370 INDEX AND NOTATION

Transcription

1 Index and Notation action of a Lie algebra on a commutative! algebra action of a Lie algebra on a chiral algebra action of a Lie algebroid on a chiral algebra 4.5.4, twisted action of a pseudo-tensor category on a category action of a tensor category on a pseudo-tensor category 1.1.6(v) admissible complex of sheaves on X S admissible D-complex on X S algebraic D X -space annihilator augmentation functor, non-degenerate 1.2.5, reliable augmentation functor in compound setting , in D-module setting augmented compound tensor category, functor augmented operad augmented pseudo-tensor category augmented pseudo-tensor functor, unit Batalin-Vilkovisky (BV) algebras BV extension of a Lie algebroid BV quantization of an odd Poisson algebra BV structure on the chiral chain complex BRST reduction, charge, differential: classical , quantum 3.8.9, BRST property , , calculus of variations cdo centralizer center of a chiral algebra central chiral modules Chern classes ch D n Chevalley-Cousin complex of a chiral algebra , relative version Chevalley complex of a Lie algebra, inner 1.4.5, chiral action of a Lie algebra , of a chiral Lie algebroid chiral action of a chiral monoid chiral algebras 3.3.3, commutative 3.3.3, non-unital 3.3.2, universal chiral algebra freely generated by (N, P ) , chiral R Dif -algebras chiral enveloping algebra of a Lie algebra 3.7.1, of a chiral Lie algebroid chiral extension of a Lie algebroid 3.9.6, rigidified chiral homology chiral lattice algebras

2 370 INDEX AND NOTATION chiral Lie algebroids chiral modules chiral L-modules chiral R Dif -modules chiral monoid chiral operations 3.1.1, for (g, K)-modules chiral A-operations chiral L ch -operations chiral product chiral pseudo-tensor structure Clifford algebra, coisson , chiral 3.8.6, linear algebra version coisson algebras , modules , D-module setting 2.6.1, elliptic commutative! algebras, modules commutative D X -algebras complementary quotients compound operad compound pseudo-tensor category 1.3.7, augmented compound pseudo-tensor functor compound tensor category, functor compound tensor product maps , binary connections for Lie algebroids connections on chiral homology 4.5 Contou-Carrère symbol convenient D X -algebra convenient R l -modules coordinate system on a D X -scheme c operations , in D-module setting correlators cotangent complex , 4.1.5, cotorsor , Cousin D-complex Cousin filtration 4.2.1, Cousin spectral sequence 4.2.3, , c-stack, c-rank D X -algebras D-algebra point D-modules: left and right 2.1.1, functoriality 2.1.2, induced D-modules: quasi-induced , maximal constant quotient D-modules: topology at a point x , universal D-modules on R(X), left D-modules on X S, right de Rham-Chevalley complex of a Lie algebroid, inner de Rham complex of a D-module de Rham homology , cohomology D X -scheme 2.3.1, formally smooth, smooth, very smooth D X -scheme: the global space of horizontal sections D X -scheme: the local space of horizontal sections Dolbeault algebras 4.1.3

3 INDEX AND NOTATION 371 Dolbeault D R(X) -algebra Dolbeault resolutions Dolbeault-style algebra Dolbeault-style D R(X) -algebra duality , duality for de Rham cohomology, global , local elliptic morphism, Lie algebroid , coisson algebra enveloping algebra of an operadic algebra , of a Lie algebroid enveloping BV algebra of a BV algebroid enveloping chiral algebra of a chiral Lie algebroid Euler-Lagrange equations factorization algebras 3.4.1, D-module setting 3.4.4, truncated factorization algebras: canonical D-module structure 3.4.7, commutative factorization algebra freely generated by (N, P ) factorization B-modules factorization structure filtration on a chiral algebra, commutative, unital flabby complex of sheaves formal groupoid formally smooth/étale morphism of D X -schemes Fourier-Mukai trnasform (g, K)-modules 2.9.7, chiral structure (g, K)-structure Gelfand-Dikii coisson algebra Gelfand-Kazhdan structure group action on a chiral algebra hamiltonian reduction handsome complexes of!-sheaves on X S Harish-Chandra pair, module Heisenberg Lie algebra Heisenberg group homotopically O X - and D X -flat complexes homotopy unit commutative algebra homotopy unital commutative algebra , BV algebra Hopf chiral algebras ind-scheme induced modules , inner Hom, inner P objects 1.2.1, in augmented sense jet scheme Kac-Moody extension Kashiwara s lemma Knizhnik-Zamolodchikov (KZ) equations lattices, c- and d Lie algebras and modules 1.4.4, D-module setting Lie algebroid 2.9.1

