indico.ph.ed.ac.uk€¦ · Building-Up the Grassmannian Correspondence: On-Shell Varieties The...

Transcript of indico.ph.ed.ac.uk€¦ · Building-Up the Grassmannian Correspondence: On-Shell Varieties The...

Building-Up the Grassmannian Correspondence: On-Shell VarietiesThe Classification of On-Shell (Cluster) Varieties

More to Explore: ‘Color-Dressed’ On-Shell Functions in sYM

Jacob L. BourjailyNiels Bohr Institute

based on work in collaboration with

S. Franco, D. Galloni, and C. Wen;

E. Herrmann, C. Langer, A. McLeod, and J. Trnka;

N. Arkani-Hamed, F. Cachazo, A. Goncharov, A. Postnikov, and J. Trnka

[arXiv:1911.09106] [arXiv:1909.09131] [arXiv:1608.00006] [arXiv:1412.8475] [arXiv:1212.5605]

3/3/2020 Cluster Algebras & Geometry of Amplitudes Higgs Centre Stratifying On-Shell (Cluster) Varieties


Enormous Advances in Understanding Scattering Amplitudes



On-Shell Physics/Grassmannian Geometry Correspondence

fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv

≡∫

ΩC δ(C, p, h)

Important Open Questions

(for math and physics)

• how many functions exist?

(how to name them?)

• what (functional) relations do they satisfy?• what are their (infinite-dimensional) symmetries?

– do these extend to entire amplitudes?• what changes with (Lie-algebra) ‘coloring’?

On-Shell Physics

• on-shell diagrams– bi-colored

, undirected, planar

bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar

bi-colored, undirected, non-planar• physical symmetries

– trivial symmetries (identities)

Grassmannian Geometry

•{strata C∈G(k, n), volume-form ΩC

}– cluster(?) variety

(?)

,(∏

idαiαi

)×JN– 4positroid variety

,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4

• volume-preserving diffeomorphisms– cluster coordinate mutations, . . .

C

⊥

≡ ()

ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv

≡∫

ΩC δ(C, p, h)




(how to name them?)



On-Shell Physics



bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar






(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡ ()

ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv

≡∫

ΩC δ(C, p, h)




(how to name them?)



On-Shell Physics



bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar






(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡ ()

ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)



On-Shell Physics



bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar






(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡ ()

ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)



On-Shell Physics



bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar






(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡ ()

ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)



On-Shell Physics



bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar






(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡ ()

ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)



On-Shell Physics



bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar






(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡ ()

ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)



On-Shell Physics



bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar






(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡ ()

ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)



On-Shell Physics



bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar






(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡ ()

ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)



On-Shell Physics



bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar






(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡ ()

ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)



On-Shell Physics



bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar






(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡ ()

ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)



On-Shell Physics



bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar






(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡ ()

ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)



On-Shell Physics



bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar






(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡ ()

ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)



On-Shell Physics



bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar






(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡ ()

ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)



On-Shell Physics



bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar






(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡ ()

ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)



On-Shell Physics



bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar






(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡ ()

ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)



On-Shell Physics



bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar






(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡ ()

ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)



On-Shell Physics



bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar






(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡ ()

ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)



On-Shell Physics



bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar






(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡ ()

ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)



On-Shell Physics



bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar



⇔Grassmannian Geometry



(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡

1 α8 α5+α8α14 α5α11 0 0 0 0 00 0 0 1 α10α4+α10α13 α4α7 0 0α3α9 0 0 0 0 0 1 α6 α3+α6α12α9 0 α1 α1α11 0 α1α2 α1α2α7 0 1

ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)



On-Shell Physics• on-shell diagrams

– bi-colored


bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar



⇔Grassmannian Geometry



(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡


ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)




– bi-colored


bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar



⇔Grassmannian Geometry•{strata C∈G(k, n), volume-form ΩC

}

– cluster(?) variety

(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡


ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)




– bi-colored


bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar




}


(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡


ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)




– bi-colored


bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar




}


(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4

• volume-preserving diffeomorphisms

– cluster coordinate mutations, . . .

