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- TL;DR Summary
- Twistors are central to "celestial" holography in flat space

Many of us have heard of "twistors", arguably Roger Penrose's biggest contribution to theoretical physics. Twistor space is a space which maps nonlocally onto physical space-time; in particular, lightlike structures in space-time, like null lines and light cones, become much more "local" in twistor space. For various reasons, Penrose thought that twistor space was possibly a more fundamental arena for theoretical physics than space-time, and for many years he and a hardy band of mostly British collaborators worked to re-describe particle physics in terms of twistors. Equations of massless particles become very simple in twistor space, because of their scale invariance; promising for physics, since all the elementary particles in the standard model are fundamentally massless.

In the 21st century, twistors got a second boost when Witten discovered that a "topological string" in supertwistor space, could explain a number of remarkably compressed formulas for Feynman diagrams in Yang-Mills theory. The use of twistor variables, with and without strings, is now standard in "amplitudeology", the quest to find the hidden reasons why amplitudes in gauge theory have all these unexpected properties. Many stalwarts of the twistor program (Hodges, Mason, Skinner) are central to these studies.

In the title, I also mention "celestial holography". Holographic duality is probably familiar to most modern readers of theoretical physics; one hears incessantly about "AdS/CFT duality", how "gravity in the AdS bulk" is holographically equivalent to a non-gravitational CFT "on the boundary"; also how anti de Sitter space is different from our universe's space-time, but that the duality still has some practical uses in condensed matter physics.

AdS/CFT has been celebrated as an explicit realization of the holographic principle that quantum gravity should behave like a quantum field theory in one less dimension, but people have still hoped to find a form of the holographic principle suitable for flat space. After all, the first glimpse of the principle was Hawking and Bekenstein's inference that the entropy of a black hole should be proportional to its surface area, not its volume. And in fact, a kind of flat-space holography has been taking shape in recent years, under the name of celestial holography.

In astronomy, the celestial sphere does for the sky, what a globe does for the Earth's surface in geography. It is a two-dimensional sphere on which all celestial phenomena can be depicted; each object located according to where in the sky it is seen, when viewed from Earth. Celestial holography implements the holographic principle in this framework: physics throughout a three-dimensional space, is described by a field theory confined to the two spatial dimensions of the celestial sphere.

The main theorist behind celestial holography has probably been Andrew Strominger, whose background is in string theory, but who has approached this topic from a more general perspective of quantum gravity. Strominger has done a lot of work on "soft" symmetries at infinity, in which there are new conserved quantities arising from the presence of arbitrary numbers of zero-energy gravitons radiating away (this extends to gravity, earlier work by Steven Weinberg on "soft pions"); this briefly came to public attention just before Stephen Hawking died, when it was claimed that the quantum information apparently lost during Hawking radiation, is actually preserved in these soft degrees of freedom.

Frankly I understand very little about celestial holography, except the bare idea of a celestial CFT being equivalent to a gravitational theory in flat space, and that some of the unexpected symmetries discovered in amplitudeology, are related to the symmetries of the celestial CFT. But it's apparent that the twistors are at work here too. It makes sense - the structure of twistor space is related to the asymptotic structure of physical space-time - recall how twistor objects correspond to lightlike entities, like light cones.

What prompts me to make this post, is the appearance in the past few months of numerous papers indicating advances in twistor theory, or celestial holography, or both. But I'll begin with an older review of twistor theory that I found:

"Twistors and amplitudes" by Andrew Hodges (2015)

"Penrose's definition of twistors can be considered as doing for the conformal group what spinors do for the Lorentz group", Hodges writes at the start of part 3. Hodges himself is best known for discovering "momentum twistors", a twistor space for momentum variables rather than position variables, an extremely simple idea which opened the door to many of the discoveries in the era of the twistor string.

