Twistor Actions, Grassmannians and Correlahedra

Twistor actions, grassmannians and correlahedra

Lionel Mason

The Mathematical Institute, Oxford [email protected]

Canterbury 11/1/2016: Total Positivity review with Adamo, Bullimore & Skinner 1104.2890, & work with Lipstein arxiv:1212.6228, 1307.1443, more recently with Eden, Heslop, Agarwala. [Also. work by Alday, Arkani-Hamed, Cachazo, Caron-Huot, Drummond, Heslop, Korchemsky, Maldacena, Sokatchev, Trnka. (Annecy, Oxford, Perimeter and Princeton IAS).] Twistors, amplitudes, Wilson loops & ’hedra

Background [Penrose,Boels, M, Skinner, Adamo, Bullimore. . . ]: 3|4 Twistor space CP space-time; Klein correspondence. • ↔ N = 4 Super Yang-Mills has twistor action in twistor space. • Axial gauge Feynman diagrams ; ‘MHV diagrams’. for • amplitudes, Wilson loops & Correlators. Planar Wilson-loop/amplitude duality is planar duality for • MHV diagrams. Super Amplitude/Correlator/Wilson-loop triality. • Focus of this talk [with Lipstein, Agarwala, Eden & Heslop]: Twistor Feynman rules give grassmannian and ’hedra • formulations. Feynman diagrams deﬁne cells in positive grassmannian. • Potentially tile amplituhedra and other correlahedra. • = 4 Super Yang-Mills The harmonicN oscillator of the 21st century?

Toy version of standard model, contains QCD and more • classically. Best behaved nontrivial 4d ﬁeld theory (UV ﬁnite, • superconformal SU(2, 2 4) symmetry, ...). | Particle spectrum • helicity -1 -1/2 0 1/2 1 # of particles 1 4 6 4¯ 1

Susy changes helicity so particles form irrep of • ‘super’-group SU(2, 2 4) like single particle. | ‘completely integrable’ in planar (large N) sector. • much twistor geometry in their amplitudes: • 1 (Ambi-)Twistor string & action descriptions, 2 Grassmannian residue formulae, 3 polyhedra volumes ; the amplituhedron, Scattering amplitudes for = 4 super Yang-Mills N 4-Momentum:

p = (E, p1, p2, p3) = E(1, v1, v2, v3), massless v = c = 1 p p := E2 p2 p2 p2 = 0. ↔ | | ⇔ · − 1 − 2 − 3 ⇔ 1 E + p3 p1 + ip2 λ0 = λ˜00 λ˜10 ⇔ √2 p1 ip2 E p3 λ1 − − | | Supermomentum: P = (λ, λ,˜ η) C4 0 C0 4 , where ∈ × ηi , i = 1,..., 4 anti-commute ; wave functions: i ij ijk Ψ(P) = a+ + ψ ηi + φ ηi ηj + ψ˜ ηi ηj ηk + a−η1η2η3η4 . Amplitude: for n-particle process is

(1,..., n) = (P ,..., Pn) A A 1 MHV degree: k Z, k + 2 = # of ve helicity particles, ∈ − susy = 0 for k = 1, 2 ⇒ A − − For k = 0, = 0, ‘Maximal Helicity Violating’ (MHV). A 6 Feynman Diagrams Ordinary Feynman diagrams When forces are weak the basic tool for describing interaction of Contributionselementary particles is Feynman diagrams.

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Trees classical, loops quantum. 10 ↔ ↔ 2 Locality: only simple poles from propagators at ( pi ) = 0. P Need for new ideas Examples of Clever Ideas

Consider the ﬁve-gluon tree-level amplitude of QCD. Enters in calculation of multi-jet production at hadron colliders. Described by following Feynman diagrams:

+ + + · · ·

If you follow the textbooks you discover a disgusting mess.