C

⊥

≡


ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)




– bi-colored


bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar

bi-colored, undirected, non-planar

• physical symmetries



}


(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4



C

⊥

≡


ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)




– bi-colored


bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar


• physical symmetries– trivial symmetries (identities)


}


(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4



C

⊥

≡


ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)




– bi-colored


bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar




}


(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡


ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)



On-Shell Physics: planar N=4• on-shell diagrams

– bi-colored


bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar




}


(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡


ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)




– bi-colored


bi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar




}


(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡


ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)




– bi-colored

, undirected, planarbi-colored

, directed

, planar

bi-colored

, undirected

, non-planar

bi-colored

, directed

, non-planar




}


(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡


ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)




– bi-colored, undirected

, planarbi-colored, directed

, planar

bi-colored, undirected

, non-planar

bi-colored, directed

, non-planar




}


(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡


ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)




– bi-colored, undirected, planar

bi-colored, directed , planarbi-colored, undirected, non-planarbi-colored, directed , non-planarbi-colored, undirected, non-planar



}


(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡


ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)








}


(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡


ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)








}


(?)

,(∏

idαiαi


,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡


ΩC ≡(dα1α1∧· · ·∧ dα14

α14

)3/3/2020 Cluster Algebras & Geometry of Amplitudes Higgs Centre Stratifying On-Shell (Cluster) Varieties



fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)








}–

cluster(?) variety

(?)

,(∏

idαiαi

)×JN– 4

positroid variety

,(∏

idαiαi

)

×JN– 4

cluster variety(?)

(?),(∏

idαiαi

)

×JN– 4


C

⊥

≡


ΩC ≡(dα1α1∧· · ·∧ dα14

α14




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)








}–

cluster(?) variety

(?)

,(∏

idαiαi

)×JN– 4

positroid variety ,(∏

idαiαi

)

×JN– 4cluster variety(?)

(?)

,(∏

idαiαi

)

×JN– 4


C

⊥

≡


ΩC ≡(dα1α1∧· · ·∧ dα14

α14




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)








}–

cluster(?) variety

(?)

,(∏

idαiαi

)×JN– 4


idαiαi

)


(?)

,(∏

idαiαi

)

×JN– 4


C

⊥

≡


ΩC ≡(dα1α1∧· · ·∧ dα14

α14




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)








}–

cluster(?) variety

(?)

,(∏

idαiαi

)×JN– 4


idαiαi

)


(?)

,(∏

idαiαi

)

×JN– 4


C

⊥

≡


ΩC ≡(dα1α1∧· · ·∧ dα14

α14




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)






• physical symmetries: the Yangian– trivial symmetries (identities)


}–

cluster(?) variety

(?)

,(∏

idαiαi

)×JN– 4


idαiαi

)


(?)

,(∏

idαiαi

)

×JN– 4


C

⊥

≡


ΩC ≡(dα1α1∧· · ·∧ dα14

α14




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)






• physical symmetries: the Yangian– trivial symmetries (identities)


}–

cluster(?) variety

(?)

,(∏

idαiαi

)×JN– 4


idαiαi

)


(?)

,(∏

idαiαi

)

×JN– 4


C

⊥

≡


ΩC ≡(dα1α1∧· · ·∧ dα14

α14




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)



On-Shell Physics: planar N



fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)






fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)






fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)



On-Shell Physics: non-planar N=4• on-shell diagrams

–

bi-colored, undirected, planarbi-colored, directed , planar


bi-colored, directed , non-planarbi-colored, undirected, non-planar

• physical symmetries: ?– trivial symmetries (identities)


}–

cluster(?) variety

(?)

,(∏

idαiαi

)×JN– 4positroid variety ,

(∏i

dαiαi

)

×JN– 4

cluster variety(?)

(?)

,(∏

idαiαi

)

×JN– 4


C⊥≡

α1 1 0 α2 0 0 0 0 00 0 α3 1 0 α4 0 0 00 0 0 α5 1 α6 0 0 00 0 0 0 0 1 0 α7 α80 0 0 0 0 α9 1 α10 0α11 0 0 0 0 0 0 1 α120 α13 α14 0 0 0 0 0 1

ΩC ≡

(dα1α1∧· · ·∧ dα14

α14




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)




–






}–

cluster(?) variety

(?)

,(∏

idαiαi


(∏i

dαiαi

)

×JN– 4

cluster variety(?)

(?)

,(∏

idαiαi

)

×JN– 4


C⊥≡

α1 1 0 α2 0 0 0 0 00 0 α3 1 0 α4 0 0 00 0 0 α5 1 α6 0 0 00 0 0 0 0 1 0 α7 α80 0 0 0 0 α9 1 α10 0α11 0 0 0 0 0 0 1 α120 α13 α14 0 0 0 0 0 1

ΩC ≡

(dα1α1∧· · ·∧ dα14

α14




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)




–






}–

cluster(?) variety

(?)

