## Final Report Summary - TSSAH (Twistor Strings, Scattering Amplitudes and Holography)

TWISTOR STRINGS, SCATTERING AMPLITUDES AND HOLOGRAPHY - PROJECT OVERVIEW

From Rutherford discovering that atoms have nuclei by observing helium atoms rebound from gold foil, to modern day searches for the Higgs and supersymmetry at the LHC, most of what we know about particle physics has been gleaned from scattering experiments. One of the most important tasks for a particle theorist is thus to understand what any proposed model of nature predicts for the outcome of these experiments and, consequently, every graduate student learning quantum #eld theory is taught how to calculate scattering amplitudes. However, while textbook methods for performing these calculations are adequate to handle relatively clean processes in which, say, two particles bounce off one another, they are hopelessly outmatched when trying to describe processes with a great morass of particles all crashing together at once – and this is exactly the situation at the LHC. Therefore, over the past few years, theorists have been developing a range of techniques aimed at improving our ability to compute these multi-particle amplitudes. These developments, far from being a mere technical toolbox, are ushering in a revolution in our understanding of what quantum #eld theory itself actually involves and what its basic structures really are.

A common theme in many of the modern techniques is that they incorporate ideas from Roger Penrose’s twistor theory. Here, light rays are regarded as more fundamental than space-time points, and space-time itself is replaced by a certain three-dimensional complex space known as twistor space. The reason twistors are a useful tool to study amplitudes is that in twistor space, the external data is unconstrained: twistor space accounts for the on-shell constraints and the helicity structure of the external states automatically. In this way, the complexity of traditional Feynman diagrams – largely attributable to gauge redundancy of individual diagrams – is avoided. Twistors are thus closely related to unitarity based techniques, and resum many millions of Feynman diagrams into a vastly simpler expressions.

PROJECT OBJECTIVES

This Marie Curie Research Project aims to study the twistor formulation of scattering amplitudes, particularly in relation to Einstein’s theory of General Relativity, building on the twistor string description of N = 8 supergravity discovered by the PI. The project aims to:

i. compute quantum corrections to the gravitational S-matrix from twistor strings,

ii. extend the scope of the twistor string to incorporate asymptotically AdS space-times,

iii. discover new twistor string descriptions for a wider range of physically interesting theories.

PROGRESS AND RESULTS

Together with collaborators including Mr. Kai Roehrig (the PI's graduate student hired under the terms of the Marie Curie Grant), the PI has constructed a new ambitwistor string theory (JHEP 1407 (2014) 048) that provides the underlying theory of the CHY formulation of amplitudes in terms of scattering equations. The higher genus version of this ambitwistor string was investigated by the PI, and a new, ambitwistor form of loop amplitudes was proposed and subjected to several non-trivial tests (JHEP 1404 (2014) 104). The proposal has subsequently been verified by a collaboration including members of the PI's group in DAMTP, by reducing it to an integral over a nodal Riemann sphere. This dramatic new approach to loop amplitudes in field theory has been further studied by the PI and K. Roehrig in JHEP 1801 (2018) 069, introducing a gluing operator (essentially the ambitwistor string propagator) whose insertion on a Riemann sphere generates multi-loop amplitudes with the appropriate higher-loop scattering equations.

The ambitwistor string has also been shown to make sense on arbitrary curved solutions to the non-linear supergravity equations (see the PI's paper JHEP 1502 (2015) 116). During the second phase of the grant, this work has been pushed further in a number of directions, firstly to study vacuum plane wave space-times (work including Adamo & Casali, former members of the PI's group) and AdS3 x S3 (work of the PI and K. Roehrig). In particular, a chiral WZW-like ambitwistor string describes Type IIB supergravity on AdS3 x S3 and ongoing work of the PI is aimed at constructing new expressions for n-point boundary CFT correlators. These correlators are a key ingredient of the AdS/CFT correspondence, but, in contrast to flat space scattering amplitudes, very little is known about them beyond n=3, even in the supergravity limit. We are optimistic that the AdS ambitwistor string can achieve real progress here.

