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**Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants.**
*(English)*
Zbl 1248.14060

The primary goal of the paper under review is to give a precise definition of the graded algebra of the BPS states which from the string theory one expects to get from a certain class of four dimensional quantum theories with \(N=2\) spacetime supersymmetry (see [J. A. Harvey and G. Moore, “On the algebras of BPS states”, Commun. Math. Phys. 197, No. 3, 489–519 (1998; Zbl 1055.81616)]). In the paper under review to each smooth algebra with potential is assigned a graded algebra called the Cohomological Hall Algebra (COHA for short). The main difference of COHA with the motivic Hall algebra is that the cohomology of the moduli stacks of objects is used instead of the constructible functions. Motivic Donaldson-Thomas series (DT-series for short) are then defined as the generating series for the Serre polynomials of the graded components of COHA.

Two different versions of COHA are introduced: “off–shell” version and “on–shell” version. The first version is related to the rapid decay cohomology of an algebraic variety with a potential. A generalization of the theory of mixed Hodge structures is obtained in the way of developing this version. In this version by using elementary topological methods strong factorization properties for the motivic DT-series are obtained and as a result integer numbers which count BPS states, and satisfy wall-crossing formulas are defined.

The second version of COHA, also called critical cohomology, is defined in terms of the sheaves of vanishing cycles. This one is related to asymptotic expansions of exponential integrals in the formal neighborhood of the critical locus of the potential. The corresponding motivic DT-series are proven to coincide with the ones introduced in [M. Kontsevich and Y. Soibelman, “Stability structures, motivic Donaldson–Thomas invariants and cluster transformations”, preprint, arXiv:0811.2435].

It is expected that this approach can be generalized to a wider class of 3-dimensional Calabi-Yau categories considered in [arXiv:0811.2435]. In the case of a smooth algebra with potential by avoiding technical difficulties of the general case a stronger version of the integrality property of the generating series conjectured in [arXiv:0811.2435] is proven.

Two more applications are considered at the end: categorification of COHA and motivic DT-invariants arising in the Chern-Simons theory with complex gauge group.

Two different versions of COHA are introduced: “off–shell” version and “on–shell” version. The first version is related to the rapid decay cohomology of an algebraic variety with a potential. A generalization of the theory of mixed Hodge structures is obtained in the way of developing this version. In this version by using elementary topological methods strong factorization properties for the motivic DT-series are obtained and as a result integer numbers which count BPS states, and satisfy wall-crossing formulas are defined.

The second version of COHA, also called critical cohomology, is defined in terms of the sheaves of vanishing cycles. This one is related to asymptotic expansions of exponential integrals in the formal neighborhood of the critical locus of the potential. The corresponding motivic DT-series are proven to coincide with the ones introduced in [M. Kontsevich and Y. Soibelman, “Stability structures, motivic Donaldson–Thomas invariants and cluster transformations”, preprint, arXiv:0811.2435].

It is expected that this approach can be generalized to a wider class of 3-dimensional Calabi-Yau categories considered in [arXiv:0811.2435]. In the case of a smooth algebra with potential by avoiding technical difficulties of the general case a stronger version of the integrality property of the generating series conjectured in [arXiv:0811.2435] is proven.

Two more applications are considered at the end: categorification of COHA and motivic DT-invariants arising in the Chern-Simons theory with complex gauge group.

Reviewer: Amin Gholampour (College Park)

### MSC:

14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |

14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |

14D07 | Variation of Hodge structures (algebro-geometric aspects) |

81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |