Symmetries of the amplituhedron

Transcription

1 Symmetries of the amplituhedron Livia Ferro Ludwig-Maximilians-Universität München Amplitudes 2017 Higgs Center, The University of Edinburgh, Based on: J. Phys. A: Math. Theor & JHEP03(2016)014 with T. Lukowski, A. Orta, M. Parisi

2 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

3 Scattering amplitudes in N=4 sym Standard methods for computing ampls involved symmetries and good (geometric) formalism help Maximally supersymmetric Yang-Mills theory planar N=4 sym N=4 sym sym massless massive (picture of L. Dixon) Interacting 4d QFT with highest degree of symmetry

4 Symmetries and ampls in N=4 sym Important in discovering the characteristic of ampls Superconformal symm. expected [Witten] Dual superconformal symm. (planar) hidden [DHKS] 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) * tree-level collinear singularities * broken at loop level [j a (0),j (0) b ]=f c ab j c (0) [j a (0),j (1) b ]=f c ab j c (1) [Drummond,Henn Plefka]

5 Symmetries and ampls in N=4 sym Important in discovering the characteristic of ampls Superconformal symm. expected [Witten] Dual superconformal symm. (planar) hidden [DHKS] 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) * tree-level collinear singularities * broken at loop level [Drummond,Henn Plefka]

6 Symmetries and ampls in N=4 sym Important in discovering the characteristic of ampls Superconformal symm. expected Dual superconformal symm. (planar) hidden Yangian symmetry! Hallmark of integrability } Integrable deformation of amplitudes (LF, T. Lukowski, C. Meneghelli, J. Plefka, M. Staudacher) }Best formalism: Grassmannian (Drummond, L.F.) Infinite-dimensional algebra Present also in spin chains Solvability of the theory Map of ampls to spin chain (R. Frassek, N. Kanning, Y. Ko, M. Staudacher; N. Kanning, T. Lukowski, M. Staudacher)

7 Grassmannian (Arkani-Hamed, Bourjaily, Cachazo, Caron-Huot, Cheung, Goncharov, Hodges, Kaplan,Postnikov,Trnka)(Mason,Skinner) In momentum twistor space: Z A i =( i, µ i, A i ) A tree n,k (Z) = I d k n c (12...k)(23...k + 1)...(n1...n + k 1)GL(k) ky =1 4 4! nx c i Z i i=1 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: } J (0)A B = X n Zi A i=1 J (1)A B = X i<j Z C i Z B i (i $ j)

8 Map to spin chain (R. Frassek, N. Kanning, Y. Ko, M. Staudacher) Tree amplitudes: states of an integrable spin chain Quantum space = space of functions (distributions) of n copies of momentum supertwistors V= V 1 V n This spin chain is integrable How to find Yangian invariants? Quantum Inverse Scattering Method

9 Map to spin chain Introduce auxiliary space V aux ( ) Define Lax operators L i (u, v i ) A B = A B +(u v i ) 1 J (0)A i B aux (,u) i (s,v i ) inhomogeneity level-zero generators Define a monodromy matrix M(u) A B = L 1 (u) A L 2 (u)...l n (u) B aux spectral parameter 1 2 n

10 Map to spin chain Yangian invariants M(u) A BA n,k = A BA n,k Expand to see generators M(u) A B = A B + 1 u J (0)A B + 1 u 2 J (1)A B +... } J (0)A B = X n Zi A i=1 J (1)A B = X i<j Z C i Z B i (i $ j) Explicit construction of Yangian invariants

11 Map to spin chain (N. Kanning, T. Lukowski, M. Staudacher) Explicit construction of Yangian invariants Ingredients: vacuum of V, 0i: k 4 4 (Z i ) s, n-k 1 s permutation and its decomposition in adjacent transpositions = 1... p bridge operator B ij (v) i ) v = Z d 1+v e i act according to decompositions

12 Map to spin chain Explicit construction of Yangian invariants n=4,k=2 vacuum: 0i = (Z 1 ) (Z 2 ) permutation: 4,2 = = (23)(34)(12)(23) A 4,2 = B 23 (0)B 12 (0)B 34 (0)B 23 (0) 0i Z d 2 4 c Y = (12)(23)(34)(41) GL(2) k 4 4 (c Z) Grassmannian

