Zoomplitudes @ Brown

Combinatorial Geometries for Scattering Amplitudes

based on collaborations w. Nima Arkani-Hamed, Thomas Lam, Giulio Salvatori, Hugh Thomas

1912.08707, 1912.11764, 1912.12948, 2005.xxxx

+ w. Zhenjie Li, Prashanth Raman, Chi Zhang: 2005.xxxx

Song He (ITP, Beijing)

motivationscombinatorial geometries (w. factorizing bd.) underlying scattering amps in config. space & kinematic space

• kinematic associahedron (bi-adjoint in Mandelstam space) & worldsheet assoc. [Arkani-Hamed, Bai, SH, Yan]

• cluster polytopes, ABCDs of amps through 1-loop & beyond (clusterhedra?) [lectures & talk by Arkani-Hamed]

• gen. string amplitudes, cluster configuration space, binary geometries (old & new)… this talk!

• positive config. space/tropical grassmannian & N=4 SYM clusterology [talks by Cachazo, Drummond, Guevara, Lam, Spradlin …]

ϕ3

ϕ3

• moduli space for conventional & (ambi-)twistor strings [Witten, Berkovits; Cachazo, SH, Yuan; Mason, Skinner; …]

• positive grassmannian , on-shell diagrams etc. for planar N=4 SYM [Postnikov; Arkani-Hamed et al]

ℳg,n

G+(k, n)

• amplituhedron: map from canonical forms in momentum twistor space [Hodges; Arkani-Hamed, Trnka + Thomas;…]

• helicity amps as differential form in momentum space [SH, Zhang] momentum amplituhedron [talk by Ferro]

G+(k, n) →

→

kinematic associahedra

hidden symmetry of amps (invisible in FD's), analog of dual conformal symmetry, manifest by geometry!ϕ3

geometric picture: FD expansion = a special triangulation, others new formula & recursion for amps→ ϕ3

canonical form =pullback of scattering form=bi-adjoint tree amps

e.g.

Ω(𝒜n−3) ϕ3

Ω(𝒜1) = (dss

−dtt

) |s+t=c = ds(1s

+1t

)

Xi,j + Xi+1,j+1 − Xi,j+1 − Xi+1,j = ci,j > 0

: subspace [ABHY]𝒜n−3 {Xi,j ≥ 0} ∩

associahedra (vertices planar cubic trees) in ↔ 𝒦n

a family of (n-3)-dim subspaces: e.g. for 1 ≤ i < j − 1 < n

wave equation & causal diamondsABHY: discrete wave eq in (1+1)-d “kin. spacetime” (mesh):

causal diamond (in-)compatible propagators/diagonals

∂u∂vX(u, v) = c(u, v)

↔

Gauss law:

continuum limit of : space of positive sol. with positive source in a region

factorizing bd. ensured by causal structure!

𝒜n−3

∂(ij)𝒜n−3 = 𝒜L × 𝒜R

cluster polytopes & amplitudes ϕ3

only finite polytopes for Dynkin diagrams gen. associahedra of finite type [Bazier-Matte et al ] (cluster polytopes)

classical: bi-adjoint through 1-loop; higher loops infinite type [talk by Arkani-Hamed]

→

ϕ3 ↔

1-loop bd.: (tree x 1-loop & forward limit)

cut in halves ; new formulas for 1-loop amps!

∂𝒟n = 𝒜 × 𝒟 & 𝒜n−1

→ 𝒟n Ω(𝒟n) → ϕ3

time evolution = “walk” on quiver (mutation on source) v → v′� : X′ �v + Xv − ∑w←v

Xw = cv

new facet ; new vector must lie outside convex hull, stop when span full spaceX′�v ≥ 0

outline

• stringy canonical forms

• binary cluster polytopes & generalized string amplitudes

• (more) binary geometries from gen. permutohedra

• grassmannian stringy integrals & config. spaces [see also SH, Lecheng Ren, Yong Zhang, 2001.09603]

stringy canonical forms [w. N. Arkani-Hamed, T. Lam]

stringy canonical formsvast generalizations of (open-)string amplitudes

• leading order , residues on “massless poles” (+ convergence): controlled by polytope P

• “stringy” properties: meromorphic with“massive poles”, exponential UV softness, “channel duality”

• scattering (saddle-point) equations & twisted (co-)homology

• dual u variables, tropical compactifications, (complex) closed-stringy integrals …

(α′� → 0)