,(∏

idαiαi


(∏i

dαiαi

)

×JN– 4

cluster variety(?)

(?)

,(∏

idαiαi

)

×JN– 4


C⊥≡

α1 1 0 α2 0 0 0 0 00 0 α3 1 0 α4 0 0 00 0 0 α5 1 α6 0 0 00 0 0 0 0 1 0 α7 α80 0 0 0 0 α9 1 α10 0α11 0 0 0 0 0 0 1 α120 α13 α14 0 0 0 0 0 1

ΩC ≡

(dα1α1∧· · ·∧ dα14

α14




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)



On-Shell Physics: non-planar N



fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)



On-Shell Physics: non-planar N



fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)




– bi-colored, undirected, planarbi-colored, directed , planarbi-colored, undirected, non-planarbi-colored, directed , non-planarbi-colored, undirected, non-planar



}


(?)

,(∏

idαiαi


(∏i

dαiαi

)×JN– 4cluster variety(?)

(?)

,(∏

idαiαi

)×JN– 4


C⊥≡

α1 1 0 α2 0 0 0 0 00 0 α3 1 0 α4 0 0 00 0 0 α5 1 α6 0 0 00 0 0 0 0 1 0 α7 α80 0 0 0 0 α9 1 α10 0α11 0 0 0 0 0 0 1 α120 α13 α14 0 0 0 0 0 1

ΩC ≡

(dα1α1∧· · ·∧ dα14

α14

)×(1+α2α4α13(α8+α7α12)

)N– 4




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)




(how to name them?)







}


(?)

,(∏

idαiαi


(∏i

dαiαi


(?)

,(∏

idαiαi

)×JN– 4


C⊥≡

α1 1 0 α2 0 0 0 0 00 0 α3 1 0 α4 0 0 00 0 0 α5 1 α6 0 0 00 0 0 0 0 1 0 α7 α80 0 0 0 0 α9 1 α10 0α11 0 0 0 0 0 0 1 α120 α13 α14 0 0 0 0 0 1

ΩC ≡

(dα1α1∧· · ·∧ dα14

α14

)×(1+α2α4α13(α8+α7α12)

)N– 4




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)

Important Open Questions (for math and physics)


(how to name them?)







}


(?)

,(∏

idαiαi


(∏i

dαiαi


(?)

,(∏

idαiαi

)×JN– 4


C⊥≡

α1 1 0 α2 0 0 0 0 00 0 α3 1 0 α4 0 0 00 0 0 α5 1 α6 0 0 00 0 0 0 0 1 0 α7 α80 0 0 0 0 α9 1 α10 0α11 0 0 0 0 0 0 1 α120 α13 α14 0 0 0 0 0 1

ΩC ≡

(dα1α1∧· · ·∧ dα14

α14

)×(1+α2α4α13(α8+α7α12)

)N– 4




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)



(how to name them?)• what (functional) relations do they satisfy?• what are their (infinite-dimensional) symmetries?






}


(?)

,(∏

idαiαi


(∏i

dαiαi


(?)

,(∏

idαiαi

)×JN– 4


C⊥≡

α1 1 0 α2 0 0 0 0 00 0 α3 1 0 α4 0 0 00 0 0 α5 1 α6 0 0 00 0 0 0 0 1 0 α7 α80 0 0 0 0 α9 1 α10 0α11 0 0 0 0 0 0 1 α120 α13 α14 0 0 0 0 0 1

ΩC ≡

(dα1α1∧· · ·∧ dα14

α14

)×(1+α2α4α13(α8+α7α12)

)N– 4




fΓ≡∏

i

(∑hi,qi

∫d3LIPSi

)∏vAv ≡

∫ΩC δ(C, p, h)


• how many functions exist? (how to name them?)







}


(?)

,(∏

idαiαi


(∏i

dαiαi


(?)

,(∏

idαiαi

)×JN– 4


C⊥≡

α1 1 0 α2 0 0 0 0 00 0 α3 1 0 α4 0 0 00 0 0 α5 1 α6 0 0 00 0 0 0 0 1 0 α7 α80 0 0 0 0 α9 1 α10 0α11 0 0 0 0 0 0 1 α120 α13 α14 0 0 0 0 0 1

ΩC ≡

(dα1α1∧· · ·∧ dα14

α14

)×(1+α2α4α13(α8+α7α12)

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