13 Map to spin chain Explicit construction of Yangian invariants n=4,k=2 A 4,2 = B 23 (0)B 12 (0)B 34 (0)B 23 (0) 0i vac

14 Map to spin chain Explicit construction of Yangian invariants n=4,k=2 A 4,2 = B 23 (0)B 12 (0)B 34 (0)B 23 (0) 0i

15 Map to spin chain Explicit construction of Yangian invariants n=4,k=2 A 4,2 = B 23 (0)B 12 (0)B 34 (0)B 23 (0) 0i

16 Map to spin chain Explicit construction of Yangian invariants n=4,k=2 A 4,2 = B 23 (0)B 12 (0)B 34 (0)B 23 (0) 0i

17 Symmetries and ampls in N=4 sym Important in discovering the characteristic of ampls Superconformal symm. expected Dual superconformal symm. (planar) hidden Yangian symmetry! Hallmark of integrability } Integrable deformation of amplitudes (LF, T. Lukowski, C. Meneghelli, J. Plefka, M. Staudacher) }Best formalism: Grassmannian (Drummond, L.F.) Infinite-dimensional algebra Present also in spin chains Solvability of the theory Map of ampls to spin chain (R. Frassek, N. Kanning, Y. Ko, M. Staudacher; N. Kanning, T. Lukowski, M. Staudacher)

18 Symmetries and ampls in N=4 sym Important in discovering the characteristic of ampls Superconformal symm. expected Dual superconformal symm. (planar) hidden Yangian symmetry! Hallmark of integrability What about the Amplituhedron? non-trivial realization! help from spin chain

19 Amplituhedron (picture of A. Gilmore) amplitude 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)

20 Amplituhedron (picture of A. Gilmore) amplitude 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) The idea: NMHV tree-level amplitudes = volume of polytope in dual momentum twistor space (Hodges) 2 a b Area = := Amplitude = area of the triangle in 1 c 3 R-invariants dual space

21 Amplituhedron Generalization of triangle in projective space: m Z 3 k Y I = nx a=1 c a Z I a Z 1 Y Z 2 Bosonized momentum twistors I =1, 2,...,k+ m =1, 2,...,k Geometric requirements { Internal" positivity (c) = interior External" positivity (Z) = convexity ordered minors > 0 physics: m=4 tree: k=1 polytope, k>1 more complicated object (which?) loops: similar, more complicated formulae

22 Amplituhedron Generalization of triangle in projective space: m Z 3 k Y I = nx a=1 c a Z I a Z 1 Y Z 2 Bosonized momentum twistors I =1, 2,...,k+ m =1, 2,...,k Geometric requirements { Internal" positivity (c) = interior External" positivity (Z) = convexity ordered minors > 0 How to compute the volume directly in dual space?

23 Amplituhedron A tree n,k (Z) = Z d m ' 1...d m ' k Z Symmetries of Ω Z mk (Y,Y 0 ) (LF, T. Lukowski, A. Orta, M. Parisi) GL(m+k) covariance d k n c (12...k)(23...k + 1)...(n1...n + k 1) ( (m) n,k ky =1 k+m (Y X c a Z a ) volume function GL(1) invariance for Z s and GL(k) covariance for Y s a Capelli differential A a µ! 1apple applek+1 1appleµapplek+1 (k+1)th order (m) n,k (Y,Z)=0 A@ZB A@Y j B@ZA 2 B@Y j A (

24 Amplituhedron A tree n,k (Z) = Z d m ' 1...d m ' k Z Symmetries of Ω Z mk (Y,Y 0 ) (LF, T. Lukowski, A. Orta, M. Parisi) GL(m+k) covariance d k n c (12...k)(23...k + 1)...(n1...n + k 1) ( (m) n,k ky =1 k+m (Y X c a Z a ) volume function GL(1) invariance for Z s and GL(k) covariance for Y s a Capelli differential equations k=1 Z+1 (m) n,1 = 0 m+1 Y A=2 dt A! m! (t Y ) m+1 ny i=m+2 (t Z i ) t 3 Θ n Θ 1 t 2

