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Hodge Correlators 86 Hodge correlators A.B. Goncharov To Alexander Beilinson for his 50th birthday Contents 1 Introduction 3 1.1 Summary ........................................ 3 1.2 Arithmetic motivation I: special values of L-functions ............... 5 1.3 Arithmetic motivation II: arithmetic analysis of periods .............. 6 1.4 Mixed motives and the motivic Lie algebra . ........ 9 1.5 Pronilpotent completions of fundamental groups of curves............. 10 1.6 Hodgecorrelators................................ 11 1.7 A Feynman integral for Hodge correlators . ......... 13 1.8 Variations of mixed R-Hodge structures by twistor connections . 14 1.9 The twistor transform, Hodge DG algebra, and variations of real MHS . 17 nil 1.10 Mixed R-Hodge structure on π1 via Hodge correlators . 19 1.11 Motivic correlators on curves . ........ 19 1.12 Motivic multiple L-values ............................... 23 1.13 Coda: Feynman integrals and motivic correlators . ............. 25 1.14 Thestructureofthepaper. ...... 25 2 Polydifferential operator ωm 27 2.1 The form ωm(ϕ1, ..., ϕm)anditsproperties ..................... 28 2.2 Twistor definition of the operator ωm. ........................ 29 3 Hodge correlators for curves 32 arXiv:0803.0297v4 [math.AG] 10 Jan 2016 3.1 GreenfunctionsonaRiemannsurface . ....... 32 3.2 Construction of Hodge correlators for curves . ............ 34 3.3 Shuffle relations for Hodge correlators . ......... 37 4 The twistor transform and the Hodge complex functor 39 4.1 The Dolbeaut complex of a variation of Hodge structures. ............. 39 4.2 The Hodge complex functor • ( ).......................... 41 CH − 4.3 The twistor transform and the DGCom structure on Hodge complexes . 44 4.4 Concludingremarks ............................... 52 1 5 Twistor connections and variations of mixed Hodge structures 53 5.1 Green data and twistor connections . ........ 54 5.2 Deligne’s description of triples of opposite filtrations ................ 56 5.3 ProofofTheorem5.7............................... 57 5.4 Conclusions ..................................... 62 5.5 Variations of real mixed Hodge structures. ........... 63 5.6 DGgeneralizations ............................... 63 6 DG Lie algebra and L -algebra of plane decorated trees 65 CH,S ∞ 6.1 A Lie algebra structure on ............................ 65 CH,S 6.2 A commutative differential graded algebra of plane decorated forests . 66 7 The Hodge correlator twistor connection 71 7.1 Preliminaryconstructions . ....... 72 7.2 The Hodge correlator twistor connection . .......... 73 7.3 ProofofTheorem7.8............................... 78 8 The Lie algebra of special derivations 82 8.1 Thelinearalgebraset-up . ..... 82 8.2 Special derivations of the algebra AH,S ........................ 83 8.3 A symmetric description of the Lie algebra of special derivations . 85 9 Variations of mixed R-Hodge structures via Hodge correlators 86 10 Motivic correlators on curves 91 10.1 Themotivicframework . 92 10.2 -motivic fundamental algebras of curves . ...... 93 C 10.3 The Lie algebra of special derivations in the motivic set-up ............ 94 10.4 Motivic correlators on curves . ........ 96 10.5 Examples of motivic correlators . ......... 99 11 Examples of Hodge correlators 102 11.1 Hodge correlators on CP1 S andpolylogarithms . 102 − 11.2 Hodge correlators on elliptic curves are multiple Eisenstein-Kronecker series . 110 12 Motivic correlators on modular curves 115 12.1 Hodge correlators on modular curves and generalized Rankin-Selberg integrals . 115 12.2 Motivic correlators in towers . ........ 118 12.3 Motivic correlators for the universal modular curve . ............... 121 13 Feynman integral for the Hodge correlators 124 13.1 TheFeynmanintegral . .. .. .. .. .. .. .. .. 124 13.2Feynmanrules ................................... 125 13.3 Hodge correlators and the matrix model . ......... 128 Abstract 2 Hodge correlators are complex numbers given by certain integrals assigned to a smooth complex curve. We show that they are correlators of a Feynman integral, and describe the real mixed Hodge structure on the pronilpotent completion of the fundamental group of the curve. We introduce motivic correlators, which are elements of the motivic Lie algebra and whose periods are the Hodge correlators. They describe the motivic fundamental group of the curve. We describe variations of real mixed Hodge structures on a variety by certain connections on the product of the variety by an affine line. We call them twistor connections. Generalising this, we suggest a DG enhancement of the subcategory of Saito’s Hodge complexes with smooth cohomology. We show that when the curve varies, the Hodge correlators are the coefficients of the twistor connection describing the corresponding variation of real MHS. Examples of the Hodge correlators include classical and elliptic polylogarithms, and their generalizations. The simplest Hodge correlators on the modular curves are the Rankin-Selberg integrals. Examples of the motivic correlators include Beilinson’s elements in the motivic cohomology, e.g. the ones delivering the Beilinson - Kato Euler system on modular curves. 