308 research outputs found
Classical BV theories on manifolds with boundary
In this paper we extend the classical BV framework to gauge theories on
spacetime manifolds with boundary. In particular, we connect the BV
construction in the bulk with the BFV construction on the boundary and we
develop its extension to strata of higher codimension in the case of manifolds
with corners. We present several examples including electrodynamics, Yang-Mills
theory and topological field theories coming from the AKSZ construction, in
particular, the Chern-Simons theory, the theory, and the Poisson sigma
model. This paper is the first step towards developing the perturbative
quantization of such theories on manifolds with boundary in a way consistent
with gluing.Comment: The second version has many typos corrected, references added. Some
typos are probably still there, in particular, signs in examples. In the
third version more typoes are corrected and the exposition is slightly
change
A Counterexample to the Quantizability of Modules
Let a Poisson structure on a manifold M be given. If it vanishes at a point
m, the evaluation at m defines a one dimensional representation of the Poisson
algebra of functions on M. We show that this representation can, in general,
not be quantized. Precisely, we give a counterexample for M=R^n, such that:
(i) The evaluation map at 0 can not be quantized to a representation of the
algebra of functions with product the Kontsevich product associated to the
Poisson structure.
(ii) For any formal Poisson structure extending the given one and vanishing
at zero up to second order in epsilon, (i) still holds.
We do not know whether the second claim remains true if one allows the higher
order terms in epsilon to attain nonzero values at zero
Moduli Spaces of Curves with Homology Chains and c=1 Matrix Models
We show that introducing a periodic time coordinate in the models of
Penner-Kontsevich type generalizes the corresponding constructions to the case
of the moduli space of curves with homology chains
\gamma\in H_1(C,\zet_k). We make a minimal extension of the resulting models
by adding a kinetic term, and we get a new matrix model which realizes a simple
dynamics of \zet_k-chains on surfaces. This gives a representation of
matter coupled to two-dimensional quantum gravity with the target space being a
circle of finite radius, as studied by Gross and Klebanov.Comment: IFUM 459/FT (LaTeX, 9 pages; a few misprints have been corrected and
the introduction has been slightly modified
Finite dimensional AKSZ-BV theories
We describe a canonical reduction of AKSZ-BV theories to the cohomology of
the source manifold. We get a finite dimensional BV theory that describes the
contribution of the zero modes to the full QFT. Integration can be defined and
correlators can be computed. As an illustration of the general construction we
consider two dimensional Poisson sigma model and three dimensional Courant
sigma model. When the source manifold is compact, the reduced theory is a
generalization of the AKSZ construction where we take as source the cohomology
ring. We present the possible generalizations of the AKSZ theory.Comment: 33 page
One-dimensional Chern-Simons theory
We study a one-dimensional toy version of the Chern-Simons theory. We
construct its simplicial version which comprises features of a low-energy
effective gauge theory and of a topological quantum field theory in the sense
of Atiyah.Comment: 37 page
Graph complexes in deformation quantization
Kontsevich's formality theorem and the consequent star-product formula rely
on the construction of an -morphism between the DGLA of polyvector
fields and the DGLA of polydifferential operators. This construction uses a
version of graphical calculus. In this article we present the details of this
graphical calculus with emphasis on its algebraic features. It is a morphism of
differential graded Lie algebras between the Kontsevich DGLA of admissible
graphs and the Chevalley-Eilenberg DGLA of linear homomorphisms between
polyvector fields and polydifferential operators. Kontsevich's proof of the
formality morphism is reexamined in this light and an algebraic framework for
discussing the tree-level reduction of Kontsevich's star-product is described.Comment: 39 pages; 3 eps figures; uses Xy-pic. Final version. Details added,
mainly concerning the tree-level approximation. Typos corrected. An abridged
version will appear in Lett. Math. Phy
QP-Structures of Degree 3 and 4D Topological Field Theory
A A BV algebra and a QP-structure of degree 3 is formulated. A QP-structure
of degree 3 gives rise to Lie algebroids up to homotopy and its algebraic and
geometric structure is analyzed. A new algebroid is constructed, which derives
a new topological field theory in 4 dimensions by the AKSZ construction.Comment: 17 pages, Some errors and typos have been correcte
BFV-complex and higher homotopy structures
We present a connection between the BFV-complex (abbreviation for
Batalin-Fradkin-Vilkovisky complex) and the so-called strong homotopy Lie
algebroid associated to a coisotropic submanifold of a Poisson manifold. We
prove that the latter structure can be derived from the BFV-complex by means of
homotopy transfer along contractions. Consequently the BFV-complex and the
strong homotopy Lie algebroid structure are quasi-isomorphic and
control the same formal deformation problem.
However there is a gap between the non-formal information encoded in the
BFV-complex and in the strong homotopy Lie algebroid respectively. We prove
that there is a one-to-one correspondence between coisotropic submanifolds
given by graphs of sections and equivalence classes of normalized Maurer-Cartan
elemens of the BFV-complex. This does not hold if one uses the strong homotopy
Lie algebroid instead.Comment: 50 pages, 6 figures; version 4 is heavily revised and extende
Odd Chern-Simons Theory, Lie Algebra Cohomology and Characteristic Classes
We investigate the generic 3D topological field theory within AKSZ-BV
framework. We use the Batalin-Vilkovisky (BV) formalism to construct explicitly
cocycles of the Lie algebra of formal Hamiltonian vector fields and we argue
that the perturbative partition function gives rise to secondary characteristic
classes. We investigate a toy model which is an odd analogue of Chern-Simons
theory, and we give some explicit computation of two point functions and show
that its perturbation theory is identical to the Chern-Simons theory. We give
concrete example of the homomorphism taking Lie algebra cocycles to
Q-characteristic classes, and we reinterpreted the Rozansky-Witten model in
this light.Comment: 52 page
- âŠ