Formality theorem for quantizations of Lie bialgebras
Using theory of props we prove a formality theorem associated with universal quantizations of (strongly homotopy) Lie bialgebras.
math.QA↗
arXiv subjects
S. A. Merkulov
Publications and source records attributed to S. A. Merkulov.
Operads, configuration spaces and quantization
We review several well-known operads of compactified configuration spaces and construct several new such operads, C, in the category of smooth manifolds with corners whose complexes of fundamental chains give us (i) the 2-coloured operad of A-infinity algebras and their homotopy morphisms, (ii) the 2-coloured operad of L-infinity algebras and their homotopy morphisms, and (iii) the 4-coloured operad of open-closed homotopy algebras and their homotopy morphisms. Two gadgets - a (coloured) operad of Feynman graphs and a de Rham field theory on C - are introduced and used to construct quantized representations of the (fundamental) chain operad of C which are given by Feynman type sums over graphs and depend on choices of propagators.
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Exotic automorphisms of the Schouten algebra of polyvector fields
Using a new compactification of the (braid) configuration space of n points in the upper half plane we construct a family of exotic Lie-infinity automorphisms of the Schouten algebra of polyvector fields on an affine space depending on a Kontsevich type propagator.
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Wheeled props in algebra, geometry and quantization
These are expanded notes of author's talk at the ECM 2008 attempting to give an elementary introduction into the main ideas of the theory of wheeled props for beginners, and also a survey of its most recent major applications (ranging from algebra and geometry to deformation theory and Batalin-Vilkovisky quantization) which might be of interest to experts.
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Wheeled pro(p)file of Batalin-Vilkovisky formalism
Using technique of wheeled props we establish a correspondence between the homotopy theory of unimodular Lie 1-bialgebras and the famous Batalin-Vilkovisky formalism. Solutions of the so called quantum master equation satisfying certain boundary conditions are proven to be in 1-1 correspondence with representations of a wheeled dg prop which, on the one hand, is isomorphic to the cobar construction of the prop of unimodular Lie 1-bialgebras and, on the other hand, is quasi-isomorphic to the dg wheeled prop of unimodular Poisson structures. These results allow us to apply properadic methods for computing formulae for a homotopy transfer of a unimodular Lie 1-bialgebra structure on an arbitrary complex to the associated quantum master function on its cohomology. It is proven that in the category of quantum BV manifolds associated with the homotopy theory of unimodular Lie 1-bialgebras quasi-isomorphisms are equivalence relations. It is shown that Losev-Mnev's BF theory for unimodular Lie algebras can be naturally extended to the case of unimodular Lie 1-bialgebras (and, eventually, to the case of unimodular Poisson structures). Using a finite-dimensional version of the Batalin-Vilkovisky quantization formalism it is rigorously proven that the Feynman integrals computing the effective action of this new BF theory describe precisely homotopy transfer formulae obtained within the wheeled properadic approach to the quantum master equation. Quantum corrections (which are present in our BF model to all orders of the Planck constant) correspond precisely to what are often called "higher Massey products" in the homological algebra.
math.DG↗
Operad of formal homogeneous spaces and Bernoulli numbers
It is shown that for any morphism, i: g --> h, of Lie algebras the vector space underlying the Lie algebra h is canonically a g-homogeneous formal manifold with the action of g being highly nonlinear and twisted by Bernoulli numbers. This fact is obtained from the study of a 2-coloured operad of formal homogeneous spaces and its minimal resolution, and is used to give a new conceptual explanation of both Ziv Ran's Jacobi-Bernoulli complex and Fiorenza-Manetti's L-infinity algebra structure on the mapping cone of a morphism of two Lie algebras. All these constructions are iteratively extended to the case of a morphism of arbitrary L-infinity algebras.
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Permutahedra, HKR isomorphism and polydifferential Gerstenhaber-Schack complex
This paper aims to give a short but self-contained introduction into the theory of (wheeled) props, properads, dioperads and operads, and illustrate some of its key ideas in terms of a prop(erad)ic interpretation of simplicial and permutahedra cell complexes with subsequent applications to the Hochschild-Kostant-Rosenberg type isomorphisms.
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PROP profile of deformation quantization and graph complexes with loops and wheels
Motivated by the problem of deformation quantization we introduce and study directed graph complexes with oriented loops and wheels. We develop some technique for computing cohomology of such graph complexes and apply it to several concrete examples such as wheeled completion of the operad of strongly homotopy Lie algebras and the wheeled completion of the dg prop of Poisson structures. We prove also a deformation quantization theorem of wheeled Poisson structures on arbitrary formal graded manifolds.
