Differential Equations and Associators for Periods

Differential Equations and Associators for Periods Stephan Stieberger, MPP München Workshop on Geometry and Physics in memoriam of Ioannis Bakas November 2-25, 26 Schloß Ringberg, Tegernsee

based on: St.St., G. Puhlfürst: Differential equations, Associators and Recurrences for Amplitudes, Nucl. Phys. B92 (26) 86-245, [arxiv:57.582] A Feynman Integral and its Recurrences and Associators, Nucl.Phys. B96 (26) 68-93 [arxiv:5.363]

Geometry and Physics numbers or functions physics: physical observables, couplings geometry: geometrical data Periods

What is a period? Kontsevich, Zagier: A period is a complex number whose real and imaginary parts are values of absolutely convergent integrals of rational functions with rational coefficients over domains in R given in polynomial inequalities with rational coefficients p 2= Z 2x 2 apple dx = Z x 2 +y 2 apple dx dy 3 = Z <x<y<z dx dy dz ( x) yz

Periods and differential equations consider family of periods (depending on parameter): e.g. hypergeometric functions 2F apple a, b c ; x = (c) (a) (c a) Z t a ( t) c a ( xt) b satisfy linear differential equations with algebraic coefficients: (generalized) Picard-Fuchs equations (Gauss-Manin systems) e.g. GKZ systems for periods of CY manifolds

systematize for generating a subclass of periods e.g.: construct non commutative words w = w w 2... in the letters w i 2 {e,e } generate series of periods (iterated integrals) w 2 {e,e } L e n (x) := n! ln n x, n2 N, L e w(x) := Z x dt t L w(t), L e w(x) := Z x dt t L w(t), L (x) =. L e = ln( x) L e m e (x) =Li m (x) multiple polylogarithms (MPLs) in one variable alphabet specifies underlying MPLs

define generating series of multiple polylogarithms (MPLs) in one variable X (x) = L w (x) w w2{e,e } unique solution to KZ equation: d dx = e x + e x Z(e,e ) = X Actually, at x= we obtain generating series of MZVs: (w) w =+ 2 [e,e ]+ 3 ([e, [e,e ]] [e, [e,e ]) w2{e,e } + 4 [e, [e, [e,e ]]] 4 [e, [e, [e,e ]]] + [e, [e, [e,e ]]] + 5 4 [e,e ] 2 +... Drinfeld Associator Z(e,e )= (x) (x)

E.g: Z z <...<z 5! 3Y dz l l=2 4Q i<j z ij s ij z 2 z 23 z 4 = 2 + + 2 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( ) z ij := z i z j alpha -expansion of tree-level N-point string scattering: iterated integrals on sphere -> MZVs periods on Riemann surface of genus with N-3 marked points alpha -expansion of one-level N-point string scattering: elliptic iterated integrals -> elliptic MZVs Z dz 4 Z z4 dz 3 Z z3 = 6 + s 2 apple (z 2, ) dz 2 (, ) 8 < : s 2 6 ln(2 ) 3 4 2 + X n 3 Li (q n ) 2 2 Li 3(q n ) 9 = ; + O( 2 ) periods on Riemann surface of genus with N- marked points

Task: find efficient and systematic procedure to gain their - expansions analytic and explicit expressions at each order closed and compact expressions systemize dependence on parameters New techniques for computing -expansions for amplitudes

Generalized hypergeometric functions relevant to both string theory and field-theory pf p (~a; ~ b; z) p F p apple a,...,a p b,...,b p ; z = X m= pq i= p Q j= (a i ) m (b j ) m z m m!, p a i,b j 2 R Pochhammer symbol (raising factorial) (a) n = (a + n) (a) = a (a + )...(a + n ) Let s first discuss: a i,b j 2 R { a i = ã i b j =+ b j Indeed any other pfq hypergeometric function with integer shifts can be expressed in terms of the above by algebraic relations (contiguous relations)

y(x) = p F q apple a,...,a p +b,...,+b q ; x n = p = q + ( + b )( + b 2 )...( + b p )y z ( + a )( + a 2 )...( + a p )y = = x d dx Fuchsian equation with three regular singularities x =,, dg dx = e x + e x g g = B @ (n y y. ) y C A representation in terms of n x n matrices are given in terms of elementary symmetric functions of a,...,a p e,e and b,...,b q

