Lectures on Twistor Theory Imperial College London, UK E-Mail: Pos(Modave2017)003 by Pen- Tim Adamo by Ward and ] (Building on 2 Spinors and Space-Time ]

Home , ADHM construction, Penrose transform, Twistor theory

PoS(Modave2017)003 https://pos.sissa.it/ ∗ [email protected] Speaker. Broadly speaking, twistor theorytime is as a geometric framework data for on encodingship a between physical complex space-time information projective and space, on twistor known space- which space as we is a twistor explore non-local space. and in has thesefour-dimensional The Minkowski some lectures. relation- space, surprising we consequences, describe Starting some of withdescribing twistor theory’s free a historic fields review successes (e.g., and of integrable thethen systems) twistor discuss as correspondence how well in for as recent someunderstanding years of how twistor many its theory historic of is shortcomings. these applied to problems We the have study been of perturbative overcome, QFT with today. a view to ∗ Copyright owned by the author(s) under the terms of the Creative Commons c Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). XIII Modave Summer School in Mathematical10-16 Physics-Modave2017 September 2017 Modave, Belgium Tim Adamo Lectures on twistor theory Imperial College London, UK E-mail: PoS(Modave2017)003 by Pen- Tim Adamo by Ward and ] (building on 2 Spinors and space-time ]. 9 1 Twistor Geometry and Field Theory ] and Dunajski [ 8 ]) that twistor theory can be combined with string perturbation theory to 3 ]. This contains more-or-less everything that happened in twistor theory and 6 , 5 ], with the long-term ambition of developing a novel approach to quantum gravity. 1 ] is also very useful, particularly for those approaching the subject from a mathematical 7 ]; this book is well-written, covers all the basics, includes many exercises, and is remarkably 4 The standard reference work in the subject is the two volume Today, twistor theory plays a prominent role in the study of interesting ‘non-standard’ struc- As such, these lectures are not designed to be a painstaking exposition of the mathematical The intended audience are theoretical and mathematical physicists, rather than pure mathe- In their original incarnation, these notes were delivered in five 1-hour lectures, but I expect For my money, the best introductory textbook for twistor theory remains that of Huggett and Twistor theory is a fascinating topic with a checkered past. It was first introduced fifty years rose and Rindler [ compact. It would be my first recommendationto to get anyone their who hands wants dirty. to learn enough twistor theory related areas up toWells the [ late 1980s.background. The book Treatments more focusedgiven on by the Mason study and Woodhouse of [ twistor theory and integrable systems are calculate the entire tree-level S-matrix of Yang-Mills theory in four space-time dimensions. tures across a range of perturbative quantum fieldtheory theories. is Yet not despite a its wide subject applicability, that twistor to most encounter graduate students in in their mathematical studies. orsenior theoretical researchers The physics who are goal are likely encountering of twistors theseand for exciting lectures the arena first is of time) to research. with provide an avenue graduate into students this (or vibrant more underpinnings of twistor theory. Nor areedge they meant aspects to of provide research an introduction whichlectures to make you the use will most of be cutting- able twistor to methods.alizations) look and at Rather, be any my able recent to hope paper understand is involving the twistor that basics theory after of (or these what some is ofmaticians. happening. its gener- Thus, I have assumed aogy, degree as of well familiarity as with a standardof QFT bit string notation of theory. and general Some terminol- background relativity.ometry in The mathematical will final subjects make such lecture your as assumes life algebraicsketchy) easier, some explanations and but exposure for differential it all to ge- is of the not the basics technical essential: tools I needed have as tried they arise. to providethat basic 90 (sometimes minute lectures wouldthe be current more research literature suited reflect to myHowever, the it own presentation would interests here. be and useful opinions, References to andtwistor throughout comment are theory, to since certainly briefly you incomplete. on will other definitely pedagogicalsubject want and from to reference scratch. refer treatments to of other sources if you are tryingTod to [ learn the ago by Penrose [ Twistor theory 0. Introduction Despite many interesting initial advances, thea subject variety stalled significantly technical by and themoved philosophical late primarily 1980s problems. into due the to For realmand of the geometry. pure following mathematics It twenty as was years, aearlier resurrected tool twistor work for for theory of physics the Nair in study 2003 [ of with integrable systems Witten’s observation [ PoS(Modave2017)003 ] 15 Tim Adamo ]; this serves as a 10 ]. 16 ] stands out as having aged 11 ]. The lecture course by Wolf [ 14 , 13 , 12 2 article by MacCallum and Penrose [ Physics Reports In the category of older review articles, the one by Woodhouse [ There have also been many review articles written about twistor theory over the years. One useful introduction and includes many ideasThe that section we on will ‘The not evaluation have of timein scattering to amplitudes’ light discuss makes of in for these the particularly lectures. modern interestingtwistor reading development theory of to the make meaningful subject; contact you with might the understand language of why particle itparticularly physics! took well. so Its long perspectives on for many aspectsin of these the lectures subject makes are use the of onesapplications Woodhouse’s used approach. in today, More and perturbative much modern QFT reviews, with canprovides a be an view towards found alternative exposition in of [ introduction many to of the application the of ideas twistortheory. theory presented Finally, to in a the these recent study historical of lectures, scattering overview as of amplitudes the well in Yang-Mills subject as was an given by [ of the most cited is the Twistor theory PoS(Modave2017)003 R = in a b ∈ (1.1) x x a d x a x signature d Tim Adamo ) x ]. These will ( 6 (at least semi- , ab 5 g C M = Euclidean 2 . This corresponds to s , C ]. Initially, each : 2 , 6 ) ) with M 3 . R − 4 x i , ⊂ d R R ( i ∈ − , M 3 ∈ − x + . In Cartesian coordinates 2 1 , , ) x 2 is defined by allowing the coordi- 2 . Complexified Minkowski space, M x , ) x ) , (+ 1 d 1 R ab . ( x − g C , ∈ , , − 1 3 2 R M holomorphically [ x ) − M , 1 ) , ( ∈ 2 x x 1 x 0 -dependence in the metric after complexifica- d ( , x a ( of − 0 ab x , . The line element x g − 1 3 ab ( 2 ) η 0 , x C . Before jumping into twistor theory itself, it is impor- are now complex-valued, holomorphic functions of the diag , d ) ) C M = x − ( , = ( M ⊂ ab b will contain the corresponding calculations in any real space- ab − 4 x ⊂ , η g C d R 2 − , a complexification , 2 M x d R (or ultra-hyperbolic) signature (+ ab η . The a split Split: = x -dimensional space-time equipped with a metric d 2 Euclidean: d s d with , equipped with the metric 2 , 4 2 C values. R , with signature , the metric is simply or be a real, ) M 3 x +) , M , to take complex values while extending 2 while the metric coefficients complex x a signature can be obtained by taking different real slices of the complexified space-time. The + , , x 1 C Note that the ‘signature’ of this complexified metric is no longer meaningful: real flat space Let’s focus on four-dimensional Minkowski space-time, We begin our study of twistor theory in the simple setting of flat, four-dimensional Minkowski Let Why do we care? Complexification means that we can study physics on x + , is then just , , ∈ any C . (By ‘holomorphic,’ we mean that there is no ¯ 0 a a x looks the same as in real Minkowskito space, take with the exception that the coordinates are now allowed of most obvious such real slicerestricting is that the of coordinates real to Minkowski takeplexification. space-time, real However, values; by taking in different other(+ real words, slices just we un-doing can the obtain process of com- x tion.) The resulting complexified space-time is denoted space-time, Twistor theory 1. Spinor and Twistor Basics tant to set the stage usingmake a our few basic life tools: substantially complexification easier and whencomplexified spinor talking space-time methods about and [ twistor phrased theory, in which termsof is of naturally the spinor defined twistor variables. for correspondence, After this, focusingspace-time. we on set the out the non-local basics relationship between twistor space1.1 and Complexified Minkowski space nates and the metric coefficients arex real functions of the these real numbers; complexifying, we allow ( M some coordinate system In this sense, complexified Minkowski spacereal is space-times. a sort of universal analytic continuation of all flat, classically), then recover results in the desiredconditions space-time later. signature A by calculation imposing on appropriate reality time signature, provided we are‘Complexify careful first, about ask how question we later.’) is restrictof a to the recurrent the day theme real we in slice. always twistor want theory. This to Of ‘moral’ wind course, (i.e., up at with the real end answers, so although later lectures often focus on PoS(Modave2017)003 1 . ) has any C with (1.4) (1.3) (1.2) , and : ˙ , α ) ˜ C C 2 a C ( α , ˙ M M 1) live in sl a 2 , ( Tim Adamo ˙ 0 × ) SL = spinor indices C ˙ , × 2 matrix, so its ) α 2 ) C ( × , C sl , 2 ( 2 ( is a 2 . Indeed, given a vector , so any vector index can a ) is translated into ˙ α σ . C α , v spinor indices. This rule (i.e., 2 ! abc ( 2 , which is locally isomorphic to T 3 ) v v i SL C representation and the other in the − , − × ) is isomorphic to 4 0 1 ) ( 0 v ) v . , C ) Weyl spinors is given by: 1 2 C 2 , 3 ˙ , , α ( v ) v 2 i ˙ 4 γ α representation of SL ( , C ( v and two conjugate SL + ) ˙ , + ˙ βγ ( α 0 0 2 vanishes. But so ˜ 1 C a , ( v ˙ v αβ ) 1 2 positive chirality α M α ˙ ( α a

2 det 4 T α 2 v = = 1 ( → √ ˙ b α v null α a = v abc v a T v spinor indices. The dotted spinor indices ( ab with respect to the metric is encoded by the determinant of ˙ α η indices: one in the 2 a 1) live in the α a v ) , √ representation of SL σ 0 C ) , 1 2 = = 2 : , ( 2 1 α ˙ α ( ): if and only if det α v 1.2 . The converse is also obviously true: any matrix of the form null ˙ α ˜ a negative chirality is , ; in other words, the Lie algebra ) α a , its representation in terms of SL a ) C v lives in the , 3 v 2 C , ( can be written as 2 M v C SL , 1 representation and will be referred to as M v × ) , ) 0 2 1 representation. v C , , 0 So the 2-spinor formalism provides an unconstrained way to represent null vectors in We can immediately observe one nice consequence of writing vectors in the 2-spinor formal- This isomorphism is easy to see if you are familiar with the classification of semi-simple Lie algebras in terms of The equivalence between a vector index on The spin group of complexified Minkowski space is SO ) 2 1 ( 2 1 ( = ( , a 0 pair of Weyl spinors of opposite chirality definethe a ‘standard’ null vector. vectorial This description, is where certainlynumbers an one improvement constrained defines over by a a null quadratic vector equation. by specifying four (complex) vanishing determinant, and hence its corresponding vector is null. for some spinors ism. Note that the norm of a vector and so forth. its spinor representation ( determinant vanishes if andvector only in if its rank is less than two. Therefore, every (non-trivial) null Dynkin diagrams. be represented by a pair of SL is nothing to be afraid of: it is given by the familiar Pauli matrices, contracting with the Pauli matrices) allowspairs us of to spinor replace indices. any number For instance, of a vector indices rank-3 on contravariant tensor This means that Twistor theory calculations in the complexified settingwill and spend ignore some time the in detailsconditions these of are early imposing actually lectures reality manifested. emphasizing conditions, such we details to make it clear1.2 how reality 2-spinors in Minkowski space SL A vector on ( v The un-dotted spinor indices ( will be referred to as the PoS(Modave2017)003 ˙ ˙ α α α α v x (1.8) (1.5) (1.6) (1.7) (1.9) (1.10) (1.11) Tim Adamo . The inner } 2 matrix ˙ α ˜ × ω , . α ˙ α ˙ β ω , ε ˙ ˙ α β . ˜ ˜ κ . The object we should ω , ! ˙ ab α α ) ; as we noted above, these ˜ η κ 2 κ C v 3 { i v = M . To summarize, in the 2-spinor ˙ + α + b , , 1 ˜ in 0 v ω v β 2 a v ˙ ( α b v -invariant tensors, which are just the ˜ , ) κ = − . , a null ab ˙ ] β ˙ αβ C β w 2 η ˜ = β , ˙ ε α ω v : αβ 3 x i 2 ε ˜ ε or its inverse v ] κ ( d [ = ˜ : + ˙ and ) = ω i α = − ab αβ ˙ 1 α ˜ α α 0 κ ε v η [ x α ! v for some spinors b a null κω v d − ˙ h ( v α ˙ β 5 ˜ ), and their inverses are defined by ˙

1 0 α ω = 0 1 ε ). , α 2 , − βα , 2det 1 α ε β ω √ null αβ

δ βα βα = − ε w 1.10 ε ε = ↔ · ˙ = = α β = = β ˙ α α a ˙ 2 ω v β γβ null αβ a null s ˙ ε αβ take complex values and are encoded in the 2 α ε α v ε d = w ε : α takes the form κ ) v 3 βα αβ α C x ε = ε a , and ˙ β 2 M α -invariant, skew-symmetric inner products on the 2-spinors of each ˙ β x α ) v ω , ˜ κ 1 C α , α x = , κ 2 κ into the language of the 2-spinor formalism. In the standard language, ˙ 0 ( α x α = C ( : ↔ v i M a null v κ ω ). h 1.2 For example, consider any two null vectors At this point, we will also introduce some notation which will make our lives easier as these Of course, in order for it to be useful, we must be able to translate everything about the usual Because they are skew-symmetric, it’s important to fix a convention for how we raise and These are the natural SL chirality. can be written as product of these two vectors is easily seen to be formalism the line element for in terms of the inner products defined by ( Sure enough, it is easy to see that according to ( lectures go along. Clearly, the Levi-Civitaand symbols positive define chirality inner spinors, products respectively. on We the will spaces denote of these negative by: (in spinor representation), this means that the dual covector is metric geometry of we raise and lower indices using the metric tensor lower spinor indices and thensign stick errors. to Our conventions it will – be otherwise, ‘lower to our the calculations right, will raise be to the inconsistent left’: due to Twistor theory used to raise and lower spinortwo-dimensional indices Levi-Civita are symbols: the natural SL These objects are skew-symmetric ( and likewise for dotted indices. with identical conventions for dotted (positive chirality) spinor indices. So given some vector where the coordinates PoS(Modave2017)003 . , ˙ α C C ¯ κ α M ∈ κ (1.14) (1.16) (1.15) (1.13) (1.12) d , c , : b Tim Adamo ˙ , α a α inside of x ]). 11 M . , where , it is easy to see that ) R ˙ α d can be written as , α ∈ c x 3 M y . , = ( ) 2 ¯ y ˙ d α . , , ) forces . ˜ ¯ 1 c ω ˙ α , y ! , where ! , α † 2 ˆ 0 ) x ! 2 = ( x 3 3 x ˙ i ¯ and x ¯ α ¯ x 2 x i ¯ 3 α = α x ) x + i ¯ ˜ x − b ˙ + − ω α 1 , null vector in 0 − 0 ¯ 1 α − x ¯ ¯ a x ¯ x x 0 x , 1 = ( 7→ − x x 2 ˙ ˙ 3 α α ! real = ( x ¯ 2 x 2 i ¯ α 3 ˜ 3 2 x 3 ω x α x i ¯ x ¯ y y i into the language of 2-spinors, we now consider x + i + : κ 0 , define the following operation on − + 1 . C 6 + + − ¯ x ¯ − 2 , we know that the appropriate reality condition is C x 1 0 1 1 0 ¯ ) ) 0 x x M ¯ y x x 3 3 , i x M

y x − )

, 3 2 2 ¯ b 2 y y 1 ,

2 i x + ( ¯ √ a 1 , 2 − 2 √ Hermitian 1 + 1 ) 1 inside = x √ 0 2 . Thus, the reality structure associated with the Lorentzian- y = ( , = 4 † i x y † 0 ) ˙ ) α = ˙ x R α ˙ : ˙ α

¯ α ( to be κ α + ( α ˙ α α 2 x ˙ 2 x α x 1 α ) to be real-valued. In terms of the matrix ( ( 7→ ˆ α √ x 1 x a y α x = κ ˆ x + ( = 2 x ) | 0 ˙ α x be preserved under this operation ( α ˙ x , and that this is compatible with the reality condition. α α α is naturally associated with a complex conjugation on 2-spinors which exchanges x κ ) = ( x C ( M 2det = 2 To fix the Euclidean real slice Suppose we wish to single out the usual, Lorentzian real Minkowski space Having translated the metric geometry of In particular, given spinors with components x : 4 for some spinor Euclidean signature In terms of the usual coordinates You can easily use this conjugation to show that any Lorentzian signature simply to force each ofthis the corresponds to requiring It is easy to see thatR this is precisely the structure required to obtain the positive definite metric on Twistor theory 1.3 Real slices and spinor conjugations how real slices of variousmeans signature finding can reality be conditions on singled the out matrix at the level of the spinor formalism. This Demanding that real slice of dotted and un-dotted spinors. Since Hermitian conjugation includestion the of transpose the matrix operation entries), (inexchanged it addition is when clear to we that compute complex positive conjuga- and negative chirality spinor representations are the induced conjugation operation acts as: which are compatible with theinduces desired a signature. natural notion As of we ‘complex will conjugation’ see, on each the choice spaces of of spinors reality (c.f., condition [ PoS(Modave2017)003 rather (1.21) (1.17) (1.20) (1.19) (1.18) ˙ α α x Tim Adamo . Any null ) with homoge- R , 3 2 ( -valued. In other CP SL , real , ∗ × R ) C ∈ . R 3 ∈ , ) x . ¯ 2 c r , ) ( , ¯ ∀ 2 ¯ d d y , , ¯ c − 1 , , . x 2 A , ! = ( = ( 0 R Z x 2 3 ˙ ˙ α α . x ¯ ∈ x ∼ i ¯ . We can describe ˜ ˜ ω ω ˙ 4 α A − + ˜ C κ 0 which is preserved under the hat-operation. 1 7→ 7→ , , we simply take the complex conjugate of , ¯ r Z x ¯ , as desired for split signature. x ˙ α C ˙ ˙ α 2 α α ! 2 ) ˜ κ 3 ˜ ˜ ω 3 M ω ω x ¯ 3 2 x i ¯ x α y x ( , + κ for 7 − + − ) 0 quaternionic − 1 ˙ α ¯ 0 be the 3-dimensional complex projective space: this 1 0 x ¯ x 2 ˜ , , , x x κ 3 ) inside . Indeed, we would need to apply the hat-conjugation 0 ) ) 1 α

¯ α 3 2 , ¯ 2 b a x , y x κ 0 , , κ 2 ( 2 CP ¯ , ¯ a b 1 − + − R , which take values in the complex numbers, are never all 0 √ − ) − 1 0 2 4 x x = = ( ) = 6= ( Z 2 = ( , α α ˙

y α ) 3 ˆ ˆ 4 κ κ α α 2 Z x Z ˆ 1 , κ + ( , √ 2 7→ 2 3 ) induces a conjugation on 2-spinors which, unlike the Lor– entzian ) Z Z 7→ α = , 0 . This forces , 1 x κ ¯ x ˙ α 2 α 1.15 Z = Z α κ are precisely those spinors whose components are x , x | 1 2 ˙ , α = ( ) = ( 2 Z = α x ( A rescalings (often called ‘projective’ rescalings) means that the homogeneous x ( R ˙ α Z ∗ α x C 2det can then be represented by 2 = , 2 2 is the set of all non-zero complex numbers. The invariance of the homogeneous co- x R ∗ C times to get back to the spinor we started from. For this reason, the reality structure associated To fix the split signature real slice One straightforward consequence of the quaternionic nature of the hat-conjugation acting on 2- Having introduced the spinor formalism for complexified Minkowski space, we are now ready The ‘hat-operation’ ( While the underlying conjugation on 2-spinors is ordinary complex conjugation, it does not , ˙ α α x neous coordinates vanishing, and are identified up to overall re-scalings: ordinates under This is simply the statement that there are no realSplit null signature vectors in Euclidean space! is the space of all complex lines through the origin in where spinors is that there is no non-trivial combination four with Euclidean signature is often referred to as to define the twistor correspondence. Let Note that this operation is qualitatively differentit from does ordinary complex not conjugation square – to in the particular, identity: Thus, 2-spinors on words, the complexified spin group in split signature is simply1.4 SL Twistor space vector on Twistor theory conjugation, does not interchange spinor representations: for which interchange the spinor representations (since we simply took the complex conjugate of and demand that than the Hermitian conjugate). So in split signature the conjugation acts on spinors as: PoS(Modave2017)003 : A 1 by Z π 3 (1.23) (1.22) . This and the CP r 4 C Tim Adamo . Since the we have the PT 1 are coefficients open subset we U ⊂ ˙ α 1 , and the map α 1 x CP ), which imposes the incidence CP × there are manifestly three 1.23 PT C i U M → . For complexified Minkowski ∼ = ) define a PS ˙ . Putting the projective scale back : ; in α and space-time. This relationship are just coordinates on : 4 } α ) PS 0 x 2 . For instance, on C PT α A π PT λ 6= C ⊂ Z , i 1 2 ˙ α and ! − Z – they are the root of everything that is , C | it is useful to divide the four homogeneous µ ) . 1 C ) i 4 for all non-zero complex numbers π while α α Z C M ( λ ˙ β λ = ( PT α , ˙ λ ∈ α α A ˙ r α x α A Z 8 : x µ Z PS ∼ C 7→ { = β ) M = ( ˙ λ α twistor correspondence = β A 2 i µ λ π on Z , } U ˙ . Suppose that we forgot about the projective scale of the ˙ α α α α α . PT M x λ x , with 1 incidence relations ( ) Z . β . In the next lecture, we’ll learn exactly is nothing but fancy notation at this point. ) / 3 λ 4 α and define a complex plane , α acts as a homogeneous coordinate on the one-dimensional complex Z λ ˙ λ α ˙ α α , CP β α λ β µ λ x ˙ of complexified Minkowski space is defined to be an open subset of α λ ( and ˙ α α , and x corresponds to a linearly and holomorphically embedded Riemann sphere 1 β ˙ α x PT Z C µ = ( / , which is just the Riemann sphere. So ˙ 3 α M 1 7→ Z µ , carry the same weight with respect to projective rescalings. In other words, the into ) 1 , the spinor CP β of the projective spinor bundle over A α Z λ into two Weyl spinors of opposite chirality: ; what does this correspond to in twistor space? From ( Z λ / , ), though. What do these relations actually tell us? First of all, suppose that we fix PS C 2 A ˙ α Z Z α M is simply the projection x has coordinates and 1.23 ( ∈ twistor space C ˙ α x PS µ M In more formal treatments of twistor theory, this relationship is often presented in terms of a The non-trivial step is defining a relationship between For our purposes, it suffices to think about twistor space purely in terms of the incidence The → equation is linear and holomorphic (i.e., thereseems are that no a complex point conjugations in appearing anywhere),in it twistor space. into the game, we find that the incidence relations (for fixed a point coordinates on twistor space for a moment; then means that on projective space in a linear equation relating incidence relations double fibration interesting about twistor theory. space, the twistor correspondence is capturedon twistor by space an and algebraic the relation coordinates between the coordinates These equations are known as the where division of the relations ( Twistor theory coordinates only contain three (complex)covering degrees it of with freedom. the In coordinate particular, we patches can chart where is non-local, and is often referred to as the PS the complex projective space coordinates coordinates should choose, but for now this is not important. On relations, well-defined complex coordinates given by taking PoS(Modave2017)003 . : C Y PT pro- M ), the ∈ being ∗ (1.28) (1.24) (1.25) (1.26) (1.27) as the and ˙ Z α C and α X 1.24 ) PT y Tim Adamo ∈ PT − . These planes x Z ( PT ∈ , then such a map is , 1 . Z α a ˙ λ α CP b ˙ ). The result is a 2-plane α α y if and only if their difference 1.28 , we denote the corresponding = C Z ˙ α in ( M µ , , ) can hold (without ˙ α ∈ a a are null separated. The point ˜ λ σ σ x C a and α a α 1.27 , c δ identified with M . ˙ α α ˙ α 0 = = λ ˜ λ α ˙ α α . After fixing this redundancy in ( α x α = a α λ λ α holomorphic linear embedding of a Riemann . We will often refer to these Riemann spheres λ α x δ λ .’), since they are linearly embedded and defined = = ) which is totally null: every tangent vector to the ˙ PT = 9 α target space. This is 4 complex degrees of freedom C any ˙ ˙ α α ˙ ) with α α 3 a α , , ⊂ ) α ˜ λ M c a µ a ) y 1 are null separated! So we discover that lines in twistor define the map. Of course, this is over-counting: we y σ σ CP ∈ 1.23 − a a ) C ˙ − x ˙ α x α CP a α are homogeneous coordinates on ( x b b M c ( ) ∼ = , 1 a is fixed by the undotted components of } ⇒ = ∈ = ˙ α X σ ˙ y ˙ α , which are the Möbius transformations, and 1 for the α α , b , 1 PT 0 ( λ µ µ x σ ∈ CP in twistor space intersect in a point Z = ( { associated to Y a , . = σ , where . X X ˙ α . Subtracting one incidence relation from the other, we discover that α Y ˜ C λ λ . If ∩ α ˙ α M ∝ planes X by varying over the choice of the spinor λ α - ˙ x ∈ α C (or an open subset thereof) can always be put into the form of the incidence α α y ) , 3 M y x − . But this means that CP ˙ x α ( ˜ λ Therefore, the lines We can be even more precise about this: What about the other way around? That is, what does a point in twistor space correspond The upshot of this is that a point in Minkowski space corresponds to a holomorphically, lin- ; since this is a 2-dimensional object, the only way that ( αβ for some plane is of the form where the 8 complex parameters haven’t taken into account the automorphism groupings of of the the homogeneous Riemann coordinates sphere of or the the projective rescal- space intersect if and only if their corresponding points in (because there are two degrees of freedom in are referred to as (3 from the automorphisms of sphere into given by jective rescalings), which can be used to fix which is precisely the incidence relations ( for two points In this equation, contraction on the undotted spinorε index is accomplished through the anti-symmetric zero) is if is described in relations for fixed to in space-time? To answer this question, it is illuminating to describe a point Twistor theory map looks like a point in space-time is described by an extended object in twistor space! intersection of two lines (that is,By holomorphic, the linearly incidence embedded relations, Riemann this spheres), means say that obeys holomorphically. This is our first taste of the non-locality of the relationship between as ‘lines’ (e.g., ‘The line early embedded Riemann sphere in twistor space. For a point Riemann sphere in twistor space by PoS(Modave2017)003 C up M ˙ α α ∈ (1.29) . This x x X . Tim Adamo 2 Z takes the form: ∧ ] 1 B 2 Z Z A [ 1 Z . The correspondence ! = PT Z X AB ⊂ X corresponds to a linearly em- X which lie on the line C . M PT ! ˙ α β αβ x ε 2 x ˙ X . Furthermore, the correspondence captures β α ˙ Twistor Space β x 1 ˙ α − ε 2 1 10

, while a point in be two points in i C 2 2 λ M Z , 1 1 λ h Z = AB -plane in The geometry of the twistor correspondence. X α x ! x . Just like a line in three real dimensions is specified by any two points X as bi-twistors Figure 1: Space-time . C i 2 M λ 1 λ h Points in So the non-locality of the twistor correspondence is manifest in both directions: a point in Using the incidence relations, show that the resulting ‘bi-twistor’ We’ve learned that a point in space-time is represented in twistor space by a linearly embedded bedded Riemann sphere in twistor space; see Figure In particular, the skew bi-twistor encodes precisely the information of the space-time point twistor space corresponds to an the conformal structure of (complexified) space-time, since points lying on the light cone of Twistor theory are uniquely identified in twistoris space also by stated the in lines purely whichtheory: holomorphic intersect holomorphic terms structures on on twistor twistor space, space which encode brings conformal us structures to on a space-time. moralExercise: of twistor means we can represent the line by taking the skew product of these two points, to a scale set by Riemann sphere, or line, which lie on that line, sotwo a points holomorphic line which in lie 3 on complex dimensions that is line. uniquely specified Let by any PoS(Modave2017)003 ) by , the (2.1) (2.3) (2.4) (2.2) ) be the α ∨ λ 1.23 , under the ˙ can be re- . Points in α PT Tim Adamo ) C µ C α M ˜ µ M , = ( ˙ α A ˜ λ Z = ( A W , i ˜ . µ λ h ) α . . ¯ ˙ µ ] + α given by contracting a twistor index against α , ˜ ˜ λ λ ˙ λ α ∨ ˙ α µ ¯ ˙ λ α α by incidence relations: ( α x PT x i C = [ 11 i 7→ − A M ) = and W α = ˙ A α λ α Z , µ PT ˜ ˙ µ α = µ : ) so that it is related to a particular real slice of ( 3 W · , recall that the natural conjugation on 2-spinors is the ordinary CP Z M , but now with homogeneous coordinates PT ) in the specific context of Lorentzian reality conditions. Let as 3 2.2 -invariant inner products on dotted and undotted spinors. are related to points in ) C CP , 2 ( (an open subset of PT To make our lives easier, in Lorentzian signature we modify the incidence relations ( We have seen that twistor space is related non-locally to complexified Minkowski space: In the previous lecture, we discussed how the various real signature slices of For real Minkowski space dual twistor space Thus, the complex conjugation naturallyin sends complimentary a representations. twistor There to is something atwistor with natural space way its (this to component is interpret indices actually this an in example terms of of something a known ‘duality’ as on projective duality). same open subset of There is a natural inner product between in terms of the SL a dual twistor index complex conjugation acts on the components as including a factor of ‘i’: this points in space-time correspond toin holomorphic, twistor linearly space. embedded Riemann The spheresof conformal these (or structure lines ‘lines’) of in space-time twistor space: isnull lines encoded separated. intersect by In if this the and lecture, only holomorphic we if explorestructures structure the further and how corresponding conformal structures space-time on structures) points space-time are are (in translated particular, into reality geometric structures on twistor2.1 space. Reality structures Twistor theory 2. Twistor Geometry covered by imposing reality conditions. Innotions the of 2-spinor formalism, complex these conjugation reality on conditions thereality induced spaces conditions of are spinors. translated Our into goal twistorimpose is on space. now to In understand other how words, these what conditions do we need to The geometry of the basicand twistor we only correspondence work is with completely ( unchanged by this modification, complex conjugation with the provisospinor that representations the are positive exchanged (dotted) under and the conjugation. negative (un-dotted) So given chirality a twistor twistor correspondence? Lorentzian signature PoS(Modave2017)003 - ) ˙ α ) 2 α PN 2 PN x , − (2.7) (2.5) (2.6) ∈ , 2 ( 2 (+ Tim Adamo – in complexified -invariant. So the A ) 2 Z which corresponds to , . 2 ˙ α ( X -plane of a twistor Z ¯ λ ) has signature α α λ 2.5 ˙ α α induces a degenerate, SU ) ). Since Lorent– zian-real Weyl † 4 and the M ? You can show that the condition x M ; in the twistor theory literature C C C − . M M M x } ( . 0 i ⊂ i = = M ¯ ¯ µ λ Z α , whose components are the complex conju- h · . Therefore, any line λ A † ¯ ˙ Z -plane – a totally null complex 2-plane whose Z be any point lying on the line in twistor space. α ) | ] + ¯ ˙ λ α α X ¯ † λ α 12 PT ) x ∈ µ ˙ α ∈ α Z correspond to in x = ( Z = [ ( { ˙ i α ¯ Z , the un-dotted spinor components of α · PN − = x α ˙ Z α λ ¯ λ PN α plane , how do we know that the corresponding space-time point λ ˙ − α corresponds to an PT α α x , this means that the inner product is an SU i in ) PT 2 1 = is the twistor space associated with ( if and only if ¯ Z CP · M PT Z ∼ = ∈ ⊂ to a point in dual twistor space, x X A PN Z The intersection between the Lorentzian real slice Recall that a point in Coming back to the Lorentzian reality structure, we can now say that the complex conjugation Given a line In other words, is often referred to as the ‘space of null twistors.’ tangent vectors are all proportional to Minkowski space. What does a point in a point in real Minkowski space-time must be contained in maps a twistor is a real null geodesic. gates of the original twistor. Thus,of complex the conjugation form induces an inner product on twistor space Using the anti-symmetry of the spinor inner products, we see that ( Using the incidence relations, it follows that But we know that when viewed non-projectively (that is, as an inner product on invariant inner product on twistor space. is valued in the real Minkowski space? Let Figure 2: Twistor theory spinors are valued inspinor SU conjugation appropriate to Lorentzian Minkowski space PoS(Modave2017)003 - → α PN con- (i.e., (2.8) (2.9) , and ∈ (2.10) 4 PT C . This ) Z R intersect M α λ Tim Adamo , , the twistor PN ˙ α 4 α R x fibration ( 1 . quaternionic -plane in ) ¯ c CP α which are preserved sending , ¯ d . Lines in 2 − PT . Thus, a point . Clearly, any line of the ) in twistor space. At the by taking the line passing ˙ with the projective spinor lines α X X 4 → ¯ 4 = ( λ is the statement that this induced a R α ˙ R α C λ ˜ ∈ PT ω PT : x . M with coordinates ) 7→ using the reality conditions, while σ α ; see Figure 1 ) is a totally null 4 ˆ λ C d , . R , ˙ CP α c M α PT ˆ µ . This means that with Euclidean reality × ˆ λ of inside of . The picture is that this real null geodesic ˙ = ( 4 α ) points in AB 4 i = ( ˙ M R µ α M σ X . R ˆ λ ˜ A ˙ ω α − ˆ -plane, namely: Z λ = , in µ α h α ˙ α in twistor space can be represented by a skew bi- 13 λ 7→ AB = ¯ λ ˙ α are separated by such a real null geodesic. ˆ ) X , X α α ˆ . µ α ) ] λ λ ¯ λ M B a , we can still ask if there are ˙ α , ˆ , = Z ¯ α ˙ b α A ˙ , or that there are no real null geodesics in positive-definite x α [ worth of points (the twistor line PT 4 − µ α Z 1 x R are any two distinct points lying on = = ( = ( CP 2 A α AB Z ˆ Z , κ is uniquely associated with a point ) and then a point on the corresponding Riemann sphere. In other X 1 : Z σ 7→ ˆ Z PT ) b ∈ , a . Although points of twistor space are mapped to points of -plane intersects the real slice Z 1 α , where ] will be preserved, since = ( B 2 CP ] α Z corresponds to a B . A ˆ × κ 4 [ Z 4 1 4 A Z id, it is clear that there are no points in twistor space which are preserved with R [ R R Z − = ∼ = and its conjugate = = AB Z 2 0 singles out a single tangent vector to X PS AB σ X = . You (hopefully) showed that any line So in Euclidean signature, a point in twistor space can be specified by fixing a point in The reality structure associated with Euclidean Even if there are no real points in The fancy way of saying this is that Euclidean reality conditions induce a ¯ σ Z : every point of twistor space gets mapped to a point of · 4 correspondence remains non-local since a fullsame Riemann point sphere on in twistor space corresponds to the is where the complex It is easy to see thatthe this incidence is relations real in with the respect sense to that the quaternionic conjugation and is compatiblea with line which is preservedwords by the Euclidean twistormeans space that is Euclidean isomorphic reality to conditions identifybundle the twistor space of corresponds to a unique real null geodesic, if and only if their corresponding points in Acting on twistor space, this conjugation induces an involution respect to this conjugation. This makes sense: a point in Twistor theory Z Euclidean signature jugation on spinors, which acts as Since the statement that there are no real (with respect to plane does not intersect the real slice signature. by form R every point of twistor conditions, every point through level of spinor variables, this fibration is given explicitly by PoS(Modave2017)003 , then (2.11) (2.12) (2.13) , which -bundle M J 1 ; then for N R determines Tim Adamo , and has the CP j PT PT V of the manifold j i ⊂ J X to TM i V . Nijenhuis tensor l i J . Take a line j . . 2 , ∂ ) 3 is determined by the intersections 2 α − R C λ j l RP , J M ˙ i are vector/covector indices on α ⊂ ∂ . µ on the tangent bundle R 0 k l ,... J j which maps a vector = ( = PT , j i TM A i + α J . This is manifest already in the basic geometry j Z k λ J → ˙ l α 14 7→ ∂ α l i ) ) J x TM α : λ − − , J k i x vice versa ˙ α J ( l µ exchange spinor representations. In other words, 2-spinors ∂ and thus the incidence relations imply that spinors. This complex conjugation acts as an involution on j l ¯ J ) Z = ( not R = A = , , so the equation is trivially satisfied. Hence, the twistor theory of Z 2 2 , we can associate an object called the k i j , Z ( is the correct twistor space for ) J 2 J R R N , we saw that the appropriate conjugation on 2-spinors was ordinary ( is a linear map 2 , PT 2 M R id. In component notation, if . To each which are (literally) real-valued: i k − δ it follows that = − PT for points in 2 X ˙ = α J j α k ∈ x J j i Z J = are manifestly real SL ˙ α 2 , α 2 It is easy to see that In general, the idea in twistor theory is to work in the complexified setting, imposing reality Recall that one of the ‘morals’ of twistor theory is that a complex structure on If you’ve had a course on complex geometry, you will have heard that an almost complex For the real slice x R a conformal structure on space-time and of the twistor correspondence: the conformal structure of conditions only at the endwere of usually a the Lorentzian calculation. ones, In while early thereality in conditions old the were ‘twistor days preferred. renaissance’ of of Nowadays, the 2004 Euclidean theful subject, reality split conditions when these signature seem performing reality to explicit conditions be calculations. the So mostcan depending use- find on what any era one of of the thetechnical literature three reasons. you reality read, In conditions you these given lectures, preference wereasons: for will a it focus combination maintains mainly of the on physical Euclidean complex-projective features signature, and correspondence of for (unlike the the split following general complexified signature); signature it twistor has the nice feature that twistor space is a structure on a manifold So the natural portion ofby twistor the space points which of is preserved by this complex conjugation isany point formed split signature Minkowski space is a theory of real variables. over space-time in Euclidean signature; and manystudy of of the perturbative recent QFT applications are of most twistor cleanly theory to phrased the in these2.2 reality conditions. Complex structures of holomorphic lines inIntuitively, twistor we have space. described itare What as holomorphic. exactly a way is of a knowing complex when structure things (e.g., on functions, twistor vectors, space? etc.) Twistor theory Split signature which obeys property complex conjugation which does of twistor space, But the almost complex structure is a rank-two tensor you should think of ascoordinates, a it sort is given of by curvature associated with the almost complex structure. In local PoS(Modave2017)003 is . ¯ ∂ M (2.16) (2.14) (2.15) (2.17) Tim Adamo 0. = J . q N ¯ a z to be our working -vectors, and those d¯ ) -vectors: ¯ i. Vector fields with 0; by the Newlander- ∂ 0 ) , , 1 − = 1 , d ( 0 2 ◦ be holomorphic is simply ( ¯ ∂ 1 ∧ ··· ∧ -forms. We can now define : the decomposition for 1- f ) 1 + if ¯ i and a q q M , z , p + d¯ anti-holomorphic form indices: p ρ ( q ∧ is the complex dimension of = p d a ) , id, the complexified tangent bundle , z M ) d ¯ a ( . − ¯ q z integrable , ) ∂ M p 1 ∂ ( , = Ω is ¯ q 0 a , | ( . 2 M p ¯ ¯ , where decomposes as: ∂ V ∂ J ¯ T A ∧ ··· ∧ d Ω (i.e., a complex structure) if ¯ Z ⊕ 1 ∂ k ⊕ M a > ∂ , the condition that a ) = z 0 ¯ z q with eigenvalues A , ∂ d q M , 1 ¯ M + ∂ , 15 q Z ( J p ¯ ) M a a p d T , then this decomposition is simply ··· on M V , which is naturally a complex manifold (of complex = 1 ( = ¯ or a 3 f M = 1 p C = ¯ integrable ∂ d a ) ) by the natural pairing between vectors and 1-forms, and + C -forms on i q 2 ··· CP , k x 1 M ∂ p a ( ∂ > TM i k 2.14 Ω ω q which squares to V be the natural projection onto Ω , anti-symmetric covariant tensors) using the wedge product. In = holomorphic form indices and → + ) k ) p p ω M TM ( M q ( , , p q has , ) p ) Ω we want to describe. Since it will be our preferred choice of reality struc- is given by M Ω M ( C , which increases the anti-holomorphic form degree of any tensor by one: : are referred to as ‘holomorphic vector fields’, or ( → q ¯ , q ∂ ¯ , M PT p ∂ J C , p /0whenever ) Ω Ω i are referred to as ‘anti-holomorphic vector fields’ or M ∈ ( − k -forms (i.e., rank- ) = k ω Ω M i under : ( q q + , , p p are local complex coordinates on ρ is an isomorphism of Ω ) ¯ a J ¯ z can be decomposed into eigenspaces of 0. In this language, an almost complex structure , Twistor space is an open subset of Let This decomposition naturally extends to differential forms on We will adopt a slightly different, but equivalent, perspective on almost complex structures. a C = z ( f given above. Dolbeault operator ¯ dimension 3 or real dimensioncomplex structure 6). on So given a notion of complex conjugation, it is clear that the Nirenberg theorem, this is equivalent to theJ vanishing Nijenhuis tensor condition for the underlying ∂ precisely the operator which distinguishesfreedom. between holomorphic For and instance, anti-holomorphic given degrees any of function where d is thedefinition usual of exterior an almost derivative. complex structure. We Indeed, take this the coincides differential with our operator intuitive definition: Clearly, a If where a section of with eigenvalue ture in subsequent lectures, wereality can conditions. explicitly write down this complex structure in the Euclidean eigenvalue in terms of the local coordinates. forms, or covectors, is induced from ( this extends to particular, this means that the bundle of We have seen that what exactly wereal mean signature by slice of the complex conjugation here depends on what sort of Twistor theory An almost complex structure is said to be TM Since PoS(Modave2017)003 . , ) of C PT , I 4 (2.20) (2.18) (2.19) (2.21) ( Tim Adamo : ABCD ε -invariants from any explicitly. ) C , , . . and future/past time-like PT 4 A ( 0 ) i ˆ ˙ α Z ∂ ˙ α α ∂ i ∂ α x ˆ . x λ A ∂ ˆ d Z D 4 λ α d α h Z λ ˆ joining them. By contrast, the con- λ C 3 = 2 Z = S = α B 2 ˙ which acts on α ˙ ˆ 0. λ α ∂ Z × ¯ . ∂ ¯ ) e ∂ A ˙ 1 , α = , R C ¯ α Z ∂ i , α 2 2 ˆ i ˙ λ 4 ¯ ˆ ˆ α λ ∂ λ : ∂ ( ¯ d ˆ e λ d ∂ ˆ ABCD λ + λ + PT α h ε h 0 ˙ 16 λ α ¯ ∂ i ˆ = µ ∂ = 0 : ˆ λ ¯ e ∂ 0 ) ¯ of topology λ e ˙ 4 α h = Z ˆ ± ( µ , ¯ ∂ = d 3 -forms on I 0 ) Z (or its real slices) in terms of holomorphic structures on ¯ = , ∂ 1 span 2 , ˙ is the projective spinor bundle, there are natural bases for the C α or its real slices are encoded in twistor space by the intersections Z 0 ¯ ∂ , ( 4 C M 1 ˙ α ) = R ¯ Z e M ( span ) by saying what the space-time looks like ‘at infinity.’ This can be PT 4 + , and it is easy to see that we can form SL ( = 0 ) 1 ¯ , ∂ 1 using only the four-dimensional Levi-Civita symbol, C 0 , 0 0 , ¯ PT e Ω 4 T ( is composed of two space-like three-spheres which form the past and future PT 4 ) and the null hypersurfaces ± i Since the twistor space of In standard language, we can make the distinction between Minkowski space and other con- It is easy to see that, as it stands, twistor space is not sensitive to the different conformal Thus far, we have been very naïve regarding the conformal structure of space-time. The null It is easy to see that this is compatible with the twistor correspondence, in the sense that four distinct points in With these bases, the complex structure on twistor space is given by: formally flat spaces (e.g., dS structures within the class offour-dimensions conformally is flat SL space-times. The complexified conformal group in Twistor theory anti-holomorphic vectors and the But light cones areto not boundary everything: conditions. these In only otherconformal capture words, class the knowing of about a conformal space-time: light structure in cones of this, is case, space-time only the up enough class to of identify conformally the flat metrics. made precise using Penrose’s notion ofMinkowski conformal space compactification. has the The structure conformal of infinity three points (space-like infinity formal infinity of dS time-like infinities. More generally, twistor spacegroup. carries One a way of natural seeing this un-broken is to action show of that twistor the indices are complexified actually spinor conformal indices of SL of the corresponding twistorabout lines. the conformal We structure happily of stated that this amounts to capturing everything cones associated with points in This follows straightforwardly from the incidenceyourself relations. that Furthermore, this you is can an easily integrable convince complex structure: 2.3 Conformal structures infinity but we can also just construct a representation of SL PoS(Modave2017)003 the ˙ α (2.24) (2.25) (2.26) (2.22) (2.23) α K Tim Adamo , ) ˙ β ∂ , ∂ µ . Crucially, we can find a B ˙ α ˙ ( α Z µ A ∂ B T ∂ µ = ˙ α ˙ , β → µ ˙ : α ˜ A J ! − which breaks conformal invariance. Z . ) in terms of twistor space quantities. ˙ α β α αβ . There is a natural line element we can x ε ∂ 1 CD linear , PT ∂λ , 2 2.24 X ˙ , α x ) α the dilatation generator. You may find it an d ˙ CP 2 β α α ˙ B β β λ x x ˙ D )) Z α ∂ ∂ AB ∼ = d x − ε ∂ ∂λ ˙ X , acting as ( α 1 2 2 1 X f A d A α B α ( ( 17 x Z T

= λ d i = D 2 ABCD = = A λ ε B 2 1 T s αβ λ = as the generators of Lorentz boosts and rotations, d J h 2 ˙ , β s ˙ α = α ˜ d J ∂ , AB ∂λ , X ˙ αβ ˙ α α J µ ∂ , ˙ α ∂ µ = α for which these generators are α ˙ P α ) λ α C , which is the conformal factor relating the metric to the flat (Minkowski) = 1. To see what structure is needed on twistor space to differentiate between K , ) carries a linear action of the conformal group means that there is no way for 4 x ˙ α ( ( α f ) = P PT x ( f are any two points lying on the line , twistor space can’t tell the difference between this metric and the true Minkowski , some additional structure is required on B 2 C Z M , A 1 Z A priori Recall that a point in conformally flat space-time is represented by a line in twistor space, and Such a representation will have generators In standard notation, the generators of the conformal group are written in twistor space as: The fact that Any conformally flat metric can be written as representation of SL conformally flat metrics, we can try to write the metric ( write in terms of the bi-twistor variables: you showed that these lines are in turn represented by skew bi-twistors one. one, for which with the identifications of for some function where us to distinguish between conformally flatspace space-times. of In particular, if we really want the twistor Twistor theory for different values of the twistor indices.be, since Note we that these already generators know are thatcaptured holomorphic, the by as causal the they structure must holomorphic (i.e., structure light on cones) twistor of space. conformally flat spaces are generator of special conformal transformations, and interesting exercise to confirm for yourself thatdo the indeed commutators generate of the these conformal operators algebra. in twistor space By comparison with thesomething space-time to do perspective, with it’s the clear ‘pointshow at that do infinity’ we this associated determine with missing the specifying structure structure the of conformal must ‘infinity’ structure. on have So twistor space? PoS(Modave2017)003 AB X (2.29) (2.27) (2.28) , which 0. So the 0, which in twistor = Tim Adamo = C CD AB X M X AB X Klein quadric AB I ABCD . But since both points ε α ) is 2 = λ 2 as space-time coordinates, X ∝ 2.26 should be the open subset of AB α are constants); is it in fact the X , 1 i PT λ 2 λ 1 ABCD , , λ ε ˙ α h . α 0. In other words, which open subset of CD x = ! X d ) is the form of the conformally flat metric, 2 6= ) d ˙ α αβ αβ α ε AB AB AB x ε 2.26 X X d ) is not homogeneous of degree zero, so it is not X αβ 0 0 0 2, such a line element will take the form: 2 , reducing the degrees of freedom to five. The fact that d , this means that AB ε 5 i AB i I 18 . In other words, I + 2

i 2 ( ) is equivalent to saying that , rather than the entire projective space itself. If the 2.26 2 λ CP λ 2 2 1 3 λ 1 PT 1 Z ABCD 2 1 λ λ ε encodes a point in space-time up to a scale, correspond- = h λ h CP h . This infinity twistor is precisely the reason why twistor satisfy and = AB = = AB of 1 I 2 = X 2 Z s s AB PT d d 0 correspond to points which lie at infinity in space-time, then AB X . This metric is singular on the hypersurface X = AB I AB ) you can show that the line element ( contains six degrees of freedom. Quotienting by projective rescalings means that AB I I AB AB X has four degrees of freedom. This is something known as the 2.25 X open subset AB infinity twistor } I 0 ) with this infinity twistor is indeed the complexified Minkowski metric. 0; since = depends upon which conformal structure we choose for space-time. ). This means that if we want to interpret the 2 = Clearly, the line element ( 2.28 X ) has homogeneous weight | PT AB 5 2 as lines in a complex projective 3-space, namely, twistor space. 2.25 X Q CP 2.26 for which AB ∈ in ( I 3 i X is the ingredient required to break conformal invariance on twistor space. It encodes 2 { CP λ = AB 1 I ⊂ λ can be treated as homogeneous coordinates on Q h X for which all lines contained in we take to be This infinity twistor also makes sense from a twistor space point ofA general view. skew-symmetric Consider a line in Indeed, working with ( So We started out with the goal of representing the flat conformal structure of AB 2 for which 3 3 X represents points in It is easy to see that is formed from the skewquadric of two vectors (i.e., the PT which is the Minkowski metric up to a scale. Thus, ( projectively well-defined. for some fixed skew bi-twistor defines a set of points ‘at infinity’ in the usual sensethe of structure conformal of compactification. the hypersurface atreason, infinity in it space-time is and known thus as the the conformal structure. For this space is required to belines an Twistor theory This metric is obviously flat (since the metric components then we must considerneous them coordinates. only up to scale; in other words, we must treat them as homoge- written in terms of thein skew bi-twistor a coordinates particular for conformal space-timefashion. structure, points. Since we In ( have order to to write get a the metric line element in a projectively invariant clearly such lines should notCP be included in CP so the line element ( Minkowski metric? The answer is no: space. We’ve now established that thisthe requires choice an appropriate choice of infinity twistor. Consider ing to PoS(Modave2017)003 – 4 are A (2.30) Z Tim Adamo ! ) if and only C ; these are the 0 M i 2.29 0 simultaneously. = α λ is 4 0. This is exactly what we 0 as well. However, = = 0 and α ˙ α λ = µ ) enters here, despite the fact that ˙ A α ¯ Z . in the spinor formalism (you’ll need µ 4 } 0 > 0, which means that the infinity twistor 0 for the infinity twistor ( . In standard Cartesian coordinates on the 4 ¯ Z = = · Z is finite, then AB | CD ˙ I . On the other hand, these points should obey the α X ) α PT 0 AB 19 x AB I , 4 I be the radial direction of Poincaré coordinates. First, ∈ ˙ α r Z µ { ABCD AdS = , and it might be useful to write down the spinor form of 0 and ε must be infinitely large if in the conformal compactification. Lines in twistor space 0 = ( ˙ 0 should correspond to points at infinity in = α x ˙ = 0 α α 0, for the infinity twistor you wrote down. Think about how A i + = x α Z α x = . This line in twistor space is precisely the space-like infinity λ PT AB AB X PT . If X in twistor space, and how this expression contracts with the infinity AB α in I AB λ in twistor space, the only way that their undotted spinor components ) to obtain the metric that you just wrote down? Finally, what is the I I AB ˙ α X X α . So the infinity twistor really does encode all the information associated x ± 2.28 have the form = I ˙ α X for which µ correspond to points in space-time which are null separated from PT I the twistor space of Euclidean is required in ( lie on the same line AB I ) corresponds to a line 2 In then end, you should find that the twistor space of Euclidean AdS This exercise involves applying both reality and conformal structures to write down the twistor Furthermore, it is easy to see that Z , 1 2.29 you write a Euclidean real if the points lying on corresponding twistor space? Surprisingly, the Lorentzian notion of complex conjugateyou (i.e., are describing a Euclidean space-time.AdS-boundary If is you’re defined having by trouble seeing why, remember that the twistor. incidence relations of Minkowski space, which is a point Twistor theory Z can be proportional is if they are both zero. So homogeneous coordinates, which means that we cannot have which intersect points of null infinity, what with the conformal structure of space-time. Exercise: general conformally flat space-time, let determine how to write the Poincaré metric on Euclidean AdS space of another conformally flat space: Euclidean AdS This means that some component of expect: lines in the unit normal to the boundary). Next, find the infinity twistor appropriate to Euclidean AdS ( to impose some reality conditions on PoS(Modave2017)003 and (3.3) (3.1) (3.2) β ; this is ) . Its field ↔ fields. In ˙ ˙ α β , while for Tim Adamo α α s β λ ( A a ∂ ↔ + ) a ˙ α A α , ( → ˙ γ β a ˙ γ A α F (ASD) portions of the field 1 2 . ˙ γ = : is skew symmetric in β ˙ γ ) ; for spin one we have the Maxwell . Clearly, there are only two ways . F α ˙ Φ α ba αβ F ( α F ˙ β F A ˙ α − ˙ β ε = β = 1 2 anti-self-dual ∂ αβ ab + − F F – a differential form which obeys some simple ˙ ˙ β β , and similarly for dotted spinor indices. So we β ˙ αγ A 20 γ αβ appearing in this expression are symmetric in their ˙ α ε , F , and so on. α ˙ F (SD) and β ∂ ab αβ h ˙ αγ ε = γ 2 1 ˙ β F 2 1 ˙ = αβ ˙ α β = self-dual F : ˙ αβ ) cohomology class ˙ β α ˙ α F ( ˜ can be translated into an object with two spinor indices, F a = A , or it must be the other way around. Anything which is skew in two un- ˙ β ˙ β ˙ α ˜ F ↔ ˙ α , for spin two the linearized metric a fields in terms of these underlying structures. A s Now that we have explored the basic geometry of twistor theory, it is natural to ask: what In physics, we often deal with free fields. For instance, if we want to compute a scattering Of course, for integer spins greater than zero this is not an invariant way of thinking about To illustrate how this works, let’s start with spin one. The usual 2-spinor yoga tells us that the which will be referredstrength, respectively. to as the symmetric under can write this decomposition as dotted spinor indices must be proportional to just the spinor version ofthat the such usual an anti-symmetry anti-symmetry can arise: either a contribution to the metric these are linearizedgauge diffeomorphisms. transformations are The the familiar objects linearized which curvature are tensors invariant associated under with the spin- By definition, this field strength is anti-symmetric under the exchange of It’s easy to see that the contracted pieces of remaining free spinor indices, so we can define the quantities is it good for?providing solutions In to this free lectureor field we half-integer equations. will spin explore in As one four-dimensionalpiece we flat of of will space-time geometric the see, data can oldest called every be applications a massless represented of free on twistor twistor field space theory: of by integer a amplitude in some quantum fieldto theory, be the free asymptotic fields; states the instates. LSZ the We reduction usually scattering formula think process imposes of are the suchmassless taken free free free fields; equations fields for in of spin terms motion zero of on this gauge is the potentials. just external a Let’s massless focus scalar on the case of free fields: different potentials canformations. describe For the the same Maxwell physical field, field these if are they the differ usual by transformations gauge trans- Maxwell gauge potential Twistor theory 3. The Penrose Transform differential equations. 3.1 Zero-rest-mass fields field four-dimensions, certain underlying structuresworking of in these the invariant 2-spinor objects formalism.spin- become This manifest enables when us to write the free field equations for massless strength is therefore PoS(Modave2017)003 ). 3.8 (3.9) (3.5) (3.6) (3.7) (3.4) (3.8) (3.10) (3.11) Tim Adamo 0. With this = . In Euclidean 1. ab − αβ F F abcd ε . A Maxwell field which , ˙ δ ˙ β helicity ε has eigenvalue ˙ γ . ˙ α . αβ γδ ε F F αβ βγ ˙ , , δ F ε ˙ γ 0 0 ˙ β ε ˙ ): α αδ = = ε , . . − ε ˙ 0 0 0 δ 3.9 + ˙ αβ αβ γ − ˙ ˙ ˜ F F β γ = = = F ˙ ˙ ˙ ˙ α β α α ˙ ˙ β β ˜ β β F ε 21 γδ ∂ ∂ ˙ ˙ α α αβ ˙ ˜ ˜ δ ε F F F ˙ + − α αβ ˙ ˙ ˙ α α α ˙ ˙ β β β ε = ε β β ∂ ∂ ∂ ˙ ˙ α α ˜ ˜ = ab , what is the condition for this to describe a SD (positive F F βδ ˙ F ˙ ˙ ˙ β ε α α β β β ˙ α ∂ ∂ ˜ ˙ F αγ αβ abcd ε α ε F 1 2 ↔ abcd describes a ASD (negative helicity) Maxwell field provided that ε 0) is identified with the positive helicity polarization, while a purely ASD 1 under the duality operation, while αβ F + = αβ F 0) field is identified with the negative helicity polarization. = ˙ has eigenvalue β 0. ˙ α ˙ β ˜ F ˙ This means that the SD and ASD parts of the field strength can be considered separately, each Working with this SD/ASD (or positive/negative helicity) decomposition of the field strength Written in terms of the SD/ASD decomposition, the Maxwell equations and Bianchi identity With this new notation, the field strength is α = ˜ F ˙ β ˙ α ˜ defining a consistent on-shell sector. Theseexpect in are four-dimensions, precisely often the referred two to on-shell as photon positive or polarizations negative we so this equation is automaticallyF satisfied. A similar argument works for the purely ASD sector, constraint, the remaining components of the Maxwell equation and Bianchi identity are equivalent: is purely SD (i.e., means that we can phrasecomponents. the Given free-field some equations symmetric ofhelicity) motion Maxwell purely field? in The answer terms is of provided the by field ( strength Similarly, a symmetric (i.e., and a straightforward calculation shows that So These two equations allow us to seeto that the purely equations SD of or motion. ASD Indeed, Maxwell a fields purely are consistent SD solutions gauge field is characterized by for the field strength are Twistor theory It is easy to seetrivial portions why of we the have field chosen strength.dard the Recall notation that names by we ‘self-dual’ can contracting and always withsignature, form ‘anti-self-dual’ the the the for Levi-Civita 4-dimensional dual symbol the field Levi-Civita is strength two symbol: translated in non- into stan- 2-spinors as: respectively. Recall that the Bianchi identity is non-dynamical: any field strength obeys ( PoS(Modave2017)003 0 = de (3.13) (3.15) (3.14) (3.12) ] bc R a ˙ Tim Adamo [ δ ˙ β ∇ ε ) which obeys ˙ γ h ˙ α ε γδ ε ˙ αβ β ˙ ε α γδ + ˙ δ Φ , . ˙ γ ˙ , δ 0 0 ε ˙ γ , 0 ˙ . β ε = = 0 ˙ α 0 | | = ε h h 0) sectors are consistent, subject to | | αβ = 2 2 = ε ˙ Φ α α βδ = ˙ the Ricci scalar. The vacuum Einstein α ε ··· ··· + 1 1 e α ˙ Ψ ˙ δ R α α αβγδ ˙ ∂ αγ γ αβγδ ˜ φ φ ˙ ε α Ψ 1 ˙ Ψ β α αβ ˙ ˙ α 1 α β ˙ α β ∂ β α Φ R ∇ 12 ∇ ∂ ∂ = ˙ γδ δ ˙ γ ε + (z.r.m.) equations: they constitute the free field 22 Φ ε ˙ any integer or half-integer) is represented by a field β ˙ 2 α , h − , , ε ˙ 0 | | δ h h ˙ | | γ + 2 2 ˙ β = ˙ (for α α ˙ α ˙ 0) and ASD (i.e., , δ ··· ··· h ˙ γ 1 1 e Ψ ˙ ˙ α α β Φ = αβγδ ˙ ˜ ˙ α β φ φ α Ψ Ψ ∇ e Ψ 0 ˙ γδ α β 0 0 zero-rest-mass γδ ε = ε ; on the support of these equations the Bianchi identity ∇ curvature tensor associated with the field. For the spin-1 case, this is > < h R s αβ h h ε = + 0 gauge field contains SD and ASD components which define consistent 2 z.r.m. equations. ˙ totally symmetric, encoding the SD and ASD portions of the Weyl curva- δ ˙ γ s = ± ˙ 0; these spinors encode the information contained in the totally trace-free β ˙ ˙ δ α ˙ γ αβ > e αβγδ Ψ Φ s Ψ γδ ε encoding the trace-free Ricci curvature; and and αβ ε ˙ ˙ δ δ ˙ ˙ γ γ dotted or un-dotted symmetric spinor indices (depending upon the sign of ˙ β | ↔ ˙ αβ α h 2) can be decomposed as | e Φ Ψ In general, a z.r.m. field of helicity Trace-free curvature tensors can always be represented as 2-forms on space-time: this was This representation associates two totally symmetric spinors (one dotted, one un-dotted) with A similar story holds for any integer or half-integer spin: the (gauge-invariant) curvature tensor = abcd s R From this we see that thethe SD Bianchi identities (i.e., a linear PDE: with 2 is equivalent to respectively. Linearizing these equations by replacingtives the gives covariant the derivatives helicity with partial deriva- equations enforce obvious in the Maxwell field case we covered above. On any 4-dimensional Riemannian manifold ture; portion of the linearized spin the entire field strength, while forcurvature spin-2 tensor it’s into the Weyl SD tensor. andheard This a ASD general more splitting parts sophisticated of geometric is explanation a ait for trace-free this here. special splitting feature before, so of it’s worth four-dimensions. mentioning You may have any field of spin with As desired, this gives astrengths. representation From of now free on, when fields wemind in refer this terms to z.r.m. of free field fields representation. their of linearized a SD given or helicity, we ASD will field implicitly have in associated to the spin- equations for Maxwell fields, formulated in terms of the SD/ASD components of the field strength. subsectors of the equations of motion.to For example, the Riemann curvature tensor (corresponding Twistor theory These equations are the spin-1 PoS(Modave2017)003 ) and (3.21) (3.23) (3.16) (3.20) (3.17) (3.18) (3.19) (3.22) 1 with 3.4 ± , 0, you can α Tim Adamo ··· = β φ βγ , so it is natural ˙ ε β ˙ αγ ˙ ˙ α α ϒ α ε ∂ αβ ε ↔ − ··· − 1, and γ ab − ··· . η α ˙ . β . φ ˙ α id . ε Z ˙ αβ ) ) = 4 ∈ . x ϒ ( 2 γ k M ∗ − 1 (just use the decomposition ( ··· Ω ∀ ( γ β 2 − . = φ ··· . → ˙ Ω s β β ab 0 ˙ β β φ η , ⊕ ˙ , α ∂ = ) ) ) k ε ˙ 4 3 x αα γ 4 yields ( Ω − ϒ ··· 2 M ˙ ˙ M β α ( Ω 23 ( ˙ Ω − α α ), which acts involutively: 2 has conformal weight αβ 2 + ∂ γ ε , φ = Ω β k ˙ ··· Ω → α γ ··· abcd − αβ β αβ α ··· α → ε ε φ ε ab Ω ∂ β 2 φ ˙ ) ) α ˆ ) = η φ k 4 1 − x 4 α ˙ , β ( ∂ Ω M C β M = Ω ( : ( ˆ ∇ 2 = 2 M γ Ω ) in terms of totally symmetric spinors of different chirality. → ˙ Ω αα ··· : ϒ β ) with ∗ αβ φ ε 3.17 1 − 3.21 ), the fact that Ω ˙ α 3.19 α ˆ ∇ Ω = γ is conformally-invariant), and similar arguments work for any other spin. ··· β ab ˆ φ F ˙ α α ˆ For concreteness, consider the negative helicity z.r.m. equation Besides allowing us to work directly with gauge-invariant representations of free fields, the ∇ . It is easy to convince yourself why this is true for 1 , the space of 2-forms has a special property: it is preserved by the Hodge star (in coordinates, Ω 4 − where hatted quantities indicate objects in the conformally re-scaled metric, and This means that any 2-formrespect can to be the decomposed Hodge into star, components which have eigenvalue Using the definitions ( deduce that This decomposition is precisely thespinor decomposition components into above! SD This and is ASD yetmanifest parts another the that advantage decomposition of we ( the worked 2-spinor out formalism: in it allows us to In the 2-spinor language, the complexified metric is represented by Under such a conformal transformation, itΩ turns out that all z.r.m.fact fields that transform with a factor of Contracting both sides of ( z.r.m. equations have anotherconsider interesting a conformal property: re-scaling of they are conformally invariant. To see this, Conformal invariance this is just the duality operator defined by Twistor theory M to declare that each factor transforms with the same weight: PoS(Modave2017)003 , ) 2 s 2 α − ··· ··· ( s wave 1 (3.27) (3.28) (3.24) (3.25) (3.26) 2 α φ − Tim Adamo is a weight f 2 and the + 2.’ The restriction . − corresponds to a line ) s of homogeneity zero. , where C ¯ λ 2 ) , X ¯ M − λ λ , ( conformally-coupled ∈ λ f , ( x ) 2 f , − s ) ··· 2 ¯ α ( − ¯ s λ ) = -form on 2 r , ) , α β 1 symmetric, un-dotted spinor indices. ··· λ , )) λ of projective weight ( s ) = 1 ˙ 2 α 1 ( ··· ¯ λ β . , ¯ 1 − r x 0 0 α s , , z.r.m. equations; this is a totally symmetric CP λ 2 α λ = = s i r λ of projective weight s − ( 2 , λ ( Φ f α β d O -form on twistor space of homogeneity ··· λ PT , ) λ 24 1 ˙ α 6 h R α 1 β , 1 φ PT x 0 + , ˙ ( ( α ( CP 1 Z 1 ¯ f α , 2 ∼ = α 0 ¯ λ X ∂ = d Ω ? -form on ) = X ) ∈ | ¯ ) λ 1 ) , x f , ¯ ( Z λ 2. Thus, we must have 0 s , 2 ( ( ) is satisfied in Minkowski space-time, then it will also be satisfied α Z ¯ α + ( ··· f which obeys s using the incidence relations. We denote such an object as f 1 is a α 1 C 3.20 . In other words, φ f 1 ) = M CP ¯ λ , CP on ∼ = λ ) ( X x f ( s 2 α ··· is the natural holomorphic measure on 1 ) to make sense, the integrand must be a -form on α ) i φ 1 λ , 3.26 d inside twistor space. So if we want to find a twistorial way of encoding the field 0 ( degrees of freedom on twistor space must be removed in some way. One way of doing this 1 λ 1 h 2 the scalar curvature of the conformally re-scaled metric. -form of homogeneity 2 is implemented by CP The z.r.m. equations are a conformally-invariant way of encoding the free field equations. In Consider a negative helicity solution to the spin Furthermore, we need to build an object which has 2 For ( − ) CP R s 0 X ∼ = , 2 1 to − Such an object is naturally provided by a which should be read as: ‘ Excluding the missing ingredients,( the portion of the integrand we have written out so far is a which we restrict to Twistor theory Thus, if the z.r.m. equation ( in any conformally flat space-time. Ations. similar argument In works the for scalar theequation positive case, in helicity it the z.r.m. conformally follows equa- flat that space-time: the massless scalar obeys the the previous lecture, we established that(and conformal invariance only is broken naturally by encoded the in choice twistoris of space then: an can additional we structure use – twistor the theory infinity to twistor). generate A solutions natural to question the z.r.m. equations? for 3.2 The Penrose transform spinor field X the is to integrate them out explicitly. This suggests some sort of twistor space construction of the form: Clearly, such a field is local on space-time, and we know that a point where stands for some other ingredients whichents are is yet tightly to constrained be simply determined. by The requiring form that of the these integral is extra well-defined. ingredi- PoS(Modave2017)003 ) 0. 0. = 3.29 (3.36) (3.35) (3.32) (3.33) (3.30) (3.34) (3.29) (3.31) α = 2. Now λ f α − ¯ ∂ λ s Tim Adamo 2 . − , ˙ , β ¯ e X | ∧ 0 X f

˙ α ˙ ˙ ¯ α α e f ¯ µ ˙ ∂ β ∂ s f ∂ of weight 2 ˙ α α 1 ¯ ∂ λ α ): . . PT ). λ + 0 X ··· ¯ | 0 imposes e ˙ α + 2 ) , ¯ 2.19 X e α Z = | i 3.26 X ( 0 ∧ f ˆ

f λ f 0 ˙ ω λ d ¯ s α s ¯ ∂ f e 2 2 -form on X ˆ 2 α λ α ∂ ) must vanish independently, since these i Z 0 ∂ µ λ λ : 1 f ˙ ˆ λ β f , . 1 ¯ ˙ = e α 0 ··· i ∧ h ˙ ··· α λ ¯ β ∂ ( 0 , 1 h ¯ 1 ∧ ¯ e λ e λ on 0 α α ˙ ˙ d − f α β X λ ¯ λ f ∂ e | ˙ ˙ is a α s α λ 0 2 f ¯ h f ∂ f α + 0 (i.e., does not depend on the complex conjugated i ∧ i ∧ ˙ ¯ α λ 0 ∂ and ¯ 25 = = λ ¯ λ ¯ e ∂ ˙ d d ˙ 0 α 0 s α ··· ¯ ¯ 2 f f e e = λ λ 1 α 0 h h α ∧ ¯ λ ). To check this, we simply compute ∂ -fibre direction of the Euclidean twistor space. So ( = X X λ 0 i ∧ ˙ 1 β Z f Z e ¯ e λ ··· ˙ d 2 β i ∧ 3.25 CP f , this holomorphicity condition can be phrased as: α ) = λ λ ) = λ holomorphic h x x d , we can expand it in the basis ( + ( ( PT s s is 0 λ = 2 2 ¯ i ∧ h e α f α PT 0 X λ ω ··· f ··· Z d 1 1 ), it is clear that only the first of these terms appears in the restriction α α λ = φ φ . h 1 ˙ s α X 2 ¯ ∂ 3.29 Z α ˙ α CP ··· -form on ¯ 1 = e ) ), we can now compute the derivative: α s 1 2 φ + , α ˙ 0 0 α -forms on twistor space. Thus, the condition ¯ ··· ( 1 2.18 ∂ 1 ) α 0 α 2 ¯ e ∂ , φ does not point along the is a 0 ˙ α ˙ ( β f 1 = e α f ∂ 0, then the terms proportional to ¯ ¯ ∂ = At this point, we have only used the fact that In the above argument, we have been a bit fast-and-loose, failing to specify what exactly we Since Putting all of the ingredients together, we are left with a proposal for the negative helicity, spin f , since ¯ ¯ ∂ X z.r.m. field of the form: | f are distinct If Using the basis ( where using the incidence relations. Clearly,So the if first our term twistor representative in the parentheses vanishes, since can be written as is the volume form on twistor variables), then it seems thatterms our of integral the formula complex does structure on indeed obey the z.r.m. equation. In mean by the anti-holomorphic dependence onto twistor be space. precise about As this, we we learned must in specifylet’s the some go reality previous through conditions lecture, the on twistor calculation space. again,on For now twistor concreteness, with space. the explicit choice of Euclidean reality conditions This results in a well-defined space-time fieldthis of field the satisfies appropriate the helicity, z.r.m. but equation it’s not ( at all clear that In the integral formula ( we can consider the action of the complex structure s Twistor theory leaving us with precisely the sort of object we need to complete ( PoS(Modave2017)003 0 = . In ]. . You f (3.41) (3.39) (3.40) (3.37) (3.38) ¯ 17 )) ∂ 2 − Tim Adamo 0.) So sure s , 2 = 0 − ( ω 0 = ¯ . O ∂ actually leads to a 0. By an argument ,