22 Result of a brute force calculation:

k k ε k ε ε ε ε 1 · 4 2 · 1 1 · 3 4 · 5

23 The Parke-Taylor MHV amplitude

However, result for helicity (+ + ) part of the amplitude is − − − 5 4 λb λc (1, 2, 3, 4, 5) = δ pa h i A ! λ1 λ2 λ2 λ3 λ3 λ4 λ4 λ5 λ5 λ1 Xa=1 h ih ih ih ih i where b and c are the + helicity particles and

i j := λ λ := λ λ λ λ h i h i j i i0 j1 − i1 j0

(similarly use [i, j] for λ˜i s). More generally ( Parke-Taylor 1984, Nair 1986)

4|8 n tree δ a=1(pa, ηaλa) MHV (1,..., n) = n A λa λ + Pa=1h a 1i Q Twistor space

3|4 Super twistor space is CP with homogeneous coords: α˙ 2 2 0|4 ∗ Z = (λα, µ , χa) T := C C C , Z ζZ , ζ C . ∈ × × ∼ ∈ T = fund. repn of superconformal group SU(2, 2 4). | | Super Minkowski space, M = G(2, 4 4) R4 8, | ⊃ 1 Incidence: a point x = (x, θ) a line X = CP PT via ↔ ⊂ α˙ αα˙ α µ = ix λα , χi = θi λα . Two points x, x0 are null separated iff X and X 0 intersect. Space-time Twistor Space

x! X x Z Supersymmetric Ward correspondence

3|4 Super Calabi-Yau: CP has weightless super volume form

D3|4Z = D3Z dχ ... dχ Ω . 1 4 ∈ Ber ‘Super-Ward’ for = 4 SYM: ¯ ¯ N 3|4 A dbar-op ∂A = ∂0 + A on bundle over CP has expansion

a ab 3a 4 A = a + χaψ + χaχbφ + χ ψ˜a + χ b and ∂¯2 = 0 solns to self-dual = 4 SYM on space-time. A ↔ N

Action for ﬁelds with self-dual interactions:[Sokatchev, Witten] SD interactions holomorphic Chern-Simons action ↔

2 3 3|4 S = tr(A ∂¯A + A )∧D Z . sd ∧ 3 ZPT Incorporating interactions of full = 4 SYM Nair, M., Boels,N Skinner, 2006

Extension to full SYM:

Sfull [A] = Ssd [A] + Sint [A] includes non-local interaction term:

2 4|8 Sint [A] = g d x log det(∂¯A X ) M | Z∞ 1 tr (A A ... A ) Dσ ... Dσ = g2 d4|8x 1 2 n 1 n n M×X n σ1σ2 σ2σ3 ... σnσ1 Xn=2 Z h ih i h i 1 4|8 th X = CPx PT for x M , σi Xi , i factor, ⊂ ∈ ∈ Dσj A = A(Z(σ )) , and K = i i ij σ σ h i j i −1 is Cauchy kernel of ∂¯ on X at σi , σj . Axial gauge Feynman rules

∂ Choose ‘reference twistor’ Z∗, impose gauge: Z¯∗ A = 0 . · ∂Z¯ Payoff: Cubic Chern-Simons vertex = 0 ; Feynman rules:

Propagator = delta-function forcing Z , Z 0, Z to be collinear. • ∗ 0 0 1 2|4 0 1 dcdc 4|4 0 0 ∆(Z , Z ) = δ¯ (Z , Z∗, Z ) := δ¯ (Z∗+cZ+c Z ) 2πi 2πi cc0 Z log-det term gives ‘MHV vertices’: • 4|4 4|4 n 3|4 d ZAd ZB δ¯ (Zr , ZA + σr ZB) V (Z1,..., Zn) = dσr . M×X n Vol GL(2) (σr−1 σr ) Z rY=1 − Vertices force Z ,... Zn to lie on line X = Z (σ) = Z + σZ 1 { A B}

Simplicity: # propagators = MHV degree +2 # loops. × Amplitudes & Wilson loops

Amplitudes: Transform rules to momentum space ; ‘MHV rules’ [CSW 2005] where vertices = off-shell MHV amplitudes. But MHV diagrams also give Wilson loops etc.: Null polygons: Supermomentum conservation for colour ordered momenta ; null polygon xi = (xi , θi ) M: { } { } ⊂ 0 0 0 (pAA , ηaλA) = (xAA xAA , θA θA ) . i i i − i+1 i − i+1

Conjecture (Alday, Maldacena)

Let W (x1,..., xn) = Wilson-loop around momentum polygon.

All loop MHV amplitude = MHV tree W (x ,..., xn) . × h 1 i Momentum polygons in twistor space

Generic polygon in PT [Hodges] null polygon in space-time. ↔

Zn X1 Z1 x1 pn p1 xn Xn X2 x2

Z2 incidence relations

x Z3 4

Change variables so that (X1,..., Xn) = (Z1,..., Zn).