In({s}) = (α′�)n−3 ∫ℳ+0,n

dn−3zz1,2⋯zn,1 ∏

a<b

(za,b)α′�sa,b

positive parametrization, e.g. z3 = 1 + x2 , z4 = 1 + x2 + x3, …, zn−1 = 1 + x2 + ⋯ + xn−2 ⟹

ℐ{p}(X, {c}) = α′�d ∫ℝd>0

d

∏i

dxi

xixα′�Xi

i ∏I

pI(x)−α′�cIintegral of the form w. positive polynomials:

long history (“Euler-Mellin”…) a diff. form exhibits geometries & remarkable properties→ ddXℐ (X)

Minkowski sum

converges iff is inside Newton polytope ,X c N(p)

single polynomial: ℐp(X, c) = α′�d ∫∞

0

d

∏i=1

dxi

xixα′ �Xi

i p(x)−α′�c

limα′�→0

ℐp(X, c) = Ω(c N[p(x)]; X)

trivial to gen. to

converges iff is inside Minkowski sumX

ℐ{p}(X, {c}) = α′�d ∫ℝd>0

d

∏i=1

dxi

xixα′�Xi

i

m

∏I=1

pI(x)−α′ �cI

𝒫 := ∑I

cIN[pI(x)]

equivalently: leading order = volume of dual polytope = tropical function (c.f. [Panzer] on Hepp bound)

limα′�→0

ddXℐ{p}(X, {c}) = Ω(𝒫; X) -deformations of canonical form of any (rational) polytope α′�

(also see [Nilsson, Passare; Berkesch et al] )

string integrals from associahedraABHY assoc. is rigid: shape independent of ; decompose as Minkowski sum ci,j > 0 𝒜n−3 = ∑

1≤i<j−1<n

ci,j 𝒜i,jXi,j + Xi+1,j+1 − Xi,j+1 − Xi+1,j = ci,j

p13 = 1 + y2, p24 = 1 + y3, p14 = 1 + y2 + y2y3 .

I5 = ∫dy2

y2

dy3

y3yα′�X25

2 yα′�X353 (1 + y2)−α′�c13(1 + y2 + y2y3)−α′�c14(1 + y3)−α′�c24

re-discover open-string integrals & moduli space from ABHY! In = ℐ𝒜n−3= ∫

n−2

∏i=2

dyi

yiyα′ �Xi,n

i ∏1≤i<j−1<n

pij(y)−α′�ci, j

Newton polynomial for 𝒜i,j : pij = 1 + yi+1 + yi+1yi+2 + ⋯ + yi+1⋯yj−1 ⟹

two classes of examples(1). stringy integral for gen. associahedra of finite type: 𝒫(Φ) ℐΦ({S}) = (α′�)d ∫A(Φ)+/T

(ω/T ) ∏γ∈Γ

xα′�Sγγ

leading order ABHY cluster polytopes:→ limα′�→0

ddXℐΦ(X, {c}) = Ω(𝒫(Φ∨, c); X)

ABCD: -extension of amps w. factorization at finite e.g. α′� ϕ3 α′� ℐDn→ ℐAm

× ℐDn−m−1, ℐAn−3

× ℐA1× ℐA1

, ℐAn−1

independent ; higher-k gen. of string integrals ( ), non-trivial to compute

[Arkani-Hamed, Lam, Spradlin] tropical [Speyer, Williams] [talks by Cachazo, Drummond, Guevara]

D = (nk) − n SI ℳ+

0,n ∼ G+(2,n)/T

𝒫k,n = ∑I

SIN[ΔI] ↔ G+(k, n)

(2). stringy integral over : G+(k, n)/T ℐk,n({S}) := (α′�)d ∫G+(k,n)/T(ωk,n/T )∏

I

Δα′�sII d := (k − 1)(n − k − 1)

generalized scattering equations for any , “scattering eqs” (saddle points) a diffeomorphism from int. domain to Mink. sum ℐ ⟹ 𝒫

SE : d log (d

∏i=1

xα′ �Xii

m

∏I=1

pI(x)−α′�cI) = 0 ⟹ map ℝd≥0 → 𝒫 : X =

m

∑I=1

cI∂ log pI(x)

∂ log x

theorem: [Arkani-Hamed, Bai, Thomas]y = f(x) diff . 𝒳 → 𝒴 ⟹ pushforward Ω(𝒴; y) = ∑x=f −1(y)

Ω(𝒳; x)

general phenomenon: field-theory limit = pushforward via saddle points @ high-energy limit [Gross, Mende]

limits deeply connected: α′� → 0 & α′� → ∞ Ω(𝒫) = limα′�→0ddX ℐ = ∑sol.