25 Amplituhedron A tree n,k (Z) = Z d m ' 1...d m ' k Z Symmetries of Ω Z mk (Y,Y 0 ) (LF, T. Lukowski, A. Orta, M. Parisi) GL(m+k) covariance Capelli differential equations d k n c (12...k)(23...k + 1)...(n1...n + k 1) ( m=4 <-> physics (m) n,k no need to think about triangulation directly in dual space ky =1 k+m (Y X c a Z a ) volume function GL(1) invariance for Z s and GL(k) covariance for Y s a

26 Amplituhedron A tree n,k (Z) = Z d m ' 1...d m ' k Z Symmetries of Ω Z mk (Y,Y 0 ) (LF, T. Lukowski, A. Orta, M. Parisi) GL(m+k) covariance Capelli differential equations d k n c (12...k)(23...k + 1)...(n1...n + k 1) ( Higher helicity (m) n,k integrand not fully fixed k+m (Y can Yangian symmetry help us? ky =1 X c a Z a ) volume function GL(1) invariance for Z s and GL(k) covariance for Y s a

27 Map to spin chain Yangian symmetry: long-standing problem Use spin chain formalism New ingredients -> bosonised twistors Z -> auxiliary k-plane Y Lax operator Monodromy matrix (LF, T. Lukowski, A. Orta, M. Parisi) S (m) k = ky i=1 Z Change of vacuum m m (Z i ) d k k (det ) k ky =1 m+k S (m) k Y A seed kx i=1 iz A i!

28 Map to spin chain Yangian symmetry: long-standing problem Use spin chain formalism k(n k) (m) n,k = Y l=1 B il j l (0)S (m) k Example n=4,k=2 (m) 4,2 = B 23(0)B 12 (0)B 34 (0)B 23 (0)S (m) 2 Y 1 k

29 Map to spin chain Example n=4,k=2 (m) 4,2 = B 23(0)B 12 (0)B 34 (0)B 23 (0)S (m) 2 Y

30 Map to spin chain Example n=4,k=2 (m) 4,2 = B 23(0)B 12 (0)B 34 (0)B 23 (0)S (m) 2 Y New on-shell diagrams transformations

31 On-shell diagrammatics New on-shell diagrams transformations Trivalent vertices and seed: 3 3 Y k Mutations for the seed: k=1 Y Y k=2 Y Y = A C = C B A B B A hold for any k B A

32 Yangian symmetry Yangian symmetry: long-standing problem Use spin chain formalism Define a set of functions A B =(J Y ) A B (m) n,k kx (J Y ) A B = Y + k B A B M(u) C B A C(Y,Z)= A B(Y,Z) Yangian generators M(u) A B = A B + 1 u J (0)A B + 1 u 2 J (1)A B +...

33 Yangian symmetry Yangian symmetry: long-standing problem Use spin chain formalism Define a set of functions (J (0) ) C B A C =0 (J (1) ) C B A C =0 A B =(J Y ) A B (m) n,k kx (J Y ) A B = Y + k B A with =1 B J (0)A B = X n Zi A i=1 J (1)A B = X i<j Z C i + k A B Z B i (i $ j) generators of Y(gl(m+k))

34 Yangian symmetry Yangian symmetry: long-standing problem Use spin chain formalism! Define a set of functions (J (0) ) C B A C =0 (J (1) ) C B A C =0 A B =(J Y ) A B (m) n,k kx (J Y ) A B = Y + k B A with =1 B J (0)A B = X n Zi A i=1 J (1)A B = X i<j Z C i + k A B Z B i (i $ j) Matrices A B are Yangian invariant

35 Conclusions Amplituhedron gives a geometric interpretation of the amplitudes for planar N=4 sym volume function possesses interesting symmetries described via spin chain suitable modification invariant under Y(gl(m+k)) how to evaluate volume for k>1 and for loops? can Yangian symmetry fix it? A lot of work still has to be done!