All right, so what are some of these recent papers that caught my attention? These are quite technical works. It's like we've arrived in the middle of the lecture and don't know everything that led up to this. But we'd better take notes and try to understand anyway, if we don't want to get left behind. :-)

"Top-down holography in an asymptotically flat spacetime" by Kevin Costello, Natalie Paquette, Atul Sharma

This paper proposes a concrete example of celestial holography: a specific 4d field theory, including gravity, and a specific 2d celestial dual. The 4d theory was actually described in a paper last year by Costello (who seems to be a leader in turning the "physicists' math" of quantum field theory into something acceptable to real mathematicians), as an example of a very rare class of quantum field theories whose space-time properties are fixed by their being local in twistor space. The fields themselves are not obviously reminiscent of anything we know from real life: an SO(8)-valued four-dimensional "WZW model", coupled to a "Kähler scalar". What may be significant, is just that a theory whose construction can be motivated by the twistor research program, happens to yield an example of celestial holography.

Meanwhile, one of the problems faced by twistor theory, arises from what was originally supposed to be a virtue: there is a natural chirality to twistor theory, in the sense that opposite chiralities are described differently, and one more naturally than the other. Penrose thought this might help to explain parity violation in the weak force. But it also means that there is a kind of bias in the twistorial description of four-dimensional Yang-Mills theory and gravity: it's easier to describe the "self-dual" part of those fields, rather than the "anti-self-dual" part (I hope I have that right). The duality in question refers to duality under the Hodge star operator, a kind of orthogonality operation in vector spaces. Four dimensions is unique in that the Hodge dual of a 2d object is itself a 2d object, and so something can be self-dual, its own dual; or anti-self-dual, the latter meaning that the dual is the original times -1.

One may approach this situation in different ways. One may, for example, use "ambitwistors", which combine twistors with dual twistors. Or one may embrace the asymmetry; we have been discussing "chiral" theories of gravity in a few other threads, recently.

But in any case, within twistor theory, one of the longstanding challenges is to describe the "nonlinear graviton", which seems to be a graviton containing a quantum of space-time curvature. A few years back, Penrose himself proposed doing this by patching together algebras of flat-space twistor field operators (analogous to the overlapping coordinate patches which appear in Riemannian geometry), in an approach called palatial twistor theory (because it was inspired by a conversation with the mathematician Michael Atiyah, that took place near Buckingham Palace). And now we have:

"Quantizing the non-linear graviton" by Roland Bittleston, Atul Sharma, David Skinner

Once again we have a kind of field that only specialists know about - "holomorphic Poisson-BF theory". But that's the field on twistor space; when we transform to physical space-time, it turns into the self-dual part of general relativity. The theory turns out to be anomalous. In physical space-time, such anomalies mean that a field theory is badly defined. But if the anomalies occur in twistor space, it just means that the space-time theory loses one of its classical symmetries (and it loses the nice property of integrability). However, the twistor anomalies can be canceled by the addition of a new field, which in space-time manifests as "a type of axion" (not the QCD axion, but one with analogous couplings) which also appeared in the twistor string.

The appearance of a new scalar is reminiscent of the previous paper by Costello et al; and there's even an author in common, Atul Sharma.

"Moyal deformations, W_{1+∞} and celestial holography" by Wei Bu, Simon Heuveline, David Skinner

"Celestial chiral algebras, colour-kinematics duality and integrability" by Ricardo Monteiro

These papers are about celestial holography: specifically, two kinds of algebra, w algebra and its quantum deformation, W algebra, which are expected to appear on both sides of some celestial dualities. On the 2d, celestial side, they should appear as algebraic relations among the operators of the celestial CFT; on the 4d side, they will show up as relations among the scattering amplitudes dual to the celestial operators. In other words, they are a bridge between amplitudeology and celestial holography.

In these two papers, one may also see, again, the emphasis on self-dual Yang-Mills and self-dual gravity, i.e. something less than the full gauge and gravity theories of the real world. The "double copy" relation between gauge theory and gravity also shows up in the second paper.

I haven't studied these papers at all.