Further progress has been made on studying the twistor space of AdS5 (JHEP 1608 (2016) 167), which can also be viewed as the ambitwistor space of the boundary S4, suggesting a twistorial approach to the AdS/CFT correspondence. This paper also constructed explicit expressions for bulk-to-boundary propagators of particles of various mass and spin, in terms of cohomology classes on the AdS5 twistor space. These will be key ingredients in constructing the vertex operators for a twistor string theory for AdS space.

In work including the PI's graduate students (JHEP 1511 (2015) 38), a wide range of new ambitwistor strings have been constructed, describing a range of physically interesting theories. In particular, all tree amplitudes of d=4 Einstein-Yang-Mills (including its supersymmetric cousins) have been shown by Roehrig (JHEP 1708 (2017) 033) to exhibit a remarkable `chiral splitting' property analogous to holomorphic factorisation of 2d CFTs, and completely invisible in the CHY formula.

The relation between ambitwistor strings and null infinity has been investigated and understood, with the PI constructing a version of ambitwistor strings that describe maps to null infinity (Class. Quant. Grav. 31 (2014) 22). The PI showed how BMS supertranslations act in this model, providing a concrete realization of the framework of Strominger et al. that relates Weinberg’s soft theorem, BMS transformations and Christodoulou’s gravitational memory. This work was extended to gauge theory by members of the PI's group in DAMTP.

The PI and his group have promoted these results at a number of international conferences, including Amplitudes 2015 (ETHZ), Eurostrings 2015 (Cambridge), Flat Holography 2016 (SCGP) and QCD Meets Gravity 2017 (UCLA). The PI has also organised two workshops - one at the Newton Institute in Cambridge during June 2016, the other at Perimeter Institute, Ontario during April 2018 - focussed on the subject of ambitwistor strings. These were attended by leading researchers from around the world. The PI also co-organised a 3 month program at the KITP, Santa Barbara. These workshops have helped to further promote the work of the PI’s group.

From Rutherford discovering that atoms have nuclei by observing helium atoms rebound from gold foil, to modern day searches for the Higgs and supersymmetry at the LHC, most of what we know about particle physics has been gleaned from scattering experiments. One of the most important tasks for a particle theorist is thus to understand what any proposed model of nature predicts for the outcome of these experiments and, consequently, every graduate student learning quantum #eld theory is taught how to calculate scattering amplitudes. However, while textbook methods for performing these calculations are adequate to handle relatively clean processes in which, say, two particles bounce off one another, they are hopelessly outmatched when trying to describe processes with a great morass of particles all crashing together at once – and this is exactly the situation at the LHC. Therefore, over the past few years, theorists have been developing a range of techniques aimed at improving our ability to compute these multi-particle amplitudes. These developments, far from being a mere technical toolbox, are ushering in a revolution in our understanding of what quantum #eld theory itself actually involves and what its basic structures really are.

A common theme in many of the modern techniques is that they incorporate ideas from Roger Penrose’s twistor theory. Here, light rays are regarded as more fundamental than space-time points, and space-time itself is replaced by a certain three-dimensional complex space known as twistor space. The reason twistors are a useful tool to study amplitudes is that in twistor space, the external data is unconstrained: twistor space accounts for the on-shell constraints and the helicity structure of the external states automatically. In this way, the complexity of traditional Feynman diagrams – largely attributable to gauge redundancy of individual diagrams – is avoided. Twistors are thus closely related to unitarity based techniques, and resum many millions of Feynman diagrams into a vastly simpler expressions.

PROJECT OBJECTIVES

This Marie Curie Research Project aims to study the twistor formulation of scattering amplitudes, particularly in relation to Einstein’s theory of General Relativity, building on the twistor string description of N = 8 supergravity discovered by the PI. The project aims to:

i. compute quantum corrections to the gravitational S-matrix from twistor strings,

ii. extend the scope of the twistor string to incorporate asymptotically AdS space-times,

iii. discover new twistor string descriptions for a wider range of physically interesting theories.

