symmetries of the amplituhedron - (indico)...symmetries of the amplituhedron livia ferro...
TRANSCRIPT
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Symmetries of the amplituhedron
Livia Ferro Ludwig-Maximilians-Universität München
Amplitudes 2017 Higgs Center, The University of Edinburgh, 10.07.2017
Based on: J. Phys. A: Math. Theor. 50 294005 & 10.1007 JHEP03(2016)014 with T. Lukowski, A. Orta, M. Parisi
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Outline
Introduction Symmetries of amplitudes in planar N=4
Yangian symmetry - map to spin chain Symmetries of the amplituhedron
Capelli differential eqs Map to spin chain - Yangian symmetry New on-shell diagrammatics
Conclusions and open questions
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
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Scattering amplitudes in N=4 sYM
Interacting 4d QFT with highest degree of symmetry
massivemassless
sYM
N=4 sYM
planar N=4 sYM
(picture of L. Dixon)
Maximally supersymmetric Yang-Mills theory
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
symmetries and good (geometric) formalism help
Standard methods for computing ampls involved
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Symmetries and ampls in N=4 sYMImportant in discovering the characteristic of ampls
Superconformal symm. Dual superconformal symm.expected
Yangian symmetry
infinite number of „levels“ of generators level-zero generators: level—one generators: + Serre relations
The Yangian Y(g) of the Lie algebra g is generated by j(0) and j(1)
(planar)
* tree-level collinear singularities * broken at loop level
[Witten]
[Drummond,Henn Plefka]
hidden[DHKS]
[j(0)a , j(0)b ] = fabcj(0)c
[j(0)a , j(1)b ] = fabcj(1)c
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
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Yangian symmetry
infinite number of „levels“ of generators level-zero generators: superconformal level—one generators: dual superconformal + Serre relations
The Yangian Y(g) of the Lie algebra g is generated by j and j(1)
Symmetries and ampls in N=4 sYMImportant in discovering the characteristic of ampls
Superconformal symm. Dual superconformal symm.expected hidden(planar)
* tree-level collinear singularities * broken at loop level
[Witten] [DHKS]
[Drummond,Henn Plefka]
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
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Yangian symmetry
!
Hallmark of integrability
Symmetries and ampls in N=4 sYMImportant in discovering the characteristic of ampls
Superconformal symm. Dual superconformal symm.expected hidden(planar)
} Best formalism: Grassmannian
Infinite-dimensional algebra Present also in spin chains „Solvability“ of the theory
}Integrable deformation
of amplitudes (LF, T. Lukowski, C. Meneghelli, J. Plefka, M. Staudacher)
(Drummond, L.F.)
(R. Frassek, N. Kanning, Y. Ko, M. Staudacher; N. Kanning, T. Lukowski, M. Staudacher)
Map of ampls to spin chain
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
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GrassmannianIn momentum twistor space:
c’s: complex parameters forming a kxn matrix (i i+1…i+k-1): determinant of kxk submatrix of c’s space of k-planes in n dimensions = Gr(k,n) Yangian invariant:
(Arkani-Hamed, Bourjaily, Cachazo, Caron-Huot, Cheung, Goncharov, Hodges, Kaplan,Postnikov,Trnka)(Mason,Skinner)
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
}Atree
n,k (Z) =
Idk⇥nc
(12...k)(23...k + 1)...(n1...n+ k � 1)GL(k)
kY
↵=1
�4|4
nX
i=1
c↵iZi
!
J (0)AB =
nX
i=1
ZAi
@
@ZBi
J (1)AB =
X
i<j
ZAi
@
@ZCi
ZCj
@
@ZBi
� (i $ j)
ZAi = (�↵
i , µ̃↵̇i ,�
Ai )
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Map to spin chain Tree amplitudes: states of an integrable spin chain Quantum space = space of functions (distributions) of n copies of momentum supertwistors
V= V1 ⊗…⊗ Vn
This spin chain is integrable How to find Yangian invariants?
