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2016 | OriginalPaper | Chapter

Remarks on Cohomological Hall Algebras and Their Representations

Author : Yan Soibelman

Published in: Arbeitstagung Bonn 2013

Publisher: Springer International Publishing

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Abstract

The aim of this paper is to discuss a class of representations of Cohomological Hall algebras related to the notion of framed stable object of a category. The paper is an extended version of the talk the author gave at the workshop on Donaldson–Thomas invariants at the University Paris-7 in June 2013 and at the conference “Algebra, Geometry, Physics” dedicated to Maxim Kontsevich (June 2014, IHES). Because of the origin of the paper it contains more speculations than proofs.

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previous chapter Elliptic Calabi–Yau Threefolds over a del Pezzo Surface

next chapter A Stratification on the Moduli of K3 Surfaces in Positive Characteristic

In fact we considered in the loc. cit. more general case of formally smooth algebras with potential.

For example, I heard from Dominic Joyce about the work in progress of Oren Ben-Bassat in which COHA would be defined for the derived moduli stack of perfect complexes on a compact Calabi–Yau three-fold.

Calabi–Yau structure which we will discuss later leads to the requirement char(k) = 0. For simplicity we will often assume that k = C.

As explained in Sect. 7 of [KoSo5], it is more convenient to work with compactly supported cohomology and then apply the duality functor.

We warn the reader that even in the case of quivers our moduli spaces are not Nakajima quiver varieties.

I thank the referee for the comments on those conditions.

I thank the referee for pointing out on this condition.

Alternatively, following Paul Seidel, one can consider the double cover of the Calabi–Yau three-fold branched along the divisor given by the complex surface. Then Lagrangian submanifolds with boundary lift to closed ones in the branched cover. One can form an equivariant Lagrangian connected sum, and then interpret it as an operation on the original Lagrangian submanifolds with boundary.

In [DiHuSo] the authors considered stable vertically framed sheaves on the compactification \(\overline{X}\). The corresponding moduli spaces were projective. Considerations with non-compact submanifold X give rise to quasi-projective moduli spaces. We ignore these technicalities here.

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Title: Remarks on Cohomological Hall Algebras and Their Representations
Author: Yan Soibelman
Publisher: Springer International Publishing
Book: Arbeitstagung Bonn 2013
Print ISBN: 978-3-319-43646-3

Electronic ISBN: 978-3-319-43648-7

Copyright Year: 2016
DOI: https://doi.org/10.1007/978-3-319-43648-7_13

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