Wall Crossings Research Articles

Wall crossing for K‐moduli spaces of plane curves

AbstractWe construct proper good moduli spaces parametrizing K‐polystable ‐Gorenstein smoothable log Fano pairs , where is a Fano variety and is a rational multiple of the anticanonical divisor. We then establish a wall‐crossing framework of these K‐moduli spaces as varies. The main application in this paper is the case of plane curves of degree as boundary divisors of . In this case, we show that when the coefficient is small, the K‐moduli space of these pairs is isomorphic to the GIT moduli space. We then show that the first wall crossing of these K‐moduli spaces are weighted blow‐ups of Kirwan type. We also describe all wall crossings for degree 4,5,6 and relate the final K‐moduli spaces to Hacking's compactification and the moduli of K3 surfaces.

The origin of Calabi-Yau crystals in BPS states counting

We study the counting problem of BPS D-branes wrapping holomorphic cycles of a general toric Calabi-Yau manifold. We evaluate the Jeffrey-Kirwan residues for the flavoured Witten index for the supersymmetric quiver quantum mechanics on the worldvolume of the D-branes, and find that BPS degeneracies are described by a statistical mechanical model of crystal melting. For Calabi-Yau threefolds, we reproduce the crystal melting models long known in the literature. For Calabi-Yau fourfolds, however, we find that the crystal does not contain the full information for the BPS degeneracy and we need to explicitly evaluate non-trivial weights assigned to the crystal configurations. Our discussions treat Calabi-Yau threefolds and fourfolds on equal footing, and include discussions on elliptic and rational generalizations of the BPS states counting, connections to the mathematical definition of generalized Donaldson-Thomas invariants, examples of wall crossings, and of trialities in quiver gauge theories.

Bridgeland stability conditions and skew lines on ℙ3

Inspired by Schmidt’s work on twisted cubics, we study wall crossings in Bridgeland stability, starting with the Hilbert scheme Hilb 2 m + 2 ( P 3 ) parametrizing pairs of skew lines and plane conics union a point. We find two walls. Each wall crossing corresponds to a contraction of a divisor in the moduli space and the contracted space remains smooth. Building on work by Chen–Coskun–Nollet, we moreover prove that the contractions are K-negative extremal in the sense of Mori theory and so the moduli spaces are projective.

Representations on the cohomology of [formula omitted

The moduli space M‾0,n of n pointed stable curves of genus 0 admits an action of the symmetric group Sn by permuting the marked points. We provide a closed formula for the character of the Sn-action on the cohomology of M‾0,n. This is achieved by studying wall crossings of the moduli spaces of quasimaps which provide us with a new inductive construction of M‾0,n, equivariant with respect to the symmetric group action. Moreover we prove that H2k(M‾0,n) for k≤3 and H2k(M‾0,n)⊕H2k−2(M‾0,n) for any k are permutation representations. Our method works for related moduli spaces as well and we provide a closed formula for the character of the Sn-representation on the cohomology of the Fulton-MacPherson compactification P1[n] of the configuration space of n points on P1 and more generally on the cohomology of the moduli space M‾0,n(Pm−1,1) of stable maps.

Stability Conditions and the Mirror Symmetry of K3 Surfaces in Attractor Backgrounds

We study the space of stability conditions on K3 surfaces from the perspective of mirror symmetry. This is done in the attractor backgrounds (moduli). We find certain highly non-generic behaviors of marginal stability walls (a key notion in the study of wall crossings) in the space of stability conditions. These correspond via mirror symmetry to some non-generic behaviors of special Lagrangians in an attractor background. The main results can be understood as a mirror correspondence in a synthesis of the homological mirror conjecture and SYZ mirror conjecture.

K-stability and birational models of moduli of quartic K3 surfaces

We show that the K-moduli spaces of log Fano pairs ({mathbb {P}}^3, cS) where S is a quartic surface interpolate between the GIT moduli space of quartic surfaces and the Baily–Borel compactification of moduli of quartic K3 surfaces as c varies in the interval (0, 1). We completely describe the wall crossings of these K-moduli spaces. As the main application, we verify Laza–O’Grady’s prediction on the Hassett–Keel–Looijenga program for quartic K3 surfaces. We also obtain the K-moduli compactification of quartic double solids, and classify all Gorenstein canonical Fano degenerations of {mathbb {P}}^3.

Geodesic Path Model for Indoor Propagation Loss Prediction of Narrowband Channels.

Indoor path loss models characterize the attenuation of signals between a transmitting and receiving antenna for a certain frequency and type of environment. Their use ranges from network coverage planning to joint communication and sensing applications such as localization and crowd counting. The need for this proposed geodesic path model comes forth from attempts at path loss-based localization on ships, for which the traditional models do not yield satisfactory path loss predictions. In this work, we present a novel pathfinding-based path loss model, requiring only a simple binary floor map and transmitter locations as input. The approximated propagation path is determined using geodesics, which are constrained shortest distances within path-connected spaces. However, finding geodesic paths from one distinct path-connected space to another is done through a systematic process of choosing space connector points and concatenating parts of the geodesic path. We developed an accompanying tool and present its algorithm which automatically extracts model parameters such as the number of wall crossings on the direct path as well as on the geodesic path, path distance, and direction changes on the corners along the propagation path. Moreover, we validate our model against path loss measurements conducted in two distinct indoor environments using DASH-7 sensor networks operating at 868 MHz. The results are then compared to traditional floor-map-based models. Mean absolute errors as low as 4.79 dB and a standard deviation of the model error of 3.63 dB is achieved in a ship environment, almost half the values of the next best traditional model. Improvements in an office environment are more modest with a mean absolute error of 6.16 dB and a standard deviation of 4.55 dB.

