Single-Valued Multiple Zeta Values and String Amplitudes

Transcription

1 Single-Valued Multiple eta Values and String Amplitudes Stephan Stieberger, MPP München Workshop "Amplitudes and Periods" Hausdorff Research Institute for Mathematics February 26 - March 2, 208

2 based on: St.St.: Closed superstring amplitudes, single-valued multiple zeta values and the Deligne associator, J. Phys. A47 (204) 5540, [arxiv: ] St.St., T.R. Taylor: Closed string amplitudes as single-valued open string amplitudes, Nucl. Phys. B88 (204) , [arxiv:40.28] Wei Fan, A. Fotopoulos, St.St., T.R. Taylor: Sv-map between Type I and Heterotic Sigma Models, to appear in Nucl. Phys. B, [arxiv:7.0582]

3 Outline Real iterated integral on x <...<x N NY 2 l=2 (RP /{0,, }) N 3 dx l! Y i<j x i x j 0 s ij (x i x j ) n ij, s ij 2 R, n ij 2 periods: MVs decomposition of motivic MVs Complex integral on (CP /{0,, }) N 3 J NY 2 d 2 z l! Y i<j z i z j 0 s ij (z i z j ) n ij (z i z j ) n ij, s ij 2 R, n ij, n ij 2 C N 3 l=2 single-valued MVs Relation: J =sv() for given n ij

4 time N 2 CFT N open strings (Tree Level) Disk A(, 2,...,N; 0 ) (N-3)! independent open string amplitudes C + z z2 z N z N

5 Actually we consider: z =0, z N =, z N = due to PSL(2, R) symmetry ( ) := D( ) Y N 2 j=2 dz j A NQ i<j z ij 0 s ij z, (2) z (2), (3)...z (N 3), (N 2) iterated real integral on (RP /{0,, }) N 3 z N 2 z N 3, 2 S N 3 z N z ij := z i z j z N z 3 z z 2 D( ) = z j 2 R 0 <z (2) <...<z (N 2) < RP \{0,, } N 3

6 Comments: = generalized Euler (Selberg) integral integrates to multiple Gaussian hypergeometric functions: Aomoto-Gelfand hypergeometric functions, GK structures

7 Comments: = generalized Euler (Selberg) integral integrates to multiple Gaussian hypergeometric functions: Aomoto-Gelfand hypergeometric functions, GK structures power series in 0 : Example: z <...<z 5 3Y l=2 dz l! 4Q i<j z ij 0 s ij z 2 z 23 z 4 = s 2 s 45 s 23 s 45 s 34 s 2 s 23 s 5 s 2 s 45 s 45 s 23 + O( 0 0 ) z ij := z i z j yields iterated integrals, which are periods of the moduli space M 0,N of genus zero curves with N ordered marked points integrate to Q-linear combinations of MVs (Brown, Terasoma)

8 ( ) := D( ) Y N 2 j=2 dz j A NQ i<j z ij 0 s ij z, (2) z (2), (3)...z (N 3), (N 2) For given all integrals can be expressed in in terms of (N-3)! dimensional basis For given all integrals can be expressed in in terms of (N-3)! dimensional basis R R 9 = ; fundamental world-sheet disk integrals

9 ( ) := D( ) Y N 2 j=2 dz j A NQ i<j z ij 0 s ij z, (2) z (2), (3)...z (N 3), (N 2) For given all integrals can be expressed in in terms of (N-3)! dimensional basis For given all integrals can be expressed in in terms of (N-3)! dimensional basis R R 9 = ; fundamental world-sheet disk integrals normalization: S := ( ) N 3 03 N F := S i.e.: F 0 =0 =

10 ( ) := D( ) Y N 2 j=2 dz j A NQ i<j z ij 0 s ij z, (2) z (2), (3)...z (N 3), (N 2) For given all integrals can be expressed in in terms of (N-3)! dimensional basis For given all integrals can be expressed in in terms of (N-3)! dimensional basis R R 9 = ; fundamental world-sheet disk integrals normalization: S := ( ) N 3 03 N F := S i.e.: F 0 =0 = F = (N 3)! (N 3)! matrix with rk(f )=(N 3)! F = period matrix of M 0,N private discussion with S. Goncharov

11 S = KLT kernel S[ ] := S[ (2,...,N 2) (2,...,N 2) ] = NY 2 j=2 s,j + Xj k=2 (j,k ) s j,k Bern, Dixon, Perelstein, Rozowsky (998) s ij = 0 (k i + k j ) 2 appears for gravitational amplitude important for double copy constructions

