arXiv subjects
Publications and source records attributed to Giovanni Felder.
We describe the asymptotic behaviour and the dependence on the regularization of logarithmically divergent integrals of products of meromorphic and antimeromorphic forms on complex manifolds. Our formula is expressed in terms of residues of Dolbeault forms, a notion introduced in this paper. The proof is based on a result on the asymptotic behaviour of Riemann mappings of small domains.
This paper is a sequel to our earlier work [BFPRW], where we study the derived representation scheme DRep_{g}(A) parametrizing the representations of a Lie algebra A in a finite-dimensional reductive Lie algebra g. In [BFPRW], we defined two canonical maps Tr_{g}(A): HC^{(r)}(A) \to \H[\DRep_{g}(A)]^G and Φ_{g}(A): H[\DRep_{g}(A)]^G \to H[\DRep_{h}(A)]^W called the Drinfeld trace and the derived Harish-Chandra homomorphism, respectively. In this paper, we give an explicit formula for the Drinfeld trace in terms of Chern-Simons classes of a canonical g-torsor associated to the pair (A, g). Our construction is inspired by (and, in a sense, dual to) the classical construction of `additive regulator maps' due to Beilinson and Feigin. As a consequence, we show that, if A is an abelian Lie algebra, the composite map Phi_{g}(A) Tr_{g}(A) is represented by a canonical differential operator acting on differential forms on Sym(A) and depending only on the Cartan data (h, W, P), where P is a W-invariant polynomial on h. We derive a combinatorial formula for this operator that plays an important role in the study of derived commuting schemes in [BFPRW].
We study the derived representation scheme DRep_n(A) parametrizing the n-dimensional representations of an associative algebra A over a field of characteristic zero. We show that the homology of DRep_n(A) is isomorphic to the Chevalley-Eilenberg homology of the current Lie coalgebra gl_n^*(C) defined over a Koszul dual coalgebra of A. We extend this isomorphism to representation schemes of Lie algebras: for a finite-dimensional reductive Lie algebra g, we define the derived affine scheme DRep_g(a) parametrizing the representations (in g) of a Lie algebra a; we show that the homology of DRep_g(a) is isomorphic to the Chevalley-Eilenberg homology of the Lie coalgebra g^*(C), where C is a cocommutative DG coalgebra Koszul dual to the Lie algebra a. We construct a canonical DG algebra map Φ_g(a) : DRep_g(a)^G -> DRep_h(a)^W, which is a homological extension of the classical restriction homomorphism. We call Φ_g(a) a derived Harish-Chandra homomorphism. We conjecture that, for a two-dimensional abelian Lie algebra a, the derived Harish-Chandra homomorphism is a quasi-isomorphism, and provide some evidence for this conjecture. For any complex Lie algebra g, we compute the Euler characteristic of DRep_g(a)^G in terms of matrix integrals over G and compare it to the Euler characteristic of DRep_h(a)^W.This yields an interesting combinatorial identity, which we prove for gl_n and sl_n (for all n). Our identity is analogous to the classical Macdonald identity, and our quasi-isomorphism conjecture is analogous to the strong Macdonald conjecture proved by S.Fishel, I.Grojnowski and C.Teleman. We explain this analogy by giving a new homological interpretation of Macdonald's conjectures in terms of derived representation schemes, parallel to our Harish-Chandra quasi-isomorphism conjecture.
Gaudin hamiltonians form families of r-dimensional abelian Lie subalgebras of the holonomy Lie algebra of the arrangement of reflection hyperplanes of a Coxeter group of rank r. We consider the set of principal Gaudin subalgebras, which is the closure in the appropriate Grassmannian of the set of spans of Gaudin hamiltonians. We show that principal Gaudin subalgebras form a smooth projective variety isomorphic to the De Concini-Procesi compactification of the projectivized complement of the arrangement of reflection hyperplanes.
We formalize the construction by Batalin and Vilkovisky of a solution of the classical master equation associated with a regular function on a nonsingular affine variety (the classical action). We introduce the notion of stable equivalence of solutions and prove that a solution exists and is unique up to stable equivalence. A consequence is that the associated BRST cohomology, with its structure of Poisson_0-algebra, is independent of choices and is uniquely determined up to unique isomorphism by the classical action. We give a geometric interpretation of the BRST cohomology sheaf in degree 0 and 1 as the cohomology of a Lie-Rinehart algebra associated with the critical locus of the classical action. Finally we consider the case of a quasi-projective varieties and show that the BRST sheaves defined on an open affine cover can be glued to a sheaf of differential Poisson_0-algebras.
Some 15 years ago M. Kontsevich and A. Rosenberg [KR] proposed a heuristic principle according to which the family of schemes ${Rep_n(A)}$ parametrizing the finite-dimensional represen- tations of a noncommutative algebra A should be thought of as a substitute or "approximation" for Spec(A). The idea is that every property or noncommutative geometric structure on A should induce a corresponding geometric property or structure on $Rep_n(A)$ for all n. In recent years, many interesting structures in noncommutative geometry have originated from this idea. In practice, however, if an associative algebra A possesses a property of geometric nature (e.g., A is a NC complete intersection, Cohen-Macaulay, Calabi-Yau, etc.), it often happens that, for some n, the scheme $Rep_n(A)$ fails to have the corresponding property in the usual algebro-geometric sense. The reason for this seems to be that the representation functor $Rep_n$ is not "exact" and should be replaced by its derived functor $DRep_n$ (in the sense of non-abelian homological algebra). The higher homology of $DRep_n(A)$, which we call representation homology, obstructs $Rep_n(A)$ from having the desired property and thus measures the failure of the Kontsevich-Rosenberg "approximation." In this paper, which is mostly a survey, we prove several results confirming this intuition. We also give a number of examples and explicit computations illustrating the theory developed in [BKR] and [BR].