Crystal melting, BPS quivers and plethystics

We study the refined and unrefined crystal/BPS partition functions of D6-D2-D0 brane bound states for all toric Calabi-Yau threefolds without compact 4-cycles and some non-toric examples. They can be written as products of (generalized) MacMahon functions. We check our expressions and use them as vacuum characters to study the gluings. We then consider the wall crossings and discuss possible crystal descriptions for different chambers. We also express the partition functions in terms of plethystic exponentials. For ℂ3 and tripled affine quivers, we find their connections to nilpotent Kac polynomials. Similarly, the partition functions of D4-D2-D0 brane bound states can be obtained by replacing the (generalized) MacMahon functions with the inverse of (generalized) Euler functions.

K-MODULI OF CURVES ON A QUADRIC SURFACE AND K3 SURFACES

AbstractWe show that the K-moduli spaces of log Fano pairs $\left(\mathbb {P}^1\times \mathbb {P}^1, cC\right)$ , where C is a $(4,4)$ curve and their wall crossings coincide with the VGIT quotients of $(2,4)$ , complete intersection curves in $\mathbb {P}^3$ . This, together with recent results by Laza and O’Grady, implies that these K-moduli spaces form a natural interpolation between the GIT moduli space of $(4,4)$ curves on $\mathbb {P}^1\times \mathbb {P}^1$ and the Baily–Borel compactification of moduli of quartic hyperelliptic K3 surfaces.

Combinatorial constructions of derived equivalences

Given a certain kind of linear representation of a reductive group, referred to as a quasi-symmetric representation in recent work of \v{S}penko and Van den Bergh, we construct equivalences between the derived categories of coherent sheaves of its various geometric invariant theory (GIT) quotients for suitably generic stability parameters. These variations of GIT quotient are examples of more complicated wall crossings than the balanced wall crossings studied in recent work on derived categories and variation of GIT quotients. Our construction is algorithmic and quite explicit, allowing us to: 1) describe a tilting vector bundle which generates the derived category of such a GIT quotient, 2) provide a combinatorial basis for the K-theory of the GIT quotient in terms of the representation theory of G, and 3) show that our derived equivalences satisfy certain relations, leading to a representation of the fundamental groupoid of a "K\"ahler moduli space" on the derived category of such a GIT quotient. Finally, we use graded categories of singularities to construct derived equivalences between all Deligne-Mumford hyperk\"ahler quotients of a symplectic linear representation of a reductive group (at the zero fiber of the algebraic moment map and subject to a certain genericity hypothesis on the representation), and we likewise construct actions of the fundamental groupoid of the corresponding K\"ahler moduli space.

Localizing virtual structure sheaves for almost perfect obstruction theories

Abstract Almost perfect obstruction theories were introduced in an earlier paper by the authors as the appropriate notion in order to define virtual structure sheaves and K-theoretic invariants for many moduli stacks of interest, including K-theoretic Donaldson-Thomas invariants of sheaves and complexes on Calabi-Yau threefolds. The construction of virtual structure sheaves is based on the K-theory and Gysin maps of sheaf stacks. In this paper, we generalize the virtual torus localization and cosection localization formulas and their combination to the setting of almost perfect obstruction theory. To this end, we further investigate the K-theory of sheaf stacks and its functoriality properties. As applications of the localization formulas, we establish a K-theoretic wall-crossing formula for simple $\mathbb{C} ^\ast $ -wall crossings and define K-theoretic invariants refining the Jiang-Thomas virtual signed Euler characteristics.

Bridgeland stability on threefolds: Some wall crossings

Following up on the construction of Bridgeland stability condition on $\mathbb{P}^3$ by Macri, we develop techniques to study concrete wall crossing behavior for the first time on a threefold. In some cases, such as complete intersections of two hypersurfaces of the same degree or twisted cubics, we show that there are two chambers in the stability manifold where the moduli space is given by a smooth projective irreducible variety, respectively the Hilbert scheme. In the case of twisted cubics, we compute all walls and moduli spaces on a path between those two chambers. This allows us to give a new proof of the global structure of the main component, originally due to Ellingsrud, Piene, and Stromme. In between slope stability and Bridgeland stability there is the notion of tilt stability that is defined similarly to Bridgeland stability on surfaces. Beyond just $\mathbb{P}^3$, we develop tools to use computations in tilt stability to compute wall crossings in Bridgeland stability.

