67 research outputs found
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
-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/ (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
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