Algebraic index theorems Polyvector fields Formality for cochains - PowerPoint PPT Presentation

Algebraic index theorems Ryszard Nest L ∞ algebras Examples Deformation functor Algebraic index theorems Polyvector fields Formality for cochains Index problem Examples Ryszard Nest Local trace density Formality for chains University of Copenhagen Local trace density I A fly in the ointment Index theorem 20th November 2010 in Lie algebra cohomology Index theorem in Lie algebra cohomology Algebraic index theorems Fedosov quantization Formal geometry Algebraic index theorem

� � L ∞ algebras Algebraic 2-groupoids index theorems Ryszard Nest Data: L ∞ algebras 1 Units G 0 • x Examples γ Deformation functor � • y 2 Arrows G 1 • x Polyvector fields Formality for cochains composable when range of one coincides with the source Index problem of the next one. Examples Local trace 3 Two-morphisms G 2 density • x Formality for chains Local trace density I A fly in the ointment Index theorem in Lie algebra θ − → γ 1 γ 2 cohomology Index theorem in Lie algebra cohomology Algebraic index • y theorems Fedosov quantization with "natural" composition structure (vertical and Formal geometry horisontal). Algebraic index theorem

L ∞ algebras Algebraic index theorems Ryszard Nest Differential graded Lie algebras (DGLA) L ∞ algebras Examples A DGLA ( L , d , [ , ]) is given by the following structure: Deformation functor Polyvector fields • a Z -graded vector space L , Formality for cochains • a differential d : L i → L i + 1 satisfying d 2 = 0, Index problem Examples • a bracket [ − , − ] : L i × L j → L i + j Local trace density These satisfy the following: Formality for chains Local trace density I 1 (graded skewsymmetry) [ a , b ] = − ( − 1 ) deg ( a ) deg ( b ) [ b , a ] . A fly in the ointment Index theorem 2 (graded Jacobi ) in Lie algebra cohomology [ a , [ b , c ]] = [[ a , b ] , c ] + ( − 1 ) deg ( a ) deg ( b ) [ b , [ a , c ]] . Index theorem in Lie algebra cohomology 3 (graded Leibniz) d [ a , b ] = [ da , b ] + ( − 1 ) deg ( a ) [ a , db ] . Algebraic index theorems Fedosov quantization Formal geometry Algebraic index theorem

L ∞ algebras Algebraic index theorems Ryszard Nest 2-groupoid of a DGLA L ∞ algebras Suppose that g is a nilpotent DGLA such that g i = 0 for Examples i < − 1. Deformation functor Polyvector fields Formality for A Maurer-Cartan element of g is an element γ ∈ g 1 satisfying cochains Index problem Examples d γ + 1 Local trace 2 [ γ, γ ] = 0 . (1) density Formality for chains Local trace density I MC 2 ( g ) 0 is the set of Maurer-Cartan elements of g . A fly in the ointment Index theorem in Lie algebra Think of MC 2 ( g ) 0 as the set of flat connections cohomology Index theorem in Lie algebra cohomology d + ad γ Algebraic index theorems Fedosov quantization Formal geometry Algebraic index theorem

L ∞ algebras Algebraic index theorems Ryszard Nest L ∞ algebras The unipotent group exp g 0 acts on the set of Maurer-Cartan Examples elements of g by the gauge equivalences. Deformation functor Polyvector fields arrows Formality for cochains MC 2 ( g ) 1 ( γ 1 , γ 2 ) is the set of gauge equivalences between γ 1 , Index problem Examples γ 2 , with action Local trace density Formality for chains d + ad γ 2 = Ad exp X ( d + ad γ 1 ) . Local trace density I A fly in the ointment Index theorem The composition is given by the product in the group exp g 0 . in Lie algebra cohomology Index theorem in Lie algebra cohomology Algebraic index theorems Fedosov quantization Formal geometry Algebraic index theorem

L ∞ algebras Algebraic index theorems Given γ ∈ MC 2 ( g ) 0 Ryszard Nest L ∞ algebras [ a , b ] γ = [ a , db + [ γ, b ]] . Examples Deformation functor is a Lie bracket [ · , · ] γ on g − 1 . With this bracket g − 1 becomes a Polyvector fields Formality for nilpotent Lie algebra. We denote by exp γ g − 1 the cochains Index problem corresponding unipotent group, and by exp γ the corresponding Examples exponential map g − 1 → exp γ g − 1 . Local trace density Formality for chains 2-morphisms Local trace density I A fly in the ointment MC 2 ( g ) 2 ( exp X , exp Y ) is given by exp γ g − 1 with action Index theorem in Lie algebra cohomology Index theorem in Lie ( exp γ t ) · ( exp X ) = exp ( dt + [ γ, t ]) exp X algebra cohomology Algebraic index theorems Fedosov quantization Formal geometry Algebraic index theorem

