23 research outputs found
Scalar extensions of triangulated categories
Given a triangulated category over a field and a field extension ,
we investigate how one can construct a triangulated category over . Our
approach produces the derived category of the base change scheme if the
category one starts with is the bounded derived category of a smooth projective
variety over and the field extension is finite and Galois. We also
investigate how the dimension of a triangulated category behaves under scalar
extensions.Comment: 15 pages, comments welcom
Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics
We discuss a conjecture saying that derived equivalence of smooth projective simply connected varieties implies that the difference of their classes in the Grothendieck ring of varieties is annihilated by a power of the affine line class. We support the conjecture with a number of known examples, and one new example. We consider a smooth complete intersection X of three quadrics in P5 and the corresponding double cover Y→P2 branched over a sextic curve. We show that as soon as the natural Brauer class on Y vanishes, so that X and Y are derived equivalent, the difference [X]−[Y] is annihilated by the affine line class
Categorification of a linear algebra identity and factorization of Serre functors
We provide a categorical interpretation of a well-known identity from linear
algebra as an isomorphism of certain functors between triangulated categories
arising from finite dimensional algebras.
As a consequence, we deduce that the Serre functor of a finite dimensional
triangular algebra A has always a lift, up to shift, to a product of suitably
defined reflection functors in the category of perfect complexes over the
trivial extension algebra of A.Comment: 18 pages; Minor changes, references added, new Section 2.
Vanishing cycles and mutation
This is the writeup of a talk given at the European Congress of Mathematics,
Barcelona. It considers Picard-Lefschetz theory from the Floer cohomology
viewpoint.Comment: 20 pages, LaTeX2e. TeXnical problem should now be fixed, so that the
images will appear even if you download the .ps fil
Remarks on quiver gauge theories from open topological string theory
We study effective quiver gauge theories arising from a stack of D3-branes on certain Calabi-Yau singularities. Our point of view is a first principle approach via open topological string theory. This means that we construct the natural A-infinity-structure of open string amplitudes in the associated D-brane category. Then we show that it precisely reproduces the results of the method of brane tilings, without having to resort to any effective field theory computations. In particular, we prove a general and simple formula for effective superpotentials
D-branes Wrapped on Fuzzy del Pezzo Surfaces
We construct classical solutions in quiver gauge theories on D0-branes
probing toric del Pezzo singularities in Calabi-Yau manifolds. Our solutions
represent D4-branes wrapped around fuzzy del Pezzo surfaces. We study the
fluctuation spectrum around the fuzzy CP^2 solution in detail. We also comment
on possible applications of our fuzzy del Pezzo surfaces to the fuzzy version
of F-theory, dubbed F(uzz) theory.Comment: 1+42 pages, 9 figures v2: references added v3: statements on the
structure of the Yukawa couplings weakened. published versio
Counting the number of Ï„-exceptional sequences over Nakayama algebras
The notion of a Ï„-exceptional sequence was introduced by Buan and Marsh in (2018) as a generalisation of an exceptional sequence for finite dimensional algebras. We calculate the number of complete Ï„-exceptional sequences over certain classes of Nakayama algebras. In some cases, we obtain closed formulas which also count other well known combinatorial objects, and exceptional sequences of path algebras of Dynkin quivers
Discrete derived categories I: homomorphisms, autoequivalences and t-structures
Discrete derived categories were studied initially by Vossieck (J Algebra 243:168–176, 2001) and later by Bobiński et al. (Cent Eur J Math 2:19–49, 2004). In this article, we describe the homomorphism hammocks and autoequivalences on these categories. We classify silting objects and bounded t-structures