X which can be written f | , g 2 which obey = ˙ f α ¯ PT ∂ , )) f ( f cohomology groups − 1 2 X s , ¯ s 6= | ∂ 2 0 2 f α − can be specified by twistor f | H . (You might worry that the λ − h h | C 1 2 2 ( α (which is suitably smooth) can ··· : M λ 2 , CP O C α 0 and , X | ··· ∼ = M f = 1 ω λ PT α X f ( λ of weight 1 ¯ 0 i ∧ ∂ , ¯ 0 ∂ λ i ∧ X d H PT Z λ 0, it follows that any λ d ∼ = h = = obeys the z.r.m. equation. X } λ 2 h s X Z C 2 | ¯ X ∂ Penrose transform ˙ α z.r.m. field on α 26 will automatically obey Z M f ··· which obey ) = 1 0 )) x -forms on α ¯ ∂ ( ) 2 )) ) = φ s 0). Thus, it seems that the space of representatives on 1 2 2 x φ every , − ( α 0 contains some trivial solutions to the z.r.m. equations | = 0 − h s λ | ( s 2 s 2 2 = α 2 α ··· − f ··· 0 ( 2 ··· − 1 1 ¯ ( α ∂ α O α . An element of a cohomology group is often referred to as a = φ , O ), we find that g φ z.r.m. fields on h , ¯ rather than the entire projective space: these cohomology groups are empty for ω λ ∂ h PT 3 X ( PT = 0 imposes that 3.33 Z 0 ( CP 0 f 1 , = Ω = 0 s < ) is missing some terms, but you can easily check that f 2 . ∈ Ω ) is the (Dolbeault) cohomology group denoted ¯ α f h ∂ helicity g ··· ∈ { 1 s which obey 3 3.37 f α f 3.38 φ ˙ α 1 α ∂ , for some g ¯ ∂ to be an open subset of any integer or half-integer. Given a cohomology class on twistor space, the corresponding = The result is an isomorphism, known as the For those who have been exposed to cohomology before, this is another place where we see that it was crucial So we have established that negative helicity z.r.m. fields on Feeding this back into ( The space of 3 h ! Physically, this is the statement that to have interesting solutions to the wave equation we need a non-compact f PT 3 for z.r.m. field on space-time can be constructedcase by we means have of already an seen; integral formula. the The other negative two helicity cases are similar: CP space-time. for should read this notation as:and the cannot set be of written‘cohomology as class.’ cohomology classes. It is straightforward tohelicity do as something well similar (we for will z.r.m.this write fields relationship the of also corresponding non-negative goes integral the formulae other momentarily).be way: represented It by turns a out twistor that cohomologydirection class is of a a bit more certain technical, weight/homogeneity. but Proving if this you other are interested then you can look at the proof in [ Twistor theory on the components of enough, the condition identical to the onevanishing used space-time above, field you (i.e., can convince yourself that any such twistor space we want to consider is actually which vanishes as a totalsecond derivative equality on in the ( Riemann sphere which we would like to getas rid of, though. Since Such spaces of differential forms(which are you well-studied may objects have in encounteredparticular, the differential in set and other ( algebraic physics geometry contexts), known as PoS(Modave2017)003 , . = i C X ˆ λ | M d a (3.45) (3.43) (3.44) (3.42) X ˆ λ | h ¯ ∂ 0), there > Tim Adamo h ), we know that the . X 3.39

f : h must trivially obey X 2 ˙ α X associated with a z.r.m. field. . But such an object is clearly ∂ | 1 ]. a ∂ µ . corresponding to a point in 18 ) CP there is not, in general, a canonical ··· O , 1 C PT -form, we need an object which looks 1 ˙ α ) potential ∂ , M 1 ⊂ , ) CP . ∂ µ 0 ( ˆ X ) λ ( 1 , , , /0using Serre duality or the Riemann-Roch O 0 i λ i ∧ i , , H λ ˆ λ x /0,it follows that ab d , such an object must be proportional to ( ) = b PT ∼ = 1 h ( ih λ ) O 27 is actually empty. 1 ih h X ˆ , , ) = λ | 0 ) ˆ X O CP 1 λ d ¯ ∂ O Z , a H O , ˆ λ h , X 1 h CP = 1 ( ∈ ( 1 ) = 1 X , a , | x CP 0 0 CP a ( ( ( h 1 H H , 2 , it is also a cohomology class on 1 4 , ˙ 0 -form on the Riemann sphere which is homogeneous of weight α 0 ∈ ) ]. 0 integral formulae obey the z.r.m. equations by using holomor- ··· H 1 PT 1 H X , ˙ | 11 α ≥ 0 ˜ a φ ( h 1 case; we want to find a way to construct a space-time Maxwell and ) to any twistor representative, which does not change the cohomology . So to form a homogeneous O g = + , and so cannot be a cohomology class. (If you know some algebraic α , ¯ 0 ∂ 1 ˆ 1 λ h > CP CP h ( 2 in 1 , + 0 : this will be a H ) ∈ O , X from a Penrose transform representative on twistor space. By ( 1 | the homogeneous coordinates of some fixed points on a ) x α is a cohomology class on CP ( b ( a a , 1 , A α So if This is an interesting and useful procedure, which we do not have the time to cover here, but Woodhouse’s paper Let’s consider the There’s a fairly intuitive way to see why this is the case. First, let’s try to construct an element We have already argued that z.r.m. fields are natural objects to study when talking about mass- 0 4 a Ω 0: zero. With the standard complex structurewhich has on weight like theorem.) for geometry, you can easily prove that is readable and you shouldthis be point! able to understand the necessary sections with the material covered in the lectures up to Consider the restriction of thisSince representative to a line not holomorphic on of field twistor representative for a positive helicity Maxwell field is a cohomology class However, the cohomology group less free fields: they arein gauge invariant four-dimensions and in manifest terms the of positive/negativepose, helicity the however, decomposition that SD/ASD you decomposition really of wanted to linearized recover curvature the tensors. gauge Sup- The Sparling transform Twistor theory Is there a way to dois a this nice directly construction from which the allows twistor us data? to do In this the due positive to helicity Sparling case [ ( You can readily check that the phicity and the incidence relations. Givenway a to z.r.m. reconstruct the field twistor on representative; this‘gauge is transformations’ partially due to the large redundancy of adding class. In Euclidean signature, there isz.r.m. a fields canonical due way to to construct Woodhouse [ twistor representatives for some PoS(Modave2017)003 1 in + (3.47) (3.46) (3.48) . This ]); you ˙ α ˜ p 19 is defined α Tim Adamo an on-shell a p a k ↔ a k , and consider , for x C · k , i ) i p . λ h ( ¯ δ z 1 1 (or its complex conjugate). This . Furthermore, since 0. It looks like the RHS of this − . . ˆ α , . λ h ¯ ∂ 0 2 , ) ) λ = i ˙ x -form which has support only where 0 α λ = ( π ) i 6 ( α 1 α 1 i i ˙ f 2 2 α λ x X , λ | α pa 0 a a p a h = A ( h ˙ h ¯ α ∂ ) = which is holomorphic and of weight α . In particular, show that z α 1 ) , ) ( λ ∂ z z 1 0 f ( ˆ λ α = ^ f , ) α λ ) = z ¯ z ) = λ ( α ∂ ( 28 ˆ ∂ ¯ = λ δ λ : , enclosing the origin. z ) = s ) ∧ d¯ λ h Γ α , z and i ˙ − α x d λ π ) which doesn’t appear on the LHS...why is this not a be the usual complex coordinate on 1 α x ( α ( D 2 ∂ α 2 h z 0), we know that we can represent this p Z ¯ ˙ a α ( α δ 2 = = α λ : ¯ δ ( ∂ 2 ) α k 1 : Let X z | ( − s λ ¯ ¯ h ∂ δ d 2 s ∗ only through the combination C ˙ α Z is a function of α x h ˙ should be interpreted as a sort of α α ) are constant 2-spinors which obey which is homogeneous of degree zero in is a disc with boundary ∂ α h α α λ be the natural extension of the holomorphic delta function to 2-spinor quantities: C ( a λ ) 2 , ⊂ ¯ α δ α λ p is precisely the Maxwell potential we set out to find. A similar story works for other D ( 2 ) ¯ momentum eigenstates x δ ( ˙ α α , it can depend on Let where where equations depends on aproblem? spinor ( Holomorphic delta functions Show that this object actsgrating like against a a holomorphic holomorphic test analogue function, of the Dirac delta function by inte- Clearly, its argument vanishes. Show that A When we do Feynman diagram calculations in perturbative QFT, we usually take the wave- 1. PT . It is clear (by an extension of Liouville’s theorem), that any such function must take the form: This positive helicity fields of higher spin (e.g., the linearized gravity case is worked out in [ λ exercise is concerned with how totransform. construct twistor representatives for such states via the Penrose on This means that may find it instructive to try this for yourself. Exercise: momentum. In the massless case ( functions of our external states to be modeled on exponential plane waves, e Twistor theory for some function is just the usual statement of the incidence relations, and implies PoS(Modave2017)003 . ) x ( ˙ α (3.49) α , A x to obtain · k ] i Tim Adamo 1 [ e gives rise to f h 2 ] ˙ h α [ ˜ p f . (Hint: treat the )) . ··· 2 x · 1 ˙ k α i − ˜ , p e h ˙ ]) 2 α i ˜ ˜ ( p p ) 0 a p O µ α [ h , ), show that a s > i h ( PT ( ( ) = 3.42 1 , x 0 ( exp ˙ α ) H )–( α α , ) and dropping any terms which vanish ∈ A i x λ · ] 1 with respect to the projective scale on s k h i [ 3.40 cb − e f − ih α , p ) x ad · correspond to gauge transformations of ( 0 k h 2 i α ¯ e δ 29 = + a ). Perform the Sparling transform on i 1 i h p ( − s h i d bd 2 λ 3.49 λ s ih ih ∗ a h C , ac Z a p x h · 1 in ( h k i = e ) = | i h = + | ) = Z 2 ( ˆ h λ ] cd α h , p [ : Consider ih f λ , ··· x ab . 1 : Let ( h : Using the integral formulae ( ˙ α α h ˜ p p α are constant 2-spinors. Show that p ) as a scaling parameter with weight ˙ α 0 ˜ = s p , ˙ < α α α h p k ( where Sparling transform Show that different choices of the spinor on the support of the holomorphic delta functions. You should find the following momentum eigenstate z.r.m. fields on space-time: a space-time gauge field. YouSchouten will identity need ( to manipulate expressions along the way, using the Penrose transform twistor space, or use the result you proved above.) parameter Twistor representatives where 4. 3. 2. Twistor theory PoS(Modave2017)003 (4.3) (4.2) (4.1) gauge , which ) x ( Tim Adamo a A . So (local) gauge- 1 – to include the gauge − g a ∂ , ab ] matrices. We know that the F g , Φ g ] , , b a ) is valued in the fundamental and → A . A x ) , , is often referred to as the ) ( x . We denote the (complexified) Lie a ab a 1 g ( + [ F A − G , D f g Φ ) a + [ PT x ∂ a ( ( 1 A , g = 0 b a ∂ ∂ Ω Φ a − − ∈ b D ) a x A , then the gauge connection acts as 30 ( a 1 G ∂ − , g , a f ] = a b A a ) D A + x , ( ¯ by an adjoint-valued connection. As we have learned, the a ∂ + g f D , which we usually talk about in terms of a 1-form = a → PT ∂ ¯ . The physics of the gauge field arises by modifying the natural D = [ a g = A ab f F a D . The resulting derivative operator, gauge field a A . It is straightforward to see that the field strength, + g a ∂ itself is not invariant; the physical information encoded in a gauge theory should be = a a A is Lie bracket, which is simply the commutator between D . ] valued in · ) , → · x [ a ( is valued in the adjoint representation of ∂ g By analogy, we should look to formulate gauge theory in twistor space by deforming the The Penrose transform gives us a way to study massless free fields in Minkowski space in What is the natural language to talk about gauge theory on twistor space? To answer this The natural objects on which the gauge connection acts are functions or tensors which are also ) x ( . Therefore, the twistor space version of a gauge connection is a deformed complex structure, ¯ which looks locally like: natural differential structure on ∂ transforms covariantly under theseinvariant quantities gauge can be transformations: formed by taking traces of combinations of the field strength. natural differential structure on twistor space is the complex structure, in the form of the operator terms of twistor data.need Of to course, be able to to studya describe interesting familiar non-linear, physical interacting or field problems interacting, theory withtheory. field with twistor We theories. obvious theory will applicability we see In tosubsector that this physics: of twistor lecture, theory Yang-Mills we non-abelian theory, provides consider Yang-Mills which atheory which natural can is description be perturbatively equivalent of used to a the to non-linear, space-time build formulation. integrable up a4.1 twistor The description Ward correspondence of the full question, it is instructive to first think aboutis the done natural by language introducing for a gauge fields in space-time. This Twistor theory 4. Gauge Theory in Twistor Space takes values in the adjoint representation of the gauge group, field algebra of the gauge groupderivative by structure on space-time – namely, the coordinate derivative connection valued in representations of the gauge group. In particular, if with invariant under gauge transformations.functions: These are just shifts of the gauge field by adjoint-valued Φ where gauge field PoS(Modave2017)003 - - 1 E ) ∼ = N 0). 1 ) , CP (4.5) (4.4) (4.6) E 0 ( ( -form, ∼ = ) = ) X 2 | X , 0 E itself. The Tim Adamo ( ( 1 E c on space-time, gauge- is just a copy of the is the partial connection ab F . Equivalently, this means vector bundle over twistor PT PT N , ∈ is restricted to a line 0; this is the condition for the ) -form taking values in the ad- ) g Z E 1 = , , ) . 0 2 ) , ( PT 0 g ( ( is a rank 0 , holomorphic F Ω on twistor space. This is a . We can’t impose the usual Yang-Mills , is a deformation of the standard complex ) PT PT ˙ 2 ¯ ∈ ( α , D ¯ 2 0 e , γ ( → 0 ˙ PT α , to be F a Ω ¯ E D complex matrices, so it follows that End + ∈ 0 entails, it’s helpful to pick a reality structure to , N 0 ] 31 ¯ e ) ¯ = D (or in the language of Chern classes, × 0 Z ) , ( a 2 ¯ , N naturally encodes the gauge transformations associated 1 X ; its fibre over a point D 0 is called a covariant almost complex structure, a − ( = E × γ ¯ F a PT = [ D ¯ N D ) , as expected. ) 2 × 1 C , . We will demand that when 0 Z − N ( N are just ( γ ∼ = C γ ) F C . As we will see later, other gauge groups arise by endowing 2 E X , ) | 0 ∼ = ∼ = → ( C E , Z ¯ F E D | γ N is best thought of as a connection on the vector bundle vector bundle ( E ¯ → N D GL ) 2 ). This means that we can expand the twistor gauge field as , 0 = ( F G 2.19 , equipped with partial connection vector bundle, we can ask what sort of field equations can be imposed on the partial E N ) and ( 0 and thus defines an integrable covariant complex structure. . Thus, the rank = ) 2.18 2 C ¯ , Having established that the natural analogue of a gauge field on A proper geometric treatment of partial connections entails the use of fibre bundles. If you are Its easy to see that To see precisely what the equation Just as gauge covariant information is packaged in the field strength D N on a rank ( ¯ with additional structures. do our calculations in. Asbases ( usual, we’ll take the Euclidean reality structure, where we can use the already familiar with these concepts, thenbe. it’s If probably clear not, to then you don’tfrom what worry: the the perspective even general of the setup twistor simplest should theory. possible We say example that captures all of essential features space if it looks locally like endomorphisms of the fibres of gl with gauge group D connection. Any reasonable field equationphrased should be in gauge terms invariant, of which the means anti-holomorphic that curvature it must be vector bundle equations, because the partial connection onlyspace. points Instead, in the we anti-holomorphic can directions simply of consider twistor the field equation you can show that dimensional vector space: in twistor space, it isThis latter trivial: requirement will means thatto information local encoded information in on this space-time. vector bundle can be translated that Under a gauge transformation connection, or a partial connection.fact These that names the can natural be notion used of interchangeably; a they gauge all connection reflect on the referred to as the ‘anti-holomorphic curvature’ of the partial connection: structure. Twistor theory In other words, the naturaljoint of gauge the field gauge on group. twistor The space operator is a covariant information is packaged in the curvature of PoS(Modave2017)003 . 2 + (4.8) (4.7) (4.9) . We (4.12) (4.10) (4.11) (4.13) ) . Now, 0 ˆ λ ¯ ∂ , self-dual 0 λ ¯ with gauge Tim Adamo e is ( 4 (i.e., a partial 4 = . R . R ˙ , X β | ¯ e 0 PT ¯ 0 (since there are ∂ ∧ = = ˙ α ¯ e 0 αβ a F . X i | ) can be used to impose ˙ instantons β ˙ β β ¯ , and the anti-holomorphic ¯ ∂ e to be a harmonic function λ a ˙ α 4.5 , , α ¯ ∧ 0 e ) ˙ α ˙ , homogeneous of weight a ˙ λ α x α ) tells us that the holomorphic a . ¯ , ˙ ( a e β h ˙ ) 0 ˙ α . α PT 0 = vector bundle on α ε ¯ + i e )) 4.11 ˙ a A ˙ β 0 β ] = N = C a a ˙ α a β , ( , ˙ α are given by a λ ¯ ˙ ¯ N α i D ∂ , 0 ( ˙ a γ β ˙ α a h = gl A is the adjoint operator of a ) = + , , 0 ∗ + ˆ . ˙ λ X ¯ is holomorphic as a function of 1 γ e ˙ α ¯ , 4 ˙ + [ 0 – that is, if the gauge field on ∂ β α ¯ ) e ] ∧ λ a R ˙ A β ˆ CP = , λ ˙ ˙ ∧ α α h ( a , x ¯ ¯ e ∂ 0 1 ˙ 32 ( α ¯ are known as Yang-Mills λ e [ + αβ ˙ α , ¯ ] H ∂ 4 F ˙ γ β x a 0 ] , and as such it must obey + ( 0 R 1 a ∈ A ˙ ˙ ˙ α γ a α 0) leads to a self-dual Yang-Mills field on , 0 ¯ ) will impose some further conditions on this space-time e a ˙ ¯ , α α e CP ˙ = ∂ α a , 0 ⇒ ∧ ) a a 0 0 [ 2 0 is therefore equivalent to two equations on the remaining , ¯ 4.11 β e + [ 0 . So the first equation in ( = ( = − ˙ 0 from the component λ α 0 ) = 0 a ˙ F X a ) α α ) | C ˙ 2 a 0 α = 2 , a are adjoint-valued functions on , λ ¯ , a ˙ ¯ ∂ 0 α ∂ 0 0 ˙ ˙ ( -form on ¯ α ¯ β N ( } ∂ ∂ ) ˙ ( a ˙ α = F α F in twistor space, where 1 0 ε − ) a , ¯ gl ∂ ) into this second equation, we find that 1 2 ˙ , , 0 α obeying 0 0 encodes a gauge field on ( a ( ] = a 0 ). So this choice of gauge actually forces CP ˙ ¯ β F 1 { ∂ 4.12 a PT PT 0 as a gauge condition. ∼ = can be removed by a gauge transformation. , CP ˙ 0. The Hodge theorem tells us that every harmonic function corresponds to a 0 = α X → = a a ) 0 2 = E : , . These SD gauge fields on a 0 a ) 0 + [ is valued in ( ] a on C ˙ ) F β X , x ¯ | a is the anti-self-dual portion of the field strength of the gauge field. This equation can D ( ¯ N ˙ ∂ α ˙ ( α [ ∗ X 0 on each -forms on ¯ ∂ α | αβ ) ¯ A F ∂ 2 = 1 respectively. We can then compute , : 0 There is another nice way of seeing this. The gauge freedom ( With this choice, the gauge field on twistor space becomes Clearly, the second equation of ( In summary, we have shown that every holomorphic rank 0 + a ( X is the component of a ∗ X | 0 ¯ where which means that encountered this situation in the previousville’s lecture theorem, in it the follows context that of the Sparling transform; by Liou- Note that all contributions to Imposing the field equation The first of these equations tells us that components of curvature is given by gauge field. Plugging ( As we already saw in theconsistently previous set lecture, this cohomology group is actually empty. Thus, we can partial connection on where connection group GL only be satisfied for non-trivial connections if ∂ Twistor theory where the coefficients and a no on cohomology class, so PoS(Modave2017)003 . ] 1 for 28 ˙ , β CP ˜ (4.14) (4.17) (4.16) (4.15) w 27 × β N λ -plane we C Tim Adamo ; recall that α = C b ∼ = . Consider the w M X α | , for complexified λ ˙ α ], Hitchin systems E ˜ v 0. Does this define α 24 PT , = λ . 23 = αβ -plane in N a F 3 [ α v C satisfying = ∼ = . d ] and was a major influence on in this way. It is easy to see that 0