"Soft Gravitons in the BFSS Matrix Model" by Noah Miller, Andrew Strominger, Adam Tropper, Tianli Wang

Here Strominger returns to his roots, and sets the stage for celestial holography within string theory, by looking for soft gravitons in flat-space M-theory. The BFSS matrix model is a sector of string theory in which everything consists of webs of strings connecting pointlike D0-branes, and the soft gravitons are found to correspond to certain small sub-webs ("matrix subblocks whose rank is held fixed... in the large-N limit", see Figure 2).

In the 21st century, twistors got a second boost when Witten discovered that a "topological string" in supertwistor space, could explain a number of remarkably compressed formulas for Feynman diagrams in Yang-Mills theory. The use of twistor variables, with and without strings, is now standard in "amplitudeology", the quest to find the hidden reasons why amplitudes in gauge theory have all these unexpected properties. Many stalwarts of the twistor program (Hodges, Mason, Skinner) are central to these studies.

In the title, I also mention "celestial holography". Holographic duality is probably familiar to most modern readers of theoretical physics; one hears incessantly about "AdS/CFT duality", how "gravity in the AdS bulk" is holographically equivalent to a non-gravitational CFT "on the boundary"; also how anti de Sitter space is different from our universe's space-time, but that the duality still has some practical uses in condensed matter physics.

AdS/CFT has been celebrated as an explicit realization of the holographic principle that quantum gravity should behave like a quantum field theory in one less dimension, but people have still hoped to find a form of the holographic principle suitable for flat space. After all, the first glimpse of the principle was Hawking and Bekenstein's inference that the entropy of a black hole should be proportional to its surface area, not its volume. And in fact, a kind of flat-space holography has been taking shape in recent years, under the name of celestial holography.

In astronomy, the celestial sphere does for the sky, what a globe does for the Earth's surface in geography. It is a two-dimensional sphere on which all celestial phenomena can be depicted; each object located according to where in the sky it is seen, when viewed from Earth. Celestial holography implements the holographic principle in this framework: physics throughout a three-dimensional space, is described by a field theory confined to the two spatial dimensions of the celestial sphere.

The main theorist behind celestial holography has probably been Andrew Strominger, whose background is in string theory, but who has approached this topic from a more general perspective of quantum gravity. Strominger has done a lot of work on "soft" symmetries at infinity, in which there are new conserved quantities arising from the presence of arbitrary numbers of zero-energy gravitons radiating away (this extends to gravity, earlier work by Steven Weinberg on "soft pions"); this briefly came to public attention just before Stephen Hawking died, when it was claimed that the quantum information apparently lost during Hawking radiation, is actually preserved in these soft degrees of freedom.

Frankly I understand very little about celestial holography, except the bare idea of a celestial CFT being equivalent to a gravitational theory in flat space, and that some of the unexpected symmetries discovered in amplitudeology, are related to the symmetries of the celestial CFT. But it's apparent that the twistors are at work here too. It makes sense - the structure of twistor space is related to the asymptotic structure of physical space-time - recall how twistor objects correspond to lightlike entities, like light cones.

What prompts me to make this post, is the appearance in the past few months of numerous papers indicating advances in twistor theory, or celestial holography, or both. But I'll begin with an older review of twistor theory that I found:

"Twistors and amplitudes" by Andrew Hodges (2015)

"Penrose's definition of twistors can be considered as doing for the conformal group what spinors do for the Lorentz group", Hodges writes at the start of part 3. Hodges himself is best known for discovering "momentum twistors", a twistor space for momentum variables rather than position variables, an extremely simple idea which opened the door to many of the discoveries in the era of the twistor string.

All right, so what are some of these recent papers that caught my attention? These are quite technical works. It's like we've arrived in the middle of the lecture and don't know everything that led up to this. But we'd better take notes and try to understand anyway, if we don't want to get left behind. :-)

"Top-down holography in an asymptotically flat spacetime" by Kevin Costello, Natalie Paquette, Atul Sharma

This paper proposes a concrete example of celestial holography: a specific 4d field theory, including gravity, and a specific 2d celestial dual. The 4d theory was actually described in a paper last year by Costello (who seems to be a leader in turning the "physicists' math" of quantum field theory into something acceptable to real mathematicians), as an example of a very rare class of quantum field theories whose space-time properties are fixed by their being local in twistor space. The fields themselves are not obviously reminiscent of anything we know from real life: an SO(8)-valued four-dimensional "WZW model", coupled to a "Kähler scalar". What may be significant, is just that a theory whose construction can be motivated by the twistor research program, happens to yield an example of celestial holography.

