Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants (1006.2706v2)

Abstract: We define a new type of Hall algebras associated e.g. with quivers with polynomial potentials. The main difference with the conventional definition is that we use cohomology of the stack of representations instead of constructible sheaves or functions. In order to take into account the potential we introduce a generalization of theory of mixed Hodge structures, related to exponential integrals. Generating series of our Cohomological Hall algebra is a generalization of the motivic Donaldson-Thomas invariants introduced in arXiv:0811.2435. Also we prove a new integrality property of motivic Donaldson-Thomas invariants.

• The paper introduces COHA as a novel algebraic structure that merges cohomology of moduli stacks with Hall algebra concepts to model BPS states.
• It details an explicit multiplication via the Künneth isomorphism and toric formulae, linking representation theory of quivers with supersymmetric gauge theories.
• The study refines motivic Donaldson–Thomas invariants using exponential Hodge structures, offering fresh insights into wall-crossing phenomena and integrality properties.

An Essay on "Cohomological Hall Algebra, Exponential"

The paper by Maxim Kontsevich and Yan Soibelman introduces the construction of Cohomological Hall algebra (COHA), a mathematical structure proposed to be intimately connected with the algebra of BPS states deriving from string theory. The work seeks to define a rigorous mathematical framework for associative algebras which are conjectured to play a central role in the context of four-dimensional quantum field theories with N=2 supersymmetry. Its relevance spans the representation theory of quivers, smooth algebras, and the paper of Donaldson-Thomas (DT) invariants.

Overview of Main Concepts

1. Cohomological Hall Algebra: COHA extends the notion of Hall algebra by incorporating the cohomology of moduli stacks of representations rather than using constructible functions. The algebra is constructed over an Abelian category of representations of a quiver or a smooth algebra and is endowed with a twisted associative product.
2. Multiplication and Associativity: The paper details construction in the context of representation stacks and their cohomology, revealing how multiplication is defined via the Künneth isomorphism and explicit Toric formulae.
3. Motivic Donaldson-Thomas Invariants: DT invariants, motivically refined, serve as the generating series for COHA. Through a stability condition, these invariants factor into products of quantum dilogarithms, aligning with BPS states' wall-crossing phenomena.
4. Monodromic Exponential Mixed Hodge Structures: A substantial innovation is the introduction of exponential Hodge structures and an associated category, showing how COHA can be enriched and providing crucial connectivity to previous work on mixed Hodge modules.
5. Factorization Systems: The work proposes a framework for understanding the integrality properties of motivic Donaldson-Thomas invariants using factorization systems, a key conceptual thrust that supports the algebraic structures inherent in COHA.

Implications and Significance

The research establishes bridges between several vital areas of mathematics and theoretical physics. Practically, COHA can model diverse systems, including quivers with potentials ubiquitous in supersymmetric gauge theories. The paper of DT invariants through COHA offers a novel approach for examining part of the string landscape, specifically the spectrum of BPS states, where the integrality and wall-crossing behaviors vividly appear.

Theoretically, the connection to cluster algebras as demonstrated via mutations and symplectomorphisms of quantum tori echoes ongoing explorations in mathematical physics regarding the categorification of algebro-geometric phenomena. These connections also provide speculative insights into potential applications in areas such as enumerative geometry and non-commutative geometry—where the quantum algebraic structures envisage rich interactions with motives and cohomological theories.

Future Directions

Future development may focus on broadening the applicability and understanding of COHA, potentially under more general conditions or alternative underpinning categories. Intriguingly, expanding the algebraic framework to encompass more general Calabi-Yau categories or their categorifications could open pathways to uncovering new invariants or algebraic phenomena. The potential to relate Cohomological Hall algebras to classical DT invariants proposed by Joyce and others remains a compelling area for ongoing research.

Furthermore, as the ideas evolve, the challenge remains to effectively implement and simulate these structures in more tangible computational frameworks or bridge quintessential gaps with physical theories. Enhanced algebraic tools for characterizing wall-crossing behaviors or integrative applications involving non-commutative motives are areas rich for exploration.

In summary, the paper by Kontsevich and Soibelman is a profound exploration of new algebraic structures with deep implications in mathematics and theoretical physics, offering a sophisticated framework that promises rich developmental prospects through its theoretical insights and operational potential.