Quantum Inverse Scattering Method
(R. Frassek, N. Kanning, Y. Ko, M. Staudacher)
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
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Map to spin chain Introduce auxiliary space Vaux(☐)
Define a monodromy matrix
aux
i (s,vi)
aux
n1 2
…
Define Lax operators
spectral parameter
level-zero generators
(☐,u)
inhomogeneity
M(u)AB = L1(u)AL2(u)...Ln(u)B
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
Li(u, vi)AB = �AB + (u� vi)
�1J (0)Ai B
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Map to spin chainYangian invariants
M(u)ABAn,k = �ABAn,k
Expand to see generators
M(u)AB = �AB +1
uJ (0)A
B +1
u2J (1)A
B + ...
Explicit construction of Yangian invariants
J (0)AB =
nX
i=1
ZAi
@
@ZBi
J (1)AB =
X
i<j
ZAi
@
@ZCi
ZCj
@
@ZBi
� (i $ j)
}
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
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Map to spin chainExplicit construction of Yangian invariants
act according to decompositions
vacuum of V, : k ’s, n-k 1’s permutation and its decomposition in adjacent
transpositions bridge operator
|0i
� = ⌧1 � ... � ⌧p�
Ingredients:
(N. Kanning, T. Lukowski, M. Staudacher)
�4|4(Zi)
Bij(v) = (Zj ·@
@Zi)v =
Zd↵
↵1+ve↵Zj · @
@Zi
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
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Map to spin chain
A4,2 = B23(0)B12(0)B34(0)B23(0)|0i
Explicit construction of Yangian invariants
n=4,k=2
�4,2 =
✓1 2 3 43 4 1 2
◆= (23)(34)(12)(23)
|0i = �(Z1)�(Z2)1314vacuum:
permutation:
=
Zd2⇥4c
(12)(23)(34)(41)GL(2)
Y
k
�4|4 (c · Z)
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
Grassmannian
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Map to spin chain
A4,2 = B23(0)B12(0)B34(0)B23(0)|0i
vac
Explicit construction of Yangian invariants
n=4,k=2
1 2 3 4
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
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Map to spin chain
A4,2 = B23(0)B12(0)B34(0)B23(0)|0i
Explicit construction of Yangian invariants
n=4,k=2
1 2 3 4L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
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Map to spin chain
A4,2 = B23(0)B12(0)B34(0)B23(0)|0i
Explicit construction of Yangian invariants
n=4,k=2
1 2 3 4L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
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Map to spin chain
A4,2 = B23(0)B12(0)B34(0)B23(0)|0i
Explicit construction of Yangian invariants
n=4,k=2
2
1 4
3
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
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Yangian symmetry
!
Hallmark of integrability
Symmetries and ampls in N=4 sYMImportant in discovering the characteristic of ampls
Superconformal symm. Dual superconformal symm.expected hidden(planar)
} Best formalism: Grassmannian
Infinite-dimensional algebra Present also in spin chains „Solvability“ of the theory
}Integrable deformation
of amplitudes (LF, T. Lukowski, C. Meneghelli, J. Plefka, M. Staudacher)
(Drummond, L.F.)
(R. Frassek, N. Kanning, Y. Ko, M. Staudacher; N. Kanning, T. Lukowski, M. Staudacher)
Map of ampls to spin chain
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
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Yangian symmetry
!
Hallmark of integrability
Symmetries and ampls in N=4 sYMImportant in discovering the characteristic of ampls
Superconformal symm. Dual superconformal symm.expected hidden(planar)
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
What about the Amplituhedron?
non-trivial realization!help from spin chain
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Amplituhedronamplitude in planar N=4 sYM
= through canonical form on the amplituhedron space
= volume of the dual amplituhedron
(N. Arkani-Hamed, J. Trnka; N. Arkani-Hamed, Y. Bai, T. Lam)
(picture of A. Gilmore)
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
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Amplituhedronamplitude in planar N=4 sYM
= through canonical form on the amplituhedron space
= volume of the dual amplituhedron
The idea: NMHV tree-level amplitudes = volume of polytope in dual momentum twistor space (Hodges)
(picture of A. Gilmore)
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
(N. Arkani-Hamed, J. Trnka; N. Arkani-Hamed, Y. Bai, T. Lam)
1
2
3
a b
c
= :=Area „Amplitude“=
area of the triangle in dual space„R-invariants“
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AmplituhedronGeneralization of triangle in projective space:
I = 1, 2, ..., k +m
Y I↵ =
nX
a=1
c↵aZIa
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
·YZ2
Z3
Z1
Bosonized momentum twistors
m
k
↵ = 1, 2, ..., k
physics: m=4 tree: k=1 polytope, k>1 more complicated object (which?) loops: similar, more complicated formulae
„Internal" positivity (c) = interior „External" positivity (Z) = convexity
Geometric requirements {
ordered minors > 0
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AmplituhedronGeneralization of triangle in projective space:
I = 1, 2, ..., k +m
„Internal" positivity (c) = interior „External" positivity (Z) = convexity
Geometric requirements {
ordered minors > 0
Y I↵ =
nX
a=1
c↵aZIa
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
·YZ2
Z3
Z1
Bosonized momentum twistors
m
k
↵ = 1, 2, ..., k
How to compute the volume directly in dual space?
