Semistable Chow–Hall algebras of quivers and quantized Donaldson–Thomas invariants

Abstract

The semi-stable ChowHa of a quiver with stability is defined as an analog of the Cohomological Hall algebra of Kontsevich and Soibelman via convolution in equivariant Chow groups of semistable loci in representation varieties of quivers. We prove several structural results on the semistable ChowHa, namely isomorphism of the cycle map, a tensor product decomposition, and a tautological presentation. For symmetric quivers, this leads to an identification of their quantized Donaldson-Thomas invariants with the Chow-Betti numbers of moduli spaces. map for fine moduli spaces of quivers. We exhibit a tensor product decomposition (Theorem 6.1) of the CoHa into all semi-stable CoHa's for various slopes of the stability, categorifying the Harder-Narasimhan, or wall-crossing, formula of [14]. We give a "tautological" presentation of the semi-stable ChowHa (Theorem 8.1), in the spirit of [7], which generalizes the algebraic description of [12] of the CoHa. Quite surprisingly, such a tautological presentation remains valid for the equivariant Chow groups of stable loci in representation varieties (Theorem 9.1). From this, we conclude that the quantized Donaldson-Thomas invariants of a symmetric quiver are given by the Poincaré polynomials of the Chow groups of moduli spaces of stable quiver representations (Theorem 9.2). This shows that quantized Donaldson-Thomas invariants are of algebro-geometric origin; compare [13] where quantized Donaldson-Thomas invariants are interpreted via intersection cohomology of moduli spaces of semi-stable quiver representations.The proofs of these structural results basically only use the Harder-Narasimhan stratification of representation varieties of [14], properties of equivariant Chow groups, and the result of Efimov [4]. All structural results are illustrated by examples in Section 10: we first give a complete description of the Hall algebra of a two-cycle quiver, which is the only symmetric quiver with known representation theory apart from the trivial and the one-loop quiver whose CoHa's are already described in [12]. Then we consider the only non-trivial (i.e. not isomorphic to the CoHa of a trivial quiver) semi-stable ChowHa for the Kronecker quiver -we observe that it is not super-commutative, but still has the same Poincaré-Hilbert series as a free super-commutative algebra; for this, the representation theory of the Kronecker quiver is used essentially. Then we illustrate the calculation of Chow-Betti numbers of moduli spaces of stable representations in the context of classical invariant theory, and finally hint at an algebraic derivation of the explicit formula of [15] for quantized Donaldson-Thomas invariants of multiple loop quivers.The paper is organized as follows: After reviewing basic facts on quiver representations (Section 2) and the definition of quantized Donaldson-Thomas invariants (Section 3), we define the semi-stable ChowHa in Section 4. In Section 5 we study the cycle map from ChowHa to CoHa, and prove it to be an isomorphism via induction over Harder-Narasimhan str...

Topological strings, strips and quivers

“…It is conjectured in [21] and proven in [28] that motivic Donaldson-Thomas invariants (identified in appendix A), or equivalently combinations (−1) d 1 +...+dm Ω d 1 ,...,dm;j , are positive integers. Motivic Donaldson-Thomas invariants Ω d 1 ,...,dm;j of a symmetric quiver can be interpreted as the intersection Betti numbers of the moduli space of its semisimple representations, or as the Chow-Betti numbers of the moduli space of all simple representations [41,42]. Interestingly, quiver generating functions (3.1) take form of generalized Nahm sums [43], which may indicate their relations to other systems in which such sums arise.…”

Donaldson-Thomas invariants, torus knots, and lattice paths

“…Motivic Donaldson-Thomas invariants Ω d 1 ,...,dm;j of a symmetric quiver Q can be interpreted as the intersection Betti numbers of the moduli space of all semisimple representations of Q, or as the Chow-Betti numbers of the moduli space of all simple representations [32,33], and they are encoded in the following product decomposition of the above series…”

“…are colored HOMFLY-PT polynomials of a knot K, which is the mirror image of the original knot K. In this work we take advantage of the fact, that coefficients of the following quotient of generating series associated to C 33) in the classical limit satisfy b l 1 ,...,lm = b l 1 ,...,lm .…”

Positivity for quantum cluster algebras

“…Cohomological wall crossing. The following "categorification" of the wall crossing formula from Donaldson-Thomas theory is a very special case of [DM16, Thm B] (see also [FR18] and the proof of [Dav18, Thm 3.21]). Theorem 6.3.…”