Meanwhile, one of the problems faced by twistor theory, arises from what was originally supposed to be a virtue: there is a natural chirality to twistor theory, in the sense that opposite chiralities are described differently, and one more naturally than the other. Penrose thought this might help to explain parity violation in the weak force. But it also means that there is a kind of bias in the twistorial description of four-dimensional Yang-Mills theory and gravity: it's easier to describe the "self-dual" part of those fields, rather than the "anti-self-dual" part (I hope I have that right). The duality in question refers to duality under the Hodge star operator, a kind of orthogonality operation in vector spaces. Four dimensions is unique in that the Hodge dual of a 2d object is itself a 2d object, and so something can be self-dual, its own dual; or anti-self-dual, the latter meaning that the dual is the original times -1.

One may approach this situation in different ways. One may, for example, use "ambitwistors", which combine twistors with dual twistors. Or one may embrace the asymmetry; we have been discussing "chiral" theories of gravity in a few other threads, recently.

But in any case, within twistor theory, one of the longstanding challenges is to describe the "nonlinear graviton", which seems to be a graviton containing a quantum of space-time curvature. A few years back, Penrose himself proposed doing this by patching together algebras of flat-space twistor field operators (analogous to the overlapping coordinate patches which appear in Riemannian geometry), in an approach called palatial twistor theory (because it was inspired by a conversation with the mathematician Michael Atiyah, that took place near Buckingham Palace). And now we have:

"Quantizing the non-linear graviton" by Roland Bittleston, Atul Sharma, David Skinner

Once again we have a kind of field that only specialists know about - "holomorphic Poisson-BF theory". But that's the field on twistor space; when we transform to physical space-time, it turns into the self-dual part of general relativity. The theory turns out to be anomalous. In physical space-time, such anomalies mean that a field theory is badly defined. But if the anomalies occur in twistor space, it just means that the space-time theory loses one of its classical symmetries (and it loses the nice property of integrability). However, the twistor anomalies can be canceled by the addition of a new field, which in space-time manifests as "a type of axion" (not the QCD axion, but one with analogous couplings) which also appeared in the twistor string.

The appearance of a new scalar is reminiscent of the previous paper by Costello et al; and there's even an author in common, Atul Sharma.

"Moyal deformations, W_{1+∞} and celestial holography" by Wei Bu, Simon Heuveline, David Skinner

"Celestial chiral algebras, colour-kinematics duality and integrability" by Ricardo Monteiro

These papers are about celestial holography: specifically, two kinds of algebra, w algebra and its quantum deformation, W algebra, which are expected to appear on both sides of some celestial dualities. On the 2d, celestial side, they should appear as algebraic relations among the operators of the celestial CFT; on the 4d side, they will show up as relations among the scattering amplitudes dual to the celestial operators. In other words, they are a bridge between amplitudeology and celestial holography.

In these two papers, one may also see, again, the emphasis on self-dual Yang-Mills and self-dual gravity, i.e. something less than the full gauge and gravity theories of the real world. The "double copy" relation between gauge theory and gravity also shows up in the second paper.

I haven't studied these papers at all.

"Soft Gravitons in the BFSS Matrix Model" by Noah Miller, Andrew Strominger, Adam Tropper, Tianli Wang

Here Strominger returns to his roots, and sets the stage for celestial holography within string theory, by looking for soft gravitons in flat-space M-theory. The BFSS matrix model is a sector of string theory in which everything consists of webs of strings connecting pointlike D0-branes, and the soft gravitons are found to correspond to certain small sub-webs ("matrix subblocks whose rank is held fixed... in the large-N limit", see Figure 2).