12 S = KLT kernel S[ ] := S[ (2,...,N 2) (2,...,N 2) ] = NY 2 s,j + Xj (j,k ) s j,k j=2 k=2 Bern, Dixon, Perelstein, Rozowsky (998) s ij = 0 (k i + k j ) 2 appears for gravitational amplitude important for double copy constructions Physics: A YM = (N 3)! dimensional vector encompassing all independent field-theory YM subamplitudes A YM ( ), 2 S N 3 A YM ( )=A YM (, (2,...,N 2),N,N) A = (N 3)! dimensional vector encompassing all independent superstring subamplitudes A( ), 2 S N 3 A = FA YM Mafra, Schlotterer, St.St. (20) Broedel, Schlotterer, St.St. (203) F has also physical meaning

13 Observation/Result 8 < X F ( 0 ) = PQexp : n 9 = 2n+ M 2n+ ; Schlotterer, Stieberger, arxiv: organization according to zeta values P =+ X 2 n P 2n, P 2n = F ( 0 ) n 2 n M 2n+ = F ( 0 ) 2n+ Q = + 5 3,5 [M 5,M 3 ] ,7 [M 7,M 3 ] ,3,5 [M 3, [M 5,M 3 ]] +... all information is kept in P and M n,...,n r := (n,...,n r )= X 0<k <...<k r ry l= k n l l, n l 2 N +, n r 2 E.g. N=5:! P 2 = 0 2 s 34 s 45 + s 2 (s 34 s 5 ) s 3 s 24 s 2 s 34 (s 2 + s 23 )(s 23 + s 34 ) s 45 s 5 this form exactly appears in F. Browns decomposition of motivic multiple zeta values coaction gives rise to factorisation of the amplitude

14 Construct closed string amplitude: need a set of complex world-sheet integrals N=4 Complex integral on (CP ) N 3 (thrice punctered sphere) J C NY 2 l=2 d 2 z l! Y i<j z i z j 0 s ij (z i z j ) n ij (z i z j ) n ij, s ij 2 R, n ij, n ij 2

15 z z 2 z z 2 Real iterated integrals vs. complex integrals Recall: we considered the real iterated integral,, 2 S N 3 ( ) := D( ) Y N 2 j=2 dz j A NQ i<j z ij 0 s ij z, (2) z (2), (3)...z (N 3), (N 2) D( ) = z j 2 R 0 <z (2) <...<z (N 2) < (RP /0,, ) N 3 J(, ) := C N 3 Y N 2 j=2 d 2 z j A NQ z ij 0 s ij i<j z, (2) z (2), (3)...z (N 3), (N 2) z, (2) z (2), (3)...z (N 2),N z N 2 z N 3 z N 2 z N 3 z N z N z N z 3 z N z 3

16 z z 2 z z 2 Real iterated integrals vs. complex integrals Recall: we considered the real iterated integral,, 2 S N 3 ( ) := D( ) Y N 2 j=2 dz j A NQ i<j z ij 0 s ij z, (2) z (2), (3)...z (N 3), (N 2) D( ) = z j 2 R 0 <z (2) <...<z (N 2) < (RP /0,, ) N 3 J(, ) := C N 3 Y N 2 j=2 d 2 z j A NQ z ij 0 s ij i<j z, (2) z (2), (3)...z (N 3), (N 2) z, (2) z (2), (3)...z (N 2),N z N 2 z N 3 z N 2 z N 3 z N z N J =sv() z N z 3 z N z 3

17 e.g. N=4: = sv C d 2 z z 2s z ( z 2u z) z =sv 0 dx x s ( x) u complex integral on (CP /{0,, }) N 3 iterated real integral on (RP /{0,, }) N 3 s (s) (u) (t) ( s) ( u) ( t) =sv (s) ( + u) ( + s + u) s = 0 (k + k 2 ) 2 t = 0 (k + k 3 ) 2 u = 0 (k + k 4 ) 2 s 2 ut 3 + O( 04 ) s u 2 ut 3 0 u (4s2 + su +4u 2 ) O( 04 )

18 e.g. N=4: = sv C d 2 z z 2s z ( z 2u z) z =sv 0 dx x s ( x) u complex integral on (CP /{0,, }) N 3 iterated real integral on (RP /{0,, }) N 3 s (s) (u) (t) ( s) ( u) ( t) =sv (s) ( + u) ( + s + u) s = 0 (k + k 2 ) 2 t = 0 (k + k 3 ) 2 u = 0 (k + k 4 ) 2 s 2 ut 3 + O( 04 ) 2 s u 2 ut 3 0 u (4s2 + su +4u 2 ) O( 04 )