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PROP profile of Poisson geometry
We argue that some classical local geometries are of infinity origin, i.e. their smooth formal germs are (homotopy) representations of cofibrant (di)operads in spaces concentrated in degree zero. In particular, they admit natural infinity generalizations when one considers homotopy representations of that (di)operads in generic differential graded spaces. Poisson geometry provides us with a simplest manifestation of this phenomenon.
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De Rham model for string topology
A De Rham model for string topology based on the theory of iterated integrals is presented.
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Nijenhuis infinity and contractible dg manifolds
We find a minimal differential graded (dg) operad whose generic representations in $R^n$ are in one-to-one correspondence with formal germs of those endomorphisms of the tangent bundle to $R^n$ which satisfy the Nijenhuis integrability condition. This operad is of a surprisingly simple origin -- it is the cobar construction on the quadratic operad of homologically trivial dg Lie algebras. As a by product we obtain a strong homotopy generalization of this geometric structure and show its homotopy equivalence to the structure of contractible dg manifold.
math.AG↗
Operads, deformation theory and F-manifolds
It is shown that every algebra over the chain operad of the little disks operad gives naturally rise to a Hertling-Manin's F-manifold, that is a smooth manifold equipped with an integrable graded commutative associative product on the tangent sheaf. In particular, moduli spaces of extended deformations of complex/symplectic structures are shown to have a canonical structure of F-manifold. With the help of the $G_\infty$-operad a strong homotopy version of the notion of F-manifold is constructed. Among natural examples of $F_\infty$-manifolds one finds formal manifolds associated with the Hochschild cohomology of an associative algebra and with the singular cohomology of an arbitrary compact topological space.
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Strongly homotopy algebras of a Kähler manifold
It is shown that any compact Kähler manifold $M$ gives canonically rise to two strongly homotopy algebras, the first one being associated with the Hodge theory of the de Rham complex and the second one with the Hodge theory of the Dolbeault complex. In these algebras the product of two harmonic differential forms is again harmonic. If $M$ happens to be a Calabi-Yau manifold, there exists a third strongly homotopy algebra closely related to the Barannikov-Kontsevich extended moduli space of complex structures.
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Semi-infinite A-variations of Hodge structure over extended Kahler cone
This is an expanded comment on Barannikov's paper math.AG/0006193. A symplectic version of his construction is discussed. It is shown that the duality transformation for mirror torus fibrations over the same Monge-Ampere manifold exchanges semi-infinite A-variations of Hodge structure introduced in this paper with Barannikov's semi-infinite B-variations of Hodge structure.
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A note on extended complex manifolds
We introduce a category of extended complex manifolds, and prove that the functor describing deformations of a classical compact complex manifold $M$ within this category is versally representable by (an analytic subspace in) $H^*(M,T_M)$. By restricting the associated versal family of extended complex manifolds over $H^*(M,T_M)$ to the subspace $H^1(M,T_M)$ one gets a correct limit to the classical picture.
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Frobenius_infinity invariants of homotopy Gerstenhaber algebras I
We construct a functor from the derived category of homotopy Gerstenhaber algebras with finite-dimensional cohomology to the purely geometric category of so-called $F_{\infty}$-manifolds. The latter contains Frobenius manifolds as a subcategory (so that a pointed Frobenius manifold is itself a homotopy Gerstenhaber algebra). If a homotopy Gerstenhaber algebra happens to be formal as a $L_{\infty}$-algebra, then its $F_{\infty}$-manifold comes equipped with the Gauss-Manin connection. Mirror Symmetry implications are discussed.
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The Moyal product is the matrix product
This is a short comment on the Moyal formula for deformation quantization. It is shown that the Moyal algebra of functions on the plane is canonically isomorphic to an algebra of matrices of infinite size.
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$L_{\infty}$-algebra of an unobstructed deformation functor
This is a comment on the Kuranishi method of constructing analytic deformation spaces. It is based on a simple observation that the Kuranishi map can always be inverted in the category of $L_{\infty}$-algebras. The $L_{\infty}$-structure obtained by this inversion is used to define an ''unobstructed'' deformation functor which is always representable by a smooth pointed moduli space. The singular nature of the original Kuranishi deformation space emerges in this setting merely as a result of the truncation of this ``naive'' $L_{\infty}$-algebra controlling the deformations to a usual differential Lie algebra.
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