114 research outputs found
Stability conditions and Stokes factors
Let A be the category of modules over a complex, finite-dimensional algebra.
We show that the space of stability conditions on A parametrises an
isomonodromic family of irregular connections on P^1 with values in the Hall
algebra of A. The residues of these connections are given by the holomorphic
generating function for counting invariants in A constructed by D. Joyce.Comment: Very minor changes. Final version. To appear in Inventione
Quadratic differentials as stability conditions
We prove that moduli spaces of meromorphic quadratic differentials with
simple zeroes on compact Riemann surfaces can be identified with spaces of
stability conditions on a class of CY3 triangulated categories defined using
quivers with potential associated to triangulated surfaces. We relate the
finite-length trajectories of such quadratic differentials to the stable
objects of the corresponding stability condition.Comment: 123 pages; 38 figures. Version 2: hypotheses in the main results
mildly weakened, to reflect improved results of Labardini-Fragoso and
coauthors. Version 3: minor changes to incorporate referees' suggestions.
This version to appear in Publ. Math. de l'IHE
Vertex Operators, Grassmannians, and Hilbert Schemes
We describe a well-known collection of vertex operators on the infinite wedge
representation as a limit of geometric correspondences on the equivariant
cohomology groups of a finite-dimensional approximation of the Sato
grassmannian, by cutoffs in high and low degrees. We prove that locality, the
boson-fermion correspondence, and intertwining relations with the Virasoro
algebra are limits of the localization expression for the composition of these
operators. We then show that these operators are, almost by definition, the
Hilbert scheme vertex operators defined by Okounkov and the author in \cite{CO}
when the surface is with the torus action .Comment: 20 pages, 0 figure
Noncommutative resolutions of ADE fibered Calabi-Yau threefolds
In this paper we construct noncommutative resolutions of a certain class of Calabi-Yau threefolds studied by F. Cachazo, S. Katz and C. Vafa. The threefolds under consideration are fibered over a complex plane with the fibers being deformed Kleinian singularities. The construction is in terms of a noncommutative algebra introduced by V. Ginzburg, which we call the "N=1 ADE quiver algebra"
Sheaves on fibered threefolds and quiver sheaves
This paper classifies a class of holomorphic D-branes, closely related to
framed torsion-free sheaves, on threefolds fibered in resolved ADE surfaces
over a general curve C, in terms of representations with relations of a twisted
Kronheimer--Nakajima-type quiver in the category Coh(C) of coherent sheaves on
C. For the local Calabi--Yau case C\cong\A^1 and special choice of framing, one
recovers the N=1 ADE quiver studied by Cachazo--Katz--Vafa.Comment: 13 pages, 2 figures, minor change
Curve counting via stable pairs in the derived category
For a nonsingular projective 3-fold , we define integer invariants
virtually enumerating pairs where is an embedded curve and
is a divisor. A virtual class is constructed on the associated
moduli space by viewing a pair as an object in the derived category of . The
resulting invariants are conjecturally equivalent, after universal
transformations, to both the Gromov-Witten and DT theories of . For
Calabi-Yau 3-folds, the latter equivalence should be viewed as a wall-crossing
formula in the derived category.
Several calculations of the new invariants are carried out. In the Fano case,
the local contributions of nonsingular embedded curves are found. In the local
toric Calabi-Yau case, a completely new form of the topological vertex is
described.