19 Complex vs. iterated integrals N=5: 0 z 2,z 3 2C z 2,z 3 2C d 2 z 2 d 2 z 3 d 2 z 2 d 2 z 3 4Q i<j z ij 2s ij z 2 z 23 z 2 z 23 z 34 4Q i<j z ij 2s ij z 2 z 23 z 3 z 32 z 24 z 2,z 3 2C z 2,z 3 2C d 2 z 2 d 2 z 3 d 2 z 2 d 2 z 3 4Q i<j z ij 2s ij z 3 z 32 z 2 z 23 z 34 4Q i<j z ij 2s ij z 3 z 32 z 3 z 32 z 24 C A 0 =sv 0<z 2 <z 3 < 0<z 3 <z 2 < dz 2 dz 3 dz 2 dz 3 4Q i<j 4Q i<j z ij s ij z 2 z 23 z ij s ij z 2 z 23 0<z 2 <z 3 < 0<z 3 <z 2 < dz 2 dz 3 dz 2 dz 3 4Q i<j 4Q i<j z ij s ij z 3 z 32 z ij s ij z 3 z 32 C A

20 Physics: double copy constructions X X M FT =( ) N 3 2S N 3 Ã YM ( ) =A YM (, (2,...,N 2),N,N ) 2S N 3 Ã YM ( ) S[ ] A YM ( )

21 Physics: double copy constructions X X M FT =( ) N 3 2S N 3 Ã YM ( ) =A YM (, (2,...,N 2),N,N ) 2S N 3 Ã YM ( ) S[ ] A YM ( ) A I ( ) =( ) N 3 X 2S N 3 X 2S N 3 ( ) S[ ] A YM ( )

22 Physics: double copy constructions X X M FT =( ) N 3 2S N 3 Ã YM ( ) =A YM (, (2,...,N 2),N,N ) 2S N 3 Ã YM ( ) S[ ] A YM ( ) A I ( ) =( ) N 3 X A HET ( ) =sv A I ( ) 2S N 3 X 2S N 3 ( ) S[ ] A YM ( ) J[ ] =sv( ( )) A HET ( ) =( ) N 3 X 2S N 3 X 2S N 3 J[ ] S[ ] A YM ( ) Taylor, St.St. (204)

23 Physics: double copy constructions X X M FT =( ) N 3 2S N 3 Ã YM ( ) =A YM (, (2,...,N 2),N,N ) 2S N 3 Ã YM ( ) S[ ] A YM ( ) A I ( ) =( ) N 3 X A HET ( ) =sv A I ( ) 2S N 3 X 2S N 3 ( ) S[ ] A YM ( ) J[ ] =sv( ( )) A HET ( ) =( ) N 3 X 2S N 3 X 2S N 3 J[ ] S[ ] A YM ( ) M N =( ) N 3 X 2S N 3 X 2S N 3 Ã HET ( ) S[ ] A YM ( ) Taylor, St.St. (204)

24 F. Brown (203): SVMVs are coefficients of the associator W Deligne introduced associator W formally as: with Ihara action providing formal multiplication rule W = on group-like formal power series in e 0 and e F (e 0,e ) G(e 0,e )=G(e 0,F(e 0,e )e F (e 0,e ) ) F (e 0,e ) =) W (e 0,e )= (e 0,We W ) (e 0,e ) (definition only uses Ihara action)

25 F. Brown (203): SVMVs are coefficients of the associator W Deligne introduced associator W formally as: W = with Ihara action providing formal multiplication rule on group-like formal power series in e 0 and e F (e 0,e ) G(e 0,e )=G(e 0,F(e 0,e )e F (e 0,e ) ) F (e 0,e ) =) W (e 0,e )= (e 0,We W ) (e 0,e ) (definition only uses Ihara action) Drinfeld associator : (e 0,e )= X Deligne associator W: w2{e o,e } (w) w =+ 2 [e 0,e ]+ 3 ([e 0, [e 0,e ]]] [e, [e 0,e ])+ W (e 0,e )=( e 0, e 0 ) (e 0,e )= W (e 0,e )=+2 3 ([e 0, [e 0,e ]]] [e, [e 0,e ])+... (e e n 0...e e n r 0 )= n,...,n r (w ) (w 2 )= (w w 2 ) and (e 0 )=0= (e ) X w2{e 0,e } sv (w) w

