539 research outputs found
Localization and traces in open-closed topological Landau-Ginzburg models
We reconsider the issue of localization in open-closed B-twisted
Landau-Ginzburg models with arbitrary Calabi-Yau target. Through careful
analsysis of zero-mode reduction, we show that the closed model allows for a
one-parameter family of localization pictures, which generalize the standard
residue representation. The parameter which indexes these pictures
measures the area of worldsheets with topology, with the residue
representation obtained in the limit of small area. In the boundary sector, we
find a double family of such pictures, depending on parameters and
which measure the area and boundary length of worldsheets with disk
topology. We show that setting and varying interpolates
between the localization picture of the B-model with a noncompact target space
and a certain residue representation proposed recently. This gives a complete
derivation of the boundary residue formula, starting from the explicit
construction of the boundary coupling. We also show that the various
localization pictures are related by a semigroup of homotopy equivalences.Comment: 36 page
D-brane categories
This is an exposition of recent progress in the categorical approach to
D-brane physics. I discuss the physical underpinnings of the appearance of
homotopy categories and triangulated categories of D-branes from a string field
theoretic perspective, and with a focus on applications to homological mirror
symmetry.Comment: 37 pages, IJMPA styl
Generalized complexes and string field theory
I discuss the axiomatic framework of (tree-level) associative open string
field theory in the presence of D-branes by considering the natural extension
of the case of a single boundary sector. This leads to a formulation which is
intimately connected with the mathematical theory of differential graded
categories. I point out that a generic string field theory as formulated within
this framework is not closed under formation of D-brane composites and as such
does not allow for a unitary description of D-brane dynamics. This implies that
the collection of boundary sectors of a generic string field theory with
D-branes must be extended by inclusion of all possible D-brane composites. I
give a precise formulation of a weak unitarity constraint and show that a
minimal extension which is unitary in this sense can always be obtained by
promoting the original D-brane category to an enlarged category constructed by
using certain generalized complexes of D-branes. I give a detailed construction
of this extension and prove its closure under formation of D-brane composites.
These results amount to a completely general description of D-brane composite
formation within the framework of associative string field theory.Comment: 31 pages, 4 figures; v2: small typos corrected, changed to JHEP styl
Graded Lagrangians, exotic topological D-branes and enhanced triangulated categories
I point out that (BPS saturated) A-type D-branes in superstring
compactifications on Calabi-Yau threefolds correspond to {\em graded} special
Lagrangian submanifolds, a particular case of the graded Lagrangian
submanifolds considered by M. Kontsevich and P. Seidel. Combining this with the
categorical formulation of cubic string field theory in the presence of
D-branes, I consider a collection of {\em topological} D-branes wrapped over
the same Lagrangian cycle and {\em derive} its string field action from first
principles. The result is a {\em -graded} version of super-Chern-Simons
field theory living on the Lagrangian cycle, whose relevant string field is a
degree one superconnection in a -graded superbundle, in the sense
previously considered in mathematical work of J. M. Bismutt and J. Lott. This
gives a refined (and modified) version of a proposal previously made by C.
Vafa. I analyze the vacuum deformations of this theory and relate them to
topological D-brane composite formation, by using the general formalism
developed in a previous paper. This allows me to identify a large class of
topological D-brane composites (generalized, or `exotic' topological D-branes)
which do not admit a traditional description. Among these are objects which
correspond to the `covariantly constant sequences of flat bundles' considered
by Bismut and Lott, as well as more general structures, which are related to
the enhanced triangulated categories of Bondal and Kapranov. I also give a
rough sketch of the relation between this construction and the large radius
limit of a certain version of the `derived category of Fukaya's category'.Comment: 31 pages, 4 figures, minor typos corrected; v3: changed to JHEP styl
On the boundary coupling of topological Landau-Ginzburg models
I propose a general form for the boundary coupling of B-type topological
Landau-Ginzburg models. In particular, I show that the relevant background in
the open string sector is a (generally non-Abelian) superconnection of type
(0,1) living in a complex superbundle defined on the target space, which I
allow to be a non-compact Calabi-Yau manifold. This extends and clarifies
previous proposals. Generalizing an argument due to Witten, I show that BRST
invariance of the partition function on the worldsheet amounts to the condition
that the (0,<= 2) part of the superconnection's curvature equals a constant
endomorphism plus the Landau-Ginzburg potential times the identity section of
the underlying superbundle. This provides the target space equations of motion
for the open topological model.Comment: 21 page
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