1 Introduction 1.1 Summary Let X be a smooth compact complex curve, v0 a non-zero tangent vector at a point s0 of X, and S∗ = s ,...,s a collection of distinct points of X different from s . We introduce Hodge { 1 m} 0 correlators related to this datum. They are complex numbers, given by integrals of certain differential forms over products of copies of X. Let S = S∗ s . When the data (X, S, v ) ∪ { 0} 0 varies, Hodge correlators satisfy a system of non-linear quadratic differential equations. Using this, we show that Hodge correlators encode a variation of real mixed Hodge structures, called below mixed R-Hodge structures. We prove that it coincides with the standard mixed R-Hodge structure on the pronilpotent completion πnil(X S, v ) of the fundamental group of X S. 1 − 0 − The latter was defined, by different methods, by J. Morgan [M], R. Hain [H] and A.A. Beilinson (cf. [DG]). The real periods of πnil(X S, v ) have a well known description via Chen’s iterated 1 − 0 integrals. Hodge correlators give a completely different way to describe them. Mixed Hodge structures are relatives of l-adic Galois representations. There are two equiv- alent definitions of mixed R-Hodge structures: as vector spaces with the weight and Hodge filtrations satisfying some conditions [D], and as representations of the Hodge Galois group [D2]. The true analogs of l-adic representations are Hodge Galois group modules. However usu- ally we describe mixed R-Hodge structures arising from geometry by constructing the weight and Hodge filtrations, getting the Hodge Galois group modules only a posteriori. The Hodge correlators describe the mixed R-Hodge structure on πnil(X S, v ) directly as a module over 1 − 0 the Hodge Galois group. We introduce a Feynman integral related to X. It does not have a rigorous mathematical meaning. However the standard perturbative series expansion procedure provides a collection of its correlators assigned to the data (X, S, v0), which turned out to be convergent finite dimen- sional integrals. We show that they coincide with the Hodge correlators, thus explaining the name of the latter. Moreover: We define motivic correlators. Their periods are the Hodge correlators. Motivic correlators lie in the motivic Lie coalgebra, and describe the motivic fundamental group of X S. The − coproduct in the motivic Lie coalgebra is a new feature, which is missing when we work just 3 The real mixed Hodge structure Hodge correlators − on the pronilpotent completion correlators of a Feynman integral of the fundamental group of X−S related to (X, S) Hodge realization Real periods The motivic fundamental group Motivic correlators − of X−S live in the motivic Lie coalgebra with numbers. We derive a simple explicit formula for the coproduct of motivic correlators. It allows us to perform arithmetic analysis of Hodge correlators. This is one of the essential advantages of the Hodge correlator description of the real periods of πnil(X S), which has a 1 − lot of arithmetic applications. It was available before only for the rational curve case [G7]. The Lie algebra of the unipotent part of the Hodge Galois group is a free graded Lie algebra. Choosing a set of its generators we arrive at a collection of periods of a real MHS. We introduce new generators of the Hodge Galois group, which differ from Deligne’s generators [D2]. The periods of variations mixed R-Hodge structures corresponding to these generators satisfy non- linear quadratic Maurer-Cartan type differential equations. For the subcategory of Hodge-Tate structures they were defined by A. Levin [L]. The Hodge correlators are the periods for this set of generators. We introduce a DG Lie coalgebra ∗ . The category of ∗ -comodules is supposed to be LH,X LH,X a DG-enhancement of the category of smooth R-Hodge sheaves, i.e. the subcategory of Saito’s mixed R-Hodge sheaves whose cohomology are variations of mixed R-Hodge structures. We show that the category of comodules over the Lie coalgebra H0 ∗ is equivalent to the category of LH,X variations of mixed R-Hodge structures. The simplest Hodge correlators for the rational and elliptic curves deliver single-valued ver- sions of the classical polylogarithms and their elliptic counterparts, the classical Eisenstein- Kronecker series [We]. The latter were interpreted by A.A. Beilinson and A. Levin [BL] as periods of variations of mixed R-Hodge structures. More generally, when X = CP1 the Hodge correlators are real periods of variations of mixed Hodge structures related to multiple polylog- arithms. For an elliptic curve E they deliver the multiple Eisenstein-Kronecker series defined in [G1]. When X is a modular curve and S is the set of its cusps, Hodge correlators generalize the Rankin-Selberg integrals. Indeed, one of the simplest of them is the Rankin-Selberg convolution of a pair f1,f2 of weight two cuspidal Hecke eigenforms with an Eisenstein series. It computes, up to certain constants, the special value L(f f , 2). A similar Hodge correlator gives the Rankin- 1 × 2 Selberg convolution of a weight two cuspidal Hecke eigenform f and two Eisenstein series, and computes, up to certain constants, L(f, 2).
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