quantum mechanical behavior and even the shape of the underlying space.

This project affects society as all physics does: we try to provide a greater understanding of the universe around us, so we can better predict its behavior as well as satisfy our curiosity. This research is fundamental in nature, improving the investigative tool of holographic dualities. Physicists have studied everything from quark gluon plasma to quantum gravity using this tool. The study of quantum gravity in general addresses our common call to understand the ""why"" and""how"" of our universe. Studies of the fundamental theories underlying nature serve this human urge to understand our world."

A holographic duality is a map between a field theory in d dimensions, and a gravity theory in d+1 dimensions. In these maps, any physical quantity in the field theory has a dual in the gravitational theory, and vice versa. ""Dual"" means that a calculation in the field theory will always give the same answer as a calculation in the gravitational theory (and vice versa). We can thus choose to do the calculation wherever it is easiest. We use these maps in two ways: to understand strongly-coupled field theories, or to study gravity when quantum mechanics is important and spacetime is highly curved. Since quantum gravity is one of the great open questions of our time, developing a tool that helps us study it is of profound importance.

The first example, AdS-CFT duality, was found twenty years ago via string theory. AdS or Anti-de Sitter is the name of the gravitational theory, while the field theory behaves the same at any scale which means it is called a CFT or conformal field theory. This original duality has a few features that aren’t like our real world: it assumes supersymmetry (which we haven’t seen), and it assumes a large number of different charges (in the theory of the strong nuclear force, this number is only 3).

Physicists have proposed new holographic dualities in order to use them for realistic systems. In the last decade, there has been particular interest in building duals to non-relativistic systems, where space and time enter on fundamentally different footings, just as in the Schroedinger equation which has one time derivative but two space derivatives.

The first major goal of the research in this project was to “stress-test” the original AdS-CFT duality, by subjecting it to various alterations in order to understand if those changes broke the duality somehow. When a change succeeds, then we have a whole new forum where the tools of holographic duality can be applied; even when a change fails, we still learn something— instead we learn about how dualities themselves work.

The second major goal of this project was to develop a novel computational method, known as the quasinormal mode method (QNM), for studying quantum effects in gravitational systems such as the Anti-de Sitter space part of AdS-CFT. The QNM method relies on studying the vibrational modes of the spacetime as it relaxes after being perturbed. These relaxation modes are known as quasinormal modes, and their behavior can tell us a great deal about the quantum mechanical behavior and even the shape of the underlying space.

This project affects society as all physics does: we try to provide a greater understanding of the universe around us, so we can better predict its behavior as well as satisfy our curiosity. This research is fundamental in nature, improving the investigative tool of holographic dualities. Physicists have studied everything from quark gluon plasma to quantum gravity using this tool. The study of quantum gravity in general addresses our common call to understand the ""why"" and""how"" of our universe. Studies of the fundamental theories underlying nature serve this human urge to understand our world."

"This work studied the underpinnings of holographic dualities for applications to non-relativistic systems like superconductors, finding that non-relativistic dualities are significantly different from relativistic ones. These differences run in both directions across the duality: the gravity dual implies the field theory has an exponentially-suppressed regime, while the field theory implies the gravitational spacetime cannot be completely reconstructed. These results both highlight the power of the altered duality (the suppressed regime can be seen in the field theory but only through the resummation of an infinite series), but also its limitations (we cannot rebuild the full gravitational theory in the same way we could for the relativistic case). So we have better understood the limitations on applying holographic dualities to non-relativistic systems.

The attached figure shows the ""entanglement wedge"" region of the proposed dual spacetime to a Lifshitz field theory where time scales twice as fast as space. Since time and space scale differently, the field theory is not relativistic, so its spacetime dual must have different asymptotics. Liight rays (rainbow lines in the figure) do not reach the boundary of the spacetime. As such the entanglement wedge does not close as in the relativistic case, and information about the bulk gravitational dual cannot get out to the boundary completely. In the figure, the spacetime boundary is at the back of the picture (at r=0); time goes vertically and the field theory's spatial directions are horizontal. We are looking at the wedge from inside the bulk dual, so we see the light rays that should close off the wedge by reaching the boundary do not do so.

We have also learned about the nature of holographic dualities themselves, via studying ""quasinormal modes"" in dualities. Quasinormal modes are the fluctuations emitted by a spacetime is it relaxes to a stable state. For example, LIGO has recently been able to hear the ringing of a black hole merger; they can actually hear the lowest quasinormal mode of the final black hole. A field theory instead hears the ringing of its dual spacetime via the pole structure of its retarded Green function, which we then use to find lowest-order quantum corrections in the dual gravity. We generalized this method to systems with lesser symmetry, particularly the so-called warped-CFT duals. We are now extending the method further, using these ringdown modes to hear the shape of more general spacetimes.

These scientific results were shared with the broader scientific community in seven invited conference talks, more than fifteen invited seminar talks, and several scientific publications."

The attached figure shows the ""entanglement wedge"" region of the proposed dual spacetime to a Lifshitz field theory where time scales twice as fast as space. Since time and space scale differently, the field theory is not relativistic, so its spacetime dual must have different asymptotics. Liight rays (rainbow lines in the figure) do not reach the boundary of the spacetime. As such the entanglement wedge does not close as in the relativistic case, and information about the bulk gravitational dual cannot get out to the boundary completely. In the figure, the spacetime boundary is at the back of the picture (at r=0); time goes vertically and the field theory's spatial directions are horizontal. We are looking at the wedge from inside the bulk dual, so we see the light rays that should close off the wedge by reaching the boundary do not do so.

We have also learned about the nature of holographic dualities themselves, via studying ""quasinormal modes"" in dualities. Quasinormal modes are the fluctuations emitted by a spacetime is it relaxes to a stable state. For example, LIGO has recently been able to hear the ringing of a black hole merger; they can actually hear the lowest quasinormal mode of the final black hole. A field theory instead hears the ringing of its dual spacetime via the pole structure of its retarded Green function, which we then use to find lowest-order quantum corrections in the dual gravity. We generalized this method to systems with lesser symmetry, particularly the so-called warped-CFT duals. We are now extending the method further, using these ringdown modes to hear the shape of more general spacetimes.

These scientific results were shared with the broader scientific community in seven invited conference talks, more than fifteen invited seminar talks, and several scientific publications."

"This project made significant progress on both of its goals: it highlighted problems that arise in holographic dualities when the system becomes nonrelativistic; and it extended the quasinormal mode method for studying quantum systems in curved spacetimes. Both of these achievements will be useful to further researchers; in the first case, we now know more about how much holographic dualities can be changed while still functioning. In the second case, we can now apply the quasinormal mode method to new spaces.

This project also resulted in the supervision of a Master’s student through successful completion of his thesis. More broadly; the grantee shared her vision behind this project with the public via a lecture series entitled ""Black Holes and Entropy"" given through the Niels Bohr Institute's Folke Universitet."

This project also resulted in the supervision of a Master’s student through successful completion of his thesis. More broadly; the grantee shared her vision behind this project with the public via a lecture series entitled ""Black Holes and Entropy"" given through the Niels Bohr Institute's Folke Universitet."