Core Concepts

This paper reviews various algebraic, combinatorial, and geometric interpretations of motivic Donaldson-Thomas invariants for symmetric quivers, highlighting their connections to moduli spaces, cohomological Hall algebras, and combinatorial formulas.

Abstract

Reineke, M. (2024). Donaldson–Thomas invariants of symmetric quivers. *arXiv preprint arXiv:2410.03219v1*.

This survey paper aims to present a comprehensive overview of different interpretations and approaches to understanding Donaldson-Thomas (DT) invariants for symmetric quivers.

The paper provides a review and synthesis of existing research on DT invariants of symmetric quivers, drawing on results from algebraic geometry, representation theory, and combinatorics. It explains key concepts, theorems, and examples to illustrate different interpretations of these invariants.

- DT invariants of symmetric quivers, despite their origins in quantum field theory, admit an elementary definition through the factorization of q-hypergeometric series.
- These invariants can be interpreted as topological invariants of moduli spaces of quiver representations, specifically their intersection cohomology or Chow groups.
- Cohomological Hall algebras (CoHa) provide an algebraic framework for studying DT invariants, revealing connections to vertex algebras and Lie superalgebras.
- Combinatorial formulas, such as counting break divisors on graphs associated with quivers, offer alternative ways to compute and understand DT invariants.

The survey demonstrates the rich interplay between different mathematical areas in understanding DT invariants of symmetric quivers. It highlights the significance of these invariants as they connect to various geometric, algebraic, and combinatorial structures.

This survey provides a valuable resource for researchers interested in DT invariants and their applications in various branches of mathematics and physics. It offers a roadmap for navigating the diverse interpretations and techniques used to study these invariants.

As a survey paper, it primarily focuses on summarizing existing knowledge. Further research could explore open questions related to DT invariants of more general quivers, connections to other areas of mathematics, and potential applications in theoretical physics.

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Stats

The moduli space of stable representations for the quiver associated with the matrix A = ((1,1),(1,1)) and dimension vector d = (2,3) is 13-dimensional, and its DT invariant is gDT_A^d(q) = q^13 + q^12 + 2q^11.
For the matrix A = ((0,m),(m,0)) and dimension vector d = (1,k) with k ≤ m, the DT invariant is given by a specific q-binomial coefficient.
The DT invariant gDT_A^d(1) for a quiver Q with enough loops and dimension vector d can be computed by counting the Wd-orbits of break divisors on the graph Γd(Q).

Quotes

"The latter have the advantage of admitting an elementary definition, via a product factorization of series of q-hypergeometric type."
"In the light of this theorem, it is natural to ask for interpretations of the numbers cd,k as dimensions of vector spaces of algebraic objects associated to A, in particular dimensions of cohomology groups, or as counts of combinatorial objects."
"The main result is that the DT invariants DT A
d (q) can be interpreted as topological invariants of the spaces Md(QA), namely as their Poincar´e polynomial in intersection cohomology with compact support."

Key Insights Distilled From

by Markus Reine... at **arxiv.org** 10-07-2024

Deeper Inquiries

While the interpretations of DT invariants for symmetric quivers offer a rich tapestry of connections between algebraic, combinatorial, and geometric perspectives, extending these interpretations to more general quivers or quivers with potential presents significant challenges and leads to sophisticated generalizations.
Quivers with Potential: For quivers with potential, which play a crucial role in studying Calabi-Yau categories, the definition of DT invariants involves intricate constructions like the Jacobi algebra and its representations. The moduli spaces become more complex, often noncommutative or derived, requiring advanced tools from noncommutative or derived algebraic geometry.
Motivic Generating Series: The motivic generating series, defined for symmetric quivers, can be generalized to quivers with potential using the formalism of vanishing cycles. However, the factorization properties and positivity results for DT invariants become more subtle and may not hold in full generality.
Cohomological Hall Algebras: The Cohomological Hall algebra construction can be extended to more general settings, including quivers with potential. However, the explicit algebraic descriptions and the connection to Lie algebras become more intricate, often involving derived categories and dg-algebras.
Geometric Interpretations: Geometric interpretations of DT invariants for quivers with potential often involve moduli spaces of objects in derived categories or noncommutative spaces. These interpretations require advanced techniques from derived algebraic geometry and noncommutative geometry.
Combinatorial Formulas: Combinatorial formulas for DT invariants of general quivers or quivers with potential are less developed. Some progress has been made using techniques from crystal bases and cluster algebras, but a general combinatorial understanding remains an active area of research.

Yes, alternative geometric interpretations of DT invariants beyond those directly tied to moduli spaces are indeed conceivable, particularly through the lens of derived categories and noncommutative geometry.
Derived Categories: DT invariants are often seen as numerical invariants of 3-Calabi-Yau categories, which are naturally associated with derived categories of coherent sheaves on Calabi-Yau threefolds. Exploring the relationship between DT invariants and categorical structures within derived categories, such as exceptional collections, stability conditions, and Bridgeland-Thomas theory, could offer new geometric insights.
Noncommutative Geometry: For quivers with potential, the relevant moduli spaces can be noncommutative, prompting the use of noncommutative geometry. Techniques from this field, such as cyclic homology, Hochschild homology, and noncommutative motives, might provide alternative ways to understand DT invariants geometrically.
Mirror Symmetry: Mirror symmetry predicts a deep relationship between complex geometry and symplectic geometry. Exploring the mirror symmetric counterparts of DT invariants, potentially through counts of holomorphic disks or Lagrangian submanifolds, could lead to novel geometric interpretations.

Combinatorial formulas for DT invariants, such as those involving break divisors or quiver diagonalization, provide a powerful lens through which to investigate the enumerative geometry of quiver representations.
Concrete Calculations: These formulas offer a concrete and often algorithmic way to compute DT invariants, bypassing the complexities of moduli spaces or derived categories. This allows for explicit calculations and the exploration of patterns and structures within DT invariants.
Positivity and Integrality: The combinatorial nature of these formulas often makes the positivity and integrality of DT invariants manifest. This provides further evidence for the deep connections between DT invariants and enumerative geometry.
Connections to Other Structures: Combinatorial formulas can reveal unexpected connections between DT invariants and other combinatorial or geometric structures, such as crystal bases, cluster algebras, or toric varieties. These connections can provide new insights into the geometry of quiver moduli spaces.
New Enumerative Problems: The combinatorial formulas themselves can suggest new enumerative geometric problems related to quiver representations. For example, the break divisor formula leads to questions about counting special types of divisors on graphs, which may have independent geometric interest.

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