## Final Report Summary - STRING (Properties and Applications of the Gauge/Gravity Correspondence)

PROPERTIES AND APPLICATIONS OF THE GAUGE/GRAVITY CORRESPONDENCE

String theory was developed as a novel framework for unifying the forces of the Standard Model with the gravitational force. As the structure of the theory has developed it has become clear that string theory provides the mathematical underpinning for addressing a much wider range of physical problems. This is based on the remarkable manner in which string theory relates Einstein’s theory of gravity (general relativity) with Yang-Mills quantum field theories, which describe the non-gravitational forces. This relationship, known as the “holographic correspondence”, or “gauge/gravity correspondence”, connects properties of quantum gravity in D-dimensional space-time to properties of non-gravitational systems in D-1 dimensions. A very striking consequence of this relationship is that it provides a new mathematical procedure for determining properties of interesting physical systems that cannot be determined in any other manner. For example, features of quantum gravity in the highly quantum domain, such as quantum properties of black holes, are described by properties of standard non-gravitational quantum field theory. Conversely, the correspondence may be applied to study properties of important strongly coupled condensed matter systems, such as strange metals, high-temperature superconductors, and the quark-gluon plasma. In such cases, rather straightforward properties of classical general relativity in the presence of Black Holes in D dimensions may be used to determine properties of strongly couple (D-1)-dimensional systems. This is a new and rapidly developing subject that relates very different disciplines. Although the correspondence principle was formulated 19 years ago, there are many features that are only now emerging. Several of these were the main subjects of this ERC research programme, which has several interconnected and complementary strands. . The project has been well-timed since it has coincided with some of the most notable developments in this area, and has made important contributions to several of them.

(1) Non-gravitational systems.

Some of the most important condensed matter systems in two or three spatial dimensions are strongly coupled, which means that their properties are not amenable to calculation with conventional methods of quantum field theory. Notable examples are the quark-gluon plasma, strange metals, high-temperature superconductivity and others. We have developed an understanding of how gravitational theory in four or five space-time dimensions can describe important features of these non-gravitational systems. Of particular significance is the understanding by the team of the manner in which atomic lattice effects can be reproduced by applying inhomogeneous background fields in the gravitational theory.

(2) Gravitational systems.

In order to understand the nature of gravitational systems within the context of the gauge/gravity correspondence it is important to develop an understanding of gravity in general numbers of space-time dimensions. In this context we have discovered a number of novel features of higher dimensional black holes and black rings that are not seen in the conventional four-dimensional case. Our understanding of the stability properties of these systems is one of the highlights of our work.

(3) Developing the structure of the gauge/gravity correspondence.

(3a) Integrability of string theory in anti de Sitter and N=4 Yang-Mills.

The main theoretical basis for the gauge/gravity correspondence is the understanding that the most supersymmetric four-dimensional Yang-Mills theory (N=4 Yang-Mills) is an integrable system, as is its holographic dual, which is superstring theory in five-dimensional anti de-Sitter space. This means that exact statements can be made about both sides of the correspondence, which is most unusual for theories in more than three space-time dimensions. The project has made significant progress in determining exact properties of the theory, such as understanding how the strings of string theory are manifested as solitons in gauge theory. These results are state of the art in this area that is central to the foundations of the gauge/gravity correspondence.

(3b) String theory extension of Einstein's theory of gravity.

The gauge/gravity correspondence depends on very special symmetries of string theory known as “dualities” that are not present in Einstein’s theory (general relativity). These symmetries, which are of central importance in determining the structure of the gauge/gravity correspondence, strongly constrain the manner in which string theory generalizes general relativity. Our programme has succeeded in identifying some mathematically intriguing relationships in string theory scattering amplitudes, which have close connections with interesting areas of number theory, a branch of mathematics that has previously had few connections with physics. Of particular significance is our understanding of how special cases of “Langlands Eisenstein series” arise in the scattering amplitudes of string theory and hence determining some special features of these series. In the latter part of the project we realized that string theory may be used to generate novel mathematical structures known as ``single-valued elliptic multiple polylogarithms”. These are connected with generalizations of multiple zeta values, which are not only of great relevance to amplitude calculations in conventional quantum field theories, but are also the focus of great activity in the mathematical areas of algebraic geometry and number theory. This symbiosis with pure mathematics should further enhance our understanding of the breadth of applicability of the gauge/gravity correspondence.

To summarize, the three strands of the project involve complementary research in a number of disciplines that include nitty-gritty condensed matter physics, the physics of QCD and other parts of the Standard Model, fundamentals of string theory, properties of black holes and elements of modern mathematics. The substantial research outcomes are a tribute to the virtues of ERC funding.