The theory of polynomials orthogonal with respect to one inner product is classical. We discuss the extension of this theory to multiple inner products. Examples include the Lamé and Heine-Stieltjes polynomials.
The third author recently proved that the Shoikhet-Dolgushev L-infinity-morphism from Hochschild chains of the algebra of smooth functions on manifold to differential forms extends to cyclic chains. Localization at a solution of the Maurer-Cartan equation gives an isomorphism, which we call character map, from the periodic cyclic homology of a formal associative deformation of the algebra of functions to de Rham cohomology. We prove that the character map is compatible with the Gauss-Manin connection, extending a result of Calaque and Rossi on the compatibility with the cap product. As a consequence, the image of the periodic cyclic cycle 1 is independent of the deformation parameter and we compute it to be the A-roof genus of the manifold. Our results also imply the Tamarkin-Tsygan index Theorem.
In this paper we prove a conjecture of B. Shoikhet which claims that two quantization procedures arising from Fourier dual constructions actually coincide.
Gaudin subalgebras are abelian Lie subalgebras of maximal dimension spanned by generators of the Kohno-Drinfeld Lie algebra t_n. We show that Gaudin subalgebras form a variety isomorphic to the moduli space of stable curves of genus zero with n+1 marked points. In particular, this gives an embedding of the moduli space in a Grassmannian of (n-1)-planes in an n(n-1)/2-dimensional space. We show that the sheaf of Gaudin subalgebras over the moduli space is isomorphic to a sheaf of twisted first order differential operators. For each representation of the Kohno--Drinfeld Lie algebra with fixed central character, we obtain a sheaf of commutative algebras whose spectrum is a coisotropic subscheme of a twisted version of the logarithmic cotangent bundle of the moduli space.
We prove a version of Kontsevich's formality theorem for two subspaces (branes) of a vector space $X$. The result implies in particular that the Kontsevich deformation quantizations of $\mathrm{S}(X^*)$ and $\wedge(X)$ associated with a quadratic Poisson structure are Koszul dual. This answers an open question in Shoikhet's recent paper on Koszul duality in deformation quantization.
Recently the first two authors constructed an L-infinity morphism using the S^1-equivariant version of the Poisson Sigma Model (PSM). Its role in deformation quantization was not entirely clear. We give here a "good" interpretation and show that the resulting formality statement is equivalent to formality on cyclic chains as conjectured by Tsygan and proved recently by several authors.
We compute the weight of a Kontsevich graph in deformation quantization. Up to rationals, the result is Zeta(3)^2/Pi^6.
A simple integral formula as an iterated residue is presented for the Baker-Akhiezer function related to $A_n$ type root system both in the rational and trigonometric cases. We present also a formula for the Baker-Akhiezer function as a Selberg-type integral and generalise it to the deformed $A_{n,1}$-case. These formulas can be interpreted as new cases of explicit evaluation of Selberg-type integrals.
Let D be a holomorphic differential operator acting on sections of a holomorphic vector bundle on an n-dimensional compact complex manifold. We prove a formula, conjectured by Feigin and Shoikhet, for the Lefschetz number of D as the integral over the manifold of a differential form. The class of this differential form is obtained via formal differential geometry from the canonical generator of the Hochschild cohomology of the algebra of differential operators in a formal neighbourhood of a point. If D is the identity, the formula reduces to the Riemann--Roch--Hirzebruch formula.
Kontsevich's formality theorem states that the differential graded Lie algebra of multidifferential operators on a manifold M is L-infinity-quasi-isomorphic to its cohomology. The construction of the L-infinity map is given in terms of integrals of differential forms on configuration spaces of points in the upper half-plane. Here we consider configuration spaces of points in the disk and work equivariantly with respect to the rotation group. This leads to considering the differential graded Lie algebra of multivector fields endowed with a divergence operator. In the case of R^d with standard volume form, we obtain an L-infinity morphism of modules over this differential graded Lie algebra from cyclic chains of the algebra of functions to multivector fields. As a first application we give a construction of traces on algebras of functions with star-products associated with unimodular Poisson structures. The construction is based on the Batalin--Vilkovisky quantization of the Poisson sigma model on the disk and in particular on the treatment of its zero modes.
We show that Fourier transforms on the Weyl algebras have a geometric counterpart in the framework of toric varieties, namely they induce isomorphisms between twisted rings of differential operators on regular toric varieties, whose fans are related to each other by reflections of one-dimensional cones. The simplest class of examples is provided by the toric varieties related by such reflections to projective spaces. It includes the blow-up at a point in affine space and resolution of singularities of varieties appearing in the study of the minimal orbit of sl(n+1).
An integral formula for the solutions of Knizhnik-Zamolodchikov (KZ) equation with values in an arbitrary irreducible representation of the symmetric group S_N is presented for integer values of the parameter. The corresponding integrals can be computed effectively as certain iterated residues determined by a given Young diagram and give polynomials with integer coefficients. The derivation is based on Schur-Weyl duality and the results of Matsuo on the original SU(n) KZ equation. The duality between the spaces of solutions with parameters m and -m is discussed in relation with the intersection pairing in the corresponding homology groups.