## Periodic Reporting for period 1 - 6D STRINGS (Tensionless strings of six-dimensional superconformal theories)

Reporting period: 2016-09-01 to 2018-08-31

String theory is a ten-dimensional theory that aims to combine the known laws of nature into a mathematically consistent framework. Our current understanding is that string theory describes interactions between multi-dimensional objects called membranes. The ultimate motivation for my project is to understand the theories that describe these membranes; this will help clarify the dynamics of string theory, and hopefully in the long run help shed light on whether string theory correctly describes the universe we live in.

One of the most powerful concepts in theoretical physics is that of symmetry. Thanks to symmetry we can explain virtually all phenomena observed at particle accelerators in terms of a handful of mathematical equations. Even mundane phenomena like water boiling are governed by an important symmetry: conformal symmetry. The same conformal symmetry also governs the behavior of membranes in string theory. Additionally, the theories of membranes also display supersymmetry, a symmetry which relates to each other the two fundamental types of particles found in nature: bosons and fermions. Theories that enjoy both conformal symmetry and supersymmetry are known as superconformal field theories, or SCFTs.

Because of the large amount of symmetry, SCFTs are described by very rigid mathematical structures. A consequence of this rigid structure is that SCFTs can only exist in six dimensions or less. Six-dimensional (6d) SCFTs are in some ways the most fundamental: for example, starting from a 6d SCFT one can obtain infinitely many superconformal theories in lower dimensions. It is even possible that all lower-dimensional theories can be obtained starting from 6d SCFTs.

At the same time, 6d SCFTs display unusual features. For example, a key role in their dynamics is played by two-dimensional, string-like objects. Remarkably, from the properties of these strings (in particular from a function known as the ‘elliptic genus’ of the strings) one can reconstruct very detailed information about the 6d theory, which is encoded in a function called the ‘6d superconformal index’. These strings have previously not been well understood, and are the subject of this research project.

The first objective of the project has been to develop a systematic understanding of the strings in arbitrary 6d SCFTs, and devise approaches to compute their elliptic genus from first principles.

The second objective has been to apply these results to a different field in string theory: topological string theory. Topological string theory is important from various perspectives: first, it is an area of string theory where very detailed computations can be made, which enable us to tackle fundamental problems in string theory. For example, it has been used to understand the internal structure of (supersymmetric) black holes, which is one of the most mysterious aspects of quantum gravity. Second, topological string theory has very deep connections to a number of active areas of research in mathematics, including geometry of so-called Calabi-Yau spaces, and knot theory; for this reason, progress in topological string theory often also leads to advances in the realm of pure mathematics.

One of the most powerful concepts in theoretical physics is that of symmetry. Thanks to symmetry we can explain virtually all phenomena observed at particle accelerators in terms of a handful of mathematical equations. Even mundane phenomena like water boiling are governed by an important symmetry: conformal symmetry. The same conformal symmetry also governs the behavior of membranes in string theory. Additionally, the theories of membranes also display supersymmetry, a symmetry which relates to each other the two fundamental types of particles found in nature: bosons and fermions. Theories that enjoy both conformal symmetry and supersymmetry are known as superconformal field theories, or SCFTs.

Because of the large amount of symmetry, SCFTs are described by very rigid mathematical structures. A consequence of this rigid structure is that SCFTs can only exist in six dimensions or less. Six-dimensional (6d) SCFTs are in some ways the most fundamental: for example, starting from a 6d SCFT one can obtain infinitely many superconformal theories in lower dimensions. It is even possible that all lower-dimensional theories can be obtained starting from 6d SCFTs.

At the same time, 6d SCFTs display unusual features. For example, a key role in their dynamics is played by two-dimensional, string-like objects. Remarkably, from the properties of these strings (in particular from a function known as the ‘elliptic genus’ of the strings) one can reconstruct very detailed information about the 6d theory, which is encoded in a function called the ‘6d superconformal index’. These strings have previously not been well understood, and are the subject of this research project.

The first objective of the project has been to develop a systematic understanding of the strings in arbitrary 6d SCFTs, and devise approaches to compute their elliptic genus from first principles.

The second objective has been to apply these results to a different field in string theory: topological string theory. Topological string theory is important from various perspectives: first, it is an area of string theory where very detailed computations can be made, which enable us to tackle fundamental problems in string theory. For example, it has been used to understand the internal structure of (supersymmetric) black holes, which is one of the most mysterious aspects of quantum gravity. Second, topological string theory has very deep connections to a number of active areas of research in mathematics, including geometry of so-called Calabi-Yau spaces, and knot theory; for this reason, progress in topological string theory often also leads to advances in the realm of pure mathematics.