4 372 INDEX AND NOTATION Lie algebroid , D-module setting , elliptic Lie coalgebroid matrix algebra middle de Rham cohomology sheaf h Miura torsor module operads modules over operadic algebras morphisms of DG D X -algebras: semi-free, elementary multijet scheme mutually commuting morphisms of chiral algebras n-coisson algebra n-poisson algebra nice complexes of!-sheaves on X S non-degenerate pairs (F, h ) normally ordered tensor product O-modules on R(X) odd coisson algebra odd Poisson algebra ope algebra, associative, commutative oper operad 1.1.4, of Jacobi type operadic algebra 1.1.6(iii), augmented operator product expansion perfect BV algebra perfect commutative DG algebra perfect complexes Poincaré-Birkhoff-Witt theorem for Lie algebras , twisted case PBW theorem for usual Lie algebroids 2.9.2, PBW theorem for chiral Lie algebroids polydifferential operations 1.4.8, D-module setting pre-factorization algebra projective connections pseudo-tensor category pseudo-tensor category: A-, k-, additive, abelian pseudo-tensor functors, adjointness pseudo-tensor k-category pseudo-tensor product pseudo-tensor structure pseudo-tensor subcategory, full 1.1.6(i) quantization of a coisson algebra , mod t quasi-factorization algebra Ran s space reasonable topological algebra representable pseudo-tensor structure resolutions of commutative D X -algebras rigidification of a Lie algebroid , of a Lie algebroid 2.9.1

5 INDEX AND NOTATION 373 rigidified B-extension A-structure algebras 1.4.1, D-module setting pairing, non-degenerate operations for D-modules 2.2.3, induced case 2.2.4(i) Schouten-Nijenhuis bracket semi-free D X -algebras 4.3.7, semi-free modules 4.1.5, smooth D X -algebras special D X I -module stress-energy tensor Sugawara s construction super complex super conventions I-topology Ξ-topologies: Ξ x , Ξ Lie x Ξ-topologies: Ξ cois x 2.6.3, Ξ as , Ξ RL x , Ξ sp x x tangent Lie algebroid , for a D-scheme: Remark (ii) in , Tate extension: D-module setting 2.7.2, 2.7.3, on a D X -scheme Tate extension: linear algebra setting 2.7.8, chiral approach Tate extension of a Lie algebra , Tate structure on a vector D-bundle Tate vector space, compact, discrete tensor product of chiral algebras tensor product of pseudo-tensor categories topological associative algebra topological commutative algebra topological Lie algebroid transversal quotients twists of chiral algebras unit object in a compound tensor category , strong vector D-bundles 2.1.5, on a D X -scheme vector D X -scheme vertex operator very smooth D X -algebras Virasoro extension Virasoro vector W -algebras Weyl algebra, chiral and coisson 3.8.1, linear algebra version Wick algebra , global θ-datum A 2.4.1, A as x 2.4.8, 3.6.2, 3.6.4, Notation

6 374 INDEX AND NOTATION A as A Lie A(P ) A w x A w (X) B(M) BV, BV u, BV C(A) C(B, A) C ch (X, A) C ch (X, A) Q, C ch (X, A) PQ C ch (X, A, {M s }), C ch (X, A, {M s }) PQ, C ch (X, A, {M s }) PQ C ch (X, A, M) PQ, C ch (X, A, M I ) PQ C ch (X, B, A) PQ CA(X) CA(X) F C l R (L, ), Cr R (L, ), C R(L) C pois (R) CM(R(X)) C cois (R) Comu 1.4.6, Comu D (X) (S) D DR DM(R(X)) End (V ) , 2.5.6(a) Exp(X) gl(v ) 2.5.6(a) gl(v ) 2.7.4, H ch (X, A) H DR (X, M) H DR (X, M) HoC JZ L L , M l, N r M const M X M r (X), M l (X), M(X) M(X)! M(X) ch 3.1.2, M(X) cl M(X, R l ) M(X, A), M(X, A) ch M(X S ) M l (R(X)) O I P ch (A, L) 3.9.6

7 INDEX AND NOTATION 375 P ch (L) P cl (L) P(F ) Q(I) Q(I, m) R(X) S Ŝ Sch D (X) Spec R, L Spf Q Tate(V ), Tate(Y) U (I) U [J/I] U(L) V(a) Z(A) λ I 2.2.2, 3.1.4, ch Ξ as x 2.4.8, Ξ x Ξ Lie x Ξ R x Ξ RL x ch = 1 ch A