∏i

dxi

xi

gen. CHY formula: limα′�→0

ℐ = Ω(𝒫) = ∫d

∏i=1

dxi

xiδ (Xi − ∑

I

cI∂ log pI

∂ log xi )

e.g. n=5, X2,5 = x (c1,3

1 + x+

c1,4(1 + y)1 + x + xy ), X3,5 = y ( c2,4

1 + y+

c1,4x1 + x + xy ) ⟹ ∑

sol.

dxdyxy

= Ω(𝒜2) = d2X m5

apply to string integral geometric origin of CHY formulas→

string integrals very special: # of sol. (huge) reduction: the smallest # for assoc. (CHY most efficient!)(n − 3)!

theorem: # of saddle points=dim of space of integral functions: twisted homology (cycles) cohomology (forms) ↔

∫d

∏i=1

dxi

xiQ(x) exp(α′�F(x)), F(x) := − ∑

I

cI log pI(x) with , = # of saddle pointsα′� → ∞ dim Hd((ℂ×)d, dF∧)

e.g. cluster case: #(ℬn) = nn, #(𝒞n) = (2n)!!/2, #(𝒢2) = 13, #(𝒟4) = 55Gr. case: #(3,6)=26, #(3,7)=1272, #(3,8)=188112 [Cachazo, Early, Guevara, Mizera; …] higher-k gen. of SE & bi-adjoint amps

all these closely related to intersection theory [Mizera; …] Feynman integrals? [c.f. Lee, Pomeransky; Mastrolia, Mizera; Frellesvig et al;…] [also see talks by Britto, Caron-Huot, Duhr, …]

pullback of : diffeomorphism {∑b≠a

sa,b

za − zb= 0} ℳ+

0,n → 𝒜n−3

complex stringy integralsℐ|⋅|2

{p}(X, {c}) = α′�d ∫ℂd

d

∏i=1

dzidzi

|zi |2 |zi |

2α′�Xi ∏I

|pI(z) |−2α′ �cIclosed strings? first mod-squared integrals:

for , convergence & leading order again controlled by 0 < α′� < ϵ 𝒫 := ∑I

cI N[pI]

more general with non-negative integer shifts ℐ|⋅|2

{p}(X, {c, n}) := ⋯ ∏i

znii ∏

I

pI(z)−nI

Minkowski sum : unbounded polyhedron ( ), canonical form/dual volume still finite!𝒫 n/α′� → ∞

similar unbounded polyhedron for bi-color amps [Herderschee, SH, Teng, Zhang]ϕ3

apply to closed & open string amps w. two orderings: [Carrasco, Mafra, Schlotterer;…]Iclosedn (α |β) & Iopen

n (α |β)

unbounded with facets @ infinity: (both) leading orders = 𝒜n−3 m(α |β)

dual u variables bounded by N facets with defines (d+m)-dim

big polyhedron ; point inside = positive comb. of vertices :

𝒫 WJaSJ ≥ 0 S := (X1, …, Xd, − c1, …, − cm) ∈ ℝd+m →

ℬ S ∈ ℬ Vα SJ = ∑α

VαJ Fα

converges iff , rewrite w. dual variables ( ) ℐ Fα > 0 uα := ∏J pVαJ

J pJ := (x1, ⋯, xd, p1, ⋯, pm)

ℐ(S) = ∫ℝd+

d

∏i=1

dxi

xi

d+m

∏J

pα′�SJJ = ∫ℝd

+

d

∏i=1

dxi

xi

Nv

∏α

uα′�Fαα := ℐ(F)

special case: = simplex ( ), , u variables multiplicative independent

each facet : true for , not for ( c.f. vs. )

ℬ N = Nv = d + m V = W−1

Fα → 0 ⟺ uα → 0 ℐΦ ℐn,k>2 𝒟4 𝒫3,6

configuration space : by u eqs, “curvy polytope” [talk by Lam] Uo, U, U≥0 {uα}1≤α≤N U≥0 := {uα ≥ 0}

binary cluster polytopes & generalized string amplitudes [w. N. Arkani-Hamed, T. Lam, H. Thomas]

cluster config. space & integrals cluster polytopes [Chapton, Fomin, Zelevinsky] Dynkin diagram

ABHY more rigid, canonical realization?