PT 0 21 = PT = ˙ → β ˙ α ∈ -planes. ˜ , E F α Z plane ab are unimodular, respectively. − F αβ ]. Furthermore, myriad integrable systems to be equipped with a positive real form, , ε α b ˙ | E β β 22 w s ˙ α corresponds to an a ˜ a λ PT F v α D ; this has a ﬁeld strength:

-plane. By definition, λ → αβ PT = C ˙ N 33 ε β α E ∈ ˜ C w M = which is topologically trivial upon restriction to lines ˙ -plane; this is given by α Z to every point plane ˜ v α ab − N F α PT = | C ], it constitutes one of the most important results from the to be trivial. These structures enable the construction of a ab ) 20 F [ E valued in plane ( whose tangent vectors are all proportional to − ) x α C | ( s holomorphic vector bundles M ab F N = gauge field on space-time which is self-dual: Z | vector bundle on twistor space? It is easy to see that this is so; indeed, ) E C vector bundle over N , , so we find N N ˙ β ( -plane is equivalent to the space of constant functions. So to each and rank ˜ w α ], and even the non-linear Schrödinger and Kortweg-de Vries equations [ C , ˙ α Ward correspondence 26 M v are any two tangent vectors to the , b on 25 w ) , instantons are described by requiring a 2 [ C v , ) = The Ward correspondence has been extremely influential in the study of classical integrable Our starting point is a SD gauge field on This establishes a one-to-one correspondence between Yang-Mills instantons with gauge group One can naturally ask if this correspondence works the other way around. That is, suppose This means that the space of covariantly constant functions valued in the fundamental repre- N N ( ( d systems. It led to early constructions of Yang-Mills instantons [ and the determinant line bundleKilling det form and ensure that the transition matrices of in lower dimensions such as the Bogomolny monopole equations in the ADHM construction of all Yang-Mills instantons [ in early years of twistorby theory. imposing further The conditions Ward on correspondenceSU the is holomorphic easily vector bundle extended on to twistor any space. gauge For group example, a holomorphic, rank we can construct the correspondingspace-time holomorphic and bundle impose over reality every conditions point at the of end of this construction. by virtue of the SD condition. Every point In particular, we can associate a copy of Known as the GL in twistor space. Furthermore, since thisbundle is is a holomorphic. totally holomorphic construction, the resulting vector we are given a GL Twistor theory this is a totally null 2-planerestriction of in the field strength to any such some spinors ˜ In other words, SD gauge fields are flat upon restriction to this leads to a rank sentation on the can assign a vector space where PoS(Modave2017)003 0. = (4.18) αβγδ non-linear solution to Tim Adamo Ψ (an in-house jour- , perturbative ab F ab replaced by the tangent bundle Twistor Newsletter . More recently Penrose proposed an- F tr PT x ]. Although we won’t have time to 4 on d 32 E Z 2 g 1 4 − 34 ) = F 5 ∧ ∗ F ( tr Z 2 ]. This gives a one-to-one correspondence between complex de- g 1 31 2 , − 30 , ] = A 29 [ ], called ‘palatial twistor theory,’ but I think it’s still unclear whether this actually solves the S 33 http://people.maths.ox.ac.uk/lmason/Tn/ construction [ On a rainy day, you can amuse yourself by looking through the archives of Although the instanton sector is important, it is a long way from the full interacting Yang- Trying to find an answer to this question was one of the major problems for twistor theory The standard Yang-Mills action in flat space is given by You might worry that this is the end of the story, but it turns out that a There is also a gravitational analogue of the Ward correspondence, known as the itself. 5 PT googly problem (and if so, in a useful way). nal published by twistorattempted theorists in at the Oxford past: fromother 1976-2000) potential to solution get [ a feel for the sort of solutions which have been Mills theory. Indeed as aintegrable QFT, (indeed, self-dual Yang-Mills the theory Ward isn’tThis correspondence very last interesting: demonstrates fact it this), can is non-unitary beonly classically seen non-vanishing and amplitudes by ‘almost’ are looking at free. at tree-levelexternal (for the gluons) one perturbative and negative scattering helicity at amplitudes and one-loopdescription two of (for of positive the full helicity all theory: Yang-Mills positive theory? helicity the external gluons). Can weduring get the a 1980s, twistor and becamekind known of as ball the which ‘googly cantheory problem,’ be is a bowled trying moniker in to derived cricket. from findthere a a The is certain twistor essence still description of no of thehard-core (fully general googly of non-linear) Yang-Mills twistor problem field solution theorists for on configurations. to Yang-Mills the the subject. To googly date, problem, despite decades of work by a graviton discuss the non-linear graviton construction inWard correspondence these with lectures, the you holomorphic can vector bundle intuitivelyT imagine it as the 4.2 Perturbative expansion around the self-dual sector the googly problem can be foundfrom which the is perspective good of enough perturbative forof QFT. computing As gauge many we quantities theory will of see, in interest amenable this terms to provides twistor of an theory. alternative a description perturbative expansion around the SD sector, which is naturally formations of twistor space and four-dimensional(holomorphic) complex metric. space-times with By a this, self-dual wedefines, conformal mean up that to the complex conformal structure equivalence, of a the space-time deformed metric twistor whose space Weyl tensor obeys Twistor theory can be viewed as symmetry reductionsvia of the the Ward instanton correspondence. equations which have twistor constructions The conformal class canthe also twistor be space fixed (namely, to a a ‘weighted SD contact Einstein structure’) metric [ by including some extra data on PoS(Modave2017)003 ) 4.23 (4.20) (4.19) (4.22) (4.23) (4.21) ]. From 34 Tim Adamo -term to the θ -term: θ )! . ). The equations ( times the . 2 g 1 . 4.18 αβ 8 . 4.21 G αβ action, presents perturbative ˙ F β αβ αβ , ˙ α 0. F ˜ 0 G αβ F ) – which is perturbatively equiva- F = ˙ β αβ = tr ˙ F α ˜ x tr F αβ 4 αβ 4.22 − x F d 4 ˙ + β G d ˙ ˙ α α Z ˜ αβ α F Z 2 4 ˙ F D g β 2 Chalmers-Siegel ˙ returns the action ( 1 α g ˜ + F αβ − F αβ 35 tr , G tr x αβ ) = ) is just the Yang-Mills action written in terms of the 4 be symmetric in its spinor indices and valued in the x ) to the Yang-Mills action, and the result will still be αβ G d 4 F ) d G x ∧ αβ Z ( 2 4.19 2 4.20 F Z 4 g F ( 2 αβ − g tr tr 1 G ) = x 2 = 4 Z F − d 2 ∧ αβ g 1 Z F 8 F ] = ). Let -term affects non-perturbative features of the gauge theory, it does ( A θ [ tr ] = ] + S G 4.21 A Z , [ , which itself acts as a covariant z.r.m. field on-shell. When the coupling S A [ αβ S G is vanishing, we recover the SD field equations: g is the dimensionless coupling constant. Expanding the field strength into its self-dual and g ) can be lifted to twistor space. This means that advantage of working with the action ( This new formulation, often referred to as the Now, recall that the Yang-Mills action can be modified by the addition of the What have we gained by doing this? The answer is easier to see by introducing a Lagrange 4.22 While the presence of the not alter the perturbative physicsfree to in add flat or space-time subtract since any it multiple of is ( a topological term. Thus, we are constant Yang-Mills action; this results in: are telling us somethingfield interesting is in encoded their by own right, though. The ASD portion oflent the to gauge the Yang-Mills action –tively is expanding that around the the SD coupling sector constant of acts thetheory theory. as in In a Minkowski other small words, space parameter we admits have for shownsomething a perturba- that which perturbative Yang-Mills is expansion not around at the all SD obvious (or from instanton) the sector usual – Yang-Mills action ( the perspective of twistor theory,the this googly is problem. just The Ward what correspondence describes wetransform the can were non-linear be hoping SD used sector, to for: and describe the the a( Penrose ASD perturbative perturbations. solution As to we will see, this means that the action So far we haven’t done anything fancy: ( The field equations of this action are: from which it is easy to see that integrating out Yang-Mills theory in terms of ASD fluctuations around a non-linear SD background [ spinor decomposition of the field strength. perturbatively equivalent to Yang-Mills theory. In particular, let us add adjoint of the gauge group, and consider the action: multiplier to re-express ( Twistor theory where anti-self-dual parts, we find that So this simplified action,perturbatively equivalent which to depends Yang-Mills theory. only on the ASD field strength of the gauge field, is PoS(Modave2017)003 , ) 0, 1 ( = U 2 (4.31) (4.32) (4.29) (4.30) (4.24) (4.26) (4.27) (4.25) (4.28) ¯ version D = G Tim Adamo . This means PT 4 given by ) in twistor space. + 4.22 . , 0 0 is described on twistor 0 by using a Lagrange multi- = , = = a . covariant, non-abelian αβ PT αβ a ) . ∧ F G , g D ˙ a ∧ α , Z . α ⊗ a + . d 0 0 ) , the Lagrange multiplier must be an ∂ 4: ) a αβ + 4 ∧ g ¯ = − = ∂ a G of projective weight − PT C ¯ ( ∂ ⊗ 3 Z ¯ ) αβ αβ Dg d , O ] = and took the abelian gauge group 4 F ∧ ¯ , G CP X ¯ D ∧ ∂ ˙ | − α g , B ( g tr α ¯ PT D Z x β tr ( O , D d 4 1 λ , , 0 36 ∧ d ¯ A , resulting in a z.r.m. field on space-time: 0 α = [ D g Z Z ) are thus λ ) = H PT Z 3 2 0? On the support of the other field equation, , ( a , 0 1 D ∈ , ( i ∧ = 0 0 ∧ 4.27 ] = g F ABCD λ , which is holomorphic: PT a Ω ε G d a = Z ¯ , Dg ⇒ + ∈ λ = + A h : 0 [ a αβ g ¯ ] = ⇔ ∂ X ¯ F ∂ Z g SD = Z 3 , , homogeneous of weight = S 0 can be enforced dynamically on a D [ 0 ¯ ¯ D Dg PT = ) = SD x = ) ( 2 S , 0 ( αβ αβ F F G with the flat complex structure the twistor gauge connection. ¯ -form on D ) ) g 1 , , 0 is in fact a cohomology class: ( PT g ( 1 , is the canonical holomorphic measure on 0 Z Ω 3 ∈ a What about the second equation, The field equation First, let’s consider how to encode the purely SD sector of the action ( Now, if we replaced then we could apply the Penrose transform to so the partial connection definesthat an on-shell, integrable (covariant) complex structure on So it seems that we getof the the correct Penrose twistor transform holds. space field equation if a The field equations of the twistor action ( the first of which is precisely the SD equation. In terms of our new perturbative expansion,by this the is action the zero-coupling limit, described on space-time with field equations with plier. Consider the action: where D Twistor theory 4.3 The twistor action By the Ward Correspondence, we know that the field equation adjoint-valued space by a partial connection, In order for this action to make sense as an integral over PoS(Modave2017)003 a for ) (4.33) (4.34) (4.35) (4.36) λ , x compare . ( γ Tim Adamo ) 2 λ acts on a rank , followed by a , . Intuitively, it’s to the case of a ¯ x D a priori ( X g PT γ 2 X in | g X ) 2 . λ ) , line λ x is a covariant z.r.m. field on , ( x in terms of 1 ( − αβ γ γ same αβ G X ) | 1 , for which the partial connection is G , . g λ ¯ ∂ ) X , | x λ ¯ ∂ ( , αβ γ x ( 1 G 1 X ) = | − g αβ λ γ . This means that we cannot , ) G 1 β 1 x looks like ( λ λ ¯ 1 , is topologically trivial. However, it need not be D tr α the partial connection over each 37 CP − x x λ ( γ X 4 | 1 ∼ = d X − ) perturbatively – that is, around ‘small’ configurations | E i ∧ ), these ASD interactions on space-time are generated X γ ¯ i Z D λ 2 ) d tr . λ 4.27 λ 4.22 γ λ d ] = , × trivializes h 2 x G X ( [ λ Z I γ provides a twistorial description of the SD sector of Yang-Mills ih ] to single out which lines are integrated over; we will assume that is ‘small’, then 1 g λ , a ) = d a x PT [ ( 1 . If λ SD ; by assumption h αβ g S 2 G i PT 2 is holomorphic with respect to the partial connection on twistor space. and λ g 1 a → λ h E trivial upon restriction to X can be holomorphically trivialized if we can find a gauge transformation 4 d X | Z be this perturbatively constructed trivialization. Then the non-abelian version of the E γ ] = g ): , This establishes that Let This non-local interaction term can be made to look a bit more twistorial by using the Eu- Now, As it turns out, this is the case. We’ll leave part of the construction as an exercise at the end a [ vector bundle I 4.34 that is, a gauge transformation which not hard to convince yourself that suchthat a we trivialization will will always exist be perturbatively. using Indeed, the we action imagine ( With such an integral formula,space-time you provided can show that the resulting theory. That such a description existsplementation is of hardly the surprising; Ward correspondence. it is Whatdescription is nothing remarkable of more is than the that a ASD we dynamical can interactions, im- Mills now theory thereby give a on completing twistorial twistor a space. perturbative description From of ( full Yang- Penrose transform integral formula is given by: automatically holomorphically trivial. integration over the four-dimensional moduli spacechoice of of these reality lines. structure This on latter integrationthe requires Euclidean a reality conditions havethrough been the chosen. holomorphic trivialization Note that this action depends implicitly on clidean reality conditions. With these reality conditions, you can show that the holomorphic volume by non-abelian gauge group is aN bit non-trivial. In particular, theholomorphically partial connection fibres of the bundle holomorphically over two different points on a line in twistor space. of the twistor fields To translate this term into( twistor data, we simply need to apply the non-abelian integral formula This integral is over two copies (labeled by subscripts 1,2) of the of the lecture, but even generalizing the integral formula for Twistor theory which PoS(Modave2017)003 ] ], ]. 39 38 35 (4.37) (4.39) (4.38) Tim Adamo . ]. Unsurpris- 4; in this case ) 4 super-Yang- 2 36 = λ = , x ( ) as N ) – at least in some N γ ) 2 4.36 Z 4.22 ]. The twistor action can ( g 37 ) 2 λ , x , ( ˙ β α 1 . x ] − d g γ , ) ∧ a 1 [ ˙ α λ I , α 2 x x 4 g ( d γ β ) on twistor space is a function of three complex λ ] + 1 α g ) Z , ( λ Z ) product of two copies of twistor space, each with 38 a g ( 4 [ ) γ R 1 i ∧ . This enables us to re-write ( SD λ 1 λ S , d x ( CP λ 1 ] = h − g × , γ = 4 . a ) [ R Z tr α S 3 2 2 , ∼ = ]. D Z 1 3 λ 41 , D PT , β ∧ 2 , 40 1 1 ) is literally equal to the space-time action in a particular choice of gauge Z λ 3 ˙ ]. α D β x 42 4.39 PT 4 R = ( Z × 2 , A 1 PT Z ] = g , a Derivation of alternative Feynman rules for Yang-Mills theory, known as ‘MHV rules’ [ All-loop integrand expressions for the scattering amplitudes of planar which substantially simplify the perturbative expansion of physicaltering observables amplitudes) (such [ as scat- Mills theory [ [ I The upshot of this is that there are gauges available on twistor space which are not readily Having demonstrated that the googly problem can be overcome perturbatively, one could ask A similar construction can be used to build twistor actions for supersymmetric Yang-Mills This leads to a proposal for the full twistor action: • • accessible on space-time. Over the lasta decade, wide this variety basic of fact interesting has results been in exploited to perturbative derive Yang-Mills theory. or A prove few examples include: variables, or six real variables.formation Compare this is to a gauge function theory offreedom on in space-time, only the where four gauge a real transformations gauge available variables. trans- on twistor So space. there is a substantially greater functional whether the twistor action isof actually the good twistor for action. anything. A gauge The transformation answer lies in the gauge invariance ingly, the most elegant of theseall is the for degrees the of maximal freedom amount canalso of be be supersymmetry, understood packaged from into the a (equivalent) singlewhich perspective of twistor may ‘Lorentz field harmonic be [ chiral something superspace’twistor [ you theory. have already encountered without knowing that it was related to (one which reduces theand remaining there gauge is freedom a to one-to-onewith correspondence that the between of values extrema space-time of of gauge the theclassically transformations), two equivalent twistor to and functionals the space-time agreeing space-time actions, action. at extrema. In other words,theories, all the of twistor which admit action a is similar perturbative expansion around the SD sector [ The twistor action ( Twistor theory measure on twistor space is given by: in keeping with the fact that Although it’s clear thatsense this – must by correspond construction, to the the space-time correspondence action between ( the two is in fact extremely precise [ Here, the integral is over thecoordinates fibre-wise (over PoS(Modave2017)003 ]). 55 (4.40) (4.41) (4.42) , 54 , Tim Adamo 53 , , 0 , 0 52 > , < h 51 h , , ) , 50 ) λ , λ x , , ( x ]. γ 0 ( γ X 44 | = X , f h | . 2 , with a holomorphic trivialization ) f ˙ ) α 4 43 ) λ g ··· be an integrable partial connection on R , 1 λ x ˙ ⊗ α , a ( ˜ ) x φ 1 ( 1 2 + − ˙ 1 α ¯ γ ∂ − α − h γ 2 h D | = ˙ h α 2 | 2 ∂ ( ¯ D α O ∂ µ 39 λ , , ··· ··· 0 PT 1 1 ( ˙ α = α 1 , λ | ∂ ¯ h 0 D | 2 ∂ µ H α i ∧ . Show that the integral formulae ··· ) λ ∈ 1 i ∧ d α λ f λ , φ λ d x ˙ α h ( 1 X λ γ ]. α h Z D X 49 Z , ) = x 48 ( ) = , | h x | given by 2 ( 47 h α 1 2 , ˙ ··· α 1 ··· 46 α CP 1 , φ ˙ α ˜ ∼ = φ 45 X the non-abelian Penrose transform Proof of the scattering amplitudes/Wilson loop duality [ Proof of various correspondencestions [ between Wilson loops and limits of correlation func- Working in Euclidean reality conditions, let • • twistor space corresponding to a SD gauge connection on Twistor theory It should be noted thatwere in first the conjectured case of usingAlthough the space-time these latter ‘traditional’ methods methods two or generated examples, substantial holography theseboth evidence strong dualities in (c.f., and favour or weak of [ correspondences the coupling, conjectures the at only known analytic proofsExercise: are provided by the twistor action! over every define space-time fields which satisfy the covariant z.r.m. equations provided that PoS(Modave2017)003 . C ) C M , ]. Yet 2 + 61 , is defined d ( Tim Adamo 35 PT ], and the self- 62 ]). This is due primar- 60 ] for a survey of various attempts in 64 ], it has not yet been possible to extend this 63 40 4? The answer is yes, although the definition is a bit technical: > ]). For massless QFTs, twistor variables have enabled perturbative calculations of d 59 , 58 , 57 Many of these issues are the subject of on-going work, and in a few yearsIt we should may be clear not by think now that the twistor formalism we’ve been using in these lectures relies Let’s start with the obvious question: can we even define a notion of twistor space for Over the last four lectures, we’ve seen that twistor theory is a useful tool for describing mass- However, it’s fair to say that twistor theory – as we’ve described it – still has many shortcom- Although we were able to provide a perturbative solution to the googly problem, this will not , 56 A pure spinor is adetermined spinor by which the obeys Clifford some quadratic algebra constraints, in the a precise given form dimension. of which The are space of projective pure spinors of them as major problemsshortcoming for of twistor twistor theory, theory. for In which there this are lecture, known we solutions: will the talk relianceintrinsically on about on 4-dimensions. another space-time obvious being 4-dimensional:spinor indices, otherwise, which we is can’tmight the split interpret foundation this vector preference indices for for into 4-dimensions everythingto as 2- we’ve a see been positive how feature doing. of to twistor make theory,Fortunately, Though the it there is some are formalism difficult generalizations people useful of for thewhich interesting basic have topics concepts proven in of themselves twistor to higher be theory numbers extremely beyond of useful 4-dimensions dimensions. in the study of5.1 perturbative QFT! From twistors to ambitwistors this direction and their shortcomings). in dimension to be the space of projective, pure spinors of the complexified conformal group, SO to a full perturbative description of Einstein gravity (see [ although general relativity can bedual classically sector embedded of into general conformal relativity has gravity a [ twistor action [ loop integrands in planar gauge theories,twistor but variables actually has performing proved the quite resulting difficult loop (though integrations in not impossible, see [ less free fields and integrable systems (suchspace. as the instanton We sector) even in saw four-dimensional that Minkowski Hopefully, this it has was convinced possible you to that twistor formulate theory perturbative is gauge good theory for in something! ings. twistor The space. ability to describethis massive could QFTs be remains overcome using outside something the[ called reach the of 2-twistor twistor description methods, of though massive particles (c.f., capture the many physicallyQFTs interesting such as non-perturbative Yang-Mills theory. phenomena Even which restrictinginteresting our occur attention massless to in theories perturbative QFT, interesting which there are stillactions. many do For not instance, have conformal satisfactorywhich gravity descriptions nonetheless – has in a many terms interesting conformally of properties invariant, – twistor non-unitary has theory a well-defined of twistor gravity action [ Twistor theory 5. Beyond Four Dimensions ily to the non-locality ofthat the relationship that between standard twistor techniques space suchvariables. and as space-time dimensional as regularization well are as hard the to fact implement in twistor PoS(Modave2017)003 ∼ = C ]. 4. ) M C 66 = (5.1) (5.2) , , 4 for- : 4 7 d ]. This ( 65 = 67 d CP Tim Adamo spinor index into a twistor ) I C , . So a twistor in ) 6 Z ( C , 8 SO ( . Heuristically, these are like ∼ = ) 6 this amounts to a single 8 considered up to overall C projective coordinates). The , = gerbes 4 . d ( ,...,