L ∞ algebras Algebraic index theorems Ryszard Nest L ∞ algebras To summarize, the data described above forms a 2-groupoid Examples which we denote by MC 2 ( g ) as follows: Deformation functor Polyvector fields Formality for 1 the set of objects is MC 2 ( g ) 0 - Maurer-Cartan elements, or cochains flat connections d + γ Index problem Examples 2 1-morphisms MC 2 ( g ) 1 ( γ 1 , γ 2 ) , are given by the gauge Local trace density transformations between d + γ 1 and d + γ 2 . Formality for chains Local trace density I 3 2-morphisms between exp X , exp Y ∈ MC 2 ( g ) 1 ( γ 1 , γ 2 ) are A fly in the ointment given by MC 2 ( g ) 2 ( exp X , exp Y ) . Index theorem in Lie algebra cohomology Index theorem in Lie algebra cohomology Algebraic index theorems Fedosov quantization Formal geometry Algebraic index theorem

L ∞ algebras Algebraic index theorems Ryszard Nest L ∞ algebras Examples Deformation functor Polyvector fields A morphism of nilpotent DGLA φ : g → h induces a functor Formality for cochains φ : MC 2 ( g ) → MC 2 ( g ) . Index problem However, there are relatively few morphisms of DGLA’s. But, Examples since we have to our disposal a differential, we can weaken our Local trace density conditions, so that they hold up to homotopy. Formality for chains Local trace density I A fly in the ointment Index theorem in Lie algebra cohomology Index theorem in Lie algebra cohomology Algebraic index theorems Fedosov quantization Formal geometry Algebraic index theorem

L ∞ algebras Algebraic index theorems Ryszard Nest L ∞ algebras Examples Deformation functor Polyvector fields Formality for cochains L ∞ algebras Index problem Examples Local trace density Formality for chains Local trace density I A fly in the ointment Index theorem in Lie algebra cohomology Index theorem in Lie algebra cohomology Algebraic index theorems Fedosov quantization Formal geometry Algebraic index theorem

L ∞ algebras Let V be a graded vectors space. and denote by CV the Algebraic index (cofree cocommutative coalgebra) theorems Ryszard Nest ⊕ n S n ( V [ 1 ]) = ⊕ n (Λ n V )[ n ] . L ∞ algebras The coalgebra structure is the one induced from the tensor Examples algebra: Deformation functor Polyvector fields � Formality for ∆( v 1 ⊗ . . . ⊗ v n ) = ( v 1 ⊗ . . . ⊗ v k ) ⊗ ( v k + 1 ⊗ . . . ⊗ v n ) cochains Index problem k Examples Local trace Definition density Formality for chains Local trace density I An L ∞ -structure on a graded vector space V is a codifferential A fly in the ointment Q of degree +1 on the graded coalgebra C(V). Index theorem in Lie algebra cohomology Such a Q is just a collection of linear maps Index theorem in Lie algebra cohomology Q n : S n ( V [ 1 ]) → V [ 1 ] , n ≥ 1 , Algebraic index theorems of degree 1 such that the coderivation Q : S ( V [ 1 ]) → S ( V [ 1 ]) Fedosov quantization Formal geometry induced by the q n ’s by imposing coLeibniz rule is a Algebraic codifferential, i.e. Q 2 = 0. index theorem

L ∞ algebras Algebraic Just to get familiarized with this notion, let us start with the case of only index two operations: theorems Ryszard Nest q 1 : V [ 1 ] → V [ 1 ]; q 2 : S 2 V [ 1 ] → V [ 1 ]; q n = 0 for n>2 . L ∞ algebras Examples Deformation functor Polyvector fields Formality for cochains Index problem Examples Local trace density Formality for chains Local trace density I A fly in the ointment Index theorem in Lie algebra cohomology Index theorem in Lie algebra cohomology Algebraic index theorems Fedosov quantization Formal geometry Algebraic index theorem

� � � � � � � � L ∞ algebras Algebraic Just to get familiarized with this notion, let us start with the case of only index two operations: theorems Ryszard Nest q 1 : V [ 1 ] → V [ 1 ]; q 2 : S 2 V [ 1 ] → V [ 1 ]; q n = 0 for n>2 . L ∞ algebras In this case Q has components: Examples Deformation functor S 2 V [ 1 ] S 3 V [ 1 ] V [ 1 ] Polyvector fields . . . Formality for cochains q 2 q 3 q 3 Q : q 1 q 2 2 2 3 Index problem Examples S 2 V [ 1 ] S 3 V [ 1 ] V [ 1 ] . . . Local trace density Formality for chains Local trace density I A fly in the ointment Index theorem in Lie algebra cohomology Index theorem in Lie algebra cohomology Algebraic index theorems Fedosov quantization Formal geometry Algebraic index theorem