Stability structures, motivic Donaldson-Thomas invariants and cluster transformations (0811.2435v1)

Abstract: We define new invariants of 3d Calabi-Yau categories endowed with a stability structure. Intuitively, they count the number of semistable objects with fixed class in the K-theory of the category ("number of BPS states with given charge" in physics language). Formally, our motivic DT-invariants are elements of quantum tori over a version of the Grothendieck ring of varieties over the ground field. Via the quasi-classical limit "as the motive of affine line approaches to 1" we obtain numerical DT-invariants which are closely related to those introduced by Behrend. We study some properties of both motivic and numerical DT-invariants including the wall-crossing formulas and integrality. We discuss the relationship with the mathematical works (in the non-triangulated case) of Joyce, Bridgeland and Toledano-Laredo, as well as with works of physicists on Seiberg-Witten model (string junctions), classification of N=2 supersymmetric theories (Cecotti-Vafa) and structure of the moduli space of vector multiplets. Relating the theory of 3d Calabi-Yau categories with distinguished set of generators (called cluster collection) with the theory of quivers with potential we found the connection with cluster transformations and cluster varieties (both classical and quantum).

• The paper introduces a framework integrating stability structures with motivic DT invariants to analyze wall-crossing in 3D Calabi-Yau categories.
• It employs motivic Hall algebras to construct enumerative invariants and elucidates the impact of quiver mutations in cluster transformations.
• The results advance mathematical understanding and offer insights into string theory by linking derived categories with BPS state counts.

Stability Structures, Motivic Donaldson-Thomas Invariants, and Cluster Transformations

The paper by Kontsevich and Soibelman explores advanced interactions between stability structures, motivic Donaldson-Thomas invariants, and cluster transformations within the field of 3-dimensional Calabi-Yau categories. The work elaborates on complex ideas stemming from mathematics and theoretical physics, particularly focusing on derived categories, wall-crossing phenomena, and the construction of invariants that serve as a bridge to understanding counts of BPS states in string theory.

Overview

The paper discusses the setup of motivic Donaldson-Thomas (DT) theory for triangulated categories, most notably those arising from Calabi-Yau environments. These structures are pivotal in understanding geometric and algebraic structures, especially in terms of moduli spaces of sheaves and their invariants. A vital aspect addressed is the integration of stability conditions into these categories, an idea that traces back to Bridgeland and has seen extensive development as a framework to understand moduli problems across algebraic geometry.

The core focus revolves around:

• Stability Conditions: The formalism provided generalizes conceptions of stability across triangulated categories. These conditions are crucial to defining moduli spaces of sheaves and to the interpretations of wall-crossing phenomena.
• Motivic Donaldson-Thomas Invariants: These invariants count stable objects in derived categories, drawing parallels to enumerative geometrics like Gromov-Witten theory. By leveraging motivic integration, they encapsulate how objects behave under stability conditions, leading to deeper insights into the algebraic structures involved.
• Cluster Transformations: Lying at the intersection of algebraic geometry and representation theory, cluster transformations allow analysis of the mutation of quivers, which are graphical representations of categories. These mutations have fascinating implications in string theory, depicting how BPS states transform across stability walls.

Main Results and Theoretical Implications

1. Ind-Constructible Categories:

• The authors introduce a framework for handling categories with infinitely many isomorphism classes, typical in derived settings.
• These generalized categories are amenable to motivic Hall algebras—a key tool in constructing numerical invariants.

2. Wall-Crossing Phenomena:

• At the heart of the paper is understanding how numerical DT-invariants change as one crosses "walls of stability" in the space of stability conditions.
• The wall-crossing formulas connect with the algebraic transformations in the motivic Hall algebra induced by tilting or mutations of the underlying categories.

3. Potential and Calabi-Yau Categories:

• The notion of potential is explored within triangulated categories, addressing how these potentials aid in constructing the motivic DT-invariants.
• The work links the potentials with critical loci in derived categories, suggesting a deep-seated relationship with mirror symmetry.

4. Cluster Transformations and Mutations:

• The paper examines the transformation (mutation) of cluster categories—categories arising from quantum cluster algebras characterized by quivers.
• These transformations mimic physical processes in theoretical physics, offering a combinatorial framework that is explained in terms of quiver mutations.

Future Directions and Speculative Insights

The intricate relationship between derived categories and motivic DT-invariants suggests several speculative insights:

• Geometry of Moduli Spaces:

The framework could lead to advanced predictions concerning the topology and the geometry of moduli spaces, particularly in examining singularities and their resolutions.

• Deeper Physical Interpretations:

Given their origins and applications in string theory, insights gained from these mathematical models offer potential to be transposed into physical phenomena—concepts like black holes and the entropy counts allied to BPS states.

• Enhanced Integrability via Stability Structures:

Stability conditions imbue categories with structural harmony that aligns with integrability, potentially linking to broader conjectures like the SYZ conjecture within mirror symmetry.

Overall, the paper serves as a crucial node connecting several areas of representation theory, algebraic geometry, and mathematical physics. The results not only push forward the boundaries of motivic DT-theory but also provide a systematic apparatus to navigate the maze of derived categories and their associated mutations through cluster transformations.