Important simpliﬁcation: Zi PT are unconstrained. ∈ What is Wilson loop in PT? Holomorphic Wilson loops

For Wilson-loop, need holonomy around polygon in PT. Vertices Z , • i Edges Xi = Z (σ) = σZi−1 + Zi , σ C . • { ∈ ∪ ∞} Global frame F (σ) of E on X with • i |Xi i

∂¯A F (σ) = 0, F ( ) = F = 1 . |Xi i i ∞ i |Zi−1 Perturbatively iterate F = 1 + ∂¯−1( F ) to get • i A i ∞ r ¯−1 ¯−1 f (σs) dσs Fi = 1+ ∂s−1 s (σs) , (∂r s f )(σr ) = A LX σr σs Xr=1 sY=1 Z i − Deﬁne • n n W = tr F = tr F (0) . i |Zi i Yi=1 Yi=1 Agrees with space-time Wilson loop on-shell. The S-matrix as a holomorphic Wilson loop

Theorem (Bullimore, M., Skinner, 2010-11) For planar = 4 SYM: N Amplitude loop-integrands = (holomorphic) Wilson loop integrand

tree (1,..., n) = W (Z ,..., Zn) . A h 1 iAMHV

Tree amplitudes Wilson-loop in self-dual sector (g=0). • ↔ Loop expansion for = g-expansion for W . • A The Axial gauge twistor space diagrams for amplitude are • planar duals of those for Wilson-loop correlator.

Proof: comparison of Feynman rules on PT (or BCFW). Examples

NMHV case: for A2 part of W , A(Z)A(Z 0) = ∆(Z, Z 0): gives h i h i W = ∆ where ∆ = h i i

P Zi

Zi 1 − =[ ,i 1, i, j 1,j] , Zj 1 ∗ − − −

5 dc1dc2dc3dc4dc5 1 4|4 [1, 2, 3, 4, 5] := δ¯ ci Zi , c1c2c3c4c5 VolGl(1) ! Z Xi=1 4 (1234)χa + cyclic = a=1 5 (1234)(2345)(3451)(4512)(5123) Q is the ‘R-invariant’. The Nk MHV tree

N2MHV: quartic terms in A in W give two Wick contractions

Zk 1 − Zj Zk

Zj 1 −

Zl 1 − Zi

Zi 1 Zl − No crossed propagators for planarity. Nk MHV tree amplitudes k propagators ; k R-invariants. ↔ • But now have ‘Boundary diagrams’ e.g. Loops

At MHV with one MHV vertex obtain i,j Kij with Kij = P

Zj 1 Zi −

3 4 3 4 = D | Z D | Z [ ,i 1, i, A, B!][ ,j 1, j, A, B!!] A ∧ B ∗ − ∗ − !Γ Zj Zi 1 X −

Loop momenta location of line X = Z Z . Recall: ↔ h A Bi dc1dc2dc3dc4 4|4 [ , i 1, i, A, B] := δ¯ (Z∗+c Z +c Z +c Z − +c Z ) ∗ − c c c c 1 A 2 B 3 i 1 4 i Z 1 2 3 4 4|4 4|4 can integrate D ZA∧D ZB against delta functions vol GL2 1 dc dc db db K = 0 1 0 1 . ij (2πi)2 c c b b Z 0 1 0 1 External data encoded in integration contour (see later). Loop integrands and correlators

Lagrangian insertions: S [ ] = d 4x (x) where int A Lint 8 (x) = d θ logR det(∂¯A ) . Lint |X Z Forming tree correlator with ` insertions gives loop integrand

W (x ,..., xn) (x ) ... (x ) tree . h 1 L 1 L l i Super BPS operators: it is invariant to integrate over just 4 θs

2 ab α 2 (x, θ, Y ) := tr(Φ(x) Y ) = (Y dθaαdθ ) log det(∂¯A ) O · b |X Z [ab cd] ab where Y Y = 0 (so depends on the 4 θs with θaαY = 0).