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Amplituhedron
Symmetries of Ω ⌦(m)n,k
Capelli differential equations
(det
@
@WA⌫aµ
!
1⌫k+11µk+1
⌦(m)n,k (Y, Z) = 0
(LF, T. Lukowski, A. Orta, M. Parisi)
�k+m(Y↵ �X
a
c↵aZa)Atreen,k (Z) =
Zdm'1...d
m'k
Z�mk(Y, Y0)
Zdk⇥nc
(12...k)(23...k + 1)...(n1...n+ k � 1)
kY
↵=1
GL(m+k) covariance
GL(1) invariance for Z’s and GL(k) covariance for Y’s
(k+1)th order
volume function
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
@2⌦
@ZAi @ZB
j
=@2⌦
@ZBi @ZA
j
@2⌦
@ZAi @Y B
j
=@2⌦
@ZBi @Y A
j
k=1:
(
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Amplituhedron
⌦(m)n,1 =
+1Z
0
m+1Y
A=2
dtA
!m!
(t · Y )m+1
nY
i=m+2
✓ (t · Zi)
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
k=1
t2
t3 𝚹n
𝚹1
Symmetries of Ω ⌦(m)n,k
Capelli differential equations
(
(LF, T. Lukowski, A. Orta, M. Parisi)
�k+m(Y↵ �X
a
c↵aZa)Atreen,k (Z) =
Zdm'1...d
m'k
Z�mk(Y, Y0)
Zdk⇥nc
(12...k)(23...k + 1)...(n1...n+ k � 1)
kY
↵=1
GL(m+k) covariance
GL(1) invariance for Z’s and GL(k) covariance for Y’s
volume function
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Amplituhedron
m=4 <-> physics no need to think about triangulation directly in dual space
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
Symmetries of Ω ⌦(m)n,k
Capelli differential equations
(
(LF, T. Lukowski, A. Orta, M. Parisi)
�k+m(Y↵ �X
a
c↵aZa)Atreen,k (Z) =
Zdm'1...d
m'k
Z�mk(Y, Y0)
Zdk⇥nc
(12...k)(23...k + 1)...(n1...n+ k � 1)
kY
↵=1
GL(m+k) covariance
GL(1) invariance for Z’s and GL(k) covariance for Y’s
volume function
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Amplituhedron
Higher helicity integrand not fully fixed can Yangian symmetry help us?
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
Symmetries of Ω ⌦(m)n,k
Capelli differential equations
(
(LF, T. Lukowski, A. Orta, M. Parisi)
�k+m(Y↵ �X
a
c↵aZa)Atreen,k (Z) =
Zdm'1...d
m'k
Z�mk(Y, Y0)
Zdk⇥nc
(12...k)(23...k + 1)...(n1...n+ k � 1)
kY
↵=1
GL(m+k) covariance
GL(1) invariance for Z’s and GL(k) covariance for Y’s
volume function
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Map to spin chain
New ingredients -> bosonised twistors Z -> auxiliary k-plane Y
Change of vacuum
S(m)k seed
S(m)k =
Zdk⇥k�
(det�)k
kY
↵=1
�m+k
Y A↵ �
kX
i=1
�↵iZAi
!
