A manifestation of the Grothendieck-Teichmueller group in geometry

Transcription

1 A manifestation of the Grothendieck-Teichmueller group in geometry Vasily Dolgushev Temple University Vasily Dolgushev (Temple University) GRT in geometry 1 / 14

2 Kontsevich s conjecture on the action of grt In 1990, V. Drinfeld introduced a (pro-nilpotent) infinite dimensional Lie algebra grt (the Grothendieck-Teichmueller Lie algebra). This Lie algebra is graded by positive integers and, odd n a non-zero element σ n grt of degree n. We call elements {σ n } n odd, Deligne-Drinfeld elements. Vasily Dolgushev (Temple University) GRT in geometry 2 / 14

3 Kontsevich s conjecture on the action of grt In 1990, V. Drinfeld introduced a (pro-nilpotent) infinite dimensional Lie algebra grt (the Grothendieck-Teichmueller Lie algebra). This Lie algebra is graded by positive integers and, odd n a non-zero element σ n grt of degree n. We call elements {σ n } n odd, Deligne-Drinfeld elements. Let X be a smooth algebraic variety over an algebraically closed field k of characteristic 0. O X is the structure sheaf, T X is the tangent sheaf, and OX T X is the sheaf of polyvector fields on X. Ch(X) is the Chern character of T X. Vasily Dolgushev (Temple University) GRT in geometry 2 / 14

4 Kontsevich s conjecture on the action of grt In 1990, V. Drinfeld introduced a (pro-nilpotent) infinite dimensional Lie algebra grt (the Grothendieck-Teichmueller Lie algebra). This Lie algebra is graded by positive integers and, odd n a non-zero element σ n grt of degree n. We call elements {σ n } n odd, Deligne-Drinfeld elements. Let X be a smooth algebraic variety over an algebraically closed field k of characteristic 0. O X is the structure sheaf, T X is the tangent sheaf, and OX T X is the sheaf of polyvector fields on X. Ch(X) is the Chern character of T X. In 1999, M. Kontsevich conjectured that grt acts naturally on the cohomology H (X, OX T X ). The action of σ n coincides (up to non-zero scalar) with the contraction with the n-th component Ch n (X) of Ch(X). Based on joint work arxiv: with Chris Rogers and Thomas Willwacher. Vasily Dolgushev (Temple University) GRT in geometry 2 / 14

5 The Grothendieck-Teichmueller group GRT formal quantization GRT= exp(grt) Kontsevich s graph complex GC Theory of motives Gal(Q/Q) Moduli of curves Vasily Dolgushev (Temple University) GRT in geometry / 14

6 To define the Lie algebra grt, we need... The family of Drinfeld Kohno Lie algebras t m, (m 2). For every m, t m is generated by {t ij = t ji } 1 i j m subject to [t ij, t ik + t jk ] = 0 #{i, j, k} =, [t ij, t kl ] = 0 #{i, j, k, l} = 4. Vasily Dolgushev (Temple University) GRT in geometry 4 / 14

7 To define the Lie algebra grt, we need... The family of Drinfeld Kohno Lie algebras t m, (m 2). For every m, t m is generated by {t ij = t ji } 1 i j m subject to [t ij, t ik + t jk ] = 0 #{i, j, k} =, [t ij, t kl ] = 0 #{i, j, k, l} = 4. and the free Lie algebra lie(x, y) in two symbols x, y. Vasily Dolgushev (Temple University) GRT in geometry 4 / 14

8 The Lie algebra grt consists of... elements σ(x, y) lie(x, y) satisfying σ(y, x) = σ(x, y), σ(x, y) + σ(y, x y) + σ( x y, x) = 0, σ(t 2, t 4 ) σ(t 1 + t 2, t 4 ) + σ(t 12 + t 1, t 24 + t 4 ) σ(t 12, t 2 + t 24 ) + σ(t 12, t 2 ) = 0. Vasily Dolgushev (Temple University) GRT in geometry 5 / 14

9 The Lie algebra grt consists of... elements σ(x, y) lie(x, y) satisfying σ(y, x) = σ(x, y), σ(x, y) + σ(y, x y) + σ( x y, x) = 0, σ(t 2, t 4 ) σ(t 1 + t 2, t 4 ) + σ(t 12 + t 1, t 24 + t 4 ) σ(t 12, t 2 + t 24 ) + σ(t 12, t 2 ) = 0. The Lie bracket on grt is the Ihara bracket: [σ, σ ] Ih := δ σ (σ ) δ σ (σ) + [σ, σ ] lie(x,y), where [, ] lie(x,y) is the usual bracket on lie(x, y) and δ σ is the derivation of lie(x, y) defined by δ σ (x) := 0, δ σ (y) := [y, σ(x, y)]. Vasily Dolgushev (Temple University) GRT in geometry 5 / 14

10 Deligne-Drinfeld elements of grt The above equations have neither linear nor quadratic solutions. The first non-trivial example of an element in grt is σ (x, y) = [x, [x, y]] [y, [y, x]]. Vasily Dolgushev (Temple University) GRT in geometry 6 / 14

11 Deligne-Drinfeld elements of grt The above equations have neither linear nor quadratic solutions. The first non-trivial example of an element in grt is σ (x, y) = [x, [x, y]] [y, [y, x]]. More generally, Proposition (V. Drinfeld, 1990) For every odd integer n there exists a non-zero vector σ n grt of degree n in symbols x and y such that σ n = ad n 1 x (y) +... where... is a sum of Lie words of degrees 2 in the symbol y. {σ n } n odd are called Deligne-Drinfeld elements of grt. Vasily Dolgushev (Temple University) GRT in geometry 6 / 14