e =........... B... @... Q p n Q p n 2... Q p 2 Q p C A, e = B @.......... p n p n... p 2 p C A elementary symmetric functions: P p = Q p = px i,...,i = i <i 2 <...<i Xp i,...,i = i <i 2 <...<i a i... a i, =,...,p, b i... b i, =,...,p, Q p p =. p = P p Q p, =,...,p, p p = P p p

generalized hypergeometric functions d dx = e x + e x linear differential equations KZ - type equations The unique solution (x) 2 Ch{e,e }i is known as the generating series of multiple polylogarithms (MPLs) in one variable (x) = X w2{e,e }? L w (x) w non-commutative word w alphabet specifies underlying MPLs: [e,e ](x) =+e ln x e ln( x)+ 2 ln2 xe 2 + Li 2 (x) e e [ Li 2 (x)+lnx ln( x) ]e e + 2 ln2 ( x) e 2 + 6 ln3 xe 3 + Li 3 (x) e 2 e [2Li 3 (x)+lnx Li 2 (x) ]e e e + Li 2, (x) e e 2 apple + Li 3 (x) ln x Li 2 (x) 2 ln2 x ln( x) e e 2 + Li,2 (x) e e e apple Li,2 (x)+li 2, (x) 2 ln x ln2 ( x) e 2 e 6 ln3 ( x) e 3 +...

e.g. 3F2: pf p apple in fact: pf p apple a,...,a p +b...,+b p ; x = [e,e ](x), a,...,a 2 +b,...,+b p ; = Z(e,e ), 3F 2 apple a, a 2, a 3 + b, + b 2 ; x = [B,B ](x), e = =+ 3 Li 3 (x)+ 3 Li 2,2 (x) Q 3 Li 4 (x)+... @ A, e = 2 Q 2 Q @ 3 3 2 2 A, = a + a 2 + a 3 b b 2, 2 = a a 2 + a 2 a 3 + a 3 a b b 2, 3 = a a 2 a 3, Q = b + b 2, Q 2 = b b 2.

pf p (~a; ~ b; z) p F p apple a,...,a p b,...,b p ; z = coefficients of - expansions for a subclass of hypergeometric functions give rise to multiple polylogarithms (MPLs) X m= pq i= p Q j= (a i ) m (b j ) m z m m! a i,b j 2 R, p { a i = ã i b j =+ b j Li n,...,n l (z,...,z l )= X z k n <k l <...<k k... zk l l... kn l l Rational values a i,b j 2 R { a i = p i q i + ã i b j = p j q j + b j

What is the underlying differential equation? At rational values (real parameters shifted by p/q): Fuchsian differential equation with q+2 regular singular points at, exp(2 ir/q),, r =,...,q roots of unity d dx = qx i= x A i x i! Schlesinger system MPLs of q-th roots of unity

d dx = qx i= x A i x i! hyperlogarithms are defined recursively from words w built from an alphabet {w,w,...} (with w i ' A i )withl +letters: l = q L w n (x) := n! ln n (x x ), n 2 N, L w n i (x) := n! ln n x x xi x i, apple i apple l, L wi w(x) := Z x dt t x i L w (t), L (x) =. L w n l w l...w n 2 w 2 w n w (x) =( ) l Li nl,...,n x x, x l x x l x,... x 3 x, x 2 x x l x x 2 x x x

A group-like solution taking values in ChAi with the alphabet A = {A,A,...,A q } can be given as formal weighted sum over iterated integrals (with the weight given by the number of iterated integrations) (x) = X L w (x) w w2a Actually: i(x)! (x x i ) A i, i =,,...,q i(x) = X w2a L i w(x) w, L b w i w(x) = Z x b t dt x i L b w(t)

q (independent) associators: Z (x k) (A,A,...,A q )= k (x) (x), k =,...,q Again: (x k ) reg. = Z (x k) (A,...,A q ) k =,...,q q regularized zeta series: Z (x k) (A,A,...,A q )= X w2a xk (w) w, k =,...,q Z (x k) (A,...,A q )=+ln(x k x ) A + X applei6=k ln xk x x i x i A i ln(x x k ) A k +....

