Self-dual quiver moduli and orientifold Donaldson-Thomas invariants

Abstract

Abstract. Motivated by the counting of BPS states in string theory with orientifolds, we study moduli spaces of self-dual representations of a quiver with contravariant involution. We develop Hall module techniques to compute the number of points over finite fields of moduli stacks of semistable self-dual representations. Wall-crossing formulas relating these counts for different choices of stability parameters recover the wall-crossing of orientifold BPS/DonaldsonThomas invariants predicted in the physics literature. In finite type examples the wall-crossing formulas can be reformulated in terms of identities for quantum dilogarithms acting in representations of quantum tori.

Cohomological orientifold Donaldson–Thomas invariants as Chow groups

“…More precisely, we show that as a graded vector space the trivial stability Hall module B Q is determined by the semistable module B θ-ss Q and algebras {A θ-ss Q,µ } µ>0 . Passing to Grothendieck groups, Theorem 3.5 recovers the motivic orientifold wall-crossing formula of [28].…”

Group actions on quiver varieties and applications

“…Contravariant involutions of a quiver were studied by Derksen and Weyman [9], Zubkov [35,36], Shmelkin [31], Bocklandt [5] and later by Young [34], where Young's motivation comes from physics and as an application he constructs orientifold Donaldson-Thomas invariants. In [34], the action of a contravariant involution is also modified using what is called a 'duality structure', which corresponds to our notion of modifying families. Motivated by questions in representation theory, Henderson and Licata study actions of so-called 'admissible' covariant automorphisms on Nakajima quiver varieties of type A and prove a decomposition of the fixed locus [17]; however, they do not use group cohomology type techniques or see phenomena such as the morphisms u f Σ failing to be injective in their setting (for example, compare [17,Lemma 3.17] with Proposition 3.11).…”

“…Therefore, one can start by studying the actions by subgroups of covariant automorphisms and actions by contravariant involutions. Since contravariant automorphisms of order two are studied in [9,31,35,36,5,34], we restrict our attention to subgroups Σ ⊂ Aut + (Q) of covariant automorphisms until §3.5 (…”

“…One can also define modifying families, which are now given by a single element u σ ∈ G Q,d (k) such that u σ Ψ σ (u σ ) ∈ ∆(k), and use these families to modify the actions Φ and Ψ, without changing the action on M θ−s Q,d ; as Z/2Z acts on k × by inversion, we have H 2 (Z/2Z, k × ) = {±1}, so there are only two possible cohomology classes for the 2-cocycle associated to a modifying family. This appears in [34] in different language, coming from physics: the duality structures in loc. cit.…”

“…Young refers to such equivariant representations as self-dual representation (cf. [34,Theorem 2.7] for an analogue of Theorem 3.38 in the contravariant setting).…”