This page serves as a list of some problems I care about with some additional remarks. For most papers referenced, I only read the reviews or abstracts.
Donaldson-Thomas type theory for CY 4-folds
The idea of Donaldson-Thomas theory originated in [1], where Donaldson and Thomas suggested some gauge-theoretic analog of 3 dimensional Chern-Simons theory and 4 dimensional Donaldson theory to higher dimensional manifolds, for example Calabi-Yau 3-fold, \(G_{2}\) manifold and \(Spin(7)\) manifold (also higher dimesional CY varieties). For a long time, only CY3 case is established in [2] and widely studied and other 2 cases remains obscure.
By the injection \(SU(4) \to Spin(7)\), the CY 4-fold can be viewed as special \(Spin(7)\) manifold. However, the problem is the obstruction theory has more than two terms and methods of [3] can not be applied. With a lot of work in derived algebraic geometry, including [4–7], the Donaldson-Thomas type invariants in CY 4-folds were established in [8] (special case) and [9] (general case) using analytic methods. More recently, algebraic virtual class is obtained in [10]. Many results for CY3 can be transported to CY4 case, including some cases of MNOP conjectures (also in [8]), the dimension zero DT generating seris in [11]. the DT/PT correspondence in [12], the relation to Gopakumar-Vafa invariants in [13–16] and maybe a lot more. It is then a natural question, whether Donaldson-Thomas invariants can be generalized to even higher dimensions, since we are not using special holonomy in construction. For some work in this direction, the counting of dimension zero subschemes is studied in [17]. As a reminder, Gromov-Witten theory is well defined for varieties of arbitrary dimension.
On the other hand, Donaldson-Thomas type sheaf counting theory is very flexible and has many generalizations. For example, via spectral equivalence, the Vafa-Witten theory, a gauge-theoretic invariants for 4-manifolds, on projective algebraic surfaces can be formulated as a special sheaf counting theory on local CY3 as in [18,19]. The similiar ideas can be used to study Kapustin-Witten theory [20] which may be formulated as a sheaf-couting theory on local CY4. The first step of which is [21], where some special CY4 was considered. Also, recently, some surface couting theory on CY4 was constructed in [22].
Categorification of Donaldson-Thomas Theory
The Calabi-Yau 3-fold case is defined by algebraic methods of [3] in [2]. Since then a lot of study in DT theory emerges. Categorification of Donaldson-Thomas theory is one of important problems in Donaldson-Thomas theory. The physicists view Donaldson-Thomas invariants as dimension of BPS states. Mathematically, the problem is raised in [23] and [24] in somehow different manner.
The approach in [23] is to defined Donaldson-Thomas invariants in motivic rings. By taking the Euler characteristic, the motivic invariants descents to numerical Donaldson-Thomas invariants. This idea has been applied in [25] to prove PT/DT correspondence. Again in confiold case [26], the motivic refinement matches refined topological vertex defined in [27] as original DT theory matches topological vertex. In [28], the result of dimension zero numerical DT invariants was recovered.
The approach in [24] is to define some perverse sheaves and realized the numerical ones as Euler characteristics again. This approach is later generalized to Saito’s mixed hodge module theory as mentioned in [28]. These perverse sheaves techniques together with critical loci was used to defined Gopakumar-Vafa invariants in [29]. A good survey of these development is [30]. Also, the categorifation of Donaldson-Thomas theory via monodromic mixed Hodge modules provides the proof of new results in quantum cluster algebra [31] and quantum enveloping algebras [32].
More recently, Toda suggest some new approaches in categorification based on derived algebraic geometry. In a seris of paper by [33–35], he define and study some version of category of moduli spaces relating to Donaldson-Thomas theory. The numerical invariants are Euler characteristic of periodic cyclic homology of the category. Further properties are studied in [36–38]. The categorical construction brings much of the results and ideas of noncommutative algebraic geometry to Donaldson-Thomas theory and recovers many classical results from a different point of view.
Donaldson-Thomas theory and Bridgeland stability
As Donaldson-Thomas theory counts (semi)stable sheaves, the notation of stability is important in the theory. [24] study the wall crossing phenomenon when changing stability condition for abelian category of coherent sheaves. A more ambitious proposal by [23] replace the abelian category by triangulated category, and Gieseker or \(\mu\)-stability by Bridgeland stability introduced in [39]. The derived category of coherent sheaves can be replaced by Calabi-Yau categories, allowing fruitful interactions with many areas, including representation of quivers in [32] and quantum cluster algebras in [31]. Its relation with mirror symmetry and wall-crossing formula was considered in [40]. More recently, Bridgeland suggesting a construction of \(\tau\) functions, a series of important functions in mathematical physics, and Joyce structrure, an analog of Frobenius structure, on stability space from Donaldson-Thomas theory of Calabi-Yau categories in [41–43], showing their integrable systematic nature.
The study of Bridgeland stability is also active. The construction of Bridgeland stability condition on derived category of coherent sheaves is a very challenging problem. The elliptic curve case is studied in [44] and K3 surface in [45] by tilting the heart. In [46], authors suggested a double-tilting methods to construct Bridgeland stability condition on 3-folds based on generalized Bogomolov-Gieseker inequality. Though the latter inequality was wrong [47], the idea is used to successfully construct the stability condition for 3-folds in various cases, including abelian threefolds in [48], some Fano threefolds in [49], quintic threefold in [50]. A remarkable property of Bridgeland stability is that the stability space is local homeomorphic to the complex linear space generated by numerical Grothendieck group of the category, and therefore admits a complex manifold structure. The study of this space has applications to moduli of sheaves [51], autoequivalence of derived category of coherent sheaves [52], Donaldson-Thomas invariants [53] and Reid-type theorem [54]. We can also construct the moduli space of Bridgeland (semi)stable objects [55], which shares similar properties of moduli space of Gieseker (semi)stable objects. More recently, in [56] the stability in families is studied to prove the deformation invariance of Donaldson-Thomas theory of Bridgeland stable objects in Calabi-Yau threefolds and to provide a new method for constructing stability condition on threefolds.
Via homological mirror symmetry, the derived category of coherent sheaves on CY3 is equivalent to the Fukaya category of its mirror. These ideas inspire the research of finding stability condition on Fukaya categories and already generate some interesting geometric applications in [57–59].
References
The references are auto generated. Due to incompatibility between biblatex and html, there may be some errors. Sorry for possible misunderstanding.