A new method toward the Landau–Ginzburg/Calabi–Yau correspondence via quasi-maps

The Landau-Ginzburg/Calabi-Yau correspondence claims that the Gromov-Witten invariant of the quintic Calabi-Yau 3-fold should be related to the Fan-Jarvis-Ruan-Witten invariant of the associated Landau-Ginzburg model via wall crossings. In this paper, we consider the stack of quasi-maps with a cosection and introduce sequences of stability conditions which enable us to interpolate between the moduli stack for Gromov-Witten invariants and the moduli stack for Fan-Jarvis-Ruan-Witten invariants.

Maximal green sequences for cluster-tilted algebras of finite representation type

We show that, for any cluster-tilted algebra of finite representation type over an algebraically closed field, the following three definitions of a maximal green sequence are equivalent: (1) the usual definition in terms of Fomin–Zelevinsky mutation of the extended exchange matrix, (2) a complete forward hom-orthogonal sequence of Schurian modules, (3) the sequence of wall crossings of a generic green path. Together with [24], this completes the foundational work needed to support the author’s work with P. J. Apruzzese [1], namely, to determine all lengths of all maximal green sequences for all quivers whose underlying graph is an oriented or unoriented cycle and to determine which are “linear”.

Perverse schobers on Riemann surfaces: constructions and examples

This note studies perverse sheaves of categories, or schobers, on Riemann surfaces, following ideas of Kapranov and Schechtman (Perverse schobers, arXiv:1411.2772 , 2014). For certain wall crossings in geometric invariant theory, we construct a schober on the complex plane, singular at each imaginary integer. We use this to obtain schobers for standard flops: in the threefold case, we relate these to a further schober on a partial compactification of a stringy Kähler moduli space, and suggest an application to mirror symmetry.

Families of Elliptic Curves in P3 and Bridgeland Stability

We study wall crossings in Bridgeland stability for the Hilbert scheme of elliptic quartic curves in the three-dimensional projective space. We provide a geometric description of each of the moduli spaces we encounter, including when the second component of this Hilbert scheme appears. Along the way, we prove that the principal component of this Hilbert scheme is a double blowup with smooth centers of a Grassmannian, exhibiting a completely different proof of this known result by Avritzer and Vainsencher. This description allows us to compute the cone of effective divisors of this component.

Multi-stream portrait of the cosmic web

We report the results of the first study of the multi-stream environment of dark matter haloes in cosmological N-body simulations in the LCDM cosmology. The full dynamical state of dark matter can be described as a three-dimensional sub-manifold in six dimensional phase space - the dark matter sheet. In our study we use a Lagrangian sub-manifold x = x(q,t) (where x and q are co-moving Eulerian and Lagrangian coordinates respectively), which is dynamically equivalent to the dark matter sheet but is more convenient for numerical analysis. Our major results can be summarized as follows. At the resolution of the simulation i.e. without additional smoothing, the cosmic web represents a hierarchical structure: each halo is embedded in the filamentary framework of the web predominantly at the filament crossings, and each filament is embedded in the wall like fabric of the web at the wall crossings. Locally, each halo or sub-halo is a peak in the number of streams field. The number of streams in the neighbouring filaments is higher than in the neighbouring walls. The walls are regions where number of streams is equal to three or a few. Voids are uniquely defined by the local condition requiring to be a single-stream flow region. The shells of streams around haloes are quite thin and the closest void region is typically within one and a half FOF radius from the center of the halo.

The birational geometry of moduli spaces of sheaves on the projective plane

We describe a close relation between wall crossings in the birational geometry of modulispaceofGiesekerstablesheaves MH (v)on P 2 andmini-wallcrossingsinthestability manifold Stab(D b (P 2 )). The general philosophy is that the birational geometry of MH (v) is closely related to the birational geometry of X.S o if−K X is ample, we hope −K MH (v) is ample as well. This turns out to be not quite right but is close. We first show that if v is a primitive topological type such that MH (v) is non-empty and irreducible, then MH (v) is smooth and −K MH (v) is big and nef. In particular, MH (v) is a Mori dream space. Therefore MH (v) provide a rich class of examples of the so called weak Fano varieties, whose classification theory is highly interesting in its own right. We achieve the statement by showing that the first Chern class

Polynomiality, wall crossings and tropical geometry of rational double Hurwitz cycles

We study rational double Hurwitz cycles, i.e. loci of marked rational stable curves admitting a map to the projective line with assigned ramification profiles over two fixed branch points. Generalizing the phenomenon observed for double Hurwitz numbers, such cycles are piecewise polynomial in the entries of the special ramification; the chambers of polynomiality and wall crossings have an explicit and “modular” description. A main goal of this paper is to simultaneously carry out this investigation for the corresponding objects in tropical geometry, underlining a precise combinatorial duality between classical and tropical Hurwitz theory.

Rank 1 Bridgeland Stable Moduli Spaces on A Principally Polarized Abelian Surface

We compute moduli spaces of Bridgeland stable objects on an irreducible principally polarized complex abelian surface corresponding to twisted ideal sheaves. We use Fourier–Mukai techniques to extend the ideas of Arcara and Bertram to express wall crossings as Mukai flops and show that the moduli spaces are projective.