The virtual enumeration of pairs is closely related to the geometry
underlying the BPS state counts of Gopakumar and Vafa. We prove that our
integrality predictions for Gromov-Witten invariants agree with the BPS
integrality. Conversely, the BPS geometry imposes strong conditions on the
enumeration of pairs.Comment: Corrected typos and duality error in Proposition 4.6. 47 page
The Quantum McKay Correspondence for polyhedral singularities
Let G be a polyhedral group, namely a finite subgroup of SO(3). Nakamura's
G-Hilbert scheme provides a preferred Calabi-Yau resolution Y of the polyhedral
singularity C^3/G. The classical McKay correspondence describes the classical
geometry of Y in terms of the representation theory of G. In this paper we
describe the quantum geometry of Y in terms of R, an ADE root system associated
to G. Namely, we give an explicit formula for the Gromov-Witten partition
function of Y as a product over the positive roots of R. In terms of counts of
BPS states (Gopakumar-Vafa invariants), our result can be stated as a
correspondence: each positive root of R corresponds to one half of a genus zero
BPS state. As an application, we use the crepant resolution conjecture to
provide a full prediction for the orbifold Gromov-Witten invariants of [C^3/G].Comment: Introduction rewritten. Issue regarding non-uniqueness of conifold
resolution clarified. Version to appear in Inventione
Quiver GIT for varieties with tilting bundles
In the setting of a variety X admitting a tilting bundle T we consider the problem of constructing X as a quiver GIT quotient of the algebra A:=EndX(T)opA:=EndX(T)op . We prove that if the tilting equivalence restricts to a bijection between the skyscraper sheaves of X and the closed points of a quiver representation moduli functor for A=EndX(T)opA=EndX(T)op then X is indeed a fine moduli space for this moduli functor, and we prove this result without any assumptions on the singularities of X. As an application we consider varieties which are projective over an affine base such that the fibres are of dimension 1, and the derived pushforward of the structure sheaf on X is the structure sheaf on the base. In this situation there is a particular tilting bundle on X constructed by Van den Bergh, and our result allows us to reconstruct X as a quiver GIT quotient for an easy to describe stability condition and dimension vector. This result applies to flips and flops in the minimal model program, and in the situation of flops shows that both a variety and its flop appear as moduli spaces for algebras produced from different tilting bundles on the variety. We also give an application to rational surface singularities, showing that their minimal resolutions can always be constructed as quiver GIT quotients for specific dimension vectors and stability conditions. This gives a construction of minimal resolutions as moduli spaces for all rational surface singularities, generalising the G-Hilbert scheme moduli space construction which exists only for quotient singularities
Solitons in Seiberg-Witten Theory and D-branes in the Derived Category
We analyze the "geometric engineering" limit of a type II string on a
suitable Calabi-Yau threefold to obtain an N=2 pure SU(2) gauge theory. The
derived category picture together with Pi-stability of B-branes beautifully
reproduces the known spectrum of BPS solitons in this case in a very explicit
way. Much of the analysis is particularly easy since it can be reduced to
questions about the derived category of CP1.Comment: 20 pages, LaTex2
Comparing imaging, acoustics, and radar to monitor Leach’s storm-petrel colonies
Seabirds are integral components of marine ecosystems and, with many populations globally threatened, there is a critical need for effective and scalable seabird monitoring strategies. Many seabird species nest in burrows, which can make traditional monitoring methods costly, infeasible, or damaging to nesting habitats. Traditional burrow occupancy surveys, where possible, can occur infrequently and therefore lead to an incomplete understanding of population trends. For example, in Oregon, during the last three decades there have been large changes in the abundance of Leach’s storm-petrels (Hydrobates leucorhoa), which included drastic declines at some colonies. Unfortunately, traditional monitoring failed to capture the timing and magnitude of change, limiting managers’ ability to determine causes of the decline and curtailing management options. New, easily repeatable methods of quantifying relative abundance are needed. For this study, we tested three methods of remote monitoring: passive acoustic monitoring, time-lapse cameras, and radar. Abundance indices derived from acoustics and imagery: call rates, acoustic energy, and counts were significantly related to traditional estimates of burrow occupancy of Leach’s storm-petrels. Due to sampling limitations, we were unable to compare radar to burrow occupancy. Image counts were significantly correlated with all other indices, including radar, while indices derived from acoustics and radar were not correlated. Acoustic data likely reflect different aspects of the population and hold the potential for the further development of indices to disentangle phenology, attendance of breeding birds, and reproductive success. We found that image counts are comparable with standard methods (e.g., radar) in producing annual abundance indices. We recommend that managers consider a sampling scheme that incorporates both acoustics and imaging, but for sites inaccessible to humans, radar remains the sole option. Implementation of acoustic and camera based monitoring programs will provide much needed information for a vulnerable group of seabirds
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