30,122 research outputs found
-Stability conditions via -quadratic differentials for Calabi-Yau- categories
We construct a quiver with superpotential from
a marked surface with full formal arc system .
Categorically, we show that the associated cluster- category is
Haiden-Katzarkov-Kontsevich's topological Fukaya category
of , which is also an
-baric heart of the Calabi-Yau- category
of . Thus
stability conditions on induces
-stability conditions on .
Geometrically, we identify the space of -quadratic differentials on the
logarithm surface , with the space of induced
-stability conditions on , with a
complex parameter satisfying . When is an
integer, the result gives an -analogue of Bridgeland-Smith's result for
realizing stability conditions on the orbit Calabi-Yau- category
via
quadratic differentials with zeroes of order . When the genus of
is zero, the space of -quadratic differentials can be also
identified with framed Hurwitz spaces.Comment: A preliminary version, 57 pages, 7 figures. Comments are welcome
-Stability conditions on Calabi-Yau- categories and twisted periods
We introduce q-stability conditions on Calabi-Yau-
categories , where is a stability condition on
and a complex number. Sufficient and necessary
conditions are given, for a stability condition on an -baric heart
of to -stability conditions on
. As a consequence, we show that the space
of (induced) open
-stability conditions is a complex manifold, whose fibers (fixing ) give
usual type of spaces of stability conditions. Our motivating examples for
are coming from Calabi-Yau- completions of
dg algebras.
A geometric application is that, for type quiver , the corresponding
space of -stability
conditions admits almost Frobenius structure while the central charge
corresponds to the twisted period , for , where
with . A categorical application is
that we realize perfect derived categories as cluster(-) categories
for acyclic quiver .
In the sequel, we construct quivers with superpotential from flat surfaces
with the corresponding Calabi-Yau- categories and realize
open/closed -stability conditions as -quadratic differentials.Comment: Updated semistable version, 43 pages, 3 figures. Comments are
welcome
C-sortable words as green mutation sequences
Let be an acyclic quiver and be a sequence with elements in
the vertex set . We describe an induced sequence of simple (backward)
tilting in the bounded derived category , starting from the
standard heart and ending at
another heart in . Then we show that
is a green mutation sequence if and only if every heart in this
simple tilting sequence is greater than or equal to ; it is
maximal if and only if . This
provides a categorical way to understand green mutations. Further, fix a
Coxeter element in the Coxeter group of , which is admissible with
respect to the orientation of . We prove that the sequence
induced by a -sortable word is a green
mutation sequence. As a consequence, we obtain a bijection between -sortable
words and finite torsion classes in . As byproducts, the
interpretations of inversions, descents and cover reflections of a -sortable
word are given in terms of the combinatorics of green mutations.Comment: Last version, to appear in PLM
Frobenius morphisms and stability conditions
We generalize Deng-Du's folding argument, for the bounded derived category
of an acyclic quiver , to the finite dimensional derived
category of the Ginzburg algebra associated
to . We show that the -stable category of is
equivalent to the finite dimensional derived category
of the Ginzburg algebra
associated to the species , which is folded from .
If is of Dynkin type, we prove that
(resp. the principal component
) of the space of the
stability conditions of (resp.
) is canonically isomorphic to
(resp. the principal component
) of the space of -stable
stability conditions of (resp. ).
There are two applications. One is for the space
of numerical stability conditions
in . We show that
consists of many connected components, each of which is
isomorphic to , for
is of type or . The other is that we
relate the -stable stability conditions to the Gepner type stability
conditions.Comment: Update versio
Contractible stability spaces and faithful braid group actions
We prove that any `finite-type' component of a stability space of a
triangulated category is contractible. The motivating example of such a
component is the stability space of the Calabi--Yau- category
associated to an ADE Dynkin quiver. In addition to
showing that this is contractible we prove that the braid group
acts freely upon it by spherical twists, in particular
that the spherical twist group is isomorphic to
. This generalises Brav-Thomas' result for the
case. Other classes of triangulated categories with finite-type components in
their stability spaces include locally-finite triangulated categories with
finite rank Grothendieck group and discrete derived categories of finite global
dimension.Comment: Final version, to appear in Geom. Topo
- β¦