Invariants of Morse complexes, persistent homology and applications.

International audienceThe algorithm for calculation of "canonical form" = "persistence barcodes/diagrams" invariants from S.Barannikov "The Framed Morse complex and its invariants" Adv. in Sov. Math., vol 21, AMS transl, (1994), is described

Solving the noncommutative Batalin-Vilkovisky equation

I show that a summation over ribbon graphs with legs gives the construction of the solutions to the noncommutative Batalin-Vilkovisky equation, including the equivariant version. This generalizes the known construction of A-infinity algebra via summation over ribbon trees. These solutions give naturally the supersymmetric matrix action functionals, which are the gl(N)-equivariantly closed differential forms on the matrix spaces, which were introduced in one of my previous papers "Noncommmutative Batalin-Vilkovisky geometry and Matrix integrals" (arXiv:0912.5484, electronic CNRS preprint hal-00102085(28/09/2006)).Comment: 17 pages, electronic CNRS preprint hal-00464794 (17/03/2010

Holomorphic potentials for graded D-branes

We discuss gauge-fixing, propagators and effective potentials for topological A-brane composites in Calabi-Yau compactifications. This allows for the construction of a holomorphic potential describing the low-energy dynamics of such systems, which generalizes the superpotentials known from the ungraded case. Upon using results of homotopy algebra, we show that the string field and low energy descriptions of the moduli space agree, and that the deformations of such backgrounds are described by a certain extended version of `off-shell Massey products' associated with flat graded superbundles. As examples, we consider a class of graded D-brane pairs of unit relative grade. Upon computing the holomorphic potential, we study their moduli space of composites. In particular, we give a general proof that such pairs can form acyclic condensates, and, for a particular case, show that another branch of their moduli space describes condensation of a two-form.Comment: 47 pages, 7 figure

Matrix De Rham complex and quantum A-infinity algebras

I establish the relation of the non-commutative BV-formalism with super-invariant matrix integration. In particular, the non-commutative BV-equation, defining the quantum A-infinity-algebras, introduced in "Modular operads and Batalin-Vilkovisky geometry" IMRN, Vol. 2007, doi: 10.1093/imrn/rnm075, is represented via de Rham differential acting on the matrix spaces related with Bernstein-Leites simple associative algebras with odd trace q(N), and with gl(N|N). I also show that the Lagrangians of the matrix integrals from "Noncommmutative Batalin-Vilkovisky geometry and Matrix integrals", Comptes Rendus Mathematique, vol 348 (2010), pp. 359-362, arXiv:0912.5484, are equivariantly closed differential forms.Comment: published versio

From Zwiebach invariants to Getzler relation

We introduce the notion of Zwiebach invariants that generalize Gromov-Witten invariants and homotopical algebra structures. We outline the induction procedure that induces the structure of Zwiebach on the subbicomplex, that gives the structure of Gromov-Witten invariants on subbicomplex with zero diffferentials. We propose to treat Hodge dGBV with 1/12 axiom as the simplest set of Zwiebach invariants, and explicitely prove that it induces WDVV and Getzler equations in genera 0 and 1 respectively.Comment: 35 page

Wheeled PROPs, graph complexes and the master equation

We introduce and study wheeled PROPs, an extension of the theory of PROPs which can treat traces and, in particular, solutions to the master equations which involve divergence operators. We construct a dg free wheeled PROP whose representations are in one-to-one correspondence with formal germs of SP-manifolds, key geometric objects in the theory of Batalin-Vilkovisky quantization. We also construct minimal wheeled resolutions of classical operads Com and Ass as rather non-obvious extensions of Com_infty and Ass_infty, involving, e.g., a mysterious mixture of associahedra with cyclohedra. Finally, we apply the above results to a computation of cohomology of a directed version of Kontsevich's complex of ribbon graphs.Comment: LaTeX2e, 63 pages; Theorem 4.2.5 on bar-cobar construction is strengthene

Mirror symmetry and quantization of abelian varieties

The paper consists of two sections. The first section provides a new definition of mirror symmetry of abelian varieties making sense also over $p$-adic fields. The second section introduces and studies quantized theta-functions with two-sided multipliers, which are functions on non-commutative tori. This is an extension of an earlier work by the author. In the Introduction and in the Appendix the constructions of this paper are put into a wider context.Comment: 24 pp., amstex file, no figure

Hypercommutative operad as a homotopy quotient of BV

We give an explicit formula for a quasi-isomorphism between the operads Hycomm (the homology of the moduli space of stable genus 0 curves) and BV/$\Delta$ (the homotopy quotient of Batalin-Vilkovisky operad by the BV-operator). In other words we derive an equivalence of Hycomm-algebras and BV-algebras enhanced with a homotopy that trivializes the BV-operator. These formulas are given in terms of the Givental graphs, and are proved in two different ways. One proof uses the Givental group action, and the other proof goes through a chain of explicit formulas on resolutions of Hycomm and BV. The second approach gives, in particular, a homological explanation of the Givental group action on Hycomm-algebras.Comment: minor corrections added, to appear in Comm.Math.Phy