Proposition (Alday, Eden, Korchemsky, Maldacena, Sokachev, Heslop, Adamo, Bullimore, M., Skinner ) n 2 (xi xi+1) 2 lim − (xi , Yi ) = W (x1,..., xn) . 2 (xi −xi+1) →0 * Yi Yi+1 O + h i Yi=1 · Gives ‘triality’ Amplitude Wilson loop BPS correlator. ↔ ↔ MHV diagrams for correlators

[Chicherin, Doobary, Eden, Heslop, Korchemsky, M., Sokatchev]

Obtain diagrams in twistor space for (x ) ... (xn) . • hO 1 O i 1 2 2 1

3 3

4 4 n n

Double line line Xl = Zl1, Zl2 PT xi . • ↔ h i ⊂ ↔ Solid lines twistor propagators. • ↔ 2 As (x x + ) 0, consecutive lines join. • i − i 1 → Diagram 0 unless consecutive lines are connected by • propagator→; Wilson loop. The momentum twistor Grassmannian

Recall from Nima’s talks:

Grassmannian contour integral formula:[M. & Skinner (2009)] For C G(k, n) consider ri ∈ d nk C k δ¯4|4(C Z ) r=1 ri i . vol GL(k)(1 ... k)(2 ... k + 1) ... (n 1 ... k 1) IΓ4k Q − Contour-integrate down to 4k-cycles and ﬁx remaining parameters against delta-functions ; terms in BCFW expansion (and leading singularities).

Loop integrands: include L lines (ZA1 , ZB1 ),... (ZAL , ZBL ) in (Z1,..., Zn) ; extend to G(k + 2L, n + 2L). 4k + 8L-cycles are characterized as positive cells. Wilson loop diagrams in positive Grassmannian

With k + 2L propagators diagram gives • k+2L dc dc r dc r dc r dc r r 0 r i1 r i2 r i3 r i4 4|4 δ¯ (Yr ) , 4 Vol GL(1) c c r c r c r c r CP r 0 r i1 r i2 r i3 r i4 rY=1 Z 4 where Y c Z c r Z r r = r 0 ∗ + p=1 r ip ip Taking this into account we can write • P n Y c Z C Z C C c r r = r0 ∗ + ri i ri = ri r ip Xi=1 The C c r deﬁne 3k-cycle in Gr k 2L n 2L . ri ( rip ) ( + , + ) c provides k extra parameters to get to 4k-cycle. • r0 4k-cycles have d-log volume forms in parameters c r . • rip Proposition (Agarwala,Marin-Amat) The 3k-cycles are 3k-cells in the positive Grassmannian. Positive Grassmannians questions

Is d-log parametrization for 4k-cycle determined by • positive 3k-cell? What is correspondence between Wilson-loop diagram • and standard classiﬁcation of positive cells? Key question: Do all 3k-cells arise? If not, how can we • characterize those that do? See Susama’s talk for more on these questions. Conjecture The 3k-cells that arise are those that project to 3k-cells in the amplituhedron. Bosonization, positivity and ’hedronizing

Take real and bosonize fermionic parts of Y , Z by • 4|4 4+k r r C R with Z = χ φ , r = 1,..., k . → · The rth fermionic delta-function arises by • δ0|4(χ) = (Y r )4 d4φr . Z Take data Z ,..., Zn positive. • { 1 } Planarity positivity of c r C Gr+(k, n) encoding • r ip ri planarity of⇒ ‘Boundary diagrams’⇒ such∈ as:

• Tiling Amplituhedra/Wilsonohedra/Correlahedra w/ Agarwala, Eden, Heslop, following Arkani-Hamed, Hodges, Trnka

Positive c r gives 4k-dimensional tiles in Gr(k, k + 4) • r ip

Yr = cr0Z∗ + Cri Zi .

where C C c r and c 0 ; positive 3k-cell. = r ip r0 = Can we tile amplituhedra? Correlahedra? •

0 < Y ... Y + Z − Z Z − Z h 1 k 2L i 1 i j 1 j i 0 < Y ... Y + Z Z Z − Z h 1 k 2L l1 l2 j 1 j i 0 < Y ... Y + Z Z Z Z h 1 k 2L l1 l2 m1 m2i Above gives W 2 (cf correlator Wilson-loop). • h i ↔ Unlike BCFW, tiles lie both inside and out for k 2. • ≥ Spurious boundaries cancel (subtle for bdy diagrams). • Gives ’hedra formulation for correlators. • [Work in progress.] Summary & conclusions

Geometry of amplituhedra and Grassmannians is built into • Feynman rules of twistor action in axial gauge. Framework extends to more general correlahedra. • MHV diagram tiling is imperfect with tiles crossing in and • out of correlahedra. Need to turn it into a better oiled machine for the actual • integrals (unitarity, motives, symbols, cluster algebras, Fuchsian differential equations, integrability, regularization....). The end

Thank You!