Brown: map: w 7! x k (w) x k (w homomorphism for the shuffle product: w) = x k (w w)+ x k ( ww) = x k (w) x k ( w) x k (w k )= ln(x x k ), x k (w )=ln(x k x ), x k (w) =L w (x k ), w /2 A w [ w k A, with w neither beginning in w k nor ending in w enough to compute the map x k for any word w 2 A because every word w can be written uniquely as a linear combination f shu e products of the words w and w k and words which neither begin in w k nor end in w. Z (x k) (A,...,A q )=+ln(x k x ) A + X applei6=k ln xk x x i x i A i ln(x x k ) A k +....

(i) q=2 (parameters shifted by half-integer): apple " a, b 2F 2 + c; z, 2 2F + a, 2 + b # " 2 2 + c ; z, 2 F + a, b # apple 2 + c ; z, 2 F 2 + a, b +c ; z {z } can algebraically be expressed in terms of e.g.: 2F apple a, b 2 + c; z apple 2 2F + a, 2 + b ; z +c {z } elliptic function elliptic function K of first kind 2F apple 2, 2 ; z = 2 K(p z) we want to determine: 2F apple a,b 2 + c ; z = X k k u k (z) coefficients functions are given in terms of harmonic polylogarithsm (HPLs)

2F apple a, b 2 + c; z With: = ' ' 2 ' (y) := 2 F apple a, b 2 + c; z ' 2 (y) :=y d dy ' y = p z p z p z + p z fullfils Fuchs system: d dy = 2X i= y A i y i! with the four singularities y =, y =, y 2 = and A = ab a + b, A = 2c, A 2 = 2c 2a 2b

(x) =+A ln x + A ln(x ) + A 2 [ ln 2 + ln(x + ) ] + A A [ 2 Li 2 (x)+i x ln x ]+A A [ 2 + Li 2 (x)+lnxln( x) ] + A A 2 [ 2 2 Li 2 ( x) ln 2 ln x ]+A 2 A [ 2 2 + Li 2 ( x)+lnx ln( + x) ] + A A 2 [ 2 2 + 2 ln2 2 ln 2 ln( x)+li, (x, ) ] + A 2 A [ 2 2 2 ln2 2+Li, ( x, ) i x {ln 2 ln( + x)} ] + 2 ln2 xa 2 + 2 ln2 (x ) A 2 + 2 [ln 2 ln( + x)]2 A 2 2 +... Z (+) (A,A,A 2 )= (x) (x) Z (+) 2 (A,A,A 2 )= i A +ln2a 2 2 A2 + 2 ln2 2 A 2 2 2 [A,A ]+ 2 2 [A,A 2 ] 2 (ln2 2 2 )[A,A 2 ] i ln 2 A A 2 +...

Z ( ) (A,A,A 2 )= 2 (x) (x) Z ( ) 2 (A,A,A 2 )=+i A +ln2a 2 A2 + 2 ln2 2 A 2 2 [A,A 2 ]+ 2 2 [A,A ]+ 2 (ln2 2 2 )[A,A 2 ] + i ln 2 A A +... ( ) reg. = Z ( ) (Z (+) ) Result: 2F apple a, b 2 + c; z = [A,A,A 2 ](y), 2F apple a, b c + 2 ; = Z ( ) (Z (+) ),

2F apple a, b c + 2 ; z =+ 2 ln2 ya 2 + 6 ln3 ya 3 +[2Li 3 (y) ln yli 2 (y) 2 ln z 2 3 ] A A A apple + 2Li 3 ( y) ln yli 2 ( y)+ 2 2 ln y + 3 2 3 A A 2 A +... universal result: - expansion is identified with normalized solution of the underlying Fuchs system compact and simple result just matrix multiplication of k matrices at degree k each order in given by some combinations of MPLs and grouplike matrix products carrying the information on parameter

generalized hypergeometric functions: generically: singularity structure described by three regular points: KZ equation monodromies are explicitly furnished by the Drinfeld associator specific parameters: after suitable coordinate transformation, properly assigning Lie-algebra and monodromy representations: Schlesinger system with q+2 regular singular points q+ associators (regularized zeta-series) generalized hypergeometric functions: singularity structure described by three regular points: KZ equation multiple Gaussian hypergeometric functions: singularity structure is generically given by divisors. whose monodromy is described by generalized KZ equations of more variables