(LF, T. Lukowski, A. Orta, M. Parisi)
Yangian symmetry: long-standing problem Use spin chain formalism
kY
i=1
�m|m(Zi)
Lax operator Monodromy matrix
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
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⌦(m)n,k =
k(n�k)Y
l=1
Biljl(0)S(m)k
Example n=4,k=2
⌦(m)4,2 = B23(0)B12(0)B34(0)B23(0)S
(m)2
Y
1 k
Yangian symmetry: long-standing problem Use spin chain formalism
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
Map to spin chain
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Example n=4,k=2
⌦(m)4,2 = B23(0)B12(0)B34(0)B23(0)S
(m)2
1 2 3 4
Y
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
Map to spin chain
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12
3
4Y
Example n=4,k=2
⌦(m)4,2 = B23(0)B12(0)B34(0)B23(0)S
(m)2
New on-shell diagrams transformations
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
Map to spin chain
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New on-shell diagrams transformations
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
On-shell diagrammatics
…hold for any k
B
Trivalent vertices and seed:
Mutations for the seed:
Y
1 2 k1 2
3
1 2
3
Y Y
=
A B AB
k=1 Y
A
B
C =
Y
C
A
k=2
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Yangian symmetry
Define a set of functions
M(u)CB ⌦AC(Y, Z) = ⌦A
B(Y, Z)
Yangian generators
Yangian symmetry: long-standing problem Use spin chain formalism
M(u)AB = �AB +1
uJ (0)A
B +1
u2J (1)A
B + ...
⌦AB = (JY )
AB ⌦(m)
n,k
(JY )AB =
kX
↵=1
Y A↵
@
@Y B↵
+ k�AB
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
![Page 33: Symmetries of the amplituhedron - (Indico)...Symmetries of the amplituhedron Livia Ferro Ludwig-Maximilians-Universität München Amplitudes 2017 Higgs Center, The University of Edinburgh,](https://reader035.vdocuments.net/reader035/viewer/2022071408/6100f87a23a28a6030006b90/html5/thumbnails/33.jpg)
Yangian symmetry
Define a set of functions
J (0)AB =
nX
i=1
ZAi
@
@ZBi
+ k�AB
J (1)AB =
X
i<j
ZAi
@
@ZCi
ZCj
@
@ZBi
� (i $ j)
with
generators of Y(gl(m+k))
⌦AB = (JY )
AB ⌦(m)
n,k
(JY )AB =
kX
↵=1
Y A↵
@
@Y B↵
+ k�AB
(J (0))CB ⌦AC = 0
(J (1))CB ⌦AC = 0
Yangian symmetry: long-standing problem Use spin chain formalism
}
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
![Page 34: Symmetries of the amplituhedron - (Indico)...Symmetries of the amplituhedron Livia Ferro Ludwig-Maximilians-Universität München Amplitudes 2017 Higgs Center, The University of Edinburgh,](https://reader035.vdocuments.net/reader035/viewer/2022071408/6100f87a23a28a6030006b90/html5/thumbnails/34.jpg)
Yangian symmetry
Define a set of functions
J (0)AB =
nX
i=1
ZAi
@
@ZBi
+ k�AB
J (1)AB =
X
i<j
ZAi
@
@ZCi
ZCj
@
@ZBi
� (i $ j)
(J (0))CB ⌦AC = 0
(J (1))CB ⌦AC = 0
Matrices are Yangian invariant⌦AB
⌦AB = (JY )
AB ⌦(m)
n,k
(JY )AB =
kX
↵=1
Y A↵
@
@Y B↵
+ k�AB
Yangian symmetry: long-standing problem Use spin chain formalism!
with
}
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017
![Page 35: Symmetries of the amplituhedron - (Indico)...Symmetries of the amplituhedron Livia Ferro Ludwig-Maximilians-Universität München Amplitudes 2017 Higgs Center, The University of Edinburgh,](https://reader035.vdocuments.net/reader035/viewer/2022071408/6100f87a23a28a6030006b90/html5/thumbnails/35.jpg)
Conclusions
Amplituhedron gives a geometric interpretation of the amplitudes for planar N=4 sYM
A lot of work still has to be done!
how to evaluate volume for k>1 and for loops? can Yangian symmetry fix it?
volume function possesses interesting symmetries described via spin chain suitable modification invariant under Y(gl(m+k))
L. Ferro (LMU München) Amplitudes 2017, 10.07.2017