12 To define Kontsevich s graph complex GC we need... a set gra n. An element of gra n is a labelled graph Γ with n vertices and with total order on the set of its edges. i 1 i Vasily Dolgushev (Temple University) GRT in geometry 7 / 14

13 To define Kontsevich s graph complex GC we need... a set gra n. An element of gra n is a labelled graph Γ with n vertices and with total order on the set of its edges. i 1 i and the graded vector space Gra(n) (for n 1) spanned by elements of gra n, modulo the relation Γ τ = ( 1) τ Γ ; Γ τ and Γ correspond to the same labelled graph but differ only by permutation τ of edges. For example: 1 i 2 = 1 2 i Vasily Dolgushev (Temple University) GRT in geometry 7 / 14

14 The full graph complex is... fgc := s 2n 2( ) Sn Gra(n) n 1 where s shifts the degree up by 1. [γ, γ ] := γ γ ( 1) γ γ γ γ, γ γ = τ Sh k,n 1 τ(γ 1 γ ), γ s 2n 2( Gra(n)) Sn, γ s 2k 2( Gra(k)) Sk. Vasily Dolgushev (Temple University) GRT in geometry 8 / 14

15 The operation 1 is... 1 : Gra(n) Gra(k) Gra(k + n 1). 2 i 1 i = iv i i 4 = Vasily Dolgushev (Temple University) GRT in geometry 9 / 14

16 The operation 1 is... 1 : Gra(n) Gra(k) Gra(k + n 1). 2 i 1 i = iv i i 4 = i 1 iv 2 i i iv 1 i 4 +

17 The operation 1 is... 1 : Gra(n) Gra(k) Gra(k + n 1). 2 i 1 i = iv i i 4 = i 1 iv 2 i i iv 1 i 4 + iv 1 2 i i 4 + iv 2 1 i i 4 Vasily Dolgushev (Temple University) GRT in geometry 9 / 14

18 Kontsevich s graph complex GC The graph Γ e = 1 2 satisfies the equation [Γ e, Γ e ] = 0. The differential on fgc is := [Γ e, ]. Definition Kontsevich s graph complex GC is the subcomplex of fgc which involves only graphs Γ satisfying these properties: Γ is connected and the complement of every vertex in Γ is also connected. Each vertex of Γ has valency. Vasily Dolgushev (Temple University) GRT in geometry 10 / 14

19 Examples of degree zero cocycles in GC are... γ = γ 5 = +... γ 7 = Vasily Dolgushev (Temple University) GRT in geometry 11 / 14

20 The Lie algebra grt versus Kontsevich s graph complex Theorem (Thomas Willwacher, 2010) We have an isomorphism of Lie algebras H 0 (GC) = grt Under this isomorphism Deligne-Drinfeld elements {σ n } n odd correspond to (non-zero scalar multiples of) classes of the above cycles {γ n } n odd. Vasily Dolgushev (Temple University) GRT in geometry 12 / 14

21 Main Results In paper arxiv: (joint with Chris Rogers and Thomas Willwacher) we introduced a resolution R of the sheaf OX T X of polyvector fields, Vasily Dolgushev (Temple University) GRT in geometry 1 / 14

22 Main Results In paper arxiv: (joint with Chris Rogers and Thomas Willwacher) we introduced a resolution R of the sheaf OX T X of polyvector fields, constructed a map of Lie algebras GC {L -derivations of R}, Vasily Dolgushev (Temple University) GRT in geometry 1 / 14

23 Main Results In paper arxiv: (joint with Chris Rogers and Thomas Willwacher) we introduced a resolution R of the sheaf OX T X of polyvector fields, constructed a map of Lie algebras GC {L -derivations of R}, proved that the action of the class [γ n ] H 0 (GC) (n odd ) on H (X, OX T X ) coincides with the contractions with the n-th component Ch n (X) of the Chern character Ch(X) of T X, and Vasily Dolgushev (Temple University) GRT in geometry 1 / 14

24 Main Results In paper arxiv: (joint with Chris Rogers and Thomas Willwacher) we introduced a resolution R of the sheaf OX T X of polyvector fields, constructed a map of Lie algebras GC {L -derivations of R}, proved that the action of the class [γ n ] H 0 (GC) (n odd ) on H (X, OX T X ) coincides with the contractions with the n-th component Ch n (X) of the Chern character Ch(X) of T X, and showed that there are examples of smooth varieties X for which Ch n (X) acts non-trivially on H (X, OX T X ). Vasily Dolgushev (Temple University) GRT in geometry 1 / 14

25 References THANK YOU! [1] V.A. Dolgushev and C.L. Rogers, Notes on algebraic operads, graph complexes, and Willwacher s construction, Mathematical aspects of quantization, , Contemp. Math., 58, AMS, Providence, RI, 2012; arxiv: [2] V.A. Dolgushev, C.L. Rogers, and T.H. Willwacher, Kontsevich s graph complex, GRT, and the deformation complex of the sheaf of polyvector fields, arxiv: [] T. Willwacher, M. Kontsevich s graph complex and the Grothendieck-Teichmueller Lie algebra, arxiv: Vasily Dolgushev (Temple University) GRT in geometry 14 / 14