In a first paper I studied the strings of a special class of 6d SCFTs: SCFTs without matter. For this class of SCFTs, I found that the theories describing the strings can be obtained starting from certain 4d theories, by performing a so-called ‘topologically twisted compactification’. This procedure can be used to determine the most important properties of the theories describing the strings (for example, their ground states, symmetries and anomaly polynomials). With this understanding, I was able to completely determine the elliptic genus of the strings for all 6d SCFTs belonging to this class. I also found an intriguing connection between the elliptic genus and an important quantity associated to the 4d theories mentioned above: the Schur index. Understanding this unexpected connection is a work in progress.

In a second paper, I employed the techniques developed in the context of 6d SCFTs to study topological string theory on certain Calabi-Yau spaces with special symmetries. I found that these symmetries are captured by a class of mathematical functions, known as Weyl-invariant Jacobi forms, whose properties have been investigated by mathematicians in recent years. The mathematical theory that underlies this class of functions can be exploited to organize computations in topological string theory in a systematic manner, which allowed us to very efficiently study the geometry of an interesting family of Calabi-Yau spaces.

In a third paper I developed a general framework to study the strings of arbitrary 6d SCFTs, and discovered that they share a number of basic properties which greatly clarify their nature. For instance, I was able to understand precisely how the data of the 6d SCFT is reflected in the theory describing the strings; I understood how to correctly describe the strings in terms of a class of theories known as ‘2d (0,4) nonlinear sigma models on instanton moduli spaces’, which I studied in detail; I also found that the elliptic genus of the strings is encoded by an auxiliary two-dimensional theory, the so-called ‘chiral algebra’, and showed that this is in turn built out of well-known mathematical structures known as Kac-Moody algebras. From this insight I was able to compute the elliptic genus for the strings of almost all 6d SCFTs. At the same time, I also found a way to generalize the ‘topologically twisted compactification’ approach discussed in point (1), so that it can be used to study the strings of arbitrary 6d SCFTs.

In ongoing work I am extending the analysis described in points (1) and (3) to bound states of strings, which also contribute to the 6d superconformal index. In order to do this, I am developing a more rigorous approach to studying the (0,4) nonlinear sigma models discussed above. I am also using the strings as a tool to study a little-understood class of 6d SCFTs: the ones that arise from so-called ‘frozen singularities’.

In the course of the project I have communicated the results of my research to other members of the string theory community by giving 7 invited workshop talks and 5 university seminars.

In a second paper, I employed the techniques developed in the context of 6d SCFTs to study topological string theory on certain Calabi-Yau spaces with special symmetries. I found that these symmetries are captured by a class of mathematical functions, known as Weyl-invariant Jacobi forms, whose properties have been investigated by mathematicians in recent years. The mathematical theory that underlies this class of functions can be exploited to organize computations in topological string theory in a systematic manner, which allowed us to very efficiently study the geometry of an interesting family of Calabi-Yau spaces.

In a third paper I developed a general framework to study the strings of arbitrary 6d SCFTs, and discovered that they share a number of basic properties which greatly clarify their nature. For instance, I was able to understand precisely how the data of the 6d SCFT is reflected in the theory describing the strings; I understood how to correctly describe the strings in terms of a class of theories known as ‘2d (0,4) nonlinear sigma models on instanton moduli spaces’, which I studied in detail; I also found that the elliptic genus of the strings is encoded by an auxiliary two-dimensional theory, the so-called ‘chiral algebra’, and showed that this is in turn built out of well-known mathematical structures known as Kac-Moody algebras. From this insight I was able to compute the elliptic genus for the strings of almost all 6d SCFTs. At the same time, I also found a way to generalize the ‘topologically twisted compactification’ approach discussed in point (1), so that it can be used to study the strings of arbitrary 6d SCFTs.

In ongoing work I am extending the analysis described in points (1) and (3) to bound states of strings, which also contribute to the 6d superconformal index. In order to do this, I am developing a more rigorous approach to studying the (0,4) nonlinear sigma models discussed above. I am also using the strings as a tool to study a little-understood class of 6d SCFTs: the ones that arise from so-called ‘frozen singularities’.

In the course of the project I have communicated the results of my research to other members of the string theory community by giving 7 invited workshop talks and 5 university seminars.

The main outcome of this project has been a systematic study of the strings of 6d SCFTs. Prior to the start of the project, these strings had only been studied by indirect methods and on a case-by-case basis. In particular, an appropriate framework that applied to the study of strings of arbitrary 6d SCFTs was not available; also, from a practical point of view, in many cases no method was known to compute the elliptic genus. The research I conducted in the course of this project has led to a much more comprehensive understanding of the strings and has highlighted many common features shared by them which had previously not been appreciated. The techniques I developed in the course of the project also allowed me to extract quantitative data about the strings (as captured by their elliptic genus) for almost all known 6d SCFTs.

The main progress in topological string theory has been to apply Weyl-invariant Jacobi forms to an interesting class of Calabi-Yau spaces, which enabled us to study them in great detail by systematically exploiting their symmetries.

The main progress in topological string theory has been to apply Weyl-invariant Jacobi forms to an interesting class of Calabi-Yau spaces, which enabled us to study them in great detail by systematically exploiting their symmetries.