not 1 0 carried a linear action of SL = 3 = I , can be charted with complex coordi- CP W 6 CD · x C Z d | carry an SL , with 7 ∼ = to flat gauge fields on space-time [ A 7 AB and Minkowski space remains non-local, but x Z C CP 6d d . Just as 6d CP (note these are ) M ∈ C 41 PT PT ) , BA on B . In these variables the purity condition is simply x ABCD 8 I ) ( ε W − B , 2 1 ], and these also induce natural twistor spaces on odd- Z A W ]. The general structure is always that of a projective = , Z 67 = A ( 69 2 AB Z s , x d will carry a natural linear action of SO = 68 = ( are automatically pure. As the space-time dimension increases, 7 6. In this case 6d I ) -dimensional definition of a twistor is consistent with the superconformal field theory. There is a notion of Ward correspondence for these C 4 and CP = Z , d PT ) 4 0 d . This can be expressed rather nicely if we split , ( I 2 ,..., ( 1 Z inside of twistor space. Similar constructions hold for Minkowski spaces = 3 B , CP A 0. Therefore, 6d twistor space takes the form of a projective quadric in = interesting chiral field configurations in 6d for structures known as ], but it requires some heavy-duty mathematics (e.g., higher category theory) to set up. W Finally, the quadric constraints appearing in the definition of these higher-dimensional · are in 4d, it’s clear that 70 , where Z 6 ) AB C = x , There Unfortunately, the utility of these higher-dimensional twistor constructions seems to be quite We still have the purity condition to worry about though; in You might wonder if this For example, consider 6 will be a homogeneous coordinate A 6 relates holomorphic vector bundles over 6 6 ( W = = A gauge connections, but where the gaugecase potential is 1-form rather is involved. replaced Since by a theimportant 2-form; field role a strength in precise of the definition a infamous in gerbeSD the is gerbes non-abelian [ a 3-form, there are self-dual gerbe in 6d, and these play an and dual twistor coordinate: As you might expect, the relationshipthe between dimensionality on either side of the correspondence is enhanced. For instance, a point in is due to the intrinsicgauge chirality field of configurations the which twistor arethe construction: chiral case. (i.e., in instantons), 4d, but there in are higher interesting dimensions non-linear this is not limited in comparison to themetric 4d spinor case. fields, these Although there do is not a correspond notion to of integer-spin Penrose z.r.m. transform for fields sym- as they do in Z dimensional anti-de Sitter spacequadric, [ thanks to the nature of the pure spinor constraints which arise. and the corresponding conformal group is SO SO d projective rescalings. quadratic constraint on It is straightforward to investigate the geometry of the twistor correspondence in 6d, see [ corresponds to a Further, non-linear constructions suchtrivial as field the configurations Ward as correspondence easily dod as not they seem do to in encode 4d. non- For example, the Ward correspondence in of increasingly higher even dimension [ Twistor theory is simply the space ofprojective scaling. spinors satisfying these quadratic constraints, considered up tomalism we’ve an been overall using. It’s clear that 4d twistors complexified Minkowski metric is given in these coordinates by which is treated projectively, but we didn’tis seem because to run all into spinors any of quadraticthe SL ‘purity’ purity constraints. condition This starts to grow teeth, though. nates PoS(Modave2017)003 , 2 − null d P (5.4) (5.5) (5.6) (5.3) Q × C Tim Adamo , M be coordinates ∼ = as a coordinate by shifts up and ) , so the space of ∗ N b a a ∗ N T P X ∂ X T , ∂ P a a X P ( 4, it follows that space-time dimensions, d = , . Geometrically, this means d . Let C d ambitwistor space P M ∂ ∂ . · by quotienting

, P C 0 1 C M / = , . CP "

2 0 P × 1 | π P 1 X ∂ C = ∂ ∂ ∂ 2 · M · CP P ∗ , up to scale. for any dimension | P ∗ P N 3, and the double fibration is given by: T C C ∼ = T C 42 2 ]. − P ∈ / M M / S M ∗ ∗ d ) A N 77 P T T × , , = 2 2 ]). ∈ π = S X 76 } ( ) . This means that you should think of , 74 A PT C ∼ = P , , 75 PA M 2 P [ M = X 73 Q ( ∗ , N is a covector specifying a direction at this point. The space of T 72 b = , P ∗ N 71 T P , while -dimensional projective quadric. For instance, in C ) 2 M is a subspace of this cotangent bundle, given by: − ambitwistor theory C d ( M is the space of complexified null directions at a point in 2 in is a , the cotangent bundle of − 2 d P C − Q P d M ∗ Ambitwistor space has many similarities with twistor space: it is a complex projective space At this point, a pessimist might conclude that twistor theory simply won’t be a useful tool be- Consider complexified Minkowski space Q T ambitwistor space has complex dimensions 2 (since the quotient by therelated complex to scale space-time of non-locally the by nullcorrespondence a scales geodesics double uniformly acts fibration. with as space-time But a dimension. unlike projective Indeed, twistor scaling) in space, and the is ambitwistor down each null direction. These shifts are generated by the the vector field Finally, we can quotient by the scale of each null geodesic to obtain which is simply the space of null geodesics in is the space of null directionswhere up to scale. Thisthat space always has the topology null geodesics is simply where yond 4-dimensions. But we are optimists,construction so instead which of mimics giving up the wespace can non-locality and try to of an look auxiliary the for projective someand twistor other space is correspondence but known as is between non-chiral. Minkowski Thankfully, such a construction exists, Twistor theory twistor spaces become increasingly byzantine,interesting making calculations, it though difficult twistors to have use beenin used the to higher-dimensions formalism (e.g. study to aspects [ perform of QFTs and string theory on labeling a point in directions We can obtain the space of (complexified) null geodesics in PoS(Modave2017)003 , a PA X (i.e., d (5.7) (5.8) (5.9) a (5.11) (5.10) P M = , while the Tim Adamo is given by θ 2 − d P PA Q is the complexified ab corresponds to a com- g : , . If PA M P ∂ M ∂ given by the functions on after quotienting by the scale · . P A PA / b → P ∂ o from -dimensional complexified space-time ∂ 0 d M L d ! = P , , 1 , PA 0 π b are the projective quadrics d ab , while a point in ω P ω D Γ a C y y / P ∗ PA P + N ∗ M F ∂ N . The natural projective scale on ab T ∂ a T V , where the subscript reminds us that this denotes ⊂ g 43 · P ) P X | → ∂ 2 M = 1 P ∂ ∗ ( − = N and space-time is clearly non-local in nature: a point d P P M T complexified space-time, . In our previous notation for line bundles of homoge- = F 2 ∗ Q c P O P d π θ PA P T : | PA ∼ = any 1 is easily seen to be a non-degenerate and closed 2-form on ac comes with a natural geometric structure, known as a sym- ∈ π g 1 in ) , since we obtain PA by L ω P P + = M , C arises naturally from a 1-form ‘symplectic potential’ ∗ 0 X T M ( D ω 1 to n through the relation: . Here + a = F ∗ X N V are un-scaled complex null geodesics. d T , P ∧ a PA P d → . denotes the inner product between vectors and differential forms. = ∗ P N T ω ω , then we can define the space of null directions up to scale by P on a symplectic manifold determines a vector field on that manifold, known as the y : P ∂ 2 is the vector field generating the flow along null geodesics in M ∂ F π · 0 D P corresponds to a projective quadric . It is also easy to see that Now, the cotangent bundle The basic correspondence between If you’ve been exposed to any symplectic geometry, you will know that every differentiable A crucial difference from the twistor construction is that this ambitwistor correspondence eas- . This means that there is a natural line bundle over until further notice. P M M ∗ plex null geodesic (considered up to scale) in assigning projective weight of which are homogeneous of weight neous functions, we would sayhomogeneity that in in plectic form: T and ambitwistor space by where M by function Hamiltonian vector field so we will just assume that we are working on a generic where Twistor theory which is the complexification ofthe the celestial space 2-sphere). of null The directionsfibres fibres at of of a point in Lorentzian-real The double fibration trivially generalizes to ily generalizes when we replace metric on PoS(Modave2017)003 . . ∗ N δθ L T , in- 0, or . 0, for θ (5.16) (5.12) (5.14) (5.13) (5.15) 1, it is = = + PA θ 0 Tim Adamo . A contact D ¯ ∂δθ . L PA 0 = on of weight and the space-time c taking values in ) P X plays the role that the ; it turns out that the f θ d , b P δθ PA a PA , the generator of the flow , it makes sense to consider is well-defined on P ( 0 on θ , which obeys θ D θ da L , this means that is sufficiently ‘small.’ In order g ∗ N T b cd δθ , Γ contact structure 0 Furthermore, it can be shown that + 7 . = . : db is homogeneous in ) . . for some function is precisely g ) 0 b , valued in θ PA f PA L P , where a cd , ¯ a L ∂ 6= Γ M suggests), and it encodes a substantial amount P , PA → ∗ 2 δθ = PA − T ab 1 2 ( d + M PA g is the contact 1-form L 1 ) 44 ∗ ( ]. , . Since θ − 0 δθ 1 θ T d b d 77 H Ω PA ( M 2 1 → X ∗ ∈ ∈ d ∧ . Upon restriction to -form on θ T − c ) 0 θ θ P 1 D θ a , δθ 0 , this implies that 0 P D ( θ c ab L Γ is a = defines a deformation of the contact structure up to infinitesimal diffeomorphisms. δθ b δθ P a ). That is, we have: P ab the natural complex structure on 5.9 obeys a non-degeneracy condition: g ¯ ; by definition, this vanishes upon restriction to the space of null directions P ∂ θ ∂ d · M from the contact structure on will also lead to something interesting on space-time. ¯ ∗ 2 1 P T d + PA PA + denotes the inner product between vectors and differential forms. Consider the function is the Lie derivative along on ω ¯ 0 X b y ∂ ω for ∂ D 0 P y · a D θ L P is preserved along the flow of null geodesics. This means that V ¯ X Such conditions ensure that Since the natural geometric structure on We want to consider a deformation Thus, the natural geometric structure on ambitwistor space is a holomorphic 1-form ab d θ 7 with its torsion-free conformal structure [ g 2 1 = ¯ is a trivial deformation ifThis it means that can a be non-trivial written deformation as of the contact structure is a cohomology class: ∂ 5.2 The Penrose transform along null geodesics ( small deformations of the contact structure.Penrose In transform twistor space, can cohomological be representatives interpreted forthat the as these small led deformations to solutions of to the freestructure complex field on equations structure, on and space-time. we Perhaps saw deformations of the contact to get something non-trivial,appropriate we conditions have are: to put some restrictions on this Twistor theory where − The Hamiltonian vector field of this function on In terms of the symplectic potential One can show that herited from the symplectic structurenatural on to think of it as valued in the line bundle of interesting geometry. From our perspective, the contact structure where that Such a 1-form is saidstructure can to be define thought of a as (weighted) antion odd-dimensional non-degenerate of analogue of a symplectic structure (as our deriva- complex structure played on twistorIndeed, space: it it can encodes be something shownM that about there the is space-time an geometry. equivalence between the data PoS(Modave2017)003 = ) is is a , we δθ 0 (5.20) (5.18) (5.22) (5.23) (5.21) (5.19) (5.17) ∗ D PA π 5.21 y 0 Tim Adamo D . ) L , 2 − d P /0,so we can write Q ( is pure diffeomorphism: 0 ) is equivalent to ) = H . and the form degrees arising M , which means that ( ) is a linear metric perturbation L ⊗ 0 5.20 P , ) ∗ D ab N δθ -dimensional projective quadrics h T ∗ , thanks to an important result in 2 ) M 2 P , 2 π 1 in ( ( − b 1 y 1 d P , + − P , 0 0 H a Q . 0 d D P ( H ) ( ) 2 is a holomorphic vector field, d M X 0 ) = and ) , ( , j L . It turns out that this fact can be used to D j 0 2 , ¯ ab ) + ∂ L ). This means that we can write the constraint ∗ ¯ − ∂ ( N h M , d P /0. T 0 2 δθ ) = = Q P . Now, we know that the projective space of null − ∗ j D 2 ( ) 45 5.19 d P /0.Furthermore, it can be proved (although we will ) = 0 π 0 × ) = ( = P Q δθ D L , H ( d ∗ ( ) implies that 2 , started life as a cohomology class defined on 1 L , ) = X ∗ M y ¯ π , N ∈ ∂ 0 δθ ( 0 2 corresponds to on space-time. on space-time. Such an T j ∗ 2 h ∼ = . H − δθ M D 5.17 0 P π ∗ P d ( ab ( N ∗ of the deformation to the projective space of null directions, = 0 N 1 ⊗ D δθ 1 h T = Q , j T is just that of a differential operator. Again using that ) D ( 0 H P 0 0, indicating that the constraint ( 1 P 0 , δθ H 0 D δθ ∗ M 2 D is homogeneous of weight ∗ ( 2 π H ] = 0 π 0 ¯ 0 ∂ into cohomology on H D , D . 0 0. Using Cartan’s formula for the Lie derivative of a differential form, ∗ ) N ∼ = L D T . Now, since X [ = ) ) ( P -form cohomology class and ) b L L has sufficiently boring topology (e.g., that it is topologically equivalent to 1 ξ , δθ , , ∗ ∗ 2 ∗ 0 N N ( π T T M 0 is also trivial: P P is holomorphic on D ( ( j is a 1 0 , L ) then the resulting metric perturbation obtained on ), we see that 0 L 0 f Ω for some 0 as D H ¯ δθ ∂ ) 5.9 ∈ ), where the action of ∗ 2 b = j ξ π = a ( δθ 5.18 . Using identical arguments, you can show that if we’d started with a trivial deformation ∇ ∗ 2 δθ Thus, the Künneth decomposition ( First, consider the pullback From ( π ; this object will be valued in = 0 M ∗ N D T ab using ( holomorphic vector field, on (i.e., h for some must have that The usual arguments for homogeneous holomorphic functions therefore indicate that for some symmetric, trace-free tensor this is L telling us that directions is a Cartesian product: Twistor theory Our task is to understand what such a P namely, that with values in split the cohomology of But since 0 and the only contribution comesfrom from the the inner exterior product derivative in between the first term of ( not show the details here) that the first cohomology of the homological algebra called the Künneth theorem. In the case at hand, this means that If we assume that flat space-time), then it follows that PoS(Modave2017)003 ]. ), is a 79 , 5.23 (5.26) (5.25) (5.27) (5.24) ab ε 78 , , 0, but we ) Tim Adamo n 76 = , . 1 the Penrose ) extend away powerful than L ) ab , 75 ε , where a X L · 5.25 k ≥ − , k PA i more ( n e 1 , -dimensional momen- PA 0 , and consider a plane ab ( d C ε 1 H , 0 and 0 M = ∼ = = H =

