combinatorial geometries for scattering amplitudes · 2020-06-03 · kinematic associahedra hidden...
Post on 05-Jul-2020
4 Views
Preview:
TRANSCRIPT
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
top related