↔

→

U space has same bd. as cluster polytope (algebraically)

binary geometry: all incompatible ( )

gen. of to finite-type cluster algebra

ua → 0 ub → 1 b | |a > 0

ℳ0,n

ua + ∏all b

ub||ab = 1, ∀a

ℐΦ(X) = α′�n ∫U+(Φ)Ω(U+(Φ))

N

∏a=1

uα′�Xaa , etc .cluster stringy integrals over (positive & complex)

binary geometries: gen. open & closed string amps

binary associahedra & string integrals

ask defines positive part

it has (curvy) shape of assoc. , e.g. is a pentagon

ui,j > 0 ⟹ 0 < ui,j < 1 U+n ∼ ℳ+

0,n

𝒜n−3 U+5

for , open-string integrals (in u variables): U+n In = ∫U+

n

Ω(U+n ) ∏

(a,b)

uα′�Xa,ba,b , Ω(U+

n ) =n−3

∏(i,j)

d logui,j

1 − ui,j

here in a triangulation without internal triangle (acyclic quiver) Parke-Taylor form!(n − 3) ui,j ⟹

u eqs: (deg.=0,1): (n-3)-dim solution space configuration space of type A {1 − ui,j = ∏(k,l) cross (i,j)

uk,l} →

any , all incompatible (binary) uk,l → 0 ui,j → 1 ⟹ ∂ui,j→0Un = U(i, i + 1,⋯, j) × U( j, j + 1,⋯, i) [see F. Brown]: algebraically same boundary structure as (holds in )Uo

n = ℳ0,n ⊂ Un ⊂ ℳ0,n 𝒜n−3 ℂ

worldsheet from U space : gauge-inv. description of (u’s cross-ratios of n points) emergence of worldsheet!Un ℳ0,n →

extended u eqs [see also F. Brown]: [a, b |c, d] + [b, c |d, a] = 1, w . [a, b |c, d] := ∏a≤i<b,c≤ j<d

ui,j

(in total eqs, follow from u eqs, which are special case w. ) (n4) ui,j = [i, i + 1 | j, j + 1]

& : all constraints satisfied by cross-ratios [a, b |c, e][a, b |e, d] = [a, b |c, d] ⟹ [a, b |c, d] =(ad)(bc)(ac)(bd)

n points ordered; how to see other orderings from u’s? sign patternsU+n : 0 < [a, b |c, d] < 1 ⟹

extended eqs forbid sign patterns w. both terms <0: exactly connected components of ∏u + ∏u = 1 (n − 1)!/2 Un(ℝ)

identical extended u eqs by monomial transf. of u’s hidden symmetry (automorphism for ) → Sn Un(ℂ)

gen. u eqs from “walk”gen. to any finite-type cluster algebra: only new ingredient compatibility degree (N by N matrix, beyond 0,1)

𝒟4

non-simply laced also by folding: 𝒜2n−3 → 𝒞n−1, 𝒟n → ℬn−1, ℬ3 → 𝒢2, ℰ6 → ℱ4

u eqs equiv. to “local” eqs (Y system [Fomin, Zelevinsky] ): each step (N eqs).YvYv′� = ∏w←v

(1 + Yw)nw, Y :=u

1 − u

concise new def. from “walk” picture (each step ): integer “Green’s function”v → v′�, X′ �v + Xv − ∑w←v

nwXw = cv b | |a := Xa

for all & a given source in initial quiver (all initial , all except ) a b X = 0 c = 0 cb = 1

cluster conf. space (N-n independent) n-dim ; algebraically same bd. structure as {ua + ∏

all b

ub||ab = 1} → Uo(Φ) & U(Φ) 𝒫(Φ)

canonical form for in any acyclic quiver: rationally solve all N u’s by Ω(U+) =n

∏α

d loguα

1 − uαα {uα}

ask all cuts out curvy polytope , e.g. “curvy” cyclohedra ua > 0 U+(Φ) ∼ 𝒫(Φ) U+(ℬ/𝒞)

binary geometries: any , incompatible (in ), factorize by removing node in Dynkin diagramub → 0 ua → 1 ℂe.g. , U(ℬ/𝒞) → U(𝒜) × U(ℬ/𝒞) U(𝒟n) → U(𝒜m) × U(𝒟n−m−1), U(𝒟n−3) × U(𝒜1)2, & U(𝒜n−1)

what about “orderings”? conjecture: connected comp. of sign patterns by extended eqs U(ℝ) ↔ UI + VI = 1

