66 research outputs found
AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories
We give a detailed exposition of the Alexandrov-Kontsevich-Schwarz-
Zaboronsky superfield formalism using the language of graded manifolds. As a
main illustarting example, to every Courant algebroid structure we associate
canonically a three-dimensional topological sigma-model. Using the AKSZ
formalism, we construct the Batalin-Vilkovisky master action for the model.Comment: 13 pages, based on lectures at Rencontres mathematiques de Glanon
200
Courant-Dorfman algebras and their cohomology
We introduce a new type of algebra, the Courant-Dorfman algebra. These are to
Courant algebroids what Lie-Rinehart algebras are to Lie algebroids, or Poisson
algebras to Poisson manifolds. We work with arbitrary rings and modules,
without any regularity, finiteness or non-degeneracy assumptions. To each
Courant-Dorfman algebra (\R,\E) we associate a differential graded algebra
\C(\E,\R) in a functorial way by means of explicit formulas. We describe two
canonical filtrations on \C(\E,\R), and derive an analogue of the Cartan
relations for derivations of \C(\E,\R); we classify central extensions of
\E in terms of H^2(\E,\R) and study the canonical cocycle
\Theta\in\C^3(\E,\R) whose class obstructs re-scalings of the
Courant-Dorfman structure. In the nondegenerate case, we also explicitly
describe the Poisson bracket on \C(\E,\R); for Courant-Dorfman algebras
associated to Courant algebroids over finite-dimensional smooth manifolds, we
prove that the Poisson dg algebra \C(\E,\R) is isomorphic to the one
constructed in \cite{Roy4-GrSymp} using graded manifolds.Comment: Corrected formulas for the brackets in Examples 2.27, 2.28 and 2.29.
The corrections do not affect the exposition in any wa
On the AKSZ formulation of the Rozansky-Witten theory and beyond
Using the AKSZ formalism, we construct the Batalin-Vilkovisky master action
for the Rozansky-Witten model, which can be defined for any complex manifold
with a closed (2,0)-form. We also construct the holomorphic version of
Rozansky-Witten theory defined over Calabi-Yau 3-fold.Comment: 12 page
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
Poisson quasi-Nijenhuis structures with background
We define the Poisson quasi-Nijenhuis structures with background on Lie
algebroids and we prove that to any generalized complex structure on a Courant
algebroid which is the double of a Lie algebroid is associated such a
structure. We prove that any Lie algebroid with a Poisson quasi-Nijenhuis
structure with background constitutes, with its dual, a quasi-Lie bialgebroid.
We also prove that any pair of a Poisson bivector and a 2-form
induces a Poisson quasi-Nijenhuis structure with background and we observe that
particular cases correspond to already known compatibilities between and
.Comment: 11 pages, submitted to Letters in Mathematical Physic
Non-Commutative Batalin-Vilkovisky Algebras, Homotopy Lie Algebras and the Courant Bracket
We consider two different constructions of higher brackets. First, based on a
Grassmann-odd, nilpotent \Delta operator, we define a non-commutative
generalization of the higher Koszul brackets, which are used in a generalized
Batalin-Vilkovisky algebra, and we show that they form a homotopy Lie algebra.
Secondly, we investigate higher, so-called derived brackets built from
symmetrized, nested Lie brackets with a fixed nilpotent Lie algebra element Q.
We find the most general Jacobi-like identity that such a hierarchy satisfies.
The numerical coefficients in front of each term in these generalized Jacobi
identities are related to the Bernoulli numbers. We suggest that the definition
of a homotopy Lie algebra should be enlarged to accommodate this important
case. Finally, we consider the Courant bracket as an example of a derived
bracket. We extend it to the "big bracket" of exterior forms and multi-vectors,
and give closed formulas for the higher Courant brackets.Comment: 42 pages, LaTeX. v2: Added remarks in Section 5. v3: Added further
explanation. v4: Minor adjustments. v5: Section 5 completely rewritten to
include covariant construction. v6: Minor adjustments. v7: Added references
and explanation to Section
On quasi-Jacobi and Jacobi-quasi bialgebroids
We study quasi-Jacobi and Jacobi-quasi bialgebroids and their relationships
with twisted Jacobi and quasi Jacobi manifolds. We show that we can construct
quasi-Lie bialgebroids from quasi-Jacobi bialgebroids, and conversely, and also
that the structures induced on their base manifolds are related via a quasi
Poissonization
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
Membrane Sigma-Models and Quantization of Non-Geometric Flux Backgrounds
We develop quantization techniques for describing the nonassociative geometry
probed by closed strings in flat non-geometric R-flux backgrounds M. Starting
from a suitable Courant sigma-model on an open membrane with target space M,
regarded as a topological sector of closed string dynamics in R-space, we
derive a twisted Poisson sigma-model on the boundary of the membrane whose
target space is the cotangent bundle T^*M and whose quasi-Poisson structure
coincides with those previously proposed. We argue that from the membrane
perspective the path integral over multivalued closed string fields in Q-space
is equivalent to integrating over open strings in R-space. The corresponding
boundary correlation functions reproduce Kontsevich's deformation quantization
formula for the twisted Poisson manifolds. For constant R-flux, we derive
closed formulas for the corresponding nonassociative star product and its
associator, and compare them with previous proposals for a 3-product of fields
on R-space. We develop various versions of the Seiberg-Witten map which relate
our nonassociative star products to associative ones and add fluctuations to
the R-flux background. We show that the Kontsevich formula coincides with the
star product obtained by quantizing the dual of a Lie 2-algebra via convolution
in an integrating Lie 2-group associated to the T-dual doubled geometry, and
hence clarify the relation to the twisted convolution products for topological
nonassociative torus bundles. We further demonstrate how our approach leads to
a consistent quantization of Nambu-Poisson 3-brackets.Comment: 52 pages; v2: references adde
Leibniz algebroids, twistings and exceptional generalized geometry
We investigate a class of Leibniz algebroids which are invariant under
diffeomorphisms and symmetries involving collections of closed forms. Under
appropriate assumptions we arrive at a classification which in particular gives
a construction starting from graded Lie algebras. In this case the Leibniz
bracket is a derived bracket and there are higher derived brackets resulting in
an -structure. The algebroids can be twisted by a non-abelian
cohomology class and we prove that the twisting class is described by a
Maurer-Cartan equation. For compact manifolds we construct a Kuranishi moduli
space of this equation which is shown to be affine algebraic. We explain how
these results are related to exceptional generalized geometry.Comment: 58 page
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