PROGRESS AND RESULTS

Together with collaborators including Mr. Kai Roehrig (the PI's graduate student hired under the terms of the Marie Curie Grant), the PI has constructed a new ambitwistor string theory (JHEP 1407 (2014) 048) that provides the underlying theory of the CHY formulation of amplitudes in terms of scattering equations. The higher genus version of this ambitwistor string was investigated by the PI, and a new, ambitwistor form of loop amplitudes was proposed and subjected to several non-trivial tests (JHEP 1404 (2014) 104). The proposal has subsequently been verified by a collaboration including members of the PI's group in DAMTP, by reducing it to an integral over a nodal Riemann sphere. This dramatic new approach to loop amplitudes in field theory has been further studied by the PI and K. Roehrig in JHEP 1801 (2018) 069, introducing a gluing operator (essentially the ambitwistor string propagator) whose insertion on a Riemann sphere generates multi-loop amplitudes with the appropriate higher-loop scattering equations.

The ambitwistor string has also been shown to make sense on arbitrary curved solutions to the non-linear supergravity equations (see the PI's paper JHEP 1502 (2015) 116). During the second phase of the grant, this work has been pushed further in a number of directions, firstly to study vacuum plane wave space-times (work including Adamo & Casali, former members of the PI's group) and AdS3 x S3 (work of the PI and K. Roehrig). In particular, a chiral WZW-like ambitwistor string describes Type IIB supergravity on AdS3 x S3 and ongoing work of the PI is aimed at constructing new expressions for n-point boundary CFT correlators. These correlators are a key ingredient of the AdS/CFT correspondence, but, in contrast to flat space scattering amplitudes, very little is known about them beyond n=3, even in the supergravity limit. We are optimistic that the AdS ambitwistor string can achieve real progress here.

Further progress has been made on studying the twistor space of AdS5 (JHEP 1608 (2016) 167), which can also be viewed as the ambitwistor space of the boundary S4, suggesting a twistorial approach to the AdS/CFT correspondence. This paper also constructed explicit expressions for bulk-to-boundary propagators of particles of various mass and spin, in terms of cohomology classes on the AdS5 twistor space. These will be key ingredients in constructing the vertex operators for a twistor string theory for AdS space.

In work including the PI's graduate students (JHEP 1511 (2015) 38), a wide range of new ambitwistor strings have been constructed, describing a range of physically interesting theories. In particular, all tree amplitudes of d=4 Einstein-Yang-Mills (including its supersymmetric cousins) have been shown by Roehrig (JHEP 1708 (2017) 033) to exhibit a remarkable `chiral splitting' property analogous to holomorphic factorisation of 2d CFTs, and completely invisible in the CHY formula.

The relation between ambitwistor strings and null infinity has been investigated and understood, with the PI constructing a version of ambitwistor strings that describe maps to null infinity (Class. Quant. Grav. 31 (2014) 22). The PI showed how BMS supertranslations act in this model, providing a concrete realization of the framework of Strominger et al. that relates Weinberg’s soft theorem, BMS transformations and Christodoulou’s gravitational memory. This work was extended to gauge theory by members of the PI's group in DAMTP.

The PI and his group have promoted these results at a number of international conferences, including Amplitudes 2015 (ETHZ), Eurostrings 2015 (Cambridge), Flat Holography 2016 (SCGP) and QCD Meets Gravity 2017 (UCLA). The PI has also organised two workshops - one at the Newton Institute in Cambridge during June 2016, the other at Perimeter Institute, Ontario during April 2018 - focussed on the subject of ambitwistor strings. These were attended by leading researchers from around the world. The PI also co-organised a 3 month program at the KITP, Santa Barbara. These workshops have helped to further promote the work of the PI’s group.