1:1 with mutation relations: for , ; eqs for , etc.𝒜n−3 I : (a, c) → (b, d) n(n − 1)(n2 + 4n − 6)/6 𝒟n

monomial transf. & beyond: new shapes for other orderings, e.g. for , tiled by 4 hexagons + 12 pentagonsℬ2 = 𝒞2

cluster string integralsU space (type A) manifest factorization @ finite : α′� ResXa,b=0In = I(a, a + 1,⋯, b) × I(b, b + 1,⋯, a)

as , : not only , (all incompatible u’s decouple)Xa,b → 0 ua,b → 0 Ω → ΩL × ΩR ∏uα′�X → ∏L

uα′ �X × ∏R

uα′�X

for , closed-string integrals (w. orderings ): Un(ℂ) α & β Icn(α |β) = ∫Un(ℂ)

Ω(α)∏uα′�X (Ω(β)∏uα′ �X)*

for any finite-type :Φ ℐΦ(X) = ∫U+(Φ)Ω(U+(Φ)) ∏

a

uα′ �Xaa , Ω(U+(Φ)) := ∏

α

d loguα

1 − uα

factorization by removing node of Dynkin diagram: factorize nicely!Ω(U+) & ∏uα′�X

factorize as 1-loop tadpoles, as 1-loop amp; exotic ones e.g. ℬn−1/𝒞n−1 𝒟n ℰ6 → 𝒟5, 𝒜5, 𝒜4 × 𝒜1 & 𝒜22 × 𝒜1

“closed stringy” integrals over well-defined for related by monomial transf.UΦ(ℂ) α & β

physical meaning of these integrals (contain string integrals)? e.g. is -deformation of 1-loop integrand for 𝒟n α′� ϕ3

arithmetic geometry for Ucount # of points in : topological prop. of Un(𝔽p) N(p) = (p − 2)(p − 3)⋯(p − n + 2) ⟹ ℳ0,n

• # of connected components/orderings:

• # of saddle points = dim of half-dim twisted cohomology: [BCJ; CHY; Mizera …]

• coef. of # of forms # of top form: [Kleiss, Kuijf]

|N(−1) | = (n − 1)!/2

|N(1) | = |χ | = (n − 3)!

N(p) ↔ d log ⟹ |N(0) | = d log (n − 2)!

for for N(p) = (p − n + −1)n ℬn, N(p) = (p − n − 1)(p − 3)(p − 5)⋯(p − 2n + 1) 𝒞n ⟹

(type : hyperplane arrangements, unknown for )𝒜 & ℬ 𝒞

beyond: (conjecture) quasi-polynomial, e.g. for , 𝒢2 N(p) = (p − 4)2 + 4δp, δp = 0 (p = 2 mod 3) or 1 (p = 1 mod 3)

25 orderings & 13 saddle points; 547 orderings, 55 saddle points for ; similarly for (some) ⟹ 𝒟4 G+(k, n)/T

(more) binary geometries from gen. permutohedra [w. Z. Li, P. Raman, C. Zhang]

stringy integrals for gen. perm.building set = collection of subsets (w. conditions) [Postnikov] d-dim as Minkowski sum of

coordinate simplices for beg for stringy canonical forms w. sum for each

B I ⊂ [0,d] → 𝒫B

ΔI I ∈ B → xI := ∑i∈I

xi ΔI

ℐB(S) := ∫ℝd+

1vol GL(1)

d

∏i=0

dxi

xi ∏I∈B

xα′�SII = ∫ℝd

+

d

∏i=1

dxi

xixα′ �Xi

i

m

∏I

x−α′�CII

projective: (fix e.g. ); leading order of , some examples:∑I∈B SI = 0 x0 = 1 ℐB = Ω(𝒫B; X)

• simplex: all singletons + [0,d], gen. beta function

• associahedron: , open-string integral (type )

• cyclohedron: all cyclic intervals, stringy integral of type !

• permutohedron: all subsets, (rigid) stringy integral for perm. !

B = ℐB = ∏i

Γ(Xi)/Γ(∑i

Xi)

B = {[i, j] | 0 ≤ i ≤ j ≤ d} Ad

B = Bd

B = ℐd := 𝒫d

remarkably, facets of labelled by subsets (factorize as ) 𝒫B I ≠ [0,d] 𝒫BI× 𝒫BI

⟹ N = d + m

d

∏i=1

xα′�Xii

m

∏I

x−α′�CII =

N=d+m

∏I

uα′�XII

ABHY realization of : (for J=1,…,m) u variables: (for I=1,…,N)𝒫B −CJ = VIJ XI ↔ uI = ∏