ab , 2 0 ∼ = h ) k ) . Sure enough, it is straightfor- 1 2

+ M ) n L b a L ξ , ··· , a 2 ∗ is a constant ( N a without ever needing to impose these X T · a ξ ∇ k 1 k P i a ( ( e = 0 PA . Indeed, for integer ∇ ab H ab ε L h on = ∈ b 0 P P ) b · / 1 a P δθ + k } P a n 46 taking values in P a j = ··· M X · 1 k h a i ( 1 on e φ − 0 ) is totally symmetric and trace-free in its indices. ab D X / ε 1 ( +

= n for some ab a j h j ) = ··· M 1 0 P a , D φ on X ( ) h resulting from a deformation of the ambitwistor contact structure is X ( 0 ab ) 1 h + n a ··· 1 -dimensional complexified Minkowski space, a ( indicates that d φ 0 ) 0 quadric to some given order. The major drawback of such a formalism is that it 1 metric perturbations , we can form + { n = a ab 2 h ··· 1 P a ( valued in different powers of the line bundle φ At first, it might seem that the ambitwistor Penrose transform is actually Before moving on to these exciting new developments, let’s first work through an instructive From Considerable effort was put towards trying to find a way to impose field equations through Thus, we have a statement for the Penrose transform on ambitwistor space: linear fields PA ward to show that: and this must be expressible as example of the ambitwistor Penrose transformspace-time to ensure to that be we see exactly what is going on. Take where the version we learned in twistorspace-time. space: it Unfortunately, makes there sense is inambitwistor a any Penrose dimension major transform and do shortcoming: on not any the obey complexified any space-time equations fields of generated motion! by Indeed, the as we saw in ( In words, this means thatby equations demanding that of the motion ambitwistor can cohomologyfrom representatives be the on imposed the on RHS of the ( is resulting very space-time difficult fields to worktheory with; until indeed, quite this led recently,transform to when was a a discovered. dearth new of strategy progress for in the obtaining study field of equations ambitwistor from the Penrose conditions. the ambitwistor Penrose transform in thebe early done, days it of requires the the subject. rather While cumbersome it turns formalism out of that formal this neighborhoods can [ unconstrained (aside from being symmetricwas and translated traceless). into space-time On fields that twistor obeyedWe space, free don’t cohomological field seem equations data to (namely, get the any z.r.m. such equations). equations of motion from the ambitwistor version of the transform. You can easily generalize this statementon to fields of alternative spin by taking cohomology classes the metric perturbation wave perturbation to theconstant, Minkowski symmetric metric. and traceless This polarization takestum. tensor, and the This form perturbation obeyswill see the that linearized we Einstein can equations construct if the corresponding Twistor theory transform reads: PoS(Modave2017)003 in . In ) z 1 d , (5.28) (5.29) , while 1 ) az ¯ z P − . Clearly, , . ( z Tim Adamo ( PA ) = z ( a P 0. Note that none be a map from the = has conformal weight PA looks like: a should be holomorphic, P e ¯ . → ∂δθ F 2 ) descends to Σ P and : e 2 ) F 0 Beltrami differentials 5.28 0 necessary for the target space , , − 0 X a · – requires the linearized Einstein = ( k X i 2 ]. The motivation for this discovery ¯ e ∂ P -form, so in order for this worldsheet b δθ 0, so ( a ) 80 P P must have conformal weight [ 1 a = Σ , P e Z 0 ). ( . This means that locally, ab ), we can construct the corresponding defor- π δθ , and it also obeys . 1 ε Σ ∗ 2 2 on the target space carry different conformal ) z z π L is a d d¯ 3.48 P ) 0 5.18 = · ¯ z z a P D e 2 k , 47 X ( P X ¯ ¯ = ∂ δ in terms of some local affine coordinates for simplicity, and let ( e 2 string theory e -form on Σ C = ) − has conformal weight j 0 . From ( ) to the worldsheet: M a , ¯ P ∂ 1 θ , then X θ ( ( Σ = ∗ 1 in F . Σ to ambitwistor space. + ) to make sense; locally, this means that δθ Σ Z ∗ 2 Σ π 0, it follows that π 1 5.29 2 = must be a -dimensional -form on with values in = ) P d ) . This suggests that a string theory governing z · S ( 1 k , a ], but we will simply proceed by looking for a string theory governing maps PA P 0 ( to ambitwistor space. What sort of properties should this map have? Well, a 82 . Likewise, the Lagrange multiplier on , Σ Σ ) is a 81 P is the complex structure on , ¯ z is simply a function on δθ X ∂ ( z ) z d¯ ( . This means that the coordinates a = X ¯ as fields on ∂ PA A natural candidate theory which has this property is one whose kinetic term is the (holomor- If On the support of Fix space-time to be The question of how to obtain field equations (even linear ones) from ambitwistor theory in ) 0 , is a Lagrange multiplier enforcing the quadratic constraint 1 You may have encountered such objects before; they are known as phic) pullback of the contact structure or chiral, in nature. weight when viewed as fields on the terminology of 2d CFT, we say that with the holomorphic delta function defined as in ( from a closed Riemann surface string worldsheet recurrent theme throughout these lectures hasspace been holomorphicity, too: and this we applies to werecoordinates ambitwistor able to say everything about ambitwistor geometry using only holomorphic Here, e to be action to make sense, ( order for the second term in ( a practical way has aconformal truly field remarkable theory answer: (CFT) we techniques must of combine ambitwistor theory with the 2d Twistor theory which has the appropriate weight mation of the contact structure: the resulting of these facts –equations. or any step in the process of constructing 5.3 Ambitwistor strings originated in a series of compactmassless expressions QFTs for [ all tree-level scattering amplitudes in a variety of PoS(Modave2017)003 . 0 z ∂ D z d (5.36) (5.30) (5.31) (5.33) (5.32) (5.34) (5.35) , so the 0. This = Σ ∂ = 0 enforced Tim Adamo 2 0, this means = Q ) is invariant: and 2 = have conformal P ) 2 ˜ 0 b , P 5.29 , ). All four of these 1 b . , − ) ( ) is invariant under the v 5.32 w 0 and conformal gauge, ∂ ( ˜ e b . Since while = ) 5.29 ) , ) e − z 0 , ) is a classical (holomorphic) ˜ c ( 0 , e ˜ , c ˜ 1 b ∂ 1 ¯ ∂ ∂α v − ∼ 5.29 − ( , ( − , it follows that the components of = ˜ = 0, under which ( c w ) ˜ . ), respectively. This can be accom- c ¯ e z , e ∂ 1 ∂ ( = − 2 δ ˜ δ b f ˜ z b PA 2 P 5.32 2 + P ˜ c 2 . ∼ c 7→ ) − ¯ z ) ∂ z + ( , , w b c ) 0 bc ( a f a ∂ b ) and ( ∂ P + = ) a f z vP z − bc a ( ) is indeed ( X 48 ∂ P ∂ c c 5.30 ∂ ¯ + ∂ δ ∂ a b = → P 5.29 as its target space, thanks to constraint cT 2 a Σ ∗ az P N Z , I − P have conformal weight T , δ a π w a ˜ 1 c = a b 2 X , P δ − c ∂ Q α z a = , P S a = ∼ − a X ) ∂ X = w . However, you can check that the action ( ( v δ e b T P = ) a z ( X a δ is only defined up to rescalings by a constant factor, which reduces the target X µ P ) are not the only transformations which preserve the worldsheet action. There are . . The gauge-fixing also results in a BRST charge given by: , this worldsheet action has ∗ ) N are the ghost and anti-ghost fields associated with holomorphic reparametrizations, and 0 T 5.30 , b is an infinitesimal transformation parameter of conformal weight P 2 , is defined only up to translations along any null direction. This is precisely the action of ( v c a another infinitesimal gauge parameter of conformal weight ): But ( To quantize this ‘ambitwistor string theory’, we must gauge fix the holomorphic reparametriza- Our gauge fixing is anomaly free provided that this BRST charge is nilpotent: A priori X are the ghost and anti-ghost fields associated with the gauge freedom ( α transform as: ˜ b a , 5.33 ˜ These transformations are infinitesimal holomorphic reparametrizations offact that the the worldsheet worldsheet model is invariant under them means that ( where space to This means that plished with the standard Fadeev-Popov procedure;then if the we resulting gauge action fix is to where fields have fermionic statistics, and c tion invariance and gauge transformations of ( by the Lagrange multiplier transformations 2d CFT. Now, under a holomorphic reparametrization also gauge transformations associated with the constraint for that in Minkowski space, so the target space of ( weight with the holomorphic stress tensor of the worldsheet theory, and normal-ordering assumed for all terms. Twistor theory P can be checked explicitly( by using the free worldsheet OPEs defined by the gauge-fixed action PoS(Modave2017)003 0. = ]. By (5.39) (5.37) (5.38) 85 QU ! In other X Tim Adamo · k i e ), for which the ab ε = ). Normal-ordering 26. 5.28 ab = h 5.39 d ] for a heuristic explanation). . ) 84 z ( X · k i e ) correspond to metric perturbations on z ] (c.f., [ , ( c b 83 . 3 0. This latter constraint comes about from P δθ ) ∂ ) = z c ( 2 δθ ) a k ( , correlators of vertex operators in ambitwistor P ) by the addition of some worldsheet fermions) : ∗ 1 26 ab F 6 49 − ε δθ Σ CP 5.29 Z d )) ( ∼ = z = ( of ambitwistor space, pulled back to the worldsheet. So Σ = P U · 2 θ k Q ( ¯ δ 0 if and only if Σ = Z = QU U ]! 87 are precisely the linearized Einstein equations for , 2 k 0, while 86 . This fact can also be extended to the non-linear level by coupling an am- ); this is the only part of the BRST charge which has a potentially anomalous = . = δθ 0 U ab 5.34 ε = a k ab ε a k term in ( 2 ), and is eliminated with the choice of critical space-time dimension It is easy to see that these conditions impose further constraints on ( The perspective of unifying ambitwistor theory with string methods has led to many exciting Although these higher genus expression are too functionally complicated (involving a localiza- But Now, vertex operators in string theories correspond to deformations of the gauge-fixed world- P 5.33 In order for this to be an admissible vertex operator, it must be normal-ordered and obey advances in recent years. Thereparticularly are exciting far one too is many related examplesin to to massless mention the QFTs. here calculation It in of turns any loop out detail, corrections that but to when one scattering amplitudes to a non-trivial background metric; quantum consistencythe of non-linear the resulting vacuum worldsheet Einstein model equations imposes on this metric [ string theories are equal to tree-levelconsidering scattering correlation amplitudes functions in on a higher variety genusfor of worldsheets, loop massless we amplitudes QFTs can [ [ obtain new representations tion problem in terms of ellipticphysicist, functions) they to can be of be practical reduced use from to the more perspective manageable of a expressions particle by degenerating the underlying contraction with bitwistor string worldsheet model (related to ( so only the gauge-fixinganomaly of is fixed the by the holomorphic holomorphic central( reparametrizations charge of is the potentially fields appearing anomalous. in the gauge fixed The action vertex operator takes the form: requires that words, quantum consistency conditions insical the Penrose ambitwistor string transform theory could havecorresponding not: done to what impose the linearized clas- field equations on the metric perturbation the sheet action which areaction annihilated is by precisely the the BRST contact structure charge. In our case, the interesting part of the Twistor theory You should try this calculationcalculation for in yourself ordinary (it’s string a theory); chiral the version result of is: the famous critical dimension vertex operators will be given by deformations We know, thanks to thespace-time. Penrose Indeed, transform, we that can such work explicitly with a plane wave deformation ( PoS(Modave2017)003 ) 5.41 (5.44) (5.43) (5.45) (5.40) (5.41) (5.42) ). Show Tim Adamo 5.33 is given by: i σ ], and looks to be a 91 , ), and 3 insertions of the 90 . , ? This set of constraints is , , ) . ) n ) ) 5.39 z 89 i n ( j p z , i p X , ( · ) k j − )( i ,..., p ) + k z e ,..., i 1 · ( i j ) 4 σ i 2 i p U ( z can be performed explicitly, and that k ( = 4 ( δ )( i = n , and show that it has only simple poles b i = ) ia i ∏ i given by ( 6= 1 ia z P j k ∑ ( σ k ) ) ( a 3 ) 1 z all i U i z CP n ( 1 = X ∑ ( i a σ n = 3 ∑ i P d ∼ = , z i V ) d¯ 0 Σ ) ab σ 0 for 2 σ ∧ ε z d z ) 50 on the Riemann sphere as on ( . Show that the solution can be written in terms of ) = z 2 1 σ ) ) ( ) = id ) = ( σ ˜ V -invariant inner product on these homogeneous coor- 2 c σ ( ) π σ ) ) σ ( 2 σ ( 1 CP 2 z ( C 2 , 2 z P ( 2 , ) = ( 1 i ( P P c ∼ = . 2 1 i σ P ]. This perspective has already led to novel representations σ σ i ( σ ) = V = ( Σ σ z = a 88 σ ( = * ) = P σ = ( a z σ P ( a Res ¯ ∂ V σ Res ) when vertex operator insertion points. What is the result of the zero-mode is the SL is some auxiliary point. Prove that this solution is independent of the n ba 1 ε 5.42 j b CP path integral? σ a i a ∈ are the σ p X scattering equations 3 insertions of the vertex operators } i σ . = z p − : { σ ) n vertex operator insertion points. Show that the residue of the pole at i j ( n the scattering equations where enforce known as the dinates, and choice of that the path integral overthe the non-zero-mode worldsheet portion fields of this integral enforces the equation homogeneous coordinates at the Why is this equivalent to Res ‘fixed’ vertex operators in the worldsheet correlation function defined by the (Euclidean) path integral with respect to the gauge-fixed action ( where the portion of the 4. Demonstrate that the remaining ingredients of the worldsheet correlation function ( 3. Compute the quadratic differential 1. Consider 2. Solve the equation ( of 1- and 2-loop scattering amplitudes in gauge theory and gravity [ Twistor theory Riemann surface into a nodal sphere [ promising route to obtaining useful new expressions for perturbative amplitudes moreExercise: generally. PoS(Modave2017)003 – in 3 of − PA n Tim Adamo (Hint: use homo- 1 CP 0 globally on simple poles, prove that setting ) = σ n ( 2 P with 1 51 CP 0 – crucial for the target space of the worldsheet theory to be ) = σ ( 2 quadratic differential on P any ) σ ( 2 P geneous coordinates.) We conclude, therefore, thatthe the constraint scattering equations are equivalent to the residues of these poles equal to zero forces the presence of vertex operator insertions. 5. For Twistor theory PoS(Modave2017)003 . 252 A43 , in . Tim Adamo J. Phys. (1988) 215–218 , . Oxford Commun. Math. , B214 (1985) 257–291 2 Commun. Math. Phys. , ]. Phys. Lett. , . Cambridge University Press, 1990. . 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