String theory was developed as a novel framework for unifying the forces of the Standard Model with the gravitational force. As the structure of the theory has developed it has become clear that string theory provides the mathematical underpinning for addressing a much wider range of physical problems. This is based on the remarkable manner in which string theory relates Einstein’s theory of gravity (general relativity) with Yang-Mills quantum field theories, which describe the non-gravitational forces. This relationship, known as the “holographic correspondence”, or “gauge/gravity correspondence”, connects properties of quantum gravity in D-dimensional space-time to properties of non-gravitational systems in D-1 dimensions. A very striking consequence of this relationship is that it provides a new mathematical procedure for determining properties of interesting physical systems that cannot be determined in any other manner. For example, features of quantum gravity in the highly quantum domain, such as quantum properties of black holes, are described by properties of standard non-gravitational quantum field theory. Conversely, the correspondence may be applied to study properties of important strongly coupled condensed matter systems, such as strange metals, high-temperature superconductors, and the quark-gluon plasma. In such cases, rather straightforward properties of classical general relativity in the presence of Black Holes in D dimensions may be used to determine properties of strongly couple (D-1)-dimensional systems. This is a new and rapidly developing subject that relates very different disciplines. Although the correspondence principle was formulated 19 years ago, there are many features that are only now emerging. Several of these were the main subjects of this ERC research programme, which has several interconnected and complementary strands. . The project has been well-timed since it has coincided with some of the most notable developments in this area, and has made important contributions to several of them.

(1) Non-gravitational systems.

Some of the most important condensed matter systems in two or three spatial dimensions are strongly coupled, which means that their properties are not amenable to calculation with conventional methods of quantum field theory. Notable examples are the quark-gluon plasma, strange metals, high-temperature superconductivity and others. We have developed an understanding of how gravitational theory in four or five space-time dimensions can describe important features of these non-gravitational systems. Of particular significance is the understanding by the team of the manner in which atomic lattice effects can be reproduced by applying inhomogeneous background fields in the gravitational theory.

(2) Gravitational systems.

In order to understand the nature of gravitational systems within the context of the gauge/gravity correspondence it is important to develop an understanding of gravity in general numbers of space-time dimensions. In this context we have discovered a number of novel features of higher dimensional black holes and black rings that are not seen in the conventional four-dimensional case. Our understanding of the stability properties of these systems is one of the highlights of our work.

(3) Developing the structure of the gauge/gravity correspondence.

(3a) Integrability of string theory in anti de Sitter and N=4 Yang-Mills.

The main theoretical basis for the gauge/gravity correspondence is the understanding that the most supersymmetric four-dimensional Yang-Mills theory (N=4 Yang-Mills) is an integrable system, as is its holographic dual, which is superstring theory in five-dimensional anti de-Sitter space. This means that exact statements can be made about both sides of the correspondence, which is most unusual for theories in more than three space-time dimensions. The project has made significant progress in determining exact properties of the theory, such as understanding how the strings of string theory are manifested as solitons in gauge theory. These results are state of the art in this area that is central to the foundations of the gauge/gravity correspondence.

(3b) String theory extension of Einstein's theory of gravity.

The gauge/gravity correspondence depends on very special symmetries of string theory known as “dualities” that are not present in Einstein’s theory (general relativity). These symmetries, which are of central importance in determining the structure of the gauge/gravity correspondence, strongly constrain the manner in which string theory generalizes general relativity. Our programme has succeeded in identifying some mathematically intriguing relationships in string theory scattering amplitudes, which have close connections with interesting areas of number theory, a branch of mathematics that has previously had few connections with physics. Of particular significance is our understanding of how special cases of “Langlands Eisenstein series” arise in the scattering amplitudes of string theory and hence determining some special features of these series. In the latter part of the project we realized that string theory may be used to generate novel mathematical structures known as ``single-valued elliptic multiple polylogarithms”. These are connected with generalizations of multiple zeta values, which are not only of great relevance to amplitude calculations in conventional quantum field theories, but are also the focus of great activity in the mathematical areas of algebraic geometry and number theory. This symbiosis with pure mathematics should further enhance our understanding of the breadth of applicability of the gauge/gravity correspondence.

To summarize, the three strands of the project involve complementary research in a number of disciplines that include nitty-gritty condensed matter physics, the physics of QCD and other parts of the Standard Model, fundamentals of string theory, properties of black holes and elements of modern mathematics. The substantial research outcomes are a tribute to the virtues of ERC funding.