J

(xJ)VIJ

e.g. simplex : Δd −C[0,d] = ∑i

Xi ↔ ui =xi

∑i xi

permutohedron : (alternating sum) 𝒫d −CJ = ∑I⊂J

( − )|J|−|I|XI ↔ uI = ∏J⊂I

x(−)|J|−|I|

J

any degeneration of with some , e.g. ℐB & 𝒫B ℐd & 𝒫d CI → 0 𝒫2 |C02→0 = 𝒜2

for d=2 ( ):𝒫2 = ℬ2 u0 =x0x012

x01x02, u1 =

x1x012

x01x12, u2 =

x2x012

x02x12, u01 =

x01

x012, u02 =

x02

x012, u12 =

x12

x012

binary geometries revisitedQ: what type of u-eqs needed for binary geometries, i.e. any , all incompatible ?ua → 0 ub → 1

trivial ex: u eqs for simplex ; , no (all compatible) u eqs for (facet)Δd ∑i

ui = 1 ui → 0 u → 1 → Δd−1

d-dim polytope with N facets: binary geometry for d-dim solution space of u eqs of the form

1 − ua = pa({u}) ∏b

ub||ab , for a = 1,⋯, N, w. polynomial satisfying pa({u}) |ua→0 = 1

cluster polytopes (& products) : “perfect” u eqs with ; more (non-trivial) binary geometries?pa({u}) ≡ 1

claim: configuration space of for any gen. permutohedron* (nestohdra & products) is binary!

(generally , but infinite families with perfect u eqs)

ℐB

pa({u}) ≠ 1

config. space of 𝒫d

we prove for : as , all incompatible , remaining u’s satisfy eqs for 𝒫d uI → 0 uJ → 1 𝒫|I|−1 × 𝒫d−|I|

the u eqs for take the form: , with 𝒫d 1 − uI = pI({u})∏J⊅I

uJ||IJ pI({u}) |uI→0 = 1

“compatibility degree”: for (0 for )J | | I = |J | − | I ∩ J | I ⊄ J I ⊂ J

for , e.g. for d=3, pI ≠ 1 | I | < d − 1 1 − u0 = u1u2u3u212u

223u

213u

3123 (1 + u0u023u013u012u03u01u02)

proof: (recall compatible iff or ) as , all for , easy to show

for (1). or (2). ; other u fall into either ( ) or ( ).

uI & uJ I ⊂ J J ⊂ I uI → 0 xi → 0 i ∈ I uJ → 1

I ∩ J = ∅ I ∩ J ≠ ∅, I, J 𝒫|I|−1 J ⊂ I 𝒫d−|I| I ⊂ J

stringy integral for factorizes at finite : ! ⟹ 𝒫d α′� ResXI→0ℐd = ℐ|I|−1 × ℐd−|I|

the proof can be extended to all gen. perm. considered: all binary, always factorizes at finite ℐB α′�

perfect u eqs from 𝒜d & ℬddegenerations of with perfect u eqs: some , new (new eqs from old ones)𝒜d = ℳ+

0,d+3 C → 0 u = ∏u

(1). for , new Ck,l = 0 i ≤ k < l ≤ n u[k,k] = ∏i≤p≤k≤q≤n

u[p,q]

1 − u[k,k] =i−1

∏j=0

u[ j,k−1]

(2). for , new Ck,l = 0 l − k > i u[0,l] =l−i+1

∏j=0

u[ j,l], u[k,n] =n

∏j=k+i−1

u[k,j]

1 − u[0,l] =l+1

∏a=l−i+2

u[a,n] ⋅l+1

∏a=l−i+2

a+i−2

∏b=l+1

u[a,b], 1 − u[k,n] =k+i−2

∏b=k−1

u[0,b] ⋅k+i−2

∏b=k−1

k−1

∏a=max(b−i+2,1)

u[a,b] .

include products of s; similar degenerations of & more examples with etc. 𝒜 ℬd 𝒜 × ℬ

Q: binary geometries/perfect u eqs from degenerating type etc. (no longer linear factors)? 𝒟

Thank you!

• integral Qs: apply to Feynman integrals? recurrence & residues at “massive” poles? beyond polytopes?

• cluster algebra Qs: taming infinity (higher loops, )? U space for infinity? “kinematic spacetime”?

• binary geometry Qs: meaning of config. space & integrals? orderings & complex integrals? classifications?

• physics Qs: geometries for YM/NLSM forms, BCJ vs. projectivity & gravity, 4d amplitude forms, loops? ……

n ≥ 8

outlook