26 F. Brown (203): SVMVs are coefficients of the associator W Deligne introduced associator W formally as: W = with Ihara action providing formal multiplication rule on group-like formal power series in e 0 and e F (e 0,e ) G(e 0,e )=G(e 0,F(e 0,e )e F (e 0,e ) ) F (e 0,e ) =) W (e 0,e )= (e 0,We W ) (e 0,e ) (definition only uses Ihara action) Drinfeld associator : (e 0,e )= X Deligne associator W: w2{e o,e } (w) w =+ 2 [e 0,e ]+ 3 ([e 0, [e 0,e ]]] [e, [e 0,e ])+ W (e 0,e )=( e 0, e 0 ) (e 0,e )= W (e 0,e )=+2 3 ([e 0, [e 0,e ]]] [e, [e 0,e ])+... (e e n 0...e e n r 0 )= n,...,n r (w ) (w 2 )= (w w 2 ) and (e 0 )=0= (e ) X w2{e 0,e } sv (w) w

27 F. Brown (203): there is a natural homomorphism: sv : P a sv m! P m per! C unipotent de Rahm MV s a 2A (Goncharov) motivic MV s m 2H (Brown) sv : n,...,n r! sv (n,...,n r ) E.g.: sv (2) = 0 sv (2n + ) = 2 2n+ sv (3, 5) = 0 3 5

28 Note: explicit representation of associators in limit mod (g 0 ) 2 (corresponds to a commutative realization of the Ihara bracket) (e 0,e )= (uv) ( u) ( v) ( u v) u = ad e,v= ad e0 ad x y =[x, y] [e 0,e ] (g 0 ) 2 =[g, g] 2

29 Note: explicit representation of associators in limit mod (g 0 ) 2 (corresponds to a commutative realization of the Ihara bracket) (e 0,e )= (uv) ( u) ( v) ( u v) u = ad e,v= ad e0 ad x y =[x, y] [e 0,e ] (g 0 ) 2 =[g, g] 2 relates to open superstring amplitude Drummond, Ragoucy (203)

30 Note: explicit representation of associators in limit mod (g 0 ) 2 (corresponds to a commutative realization of the Ihara bracket) (e 0,e )= (uv) ( u) ( v) ( u v) u = ad e,v= ad e0 ad x y =[x, y] [e 0,e ] (g 0 ) 2 =[g, g] 2 relates to open superstring amplitude Drummond, Ragoucy (203) ( u) ( v) (u + v) W (e 0,e )=+(uv) (u) (v) ( u v) + [e 0,e ] relates to closed superstring amplitude St.St. (203)

31 Sv-map between type I and heterotic sigma models sv-map can also be anticipated at the D=2 sigma models describing the world-sheet couplings of open and heterotic strings mapping respective interaction terms (or individual Feynman diagrams) S I = d g A µ X µ 2 µ F µ +... background F l = F l (F, D) I = X l F l I I l ln beta-function corresponding to boundary coupling A X µ

32 Sv-map between type I and heterotic sigma models sv-map can also be anticipated at the D=2 sigma models describing the world-sheet couplings of open and heterotic strings mapping respective interaction terms (or individual Feynman diagrams) S I = d g A µ X µ 2 perturbation theory in D=2: µ F µ +... background F l = F l (F, D) I = X l F l I I l = 0 gdf+ O( 0 2 ) beta-function corresponding to boundary coupling A X µ ln simplest case l=: S I = d D F X µ G (, 0 )! 0 G (, 0 )= ln 0 A X µ

33 type I open string sigma-model S I = d g A µ X µ 2 µ F µ +... I = X l F l I I l ln simplest case l=: A X µ

34 type I open string sigma-model closed heterotic string sigma-model S I = d g A µ X µ 2 µ F µ +... Actually the heterotic sigma-model action can be reorganized and mapped onto open string one: S HET = d 2 zg A µ z X µ µ F µ I = X l F l I I l ln simplest case l=: A X µ

35 type I open string sigma-model closed heterotic string sigma-model S I = d g A µ X µ 2 µ F µ +... Actually the heterotic sigma-model action can be reorganized and mapped onto open string one: S HET = d 2 zg A µ z X µ µ F µ After this reorganization the heterotic string perturbation expansion is diagrammatically equivalent to the open string case I = X l F l I I l HET = X l F l I HET l ln ln simplest case l=: simplest case l=: A X µ A z X µ

36 type I open string sigma-model closed heterotic string sigma-model S I = d g A µ X µ 2 µ F µ +... Actually the heterotic sigma-model action can be reorganized and mapped onto open string one: S HET = d 2 zg A µ z X µ µ F µ After this reorganization the heterotic string perturbation expansion is diagrammatically equivalent to the open string case I = X l F l I I l HET = X l F l I HET l ln ln simplest case l=: simplest case l=: A X µ A z X µ Proposal: HET =sv( I )

37 <latexit sha_base64="kcrdntrmtsj4p/sl2e9fs57rreu=">aaafv3icdrbt9swfmcno4xn9ge92ktmrahquqge/aywamklhdabuplem4qyutr7bbqvj5nvs0e9etk8z5+jwevhye87vnbp7/g3txmvynh+zsw+mwvmp4nz+4uwr4tlb84khwtmupgzli58jamjcekqqhi5savbsc/iux+lcfpj0riypntnu3jierqkokktjouphve8vh+ilyb9ppr627k07ns9n2o8srxylo7pmrmfqg3f+jwg0ukgi/sod3y872xcxwugryg2brptpxgtui3d4dlcrbdwpi5jojbduvdjudjysimjgs6y0lsrg+qhhpgznbmedxewxgx8mcwdihmjggw9xafrlou09kmjnri3o/l8kfyf6zctygmstpwjmhlqugyqcvhrhomqcbysakxebbu7bxieriik6ptnvxkabyg9xrrfpwm5aqo+f2mmkvfnpcqq9qtpkl8axnipr5ydye6fpni3stbnhrjczcosso0fvkymtawafuvmme+0mutk6jcsphaeumlbopru5rehbvimnyfvogmawdfuierae9de2w3hy23bepdcm3oxs/sy/g2n5bjhmaoiribldcthvvfd23puzb/wjkdxztbqsgipsp0mxtivbkdgllpbsmtrtrmmz5iauqp5txxnbs9mxn/bd2sqz0j0r/yk6zswcluwx/r7dyldyi7noc4zjmsy6utvretske5sskn3v6rkwbq4payn8vyho0d8rn9hitqs9up2etti3tmsx6vouaf2jctuxlcool7fzo8v/7evlnisiwfmekiz8bbpa3zc6c639vouae8a6phbao/aefaiuwaubybscgi7a4cf4bx6dp3p/gqax3gou2dnqpq34m5olp0h0tfq4a==</latexit> three-loop l =3 (DF)F l,! F l+ Feynman diagram corresponding to D µ F µ 3 F µ 4 µ 3 F µ µ 4,! F 4 DF F F I I 3 = <t <t 2 <t 3 < ln t 2 dt dt 2 dt 3 = t 3 t 32 dt 2 ln +... I HET 3 = C 3Y j= d 2 z j z 2 z 23 ln z 2 2 z 23 z 3 = d 2 z 0 ln +...

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F µ 3 F µ 5 µ 4 F µ 4 µ 3 F µ µ 5,! F 5 DF () F (2) F (3) F (4) <t <t 2 <t 3 <t 4 < ln t 3 dt dt 2 dt 3 dt 4 = t 4 t 42 t 32 dt 3 ln +... C j=4 Y j= d 2 z j z 2 z 23 z 34 ln z 3 2 z 4 z 24 z 23 = d 2 z 2 3 ln +...

39 Remarks sv-map acts on individual graphs (associated to individual terms in effective action) reformulated perturbation theory in terms of identical Feynman diagrams proposal relies on a sv-compatible regularization scheme (tangential base point regularization)

40 Addon: coefficients of an associator W: (reduced) K equation: d dz L e 0,e (z) =L e0,e (z) e0 z + e z with generators e 0 and e of the free Lie algebra g its unique solution can be given as generating series of multiple polylogarithms: L e0,e (z) = X w2{e 0,e } L w (z) w with the symbol w 2 {e 0,e } denoting a non-commutative word w w 2... in the letters w i 2 {e 0,e } L =, L e n 0 = n! lnn z, L e n = n! lnn ( z) Drinfeld associator : (e 0,e ):=L e0,e () = X (e e n 0...e e n r 0 )= n,...,n r (w ) (w 2 )= (w w 2 ) and (e 0 )=0= (e ) w2{e 0,e } (w) w =+ 2 [e 0,e ]+ 3 ([e 0, [e 0,e ]] [e, [e 0,e ])+... F. Brown (2004) defines generating series of SVMPs: L e0,e (z) =L e0, e 0 (z) L e0,e (z) e 0 determined recursively by fixed point equation: ( e 0, e 0 ) e 0 ( e 0, e 0 ) = (e 0,e ) e (e 0,e ) Deligne associator W: W (e 0,e ):=L() = ( e 0, e 0 ) (e 0,e )= X w2{e 0,e } sv (w) w W (e 0,e )=+2 3 ([e 0, [e 0,e ]] [